ABSTRACT
Title of Dissertation: THE COMPLEXITY OF SIMULATING
QUANTUM PHYSICS:
DYNAMICS AND EQUILIBRIUM
Abhinav Deshpande
Doctor of Philosophy, 2021
Dissertation Directed by: Professor Alexey V. Gorshkov
Department of Physics
University of Maryland
Professor Bill Fefferman
Department of Computer Science
University of Chicago
Quantum computing is the offspring of quantum mechanics and computer science,
two great scientific fields founded in the 20th century. Quantum computing is a relatively
young field and is recognized as having the potential to revolutionize science and technology
in the coming century. The primary question in this field is essentially to ask which problems
are feasible with potential quantum computers and which are not. In this dissertation, we
study this question with a physical bent of mind. We apply tools from computer science and
mathematical physics to study the complexity of simulating quantum systems. In general,
our goal is to identify parameter regimes under which simulating quantum systems is easy
(efficiently solvable) or hard (not efficiently solvable). This study leads to an understanding
of the features that make certain problems easy or hard to solve. We also get physical insight
into the behavior of the system being simulated.
In the first part of this dissertation, we study the classical complexity of simulating
quantum dynamics. In general, the systems we study transition from being easy to simulate at
short times to being harder to simulate at later times. We argue that the transition timescale
is a useful measure for various Hamiltonians and is indicative of the physics behind the change
in complexity. We illustrate this idea for a specific bosonic system, obtaining a complexity
phase diagram that delineates the system into easy or hard for simulation. We also prove that
the phase diagram is robust, supporting our statement that the phase diagram is indicative
of the underlying physics.
In the next part, we study open quantum systems from the point of view of their po-
tential to encode hard computational problems. We study a class of fermionic Hamiltonians
subject to Markovian noise described by Lindblad jump operators and illustrate how, some-
times, certain Lindblad operators can induce computational complexity into the problem.
Specifically, we show that these operators can implement entangling gates, which can be
used for universal quantum computation. We also study a system of bosons with Gaussian
initial states subject to photon loss and detected using photon-number-resolving measure-
ments. We show that such systems can remain hard to simulate exactly and retain a relic of
the ?quantumness? present in the lossless system.
Finally, in the last part of this dissertation, we study the complexity of simulating a
class of equilibrium states, namely ground states. We give complexity-theoretic evidence to
identify two structural properties that can make ground states easier to simulate. These are
the existence of a spectral gap and the existence of a classical description of the ground state.
Our findings complement and guide efforts in the search for efficient algorithms.
THE COMPLEXITY OF SIMULATING QUANTUM PHYSICS:
DYNAMICS AND EQUILIBRIUM
by
Abhinav Deshpande
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2021
Advisory Committee:
Professor Mohammad Hafezi, Chair
Professor Alexey V. Gorshkov, Co-chair/Advisor
Professor Bill Fefferman, Co-advisor
Professor Andrew M. Childs, Dean?s Representative
Professor Christopher Monroe
? Copyright by
Abhinav Deshpande
2021
Dedication
To my family, ???? (Amma), ? ??? (Pappa), ????? (Ajji), ???? (Akka), and ?w???? (Poorvi).
ii
Acknowledgments
I would like to first of all thank my advisor Alexey Gorshkov for his constant support
throughout my Ph.D. life. I truly enjoy working with him and have learned a lot from him
over these years. His infectious enthusiasm and the constant flow of new ideas have always
been motivating.
Bill Fefferman, my co-advisor, has also been unwavering in his support throughout
my Ph.D. life. His enormous patience when helping me learn is admirable. I have also been
fortunate to learn through example his philosophy on writing and presentations, something
I wish to emulate one day.
Thanks also go to other professors with whom I have had the fortune to collaborate,
such as members of my committee Andrew Childs, Mohammad Hafezi, and Christopher
Monroe. I have also benefited immensely from the research environment at QuICS, JQI and
CMTC thanks to Brian Swingle, Gorjan Alagic, Jacob Taylor, Michael Gullans, Paul Julienne,
Stephen Jordan, and Yi-Kai Liu, and benefited from a similar environment at IQIM during
my visit there thanks to Fernando Brand?o, John Preskill, Thomas Vidick and the graduate
students and postdocs there. I am very grateful to my teachers at UMD such as Andrew
Childs, Aravind Srinivasan, Ian Appelbaum, Ian Spielman, Paulo Bedaque, and Ted Jacob-
son, whose dedication toward teaching is readily apparent in their lectures. I am also indebted
to postdocs Aarthi Sundaram, Cedric Lin, Chris Baldwin, Dominik Hangleiter, Jim Garri-
son, Michael Foss-Feig, Oles Shtanko, Paraj Titum, Przemek Bienias, Shelby Kimmel, and
iii
Zhexuan Gong, and graduate students such as Aaron Ostrander, Ali Hamed Moosavian, Ali
Lavasani, Alireza Seif, Andrew Glaudell, Arthur Mehta, Chiao-HsuanWang, Eddie Schoute,
Gina Quan, Nishad Maskara, Steve Ragole, Tongyang Li, Troy Sewell, Wen-Lin Tan, Yim-
ing Huang, and Yuan Su for being great friends and teaching me many things along the
way. I am also grateful to the members of the Gorshkov group for the camaraderie: Adam
Ehrenberg, Andrew Guo, Ani Bapat, Dhruv Devulapalli, Fangli Liu, Igor Boettcher, Ja-
cob Bringewatt, Jeremy Young, Joe Iosue, Lucas Brady, Luispe Garc?a-Pintos, Minh Tran,
Pradeep Niroula, Seth Whitsitt, Su-Kuan Chu, Yidan Wang, and Zach Eldredge.
Life at UMD was very smooth thanks to the tireless efforts of Andrea Svejda, Donna
Hammer, Javiera Caceres, Jessica Crosby, Josiland Chambers, Kelly Phillips, Melissa Britton
and the numerous other staff at UMDwho kept it running. I would like to single out Josiland
for her commitment and willingness to always help with anything.
No amount of thanks suffices to express my gratitude to my friends who have been there
throughout: my roommates and circle of friends Ankan Bansal, Ankit Gargava, Atul Singh,
Faez Ahmed, Kiran Burra, Lovlesh Kaushik, Sarthak Chandra, Saurabh Saxena, Shashank
Ganesh, Wei Chen, and Yidan Wang, and my friends from undergrad who?ve also been in
graduate school Abhishek Gupta, Akshansh Singh, Anupam Kumar, ReevuMaity, Shamreen
Iram, and Sumit Sinha.
I would also like to acknowledge the schooling system in place in India, my school?s
library where I first encountered and fell in love with physics, and the encouragement given to
me by my teachers at every level. Institutions and programs such as the National Initiative for
Undergraduate Science and the olympiads at the Homi Bhabha Centre for Science Education,
the Visvesvaraya Industrial & Technological Museum, and the Kishore Vaigyanik Protsahan
iv
Yojana of the Department of Science and Technology furthered my interest in science and
helped me in my decision to study physics as a major. My alma mater the Indian Institute of
Technology Kanpur gave me a lifetime of experience and learning that I will always cherish.
Finally, my family is the sole reason I am here today. This dissertation is dedicated
to them. My parents have made innumerable sacrifices to their own well-being in order to
prioritize my sister?s and my education. My grandmother and sister have been as important
as my parents at shaping me and teaching me to be a good person. I also thank my aunts,
cousins, brother-in-law, and other relatives and well-wishers for their support and encour-
agement. Special thanks and adulation to my one-year-old niece Poorvi, whose presence has
lent added color and melody to life.
v
Table of Contents
Dedication ii
Acknowledgments iii
Table of Contents vi
List of Tables vii
List of Figures viii
List of Abbreviations ix
Citations to Previously Published Work x
Chapter 1: Introduction 1
1.1 Why Complexity theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Sampling Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Chapter 2:Dynamical phase transitions in sampling complexity 18
2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Easiness at short times . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Hardness at longer times . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Expression for output probabilities . . . . . . . . . . . . . . . . . . . . . . 27
2.6 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.7 Bound on variation distance . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.8 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Chapter 3:Complexity phase diagram for interacting and long-range bosonic Hamil-
tonians 44
3.1 Setup and summary of results . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Easy-sampling timescale . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3 Sampling hardness timescale . . . . . . . . . . . . . . . . . . . . . . . . . 51
vi
3.4 Sharp and coarse transitions . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.5 Related Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.6 Approximation error under HHKL decomposition . . . . . . . . . . . . . 56
3.7 Closeness of evolution under H and H ?. . . . . . . . . . . . . . . . . . . . 61
3.8 Extended Easiness timescale for 1D . . . . . . . . . . . . . . . . . . . . . 67
3.9 Hardness timescale for interacting bosons . . . . . . . . . . . . . . . . . . 70
3.9.1 One dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.9.2 Hardcore limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.10 Hardness timescale for free bosons . . . . . . . . . . . . . . . . . . . . . . 80
3.10.1 Optimizing hardness time . . . . . . . . . . . . . . . . . . . . . . 85
3.10.2 Almost free bosons, V = o(1) . . . . . . . . . . . . . . . . . . . 87
3.11 Types of transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.11.1 Defining phase transitions . . . . . . . . . . . . . . . . . . . . . . 89
3.12 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Chapter 4:Complexity of Fermionic Dissipative Interactions and Applications toQuan-
tum Computing 93
4.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.2 Free-fermion sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.3 Easy Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.3.1 Efficient classical algorithms . . . . . . . . . . . . . . . . . . . . . 106
4.3.2 Performance of the classical algorithms . . . . . . . . . . . . . . . 111
4.4 Hard class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.4.1 Reduction from a generic quantum circuit . . . . . . . . . . . . . . 115
4.4.2 Robustness of the hardness result . . . . . . . . . . . . . . . . . . 118
4.5 Easy Class 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.6 Easy Class 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.7 Easy Class 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.8 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.8.1 Proof of the Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Chapter 5:Quantum Computational Supremacy via High-Dimensional Gaussian Bo-
son Sampling 133
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.2 Hardness of approximate GBS . . . . . . . . . . . . . . . . . . . . . . . . 135
5.2.1 Recap: Gaussian boson sampling . . . . . . . . . . . . . . . . . . 136
5.2.2 Recap: Approximate sampling hardness of boson sampling . . . . . 138
5.2.3 Average-case hardness of computing GBS probabilities . . . . . . . 140
5.2.4 Hardness of computation of output probabilities for noisy GBS . . . 142
5.2.5 The complexity of noisy and approximate GBS . . . . . . . . . . . 145
5.2.6 Hardness for computing noisy probabilities in high-dimensional GBS147
5.3 Average-case hardness of computing GBS output probabilities . . . . . . . . 148
5.3.1 Worst-case hardness . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.3.2 Worst-to-average equivalence . . . . . . . . . . . . . . . . . . . . 151
vii
5.4 Average-case hardness of computing noisy GBS output probabilities . . . . . 155
5.5 Average-case hardness of computing noisy probabilities in high-dimensional
GBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.6 Open problems and discussions . . . . . . . . . . . . . . . . . . . . . . . 159
5.6.1 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.6.2 Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . . 161
Chapter 6:The importance of the spectral gap in estimating ground-state energies 163
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.1.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.1.2 Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.1.4 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.2 Definitions and complete problems . . . . . . . . . . . . . . . . . . . . . 183
6.2.1 Complete problems . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.3 Problems characterized by PP . . . . . . . . . . . . . . . . . . . . . . . . 192
6.4 Problems characterized by PSPACE . . . . . . . . . . . . . . . . . . . . . 196
6.5 Other related classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.5.1 Amplification for postQMA . . . . . . . . . . . . . . . . . . . . . 201
6.5.2 Asymmetric promises on spectral gap and uniqueness . . . . . . . . 203
6.5.3 Complexity of PGQCMA, EGQCMA, and PreciseEGQCMA . . . . 205
6.6 The Schrieffer-Wolff transformation . . . . . . . . . . . . . . . . . . . . . 209
6.7 Modified clock constructions with spectral gaps . . . . . . . . . . . . . . . 211
6.8 Precise phase estimation of gapped Hamiltonians . . . . . . . . . . . . . . 222
6.9 Phase estimation in the presence of efficient circuit descriptions . . . . . . . 226
6.10 Details of PP algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
6.11 Turing machine construction for PSPACE-hardness . . . . . . . . . . . . 240
6.12 Complexity of PrecisePGQMA and PreciseEGQMA with asymmetric spec-
tral gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
Appendix A: Mathematical Preliminaries 247
A.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
A.2 Notions of error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
A.3 Notions of simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
Appendix B: Complexity-theoretic basics 251
B.1 Quantum classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
viii
List of Tables
3.1 Easiness and hardness timescales . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Easiness and hardness timescale exponents with ? = 1. . . . . . . . . . . . 49
3.3 Summary of easiness timescales in different regimes . . . . . . . . . . . . . 67
4.1 Complexity classification of dissipative processes . . . . . . . . . . . . . . . 99
6.1 Complexity of variants of the LOCALHAMILTONIAN problem as a function of
the promise gap and the spectral gap . . . . . . . . . . . . . . . . . . . . . 171
ix
List of Figures
2.1 Example of the initial state . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Input and output basis states in 1D . . . . . . . . . . . . . . . . . . . . . 28
2.3 Representation of lattice of vectors with integer coordinates . . . . . . . . . 38
2.4 Complexity phase diagram for free bosons . . . . . . . . . . . . . . . . . . 40
3.1 Slices of the complexity phase diagram for the long-range bosonic Hamiltonian 45
3.2 Illustration of Haah-Hastings-Kothari-Low decomposition algorithm . . . 57
3.3 Illustration of cluster distances . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4 Decomposition scheme for longer easiness timescale in 1D . . . . . . . . . 68
3.5 Protocol to implement a logical circuit . . . . . . . . . . . . . . . . . . . . 72
3.6 Snaking scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.7 Strategies for implementing a SWAP . . . . . . . . . . . . . . . . . . . . 77
3.8 Types-I and -II complexity phase transitions . . . . . . . . . . . . . . . . . 89
4.1 Classical simulability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.2 Complexity phase diagram for a fermionic system with simultaneous pair
losses and gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.1 Graphs representing states of Turing machines . . . . . . . . . . . . . . . 198
6.2 One-qubit phase-estimation circuit . . . . . . . . . . . . . . . . . . . . . 223
6.3 Graphs representing a rejecting computation . . . . . . . . . . . . . . . . 240
B.1 Major complexity classes featuring in the dissertation . . . . . . . . . . . . 256
x
List of Abbreviations
AA Aaronson and Arkhipov
BPP Bounded-error Probabilistic Polynomial time
BQP Bounded-error Quantum Polynomial time
CH Counting Hierarchy
DMRG Density Matrix Renormalization Group
EC1 Easy Class 1
EC2 Easy Class 2
EC3 Easy Class 3
GBS Gaussian Boson Sampling
HHKL Haah-Hastings-Kothari-Low
HOG Heavy Output Generation
IQP Instantaneous Quantum Polynomial time
MPS Matrix Product States
NP Nondeterministic Polynomial Time
P Polynomial Time
#P Sharp P
PGQMA Polynomially Gapped QMA
PH Polynomial Hierarchy
PMP Perfect Matching Permutation
PNR Photon Number Resolving
PP Probabilistic Polynomial Time
QCMA Quantum Classical Merlin Arthur
QCS Quantum computational supremacy
QMA Quantum Merlin Arthur
RCS Random circuit sampling
VQE Variational Quantum Eigensolver
xi
Citations to Previously Published Work
Most of the work appearing in this thesis is either published or has appeared on the
preprint server arXiv. We mention these here.
? Chapter 2: ?Dynamical Phase Transitions in Sampling Complexity?, A. Deshpande,
B. Fefferman, M. C. Tran, M. Foss-Feig, and A. V. Gorshkov, Phys. Rev. Lett. 121,
030501 (2018).
? Further work related to this problem appeared in ?Hierarchy of Linear Light
Cones with Long-Range Interactions?, M. C. Tran, C.-F. Chen, A. Ehrenberg,
A. Y. Guo, A. Deshpande, Y. Hong, Z.-X. Gong, A. V. Gorshkov, and A. Lucas,
Phys. Rev. X 10, 031009 (2020).
? Chapter 3: ?Complexity Phase Diagram for Interacting and Long-Range Bosonic
Hamiltonians?, N. Maskara, A. Deshpande, A. Ehrenberg, M. C. Tran, B. Fefferman,
and A. V. Gorshkov, arXiv:1906.04178 (2019).
? Chapter 4: ?Limits on Classical Simulation of Free Fermions with Dissipation?, O.
Shtanko, A. Deshpande, P. S. Julienne, and A. V. Gorshkov, arXiv:2005.10840 (2020).
? Chapter 5: ?Quantum Computational Supremacy via High-Dimensional Gaussian
Boson Sampling?, A. Deshpande, A. Mehta, T. Vincent, N. Quesada, M. Hinsche,
M. Ioannou, L. Madsen, J. Lavoie, H. Qi, J. Eisert, D. Hangleiter, B. Fefferman, and
I. Dhand, arXiv:2102.12474 (2021).
Other work relating to quantum advantage in experimental contexts has appeared in
? ?Interference of Temporally Distinguishable Photons Using Frequency-Resolved
Detection?, V. V. Orre, E. A. Goldschmidt, A. Deshpande, A. V. Gorshkov, V.
Tamma, M. Hafezi, and S. Mittal, Phys. Rev. Lett. 123, 123603 (2019).
? ?Quantum Approximate Optimization of the Long-Range Ising Model with a
Trapped-IonQuantum Simulator?, G. Pagano, A. Bapat, P. Becker, K. S. Collins,
A. De, P. W. Hess, H. B. Kaplan, A. Kyprianidis, W. L. Tan, C. Baldwin, L. T.
Brady, A. Deshpande, F. Liu, S. Jordan, A. V. Gorshkov, and C. Monroe, Proc.
Natl. Acad. Sci. USA 202006373 (2020).
? Chapter 6: ?The Importance of the Spectral Gap in Estimating Ground-State Ener-
gies?, A. Deshpande, A. V. Gorshkov, and B. Fefferman, arXiv:2007.11582 (2020).
xii
Chapter 1: Introduction
?The theory of computation has traditionally been studied
almost entirely in the abstract, as a topic in pure mathematics.
This is to miss the point of it. Computers are physical objects,
and computations are physical processes. What computers can
or cannot compute is determined by the laws of physics alone,
and not by pure mathematics.?
? David Deutsch [1]
Physics, to slightly overgeneralize, is humanity?s quest to understand the natural world.
Numerous times in the history of this quest, ideas and techniques from disparate fields in
the natural sciences have found their way into the physicist?s toolkit because of their utility.
Conversely, important mathematical tools have been invented primarily in order to address
problems arising in physical contexts. This cross-fertilization of ideas between physics and
the other sciences is in of itself a valuable cultural activity, notwithstanding its scientific,
technological, and ultimately economic value.
Similarly overgeneralizing, computer science is the quest to understand what is (ef-
ficiently) computable in our world. A cornerstone of theoretical computer science is the
extended Church-Turing thesis. This is the observation that various models of computa-
tion modeled on the physical world seem able to simulate each other with small overhead,
1
and implies that a problem efficiently solvable according to one computational model is also
efficiently solvable according to any other model. This observation enables computer sci-
entists to reason about the efficiency of computation by abstracting away the details of the
computational model at hand. Ultimately, the thesis also lends meaning to the question of
what is efficiently computable or not. This is because if the notion of efficiently computable
depended heavily on the computational model, it would cease to be a universal one.
The goals of physics and computer science are very much aligned. The question of
what can be efficiently computed in our world depends on what kinds of devices are physically
possible to realize in our world, which ultimately must obey the laws of physics. Therefore,
this question is as much a question in physics as it is one of interest to computer science, as
is elegantly argued for in Refs. [1; 2], for example.
When quantum mechanics enters this picture, the relation between physics and com-
puter science becomes much more interesting. Despite quantum mechanics already having
been formalized by 1930, it was not realized until the 1980s that quantum-mechanical com-
puters could potentially be faster than classical (i.e., ordinary) computers at certain tasks,
such as simulating physics [3]. This realization stemmed from the observation that tasks
such as simulating quantum systems seem hard on classical computers. Therefore, the exis-
tence of quantum computing changes the meaning of what is efficiently computable in our
physical world, which is an exciting development for both computer science and physics. We
elaborate on this point of view in the coming sections.
2
1.1 Why Complexity theory
One branch of computer science that acquires new life on account of quantum com-
puting is complexity theory. This is the branch of theoretical computer science that deals
with quantifying the resources required to solve various problems and classifying problems
into various complexity classes. The efficiency of an algorithm is measured in terms of the
asymptotic scaling of the algorithm?s runtime as a function of the size of the problem. It
is of interest to determine the optimal algorithm for solving a particular problem, i.e. the
algorithm with minimum asymptotic scaling of the runtime. However, the runtime for a
problem can depend on the model of computation used by virtue of the cost associated with
elementary operations. This issue is addressed by the extended Church-Turing thesis, which
asserts that differing models of computation can simulate each other with an overhead at
most polynomial in the problem size. Due to this reason and the fact that polynomials
are closed under composition, we take a problem to be efficiently solvable if the optimal
algorithm has runtime polynomial in the problem size (under any model of computation).
Classifying the complexity of problems becomes even more interesting in the context of
quantum computing, since quantum computing can change the meaning of what is efficiently
computable in our world. This is because of the possibility that quantum computers are not
efficiently simulable by classical computers, which would falsify the extended Church-Turing
thesis in the quantum setting, meaning that tasks efficiently solvable by quantum computers
are not necessarily efficiently solvable on classical computers. A major undertaking in the field
of quantum computer science is thus to precisely characterize the class of problems efficiently
solvable by quantum computers. We turn to the potential benefits such an undertaking can
3
have.
First, as has been emphasized above, understanding the power and limitations of quan-
tum computers is tantamount to understanding the fundamental limitations of computing
according to the laws of physics. This is because, according to our best guess for how the
laws of physics would work, quantum field theories and as-yet-undiscovered theories gov-
erning quantum gravity might be efficiently simulable with quantum computers1[4], which
can be regarded as a statement of the quantum extended Church-Turing thesis. Apart from
being an extremely interesting physics question in itself, it also gives rise to other questions
in physics. An example is that of black hole physics, where a complexity-theoretic point of
view has led to several creative ideas and proposals in the field, see for instance Refs. [5?8].
Second, this quest can lead to the identification of new, practical problems that quan-
tum computers can efficiently solve. Indeed, the whole premise of the quest is that the set
of problems efficiently solvable on quantum computers (BQP) is potentially larger than that
solvable on randomized classical computers (BPP)2. In order to understand the power of
BQP, it is necessary to identify these problems. It is possible that at least a few of these are
of practical use, paving the way for applications of quantum computers.
Third, another practical benefit from understanding the boundaries of BQP is know-
ing the limitations of quantum computers. As a ?meta-benefit?, knowing the limitations
of quantum computing saves time for practitioners and quantum algorithms researchers.
Other applications are in designing protocols for classical communication and verification of
quantum computing that are secure against quantum-capable adversaries [9].
1Meaning that a hypothetical ?quantum gravity computer? is possibly no more powerful than a quantum
computer.
2For an overview of some of the complexity classes discussed in this dissertation, refer to Appendix B.
4
This dissertation attempts at capturing some of the rich physics coming out of ex-
amining quantum systems from a complexity-theoretic point of view. Specifically, we will
consider the problem of simulating quantum systems in the sense of both dynamics and
equilibrium. The complexity of solving this problem depends on features of the quantum
system and can be classified into various classes. Studying and classifying the dependence
of the complexity on system parameters often yields a ?complexity phase diagram?, which
we elaborate upon later in the dissertation. These phase diagrams are also associated with
complexity phase transitions, illustrative of the physics underlying the quantum system at
hand.
1.2 Sampling Complexity
When we study the classical complexity of simulating quantum systems, it is prudent
to pay attention to the precise notion of the task being considered. First off, we note that we
cannot demand that the classical computer output an explicit representation of the quantum
state of the system, since it is infeasible even to write out 2n complex amplitudes of an n-qubit
system for large n ? 60. Moreover, for large system sizes, an experiment cannot feasibly
access any particular amplitude or probability, let alone 2n of them. Therefore, we should
phrase the problem in a way that captures what an experimentalist can access. An alternative
is to demand that a classical computer be able to efficiently compute (to a suitable accuracy)
all k-body correlation functions (k-local observables) for any constant k. This task can
be efficiently solved by running the experiment. However, this does not capture everything
about a quantum experiment, as we are about to elucidate. Such correlation functions cannot
5
capture correlations present across all n particles of a system that are vital to the physics.
One way to put the classical simulator and a real quantum experiment on the same
footing is to notice that every such quantum experiment is inherently outputting a sample
from an underlying distribution describing the probabilities of the 2n different outcomes.
We also allow for the classical algorithm to make some slight error in the distribution it is
sampling from, measured in terms of the total variation distance between distributions3. This
defines the task of approximate sampling. Approximate sampling is a nice way of characterizing
the task of simulating quantum systems because it satisfies both desiderata of being not too
powerful and being able to capture all that an experiment can capture. This is because
the total variation distance characterizes the ability of a referee to distinguish between two
distributions. Therefore, a small total variation distance between the simulated and the
actual distributions implies that the two cannot be reliably distinguished, constituting a
successful simulation. Moreover, approximate sampling from the output distribution when
measuring in a certain basis also enables one to compute local observables diagonal in that
basis4. Therefore, it is appropriate to study the classical complexity of approximate sampling5.
Fortunately, for the notion of sampling from a distribution, there is some evidence
that classical computers cannot efficiently simulate certain quantum systems that are oth-
erwise simulable on quantum computers. This statement constitutes evidence against the
extended Church-Turing thesis. For the case of sampling exactly from the target distribu-
tion, early work by Terhal and DiVincenzo [11] gave some complexity-theoretic evidence
3For two distributionsD1 andD2 o?ver a space of outcomesX with associated probabilities p1(x) and p2(x),
the total variation distance is given by x?X |p1(x)? p2(x)|.
4In fact, sampling from a distribution is exactly what many experiments are set up to do, and local observables
are computed a posteriori from the observed samples.
5There is, however, one significant drawback to the total variation distance measure. Verifying that two
distributions are close under this measure can take exponentially many samples [10].
6
for the hardness of this task. Subsequent work by Bremner et al. [12] on Instantaneous
Quantum Polynomial time (IQP) circuits and by Aaronson and Arkhipov [13; 14] on boson
sampling strengthened this complexity-theoretic evidence. They showed that the existence
of a classical algorithm for exact sampling implies the falsity of a widely-believed conjecture
in complexity theory, namely the non-collapse of the so-called Polynomial Hierarchy (PH).
The non-collapse of the PH is a conjecture that generalizes the P ?= NP conjecture. Briefly
speaking, the polynomial hierarchy is defined to be an infinite tower of complexity classes
generalizing NP, with ?levels? starting from P, NP, and so on, each level more complex than
the previous. The conjecture asserts that the tower is truly infinite, in that no two levels are
equivalent to each other. Aaronson and Arkhipov also proved, modulo a few other conjec-
tures, that a classical algorithm to approximately sample from certain quantumly sampleable
distributions would imply the collapse of the PH.
At a high level, in order to prove the hardness of exact sampling, it suffices to show that
computing any single output probability to multiplicative precision6 is #P-hard. If there is a
classical sampling algorithm, then one can, using the tool of Stockmeyer counting [15; 16],
compute a multiplicative approximation to the output probability within the third level of
the polynomial hierarchy. This thereby solves any P#P problem7 using access to resources
in the third level of the polynomial hierarchy. The contradiction then follows from the fact
that the entirety of the polynomial hierarchy (i.e. each level of the PH) is contained in P#P
[17], which causes the PH to collapse.
The proof of Aaronson and Arkhipov for the hardness of approximate sampling is along
6Refer to Appendix A for an overview of these mathematical definitions.
7For two complexity classes A and O, the notation AO refers to the class of problems solvable in the class
A with access to an oracle for any problem in the class O. An oracle for a class correctly solves problems in
that class in one time step.
7
the same lines but involves new ideas. They show that hardness of approximate sampling
follows from the #P-hardness of approximately computing a random output probability to a
certain additive error. In other words, showing the average-case hardness of estimating out-
put probabilities (as opposed to worst-case hardness in the previous case) is enough. This step
necessitates considering a family of random instances, which serves to define the ensemble of
output probabilities. The additional conjectures made by Aaronson and Arkhipov serve to
give evidence for this statement on average-case hardness. The first conjecture, namely anti-
concentration, asserts that a randomly selected output probability is not too small compared
to the average output probability. The second conjecture states that there is an equivalence
between average-case and worst-case hardness of computing to multiplicative error the out-
put probability of the class of quantum circuits. These conjectures have since been ported
to other contexts and are by now standard assumptions in the so-called ?quantum compu-
tational supremacy? literature [18?24]. While anticoncentration has been proven in some
contexts (but not yet for boson sampling) [19; 25?29], the average-case hardness conjec-
ture has remained unproven, although recent work has come tantalizingly close to a proof
[30?32].
In order to conclude that the extended Church-Turing thesis is violated, we need
to demonstrate that it is possible to quantumly sample from the distributions considered
above. Assuming that quantum mechanics as we know it, with its exponentially large Hilbert
space and the possibility for quantum error correction, is valid, we can theoretically infer
that the extended Church-Turing thesis is violated. However, a skeptic may ( justifiably)
raise the issue that quantum mechanics is not sufficiently validated in the so-called ?high
complexity? regime, and that the exponentiality of Hilbert space has not been proven to
8
be necessary. Therefore, only by performing an actual experiment and demonstrating that
quantum mechanics allows one to sample from the distributions mentioned above can we
satisfactorily claim to have refuted the extended Church-Turing thesis. This is what recent
notable experiments have claimed to do [33; 34]. The situation is very reminiscent of Bell
tests, which aim to give experimental evidence for quantum nonlocality and only recently
were performed in a loophole-free manner.
We now turn to noise, an ever-present feature of any quantum experiment. In order
to implement an experiment that achieves a small total variation distance as the system size
increases, one should either depend on quantum error correction or ensure that the noise
strength decreases with system size [35], an unrealistic assumption. Recent developments
in the field have nevertheless attempted to demonstrate a quantum advantage over classical
computers via different means. One route is to move away from the task of approximate
sampling to the task of scoring high on a test intended to capture the similarity of the sampled
distribution and the original distribution via the so-called (linear) cross-entropy measure in
the context of random circuit sampling [33; 36]. Outputting samples that achieve high scores
on this test is conjectured to be classically hard [37; 38], although it can be potentially simpler
than the task of approximate sampling [39]. There is another nice feature of the linear cross-
entropy measure. Under the (strong) assumption that the noise model is given exactly by
global depolarization, the linear cross-entropy measures the fidelity of the experiment [36].
Under the same assumption, sampling exactly from the noisy distribution with any nonzero
fidelity is also classically hard [33].
9
1.3 Dynamics
We now focus on the problem of simulating quantum dynamics. By the quantum ex-
tended Church-Turing thesis, efficiently simulating quantum dynamics seen in nature should
always be possible on a universal quantum computer. This is indeed true by virtue of im-
pressive advances in quantum algorithms for Hamiltonian simulation over the last decade
[40?48]. Therefore, studying the classical complexity of this problem is perhaps a more
interesting venture if one is only concerned with the existence of (in)efficiency and not its
degree.
In this dissertation, we study the classical complexity of simulating quantum dynamics
for some quantum systems and investigate when the problem is either solvable in time poly-
nomial in the problem size (easy) or not (hard). Studying the boundary between easy and
hard with regard to the problem of simulating quantum dynamics gives a characterization
of when the problem is classical or has non-classical, quantum features. In other words, the
classical simulability of a quantum system is a measure of non-classicality, or ?quantumness?.
Several quantum experiments, at their heart, involve preparing a fixed initial quantum
state |?0?, evolving it under a potentially time-dependent sys{tem H?amiltonian}H for time
t, and making measurements on the final state |?t? = T exp ? t[ i dsH(s)] |?0?. The0
initial state is often simple to prepare, which we take to be a computational basis state for
simplicity. Similarly, the measurement at the end is also a simple-to-implement one, and
we take the measurement to be in the computational basis. These choices only mildly affect
the analysis of the complexity, if at all, since other initial states and choices of measurement
10
bases can be accounted for with small-depth unitaries8.
With this setup, we consider the question of how hard it is to approximately sample
from the state9 |?t? as a function of time t, which may be a function of the system size n.
Depending on the quantum system and Hamiltonian, for long enough times the dynamics
can prepare states that are classically hard to sample from, assuming the conjectures in the
previous section. Since, as we have defined it, the notions of ?easy? and ?hard? are mutually
exclusive, at each time t(n) the system is either easy or hard to simulate. Further, since the
system evolves from an easy-to-sample-from state to a hard-to-sample-from one, it must
have transitioned from easy to hard at some timescale t?(n). This timescale t?(n) is what
we call the transition timescale, and we call the transition between easy and hard a dynamical
phase transition in sampling complexity.
We propose using the transition timescale as a measure of how ?quantum? or how
?complex? a class of Hamiltonians is. This makes sense because different Hamiltonians may
have different timescales for the transition from easy to hard. This transition timescale
may be compared to other indicative timescales of quantum dynamics such as the Ehrenfest
time, scrambling time, time to reach maximal entanglement, time to reach maximal circuit
complexity, and equilibration time. The transition timescale defined here may differ or
coincide with some of these other timescales.
Is the phase transition between classically easy and classically hard cases a ?physical?
one? In other words, is there a physically interesting characterization of the timescale or some
8In restricted models of quantum computation, these choices can and do affect the complexity. An example
is that of IQP circuits [12], where there is a strong basis-dependence in the classical complexity. When this
happens, we take care to clarify the set of initial states and measurement bases our result applies to.
9We have in mind an infinite family of initial states and Hamiltonians defined for an infinite number of
system sizes.
11
physics underlying the transition? We seek to answer these questions in this dissertation
by taking as example a system of bosonic particles evolving under lattice Hamiltonians. In
Chapter 2, we study the transition timescale as a function of system parameters for a system of
free bosons hopping on a lattice. We also study the case where the Hamiltonian is promised
to be Anderson-localized [49], and show that the system stays classically simulable for all
times, or in other words, the transition timescale diverges. In Chapter 3, we continue this
study by considering interactions and long-range hopping. For a wide class of interactions,
we see that there is almost no dependence of the transition timescale on the presence of
interactions or their strength. This illustrates that the complexity phase transition is robust
to perturbations and has physical meaning.
The examples we have seen here do not explicitly model possible experimental noise;
so far, we have only accounted for noise in terms of the total variation distance error be-
tween the target distribution and the actual distribution. As mentioned above, in order to
experimentally realize approximate sampling with small total variation distance, one needs
quantum error correction. One can make strong assumptions on the noise model in order
to weaken the experimental requirement of achieving a high fidelity. However, this state of
affairs may be unsatisfactory if the intention of the experiments is to provide a refutation of
the extended Church-Turing thesis. This is because the assumption of the noise model is
justified by quantum mechanical models of the experiment, however it is these very models
whose validity is being tested in an experiment purporting to demonstrate a quantum advan-
tage, as argued in the previous section. Nevertheless, our purpose here is not to invalidate
the extended Church-Turing thesis but merely to study, assuming the validity of quantum
mechanics in all regimes, what features make a quantum system easy or hard to simulate.
12
Therefore, considering the hardness of sampling from the noisy distribution under a suitable
noise model is acceptable. Moreover, such a study is more realistic than that of the idealized
noiseless case.
We take such an approach in Chapters 4 and 5. In Chapter 4, we consider the com-
plexity of simulating otherwise-free fermions subject to Markovian noise described by lin-
ear/quadratic Lindbladian jump operators. We classify the complexity of simulating the
dynamics according to the type of jump operators present. Interestingly, in some cases it
is the dissipation that leads to classical hardness of simulation, since free fermions without
dissipation are known to be classically simulable [50?52]. In Chapter 5, we study the com-
plexity of Gaussian boson sampling, the problem of sampling from the output distribution of
a linear-optical network when certain Gaussian states are sent in as input [53; 54]. Our main
aim in this chapter is to gather evidence for the persistence of a computationally complex
quantum signal in the output distribution. To this end, we consider the problem of simulat-
ing lossy Gaussian boson sampling instances and prove that there is a sign of computational
hardness in the output distribution. On the way, we also improve the theoretical analysis for
idealized (lossless) Gaussian boson sampling and bring it up to the same standard as that of
boson sampling.
1.4 Equilibrium
Simulating and predicting dynamics is far from being the only activity of interest to a
physicist; predicting equilibrium properties of a wide variety of physical systems is an equally
important task. This task involves computing properties of Hamiltonians with respect to the
13
( )
thermal (Gibbs) state at temperature T , ? = e??H/Tr e??H (. Spec)ifically, tasks such as
computing correlation functions, the partition function Z = Tr e??H , the free energy and
entropy, and Gibbs sampling are all of interest in various fields of physics. The last of these,
Gibbs sampling, or sampling from the thermal state ?, is a task whose solution allows for
solving the other tasks as well [55; 56].
In computer science as well, Gibbs sampling is of interest in several contexts. One of
these is in machine learning, through the use of restricted Boltzmann machines. Another is
the matrix multiplicative weights update method, which is used in conjunction with Gibbs
sampling [57] and is a useful subroutine in numerous algorithms (see [58] for a survey).
A special case of Gibbs states is that of states at zero temperature, or ground states.
These are the minimal-energy eigenstates of the Hamiltonian. It is not an overstatement
to claim that a majority of condensed matter physics deals with ground states, ground-state
properties, excitations above the ground state and ground-state phase diagrams. Even in
high-energy physics, finding ground-state (vacuum) properties and the spectral gap (mass
gap) is among the first steps in analyzing a theory.
In computer science, constraint satisfaction problems can bemapped to classical Hamil-
tonians so that the ground state of the Hamiltonian encodes the answer to the constraint
satisfaction problem. In turn, constraint satisfaction problems encode a wide variety of op-
timization problems across science, industry, and more generally, all aspects of human life.
Hence, developing and improving algorithms for finding ground states, or low-energy states,
of Hamiltonians would be immensely useful.
In this dissertation, we seek to understand the complexity of computing ground-state
properties and preparing ground states. Unlike the case of quantum dynamics, this problem
14
can also be hard for quantum computers since it is not necessarily the case that nature can
efficiently find ground states. There can be three different ?phases? for this problem, namely
i) classically (and quantumly) easy, ii) classically hard but quantumly easy, and iii) quantumly
(and classically) hard. Understanding the boundaries between these phases would help give
insight into what features of a Hamiltonian enable efficient preparation of its ground states.
Additionally, akin to our studies in Chapters 2 to 4, this endeavor would result in a complexity
classification.
The easiness/hardness of preparing ground states is studied in the field of Hamiltonian
complexity, which is at the intersection of quantum information, many-body physics, and
complexity theory. As a proxy for the hardness of preparing the ground state, we study the
problem of finding ground-state energies10. Kitaev [59] showed that finding the ground-
state energy of a local Hamiltonian is QMA-hard in general. Since then, there has been an
effort to understand the complexity of this problem in more physically natural settings [60?
63]. There has also been some effort on the algorithms side to provide heuristic quantum
algorithms for the problem, such as the variational quantum eigensolver (VQE) [64]. It is
an open question to rigorously formulate conditions under which these algorithms can be
successful.
In Chapter 6, we take some initial steps toward finding conditions that provably affect
the complexity of computing ground-state properties. Specifically, we identify two structural
properties of Hamiltonians that affect the complexity of finding their ground-state energies.
These are the spectral gap and the existence of a useful classical description of the ground
10Since it is possible to efficiently find the ground-state energy with access to the ground state, the hardness
of finding ground-state energies implies the hardness of preparing ground states.
15
state. Our results lead us to make an interesting conjecture regarding the preparation of
gapped ground states. This conjecture, if true, would explain the efficacy of the variational
quantum eigensolver at preparing ground states of Hamiltonians.
1.5 Outline
The outline of this dissertation is as follows.
In Chapter 2 titled ?Dynamical phase transitions in sampling complexity?, we study the
dynamics of n bosonic particles evolving under a class of nearest-neighbor bosonic Hamilto-
nians. We identify upper and lower bounds on the transition timescale by giving a sampling
algorithm for short times and proving the hardness of the dynamics at longer times. As
a corollary, we also obtain results for Anderson localized systems, for which the transition
timescale diverges.
In Chapter 3 titled ?Complexity phase diagram for interacting and long-range bosonic
Hamiltonians?, we improve upon these results in two ways. First, we tighten the bounds on
the transition timescale from the previous chapter using a new technique. The same tech-
nique also allows us to generalize to the case of interacting bosons with long-range hopping.
We observe that the phase transition is stable to perturbations such as the addition of inter-
actions and change in their strength. Considering the transition timescale as a function of
how long-range the system is also allows us to draw a complexity phase diagram.
In Chapter 4 titled ?Complexity of fermionic dissipative interactions and applications to
quantum computing?, we classify the complexity of simulating the dynamics of free fermions
subject to dissipation described by linear and quadratic Lindbladian jump operators. We
16
classify different jump operators based on the classical complexity of the dynamics, which is
either classically easy or universal for quantum computing. We elaborate on the design of an
entangling gate using dissipation by giving an example with fermionic atoms.
In Chapter 5 titled ?Quantum computational supremacy via high-dimensional Gaus-
sian boson sampling?, we improve the available evidence for hardness of Gaussian boson
sampling. We also show that computing output probabilities of a lossy Gaussian boson sam-
pling instance is hard on average. We propose an architecture for Gaussian boson sampling
designed to balance the competition between loss and connectivity.
Finally, in Chapter 6 titled ?The importance of the spectral gap in estimating ground-
state energies?, we give a complexity classification of the problem of precisely estimating
ground-state energies of Hamiltonians with or without a spectral gap. We obtain evidence
that the spectral gap can make the problem easier. We also make an interesting conjecture
on the relation between spectral gaps and the existence of polynomial-sized quantum circuits
to prepare ground states.
The appendices contain relevant mathematical definitions and notation used in the
dissertation and an overview of the basics of complexity theory.
17
Chapter 2: Dynamical phase transitions in sampling complexity
In the quest towards building scalable and fault-tolerant quantum computers, demon-
stration of a quantum speedup over the best possible classical computers is an important
milestone and is termed quantum computational supremacy [65]. There are several candidates
for tasks where such a speedup could be demonstrated [11?13; 18?22; 37; 51; 66; 67], where
the problem is to simulate a quantum system in the sense of approximate sampling. How-
ever, there has also been some debate about the required system size before one can claim
quantum computational supremacy, due to improved simulation techniques and algorithms
[68?70]. In this Chapter, we consider the impact of the field of quantum computational
supremacy on other areas of physics and show that studying the complexity of simulating
physical systems is useful for understanding phase transitions.
Here we consider the classical complexity of approximate sampling, referred to as ?sam-
pling complexity?. This is the task of producing samples from a distribution close to the
probability distribution occurring in a quantum system upon measurement in a standard ba-
sis. This task is a good notion of what it means to simulate physics on a classical computer
since it captures how well a computer can mimic an experiment in which one can measure
the output at several sites. When we consider sampling complexity as a function of system
parameters, the system can be classified as easy in some regimes and hard in some others.
18
Since the designations ?easy? and ?hard? are exhaustive and there is no smooth way to go from
one regime to another, we posit that this transition from easy to hard happens abruptly, a
phenomenon very reminiscent of phase transitions. Just like order parameters are zero on
one side of a phase transition and nonzero on another, sampling complexity is different on
either side of the transition, and can be used to draw phase boundaries as a function of time
and other system parameters. These boundaries can be different from those drawn by more
conventional order parameters, signifying new physics in otherwise well-studied systems.
Indeed, phase transitions in average-case complexity have been studied both in the classical
[71] and quantum regime [72].
In this Chapter, we show that transitions in sampling complexity [73] are indicative
of physical transitions. We consider a system of n bosons hopping from one site to another
on a lattice of m sites and study the sampling complexity as a function of time for an initial
product state. We show that it goes from easy to hard as the scaling of evolution time
t with the number of bosons n increases, thereby exhibiting a dynamical phase transition
[74; 75]. We find that the timescale at which the complexity changes is the timescale when
interference effects start becoming relevant. We conjecture that in general, this is linked to
the Ehrenfest timescale at which quantum effects in a system become considerable [76]. We
also show that systems in the Anderson-localized phase are always easy to simulate. We use
the Lieb-Robinson bound [77] as an ingredient in our proof of easiness.
19
Figure 2.1: An example of the initial state in d = 2 dimensions. Here m = 96, n = 4,
? = 3 and c1 = 3/2. The black circles represent sites with a single boson. The cyan circles
represent the ancillas.
2.1 Setup
The model consists of free bosons hopping on a lattice in d dimensions (denoted dD)
with sites labeled by indices i, j. Our results in this Chapter can be applied to linear optics
as well, wi?th the bosonic sites being replaced by photonic modes. The Hamiltonian is given
by H = mi,j=1 Jij(t)a
?
iaj , where a
?
i is the creation operator of a boson at the i?th site.
J(t), which can be time-dependent in general, is an m?m Hermitian matrix that encodes
the connectivity of the lattice. One way to show hardness in the boson sampling proposal
by Aaronson and Arkhipov (AA) [13] is to generate any linear optical unitary U acting on
the bosonic sites, in particular, Haar-random unitaries. Any U can be generated through a
free boson Hamiltonian by taking H = i logU and evolving it for unit time. However, this
20
Hamiltonian can require arbitrarily long-range hops on the lattice in general. Since such
long-ranged Hamiltonians may not be realistic, we consider the sampling complexity of a
bosonic Hamiltonian with nearest-neighbor hops. Jij is nonzero only if i = j or i and j
label adjacent sites. We further restrict Jij to satisfy |Jij| ? 1 in order to set an energy scale.
Our proofs remain valid even if we allow the diagonal terms Jii to be unbounded.
One can efficiently solv?e the equations of motion ia?
?
i (t) = [a
?
i (t), H(t)] on a classical
computer to obtain a?i (t) = k a
?
k(0)Rki(t) for some transformation matrix R. From here
onward, we shall take R(t) to be the input to the problem, since it can be determined from
the input Hamiltonian H and time t in time poly(m, log t).
The m sites in the problem are numbered from 1 to m, and together with n ancilla
sites, are arranged in a lattice of side length (m + n)1/d in d dimensions. The initial state
has n bosons equally spaced in the lattice as shown in Fig. 2.1. We take m = c n?1 , where
? controls the sparsity of occupied sites in the lattice and can be set to 5 as required for
the hardness of boson sam(pling)[13]. The minimum spacing between any two bosons in
the initial state is 1/d 1/d ??12L = m+n > c n d1 . The quantity L is an important lengthn
scale in the problem. The ancillas in the lattice, marked in cyan, are not counted as part of
the m sites and are present in order to accelerate the time required to construct an arbitrary
unitary, which is useful for the hardness result. Their presence does not change the scaling
of quantities like L with n.
The input states are described by vectors of the form r = (r1, . . . , rm), specifying the
number of bosons on each site, so that r1+ . . . rm = n. Measurement in the boson number
basis defines a distribution DU , which we aim to sample from. The probability of finding an
21
output state s = (s1, s2, . . . , sm) is given by
Pr 1[s] = |Per(A)|2, (2.1)
DU r!s!
where r! := r1! . . . rm! (with s! defined similarly), An?n is a matrix formed by taking si
copies of the i?th column and rj copies of the j?th row of R in any order, and Per(A) denotes
the permanent of A (see Ref. [78] or Section 2.5 for details).
For the particular choice of initial states described in Fig. 2.1, the task is to sample
from a distribution that is close to DU in variation distance when given a description of the
unitary R(t). We now formalize the notion of efficient sampling.
Definition 1. Efficient sampler: An efficient sampler is a classical randomized algorithm that
takes as input the unitary Rij and outputs a sample s from a distribution DO such that the
variation distance between the distributions ? = ?DO ? DU? ? O( 1poly ), in runtime(n)
poly(n)1.
We call the sampling problem easy if there exists an efficient sampler for the problem
in the stated regime. Conversely, the problem is hard if there cannot be an efficient sampler.
Since a negative statement such as the inexistence of an algorithm is difficult to prove, our
practical definition of hardness of a sampling problem is if it is at least as hard as boson
sampling. This enables us to use the results of AA [13] to claim the hardness of sampling in
some regimes. In doing so, our hardness results ultimately rely on the truth of AA?s conjec-
tures. One of these conjectures concerns the hardness of additively approximating |Per(G)|2
1Our easiness results also apply to a stronger notion of approximate sampling, namely that the algorithm
samples from an ?-close distribution in runtime poly(n, 1/?).
22
for Gaussian-random G with high probability (for which AA give reasonable evidence). The
second, more widely believed conjecture is that the polynomial hierarchy, an infinite tower
of complexity classes, does not ?collapse?, i.e. is truly infinite.
We restrict our attention to two special cases where we can show the existence of an
efficient sampling algorithm: i) when the system evolves for a time smaller than the system
timescale L/v (where v is the Lieb-Robinson velocity of information spreading in the lattice,
defined more precisely in Eq. (2.2)), and ii) when there is Anderson localization in the system
[49]. These two cases correspond to a promise on the input unitary R. We now state our
main results.
Theorem 1.A (Easiness of simulation at short times). For ? > 1 and for all dimensions d, the
sampling problem is easy for all t ? 0.9L/v, i.e. ?t ? c n(??1)/d2 for some constant c2.
The intuition behind this Theorem is that when the time is smaller than the Lieb-
Robinson timescale of particle interference L/v, the dynamics is approximately classical (in
the sense that the particles are distinguishable).
An important technical achievement of this Chapter is to prove rigorously that this
intuition is correct. This is done in Lemma 3 by showing that the approximation works.
Theorem 1.B (Hardness of simulation at longer times). (based on Theorem 3 of Ref. [13])
When t = ?(n1+?/d)), the sampling problem is hard in general.
For this result, we show that we can apply any unitary R after the stated time. There-
fore, if an efficient sampler exists for this problem, then we have an efficient boson sampler,
which cannot exist by our assumption.
23
The result for the case of Anderson localization comes out as a corollary from Theo-
rem 1.A:
Corollary 2 (Easiness of Anderson-localized systems). For Anderson-localized systems in any
dimension d, the sampling problem is easy for all times.
The easiness of sampling for Anderson-localized systems is analogous to results show-
ing efficient simulation of localized systems according to various definitions [79?83].
2.2 Easiness at short times
In this section, we prove Theorem 1.A and Corollary 2. First, let us examine the
promise we have on the unitary R in both cases. We use the Lieb-Robinson bound [77] on
the speed of information propagation in a system. Applying the bound to our Hamiltonian,
we get
( ( ))
| vt? ?ij[ai(t), a?j(0)]| = |Rij(t)| ? min 1, exp , (2.2)?
where ?ij is the distance between two sites i and j, v is the upper bound to the velocity of
information propagation called the Lieb-Robinson velocity and ? is a length scale). Note
that the results from Ref. [84] do not apply since we have free bosons here and we work in
the single-particle subspace. The Lieb-Robinson velocity is at most 4(1 + 2de) [85] when
|Jij| ? 1 and ? = 1.
When the Hamiltonian is Anderson-localized, the unitary R satisfies the following
24
promise at all times [86]:
( )
|Rij| ? exp
??ij
. (2.3)
?
Here, ? is the maximum localization length among all eigenvectors. Equation (2.3) can be
viewed as a consequence of a Lieb-Robinson bound with zero velocity [87]. On account of
the zero-velocity Lieb-Robinson bound, all results for the time-dependent case can be ported
to the Anderson localized case, setting v = 0.
We give an algorithm that efficiently samples from the output distribution for short
times t < 0.9L = O(n(??1)/d), given the promise in Eq. (2.2). The algorithm outputs a
v
sample from a distribution DDP , obtained by assuming that the bosons are distinguishable
particles, ignoring the effects of interference. The algorithm is described in more detail in
Section 2.6.
2.3 Analysis
We prove the correctness of the algorithm by showing that the variation distance be-
tween the distributions is upper bounded by an inverse exponential in n. When the bosons
are distinguishable, their dynamics is given by a Markov process, described by the matrix
Pkl = |R(t)|2kl. The probability of getting an outcome s is given by
?
Pr 1[s] = Pin1,out P?(1) in2,out . . .PD s! ?(2) inn,out , (2.4)?(n)DP
?
25
where the sum is over all permutations ? mapping the input bosons to the output ones. In
the above equations, ini is the site index of the i?th boson in nondecreasing order in the input
and out is defined similarly at the output (see Section 2.5 for an example). We now state a
result on how close DDP is to the true distribution DU .
Lemm[a 3. When ? > 1 and t]? 0.9L/v, the variation distance satisfies ?DDP ? DU? =
O(exp 2vt?L + 2(d? 1) logL ).
?
This Lemma makes intuitive sense: because of the Lieb-Robinson bound Eq. (2.2)
and the fact that the initially occupied sites are separated by a minimum distance ?(L) from
each other, it takes a time t = ?(L/v) for the bosons to start interfering considerably.
Therefore, the classical and quantum distributions agree exponentially closely in L when
t ? 0.9L/v. For a proof of Lemma 3, see Section 2.6. Assuming this Lemma, we now
show Theorem 1.A.
Proof of Theorem 1.A. Lemma 3 shows that the algorithm samples from a distribution with
exponentially small error in n, since L = ?(n(??1)/d). To complete the proof of Theo-
rem 1.A, we need to show that the runtime of the algorithm is polynomial in n. This is
true because the corresponding Markov process of n distinguishable bosons walking on m
sites for one step is efficiently simulable: for each of the n particles, we select one among
the m sites to walk to, based on the matrix elements of P . This takes time O(n ? poly(m))
= O(poly(n)) to simulate on a classical computer.
26
2.4 Hardness at longer times
If we allow the system to evolve for a longer amount of time, we can use the time-
dependent control to effect any arbitrary unitary and implement any boson sampling instance
in the system. We can perform phase gates on a site k by setting Jkk to be nonzero for a
particular time, with the hopping terms and other diagonal terms set to zero. We can apply a
nontrivial two-site gate betwee?n adjacen?t sites, for example the balanced beamsplitter unitary?1 ?1?
on the sites 1 and 2, U = ?1 ?? ??, by setting H = ?i(a?a ?1 2 ? a1a2) for time t = ? .2 4
1 1
One can also apply arbitrary unitaries using arbitrary on-site control Jii(t) and fixed, time-
independent nearest-neighbor hopping Jij(t) = 1.
Using the constructions in Refs. [88; 89], we can effect any arbitrarym?m unitary on
the m sites with O(m2) depth from a nontrivial beamsplitter and arbitrary single-site phase
gates. A construction of AA that employs ancillas to obtain the desired final state rather than
applying the full unitary on all the sites can be used to achieve a depth O(nm1/d). Each of
the n columns of the unitary are implemented in time O(m1/d), which corresponds to the
timescale set by the Lieb-Robinson velocity and the distance between the furthest two sites
in the system2.
Proof of Theorem 1.B. From the above, when ?t = ?(n1+ d ), we see that we can effect any
arbitrary unitary. This implies that an efficient sampler for this regime can also be used
as an efficient boson sampler, which is widely believed not to exist because of AA?s results
[13].
2AA?s construction applied each column of the unitary in O(logm)-depth, whereas we can only apply it in
O(m1/d) depth because of the spatial locality of the Hamiltonian and the Lieb-Robinson bounds that follow.
27
In Chapter 3, we improve upon this bound by applying a different technique.
2.5 Expression for output probabilities
In this section, we describe the standard boson sampling set-up and derive an expres-
sion for the output probabilities of a boson sampling experiment that define the distribution
DU . First, let us represent the input and output states pictorially and develop some notation.
1 2 3
|r? = 1 2 3 4 5 6 7 8 9
1 2 3
|s? = 1 2 3 4 5 6 7 8 9
Figure 2.2: A representation of input and output basis states in 1D.
In Fig. 2.2, the top line denotes the input state |r? and the bottom line the output |s?.
Each filled circle denotes a boson occupying the corresponding lattice site, which is labeled
below the circles. The numbers marked in orange above each boson label the bosons from
left to right (more generally, this is in a nondecreasing order of the site index). We will call
this the boson index.
A given configuration (basis state) is completely specified by specifying the boson num-
ber in each site, such as r = (1, 0, 0, 0, 1, 0, 0, 0, 1) and s = (0, 1, 0, 0, 1, 1, 0, 0, 0) in the
above. It can also be specified by listing the site index for every boson index, i.e. the oc-
cupied sites. Thus the input state can be represented as in = (1, 5, 9), the output state as
out = (2, 5, 6).
All n! permutations of the boson indices represent valid paths that the bosons can take
to the output state, and correspond to the n! terms in the permanent of the matrix. In cases
28
where there are two or more bosons in a particular site at the input or output, there are n!
r!s!
paths (and terms in the amplitude). Here, r! := r1!r2! . . . rm! and similarly s!. By taking
repeated rows and columns of R, this has the effect of still giving n! terms in total, which we
identify with the n! permutations in the boson indices. The expression for the probability of
an outcome s is (here, bi := ai(t)):
? ?2
Pr 1 ?? r r[s] = ? ?vac|
s1 s2 sm ? 1 ? 2 ?rm ?b1 b2 . . . bD r !r ! . . . r !s !s ! . . . s ! m a1 a2 . . . am |vac??? (2.5)U 1 2 m 1 2 m1 ?? r r r 2= ???vac|(
? 1 2 m ?(U? as1U?)(U? ?as2U?) . . . (U? ?asmU?)a? a? . . . a? |vac?? (2.6)
r!s!
1 ??? ?
1 )2 ( m 1 )2 m ?
m s1 ?m sm ?2
vac r1 r? | 2 rm ?= R?1k ak1 . . . R? a ? ?mk km a1 a2 . . . a?m |vac?? ,r!s! 1 m ?
k1=1 km=1
(2.7)
w?here Rij describes the action of U? on the annihilation operators at a site: bi = ai(t) =
R?k ik(t)ak(0). Now define the matrix A? to be the one obtained by taking si copies of
the i?th row and r copies of the j?th column of R?j . For concreteness, this can be done by
first considering the rows and repeating a row i of R? whenever si > 1, or not picking it
if si = 0, to convert it into an n ? m matrix. We can then do the same with columns to
convert it into an n ? n matrix. However, the order ultimately does not matter since the
quantity that emerges, the permanent, is symmetric under exchange of rows or columns. We
have
????? ???2Pr 1[s] = ? R? ? (2.8)
DU r!s! ?? out?(i?),ini? ?? ? i 21
= ?
r!s! ??? ? ??A?(i),i?? , (2.9)
? i
29
where the sum is over all permutations ?. This finally gives us
? ?
Pr 1 ?Per ? ?2 1[s] = (A ) = |Per(A)|2, (2.10)
DU r!s! r!s!
where Per(A) is the permanent of A.
2.6 Algorithm
The sampling algorithm is given below. It is easy to see that it implements one step
of a Markov process of n distinguishable bosons walking on a lattice.
Algorithm 1: Sampling algorithm
Input: Unitary R(t), tolerance ?
Output: Sample s drawn from DDP , a distribution that is close to DU .
1 Pkl = |R(t)|2kl
2 for i ? {1, 2, . . . n}, do
3 Select site l from the distribution Pin ,l for the boson at ini to hop to.i
4 Increment output boson number of site l by 1: sl ? sl + 1 (or equivalently,
assign outi = l)
5 end
6 return configuration s (or out), a sample from DDP .
Note that P from line 1 is a doubly stochastic matrix describing the classical Markov
process. To see that the runtime is polynomial in n, note that the loop is over n boson indices.
Line 3 takes time O(m logm) = O?(n?), giving a total runtime of O?(nm) = O?(n1+?). The
notation O? suppresses factors of logn.
30
2.7 Bound on variation distance
?
Here we derive a bound on the variation distance ?DU ?D 1DP? = 2 s |PrD (s)?U
PrD (s)|. Rewriting the actual probability in terms of the amplitudes, we haveDP
Pr(s) = |?|2, with (2.11)
?DU
? ?1? = Rin1,out R?(1) in2,out . . . R?(2) inn,out (2.12)?(n)
r!s! ?
1 ?
= ? A1,?(1)A2,?(2) . . . An,?(n), (2.13)
s! ?
where A is the n?n matrix formed by taking the appropriate number of copies of each row
and column of Rm?m. We have set r! = 1 since our input state has bosons in distinct sites.
Continuing,
Pr 1
?
(s) = |R 2 2 2in1,out | |Rin2,out | . . . |R?(1) ?(2) inn,out | +D s! ?(n)U
?
1 ?
R R . . . R (R R ?in1,out?(1) in2,out?(2) inn,out?(n) in1,out?(1) ins! 2
,out . . . Rinn,out ) . (2.14)?(2) ?(n)
? ?=?
The probability distributionDDP that the algorithm samples from is given by the first
line of Eq. (2.14):
?
Pr 1(s) = Pin1,out P?(1) in2,out . . .P?(2) inn,out , (2.15)D ?(n)DP s!
?
where the sum is over all the n! ways of assigning the n input states to the n output states.
As before, the s! is to account for overcounting when two distinct permutations in the boson
31
index refer to the same site index in the output state.
The expression for the probability is proportional to the permanent of the matrix with
the positive entries Pin ,out , and can hence be efficiently approximated [90]. Note that thei j
algorithm does not explicitly calculate this probability but only samples from the distribution.
We can now prove Lemma 3.
Proof of Lemma 3. The variation distance is given by
? ?1 ????? ?
??
? = Rin1,out . . . Rinn,out (Rin1,out . . . R
?
?(1) ?(n) ?(1) inn,out ) (2.16)2s! ?(n) ?? ?s ?? ?=?
? 1 |Rin1,out . . . R?(1) inn,out ||Rout ,in1 . . . Rout ,inn | (2.17)?(n) ?(1) ?(n)
? 2s 1?? ?=?
= |Rin ,out R | . . . |R R |, (2.18)
2 1 ?(1)
out?(1),in?(1) inn,out?(n) out?(n),in?(n)
s ?,?
where ? = ??1?? ?= Id, the identity permutation. The last equality comes from rearranging
the terms in |Rout ,in1 . . . Rout ,inn | so that the terms involving R and R?(1) ?(n) ini,out?(i) out?(i),j
(for some j) are together:
??? ??? rearrange ???|Rout ,in | = |R?(i) i outi,in ?1 | ????? |R? (i) out |.? ?(i),ini ?(i) ??1(?(i))
? ? i ? ? i ? ? i
(2.19)
Summing over all outcomes s (or configurations out), Eq. (2.18) is equivalent to
? 1
???
? |Rin ,j ||Ri i ji,in |, (2.20)2 ?(i)
j ?=? Id i
where the sum j is over ordered tuples (j1, . . . jn), representing the intermediate lattice
32
sites that the bosons in positions (in1, . . . inn) jump to, before jumping back to positions
(in?(1), . . . in?(n)). We can proceed to break the sum in Eq. (2.20) based on the number of
fixed points of the permutation ?, that is, the number of indices i such that ?(i) = i. We
bound these quantities separately as follows:
?
|Rin ,j ||Rj ,in | = Ci ? c ? i andi i i ?(i)
ji,?
??(i)=? i ?
|Rin ,j ||Rj ,in | = |R 2in ,j | = Di = 1 ? i. (2.21)i i i ?(i) i i
ji,? ji
?(i)=i
The variation distance is therefore bounded above:
? 1
?? ?
? Ci Dk, (2.22)
2
i?IC k?ID
where the sum is over subsets IC of the indices representing the input state, in. ID is the
complement of IC and |ID| is the number of fixed points. Suppose we find an upper bound
c for Ci in Eq. (2.21), we then have
1 ?n ( )
? ? n cl = (c+ 1)n ? nc? 1. (2.23)
2 l
l=|IC |=2
In Lemma 4, we show that c = ?Ld?1e(vt?L)/? for some constant ?. Continuing from
33
Eq. (2.23),
1
? ? ([()c+ 1)n ? nc? 1] (2.24)2
n
= (1 + h)n?2c2 for some h ? [0, c] (by Taylor?s theorem) (2.25)
2
? ? exp [2 logn+ (n? 2) log(1 + c) + 2 log c] . (2.26)
Now, plugging in the value of c and assuming that vt ? 0.9L and ? > 1, we get
( [ ])
? exp ? ? d?1e(vt?L)/? vt? L? O( [(n 2) ?L ]+)2 + 2(d? 1) logL (2.27)?
? exp vt? LO 2 + 2(d? 1) logL . (2.28)
?
In the first line, we use the inequality log(1 + x) ? x. In the second, we use
[ ]
exp (n? 2)? ?Ld?1e(vt?L)/? = O(1) (2.29)
since vt < L and |vt? L| = ?(n??1). This completes the proof of Lemma 3.
Lemma 4. For all constant dimensions d, c = ?Ld?1e(vt?L)/?.
Proof. Recall that
? ? ? ?
Ci = |Rin ,j |
? ?
R
i i ? ji,in . (2.30)?(i)?
ji ?(i)=? i
Since we are looking for a bound that applies for all i, let us, for convenience, make the
following changes in notation: ini ? i, ji ? j, in?(i) ? k, denoting a boson starting at
34
position i, jumping to j and then to k, where i and k ?= i are site indices belonging to in.
We split the sum in Ci based on the distance between i and j, ?ij =: ?.
?? ??
Ci = |Ri,j| |Rj,k|+ |Ri,j| |Rj,k| . (2.31)
j k?in j k?in
??L k=? i ?>L k ?=i
Consider the first term:
? ? ? ?
|Ri,j| |Rj,k| ? |R (vt??kj)/?i,j| e (2.32)
j k?in ??
j? k?in??L k ?=i ???L k=? i??? ??? ????? 12???? ?? ?|R |2 evt/? e(?2L?x?k?+L)/?i,j (2.33)
j j k:?x?k??1
??L ???L
? aLd/2e(vt+L)/? e?2L||x?k||/? (2.34)
k:||x?k||?1
? abe(vt?L)/?Ld/2. (2.35)
Here in the first line, we have used the Lieb-Robinson bound Eq. (2.2). In the second
line, we use the Cauchy-Schwarz inequality and the fact that ? = ?ij ? L. In the second
line, x?k is the position vector of site k relative to site i, re-scaled by 2L. Therefore the sum
over x?k is over all vectors with integer coordinates. In the last line, we use Lemma 6, to be
proven below. a and b are constants independent of n that depend on the dimension d and
the length scale ?.
Now, in the second term for Ci in Eq. (2.31), the intermediate site j is not necessarily
close to i. Therefore, there are terms where j is close to k ?= i and one has to treat these
terms carefully. For these terms, we use the trivial Lieb-Robinson bound of 1 in Eq. (2.2)
35
( )
rather than exp vt??jk > 1.
?
? ? ? ?
|R | |R | ? |R | min(1, e(vt??k,j)/?i,j j,k i,j ) (2.36)
j k?in j k?in
?>L k ?=i ??>L ?k=? i ? ?
? |R ?i,j| 1 + evt/? e(L?2L?x?k?)/?? (2.37)
j k:||x? ||?1
( k?>L )?
? 1 + be(vt?L)/? |Ri,j| (2.38)
j
( ) ?>L ?
? 1 + be(vt?L)/? evt/? e??/? (2.39)
j
( ) ?>L
? 1 + be(vt?L)/? b?Ld?1e(vt?L)/?. (2.40)
In the second line, we use 1 as a Lieb-Robinson bound for |Rj,k| when k = k?, the site
belonging to in that is closest to j. All other k?s have distances from j bounded below by
2L?x?k? ?L, where x?k is now the re-scaled position vector of a site k with the origin at k?.
We apply Lemma 6 in the third line and Lemma 5 in the fifth. Collecting everything, we
have
( )
C ? abe(vt?L)/?Ld/2 + 1 + be(vt?L)/? b?Ld?1e(vt?L)/?i (2.41)
? e(vt?L)/??Ld?1 for large enough L. (2.42)
?
Lemma 5 (d-dimensional sum). The sum ??x??/??x???L e over points x? with integer coordinates
36
? ( )is upper bounded by a e L/?d ?Ld?1 for large enough L for some dimension-dependent constant?
ad.
Proof. We can view the sum over a lattice of vectors with integer coordinates as a Riemann
sum and bound the corresponding d-dimensional integral. Consider the quantity
? ( )
e??x??/?dd 2?
d/2 L
g = x? = ?d? d, , (2.43)
d
?x???L ?( ) ?2
?
where ??(d, x) = yd?1e?y?dy is the incomplete ? function. We can lower bound thex
integral by the Riemann sum V e??y??/?? ? , where the sum is over cells ? with volume
V? centered at lattice points x?. y? is the point in the cell ? with the highest norm ?y??.
Further, the point with the highest norm is not too distant from the one at the center:
?
?y?? ? ?x??+ d . Therefore, we have
2
? ?
f := e?x??/? ? g ? e d/(2?). (2.44)
?x???L
Wenow need an upper bound on the incomplete? function?(d, x) for large x [91, Eq. 8.11.i]:
( ( ))
1
?(d, x) ? xd?1e?x 1 +O as x ? ?. (2.45)
x
Combining Eq. (2.43) and Eq. (2.44), we get:
( [ ])
f ? LO ?Ld?1 exp ? . (2.46)
?
37
(?2, 2) (?1, 2) (0, 2) (1, 2) (2, 2)
(?2, 1) (?1, 1) (0, 1) (1, 1) (2, 1)
(?2, 0) (?1, 0) (0, 0) (1, 0) (2, 0)
(?2,?1) (?1,?1) (0,?1) (1,?1) (2,?1)
(?2,?2) (?1,?2) (0,?2) (1,?2) (2,?2)
Figure 2.3: Part of the lattice of vectors with integer coordinates. The black dots are the
points in the cell with the maximum norm ?y?? and the exponential is evaluated at these
points. The white ones do not enter the Riemann sum and are related to fd?1, the cor-
responding quantity in one lower dimension. The arrows show which point in the cell is
picked to lower bound the Riemann sum.
Using a similar method, we also get bounds on a related sum.
? [ ]
Lemma 6. For x? ? Zd, fd := ?x???1 e?2L?x??/? ? b exp ?2Ld for some dimension-?
dependent constant bd.
Proof. We prove the statement by induction on the dimension d. For d = 1, the statement
is seen to be true since the sum evaluates exactly:
? ??
e?2L|x|/? e?2Lx/? 2e
?2L/?
f1 = = 2 = e? ? 2.1? e
?2L/?. (2.47)
1? 2L/?
|x|?1 x=1
?
For the inductive step, consider the integral ?g(d) = ? ?? e?2L?y??/?ddy?. This is lowery? 1
bounded by the Riemann sum represented in Fig. 2.3. The white dots represent vectors with
at least one zero coordinate and do not enter the Riemann sum according to this way of
dividing the region of integration into cells. In the following, the set of points with at least
38
one zero coordinate is denoted cc. We have:
? ? ?
g(d) = e?2L?y??/?ddy? ? e?2L?x??/??x?, (2.48)
?y???1 x??/cc
where?x? is the volume of the cell associated with the lattice vector x?. In Fig. 2.3, the volume
of most cells (whose center is at distance 1.5 or beyond from the origin) is 1. The cells near
the origin have some volume ?d < 1 that depends on the dimension. Lower bounding all
volumes ?x? by ?d,
?
e?2L?x??/? g(d)< (2.49)
?
x?? d/cc ( )
d/2 d
( )
2? ? 2L
= ? d, . (2.50)
?d?(
d) 2L ?
2
Now it remains to upper bound the contribution from summing over the points cc. Notice
that the sum over these points is upper bounded by the sum over d hyperplanes of dimension
d? 1. From the inductive hypothesis,
?
e?2L?x??/? ? dfd?1 (2.51)
x??cc
39
Figure 2.4: Complexity phase diagram of free bosons, illustrating that sampling complexity
can delineate boundaries of a physical system as a function of system parameters, including
time. Our results indicate that ??1 ? c ? ?+d , where c is the transition point for the
d d
scaling exponent of t with n.
since there are d hyperplanes of dimension d? 1. Adding Eqs. (2.50) and (2.51), we get
? ( )d ( )
e?2L?x??/? 2?
d/2 ? 2L
= fd < dfd?1 + ? d, . (2.52)
?[?(d)x? d 2 ] 2L ( ? ) [ ]
d/2
? 2L 2? ? 2Ldbd?1 e[xp ?] + exp ? (2.53)? ?d?(d) 2L ?2
exp ?2Lfd < bd , (2.54)
?
proving the lemma. In the second line we have expanded the incomplete ? function for large
L/?.
2.8 Outlook
We have defined the sampling problem for local Hamiltonian dynamics and given upper
and lower bounds for the scaling of time t(n) with the number of bosons n for which the
problem is efficiently simulable or hard to classically simulate, respectively. Our results are
captured in Fig. 2.4 that illustrates the complexity phase diagram of the system. For time-
independent systems, we observe that boson sampling is classically easy for all times if the
system is Anderson-localized. In a future work [92], we show that there is a class of static,
local Hamiltonians that generate a hard-to-sample output distribution at some time that is
40
not formally infinity. This means that sampling complexity distinguishes Anderson-localized
and delocalized systems, which makes it similar to an order parameter that distinguishes
different phases.
The case with nonzero Lieb-Robinson velocity [Eq. (2.2)] shows two regimes of the
scaling of t with n where sampling is provably easy/hard. We have shown that sampling is
easy when ??1 ?t ? teasy = ?(n d ) and hard when t ? thard = ?(n1+ d ). Since our definitions
of easiness and hardness are exhaustive, we argue that there must exist a constant c such
that sampling is efficient for t < ?(nc) and hard otherwise, illustrating a dynamical phase
transition. Our proof implies that c ? [??1 , ?+d ] and we show in the next chapter (based on
d d
Ref. [93]) that c is given by ??1 . The transition is between two regimes, one for short times
d
in which the system?s dynamics is essentially indistinguishable from classical dynamics; and
the other in which quantum mechanical effects dominate to such an extent as to forbid an
efficient classical simulation.
This result may be viewed as a generalization of a similar result in Ref. [94], where it
was shown that exact boson sampling for depth-4 circuits is hard. The results there are not
directly comparable to ours, since Ref. [94] assumes ? = 1, whereas our results need ? > 1
(easiness) and ? ? 2 or 5 (hardness). The reason we get easiness even after polynomial time
is that we deal with approximate sampling, a less stringent notion of simulation. In addition,
adapting the hardness proof of Ref. [94] into our setup, we obtain that the system is hard
to exactly sample from at timescale ??1?(8L/v) = ?(n d ) [92], partially answering what
happens in the subregion with the question mark in Fig. 2.4. Another work [95] roughly
concurrent to ours [96] showed that if the initial state is not well separated (L = 1 in our
language), the transition timescale is logarithmic in the number of bosons.
41
Recent studies [76; 97?99] have also studied transitions based on time-dependent fea-
tures of the out-of-time-ordered correlator. This raises the question of the connection be-
tween sampling complexity and scrambling time [100; 101] in quantum many-body systems
[102?105] and fast scramblers like the Sachdev-Ye-Kitaev [97; 106?109] models, and black
holes [110?114], where one can explore the connection to recent conjectures on complex-
ity in the dual conformal field theory [113?116]. Further, it would be interesting to study
sampling complexity in various other physical settings [117], like systems with topological
or many-body-localized phases and systems in their ground state to explore the connection
with Hamiltonian complexity.
We therefore propose another motivation for considering sampling complexity beyond
the field of quantum computational supremacy: complexity provides a natural way to classify
phases of matter that is complementary to traditional approaches based on symmetries and
topology. This is akin to how the study of entanglement in many-body physics has helped
us understand phases of matter [118; 119] and characterize thermalization and localization
[120; 121].
Coming to experimental implementations, platforms such as ultracold atoms in opti-
cal lattices [122; 123] and superconducting circuits [124; 125] are ideal for experimentally
studying the transition by comparing the distribution sampled by the algorithm and the
experimental distribution. The ingredients required, which have been realized in several
groups [126?131], are: 1. Preparation of the initial state [132; 133] of the type shown in
Fig. 2.13. 2. Evolution under a Hamiltonian with either arbitrary time-dependent nearest-
3All of our arguments go through even if the initial state is not regular as in Fig. 2.1 but still has the
property that the initially occupied sites are separated by a minimum distance of ?(L) from each other.
42
neighbor hopping strength or fixed nearest-neighbor hopping strength Jij(t) = 1 together
with arbitrary time-dependent on-site potential Jii(t) [132; 134], and 3. Single-site resolved
measurement of occupation number of the sites [127; 128]. Cold atoms in quantum gas
microscopes can be controlled at the single site level, enabling all three ingredients above.
To maintain integrability, we can turn off the Hubbard interaction for the bosonic atoms by
tuning to a Feshbach resonance [135?138].
New architectures like optical tweezers [139] are also promising since they allow for
deterministic creation of desired initial states and feature tunable interactions in time [140;
141]. Similarly, superconducting circuits have been proposed for quantum simulation of
quantum walks [142] and the Bose-Hubbard model [143?145]. The gmon qubit architec-
ture, which was used in Ref. [145], naturally allows for time-dependent variation of coupling
strengths [146], and raises the prospect of an experimental study of the transition.
43
Chapter 3: Complexity phase diagram for interacting and long-range bosonic
Hamiltonians
A major effort in quantum computing is to find examples of quantum speedups over
classical algorithms, despite the absence of general principles characterizing such a speedup.
The study of classical simulability of quantum systems evolving in time allows one to iden-
tify features underlying a quantum advantage. Studying the classical simulability of both
quantum circuits [11?13; 18; 19; 50?52; 147?152] and Hamiltonians [153; 154], especially
under restrictions such as spatial locality [24; 27; 95; 96; 155], allows one to understand the
classical-quantum divide in terms of their respective computational complexity. Computa-
tional complexity has been closely linked to phases of matter in contexts such as dynamical
phase transitions [96], measurement based quantum computing [156], thermal phase tran-
sitions [157], and entanglement phase transitions [158].
In this Chapter, we characterize the worst-case computational complexity of simu-
lating time evolution under bosonic Hamiltonians and study a dynamical phase transition
in approximate sampling complexity [95; 96]. Previous work [96] studied free bosons with
nearest-neighbor hopping but did not consider the robustness of the transition to perturba-
tions in the Hamiltonian, a crucial question in the study of any phase transition. We gener-
alize Ref. [96] to include number-conserving interactions and long-range hops and conclude
44
Figure 3.1: Slices of the complexity phase diagram for the long-range bosonic Hamiltonian
in a) 1D, b) 2D, and c) 3D with n bosons when the number of sites is m = ?(n2). Colors
represent whether the sampling problem is easy (yellow), hard (magenta), or not currently
known (hatched). The X-axis parametrizes the evolution time as a polynomial function of
n, and the Y -axis is ?, the exp?onent characterizing the long-range nature of the hopping
Hamiltonian (with scale y=1/ ? except for the point ?=0).
that the phase transition survives under perturbations. These kinds of interactions are ubiq-
uitous in experimental implementations of hopping Hamiltonians with ultracold atoms and
superconducting circuits [159; 160]. Long-range hops that fall off as a power law are also
native to several architectures [161?165]. We study the location of the phase transition and
its dependence on various system parameters, constructing a complexity phase diagram, a slice
of which is presented in Fig. 3.1.
3.1 Setup and summary of results
Consider a system of n bosons hopping on a cubic lattice of m sites in D dimen-
sions with real-space bosonic operat?ors a . We let m=?(n
?
j ?) and assume sparse fill-
ing: ?? 1. The Hamiltonian H = i,j Jij(t)a
?
iaj + h.c. + i f(ni) has on-site inter-
actions f(ni) and time-dependent hopping terms bounded by a power-law in the distance
d(i, j) as |Jij(t)| ? 1/d(i, j)?. The parameter ? governs the degree of locality. When
?=0, the system has all-to-all couplings, while ??? corresponds to nearest-neighbor
45
hops. The on-site terms Jii(t) can be large, and the interaction strength is parametrized
by V . For concreteness, our hardness results are derived using a Bose-Hubbard interaction
f(ni)=V ni(ni ? 1)/2, but the timescales we present are valid for generic on-site inter-
actions [166]. The bosons in the initial states considered are sparse and well-separated.
Specifically, partition the lattice into K clusters C1, . . . , CK containing b1, . . . , bK initial
bosons, respectively, such that b :=max bi =O(1) does not scale with lattice size. Define
the width Li of a cluster Ci as the minimum distance between a site outside the cluster and
an initially occupied site inside the cluster and let L=mini Li. As in Ref. [96], we consider
states with the bosons roughly equally spaced throughout the lattice, so that m=?(nLD),
giving L=?(n(??1)/D).
The computational task of approximate sampling is to simulate projective measure-
ments of the time-evolved state in the local boson-number basis. The approximate sampling
complexity measures the classical resources needed to produce samples from a distribution
D? that is ?=O(1/poly(n))-close in total variation distance to the target distribution D1.
Sampling from a distribution D? satisfying the above takes runtime T (n, t) in the worst case
on a classical computer, where t is the evolution time. Like thermodynamic quantities,
the complexity is defined asymptotically as n??, so we consider the scaling of T along
a curve t(n). Along any curve t(n) = cn? , sampling is easy if there exists a polynomial-
runtime classical algorithm for all n, or hard if such an algorithm cannot exist. Since the
problem is either easy or hard for a particular function t(n), there is always a transition in
complexity as opposed to a smooth crossover. We prove upper and lower bounds on the
1This is a weaker requirement than demanding a classical sampler that works for arbitrary ? and has runtime
poly(n, 1/?).
46
? Easiness exponent Hardn(ess expo)nent4
? < D ? ?? D
2 D 2
D ? ? ? D
2 ? 0
D ? ? ? D + 1 ??1(??D)
D
D + 1 ??? 2D + D 5
?? 01 ( ) ??1D
2D + D? ? ? < ?
??1 ??2D ? 1
? 1 D ??D ??D
??1
? ? ? ??1 D if D? 2 or V <?
D
? otherwise.
Table 3.1: Leading exponents ?easy and ?hard in the easiness and hardness timescales for various
regimes of ?. Some of these transitions are likely sharp and some coarse.
transition timescale by presenting sampling algorithms on the easiness side, and performing
reductions to quantum supremacy proposals on the hardness side. Specifically, we show that
approximate sampling is easy for all times t< c ? ?easyn easy and hard for all times t> chardn easy .
We find that the transition comes in two types, which we call ?sharp? and ?coarse?. For
sharp transitions, these bounds coincide in the exponent ?easy= ?hard and the transition occurs
in the coefficient2 ceasy? chard. For coarse transitions, however, the transition occurs in the
exponent. In our results, we will show that sampling is easy for any time t=O(n?easy), but
hard along any curve with exponent ? 3hard>?easy (see [167] for more precise definitions).
An example of a sharp transition is when the transition timescale is t? =2n, so that the
problem is easy for all times t? 1.99n and hard for all times t? 2.01n. An example of a
coarse transition is when the transition timescale is t? =?(n logn), so that the problem is
easy for all times t? cn and hard for all times t? cn1.01.
We summarize our main result in Table 3.1. The easiness result comes from applying
2More precisely, for sharp transitions, we have thard = ?(teasy), while for coarse transitions, thard = ?(teasy).
3In principle, there can be other kinds of coarse transitions where sampling is hard for any time t = ?(n?hard)
and easy for any ?easy < ?hard.
4Up to an additive constant ? > 0 that is present when indicated in the text.
5The easiness timescale for this case is teasy = logn.
47
classical algorithms for quantum simulation, and depend on Lieb-Robinson bounds on in-
formation transport [77; 168?171]. The hardness results come from reductions to families of
quantum circuits for which efficient approximate samplers cannot exist, modulo widely be-
lieved conjectures in complexity theory [13; 24; 27; 155], and from fast protocols to transmit
quantum information across long distances [172; 173].
Note that the hardness exponents in Table 3.1 sometimes come with an arbitrarily
small additive term ? > 0. This happens whenever at least one of the following cases
hold: ?<D / 2, V = o(1), or D=1. When the easiness and hardness timescales coincide,
we interpret the presence of this term ? as signifying a coarse transition, since it ensures
?hard>?easy.
We examine the various limits: ??? (nearest-neighbor), ?? 0 (all-to-all con-
nectivity), V ? 0 (free bosons), and V ?? (hardcore bosons). First, when ???, the
hardness timescale upper bound is O(L) in all cases except when V ?? and D=1, which
we discuss later. This matches the easiness timescale t = ?(L), which corresponds to the
distance L between clusters. We therefore pin down the transition timescale to?(L), which
is when interference between clusters become relevant [96]. In the opposite limit when the
model is sufficiently long-range (?<D/2), the role of the dimension is unimportant, giving
?hard< 0 in all cases. This suggests a hardness timescale close to 0, signifying the immediate
onset of hardness.
Next, we observe that the transition timescale does not heavily depend on the inter-
action strength V , showing that the location of the complexity phase transition is robust
to the presence of interactions. The only exception is the limit of hardcore interactions and
nearest-neighbor hops (V, ? ? ?) in 1D. In this case, there is no hardness regime. This is
48
? Easiness exponent Hardn(ess expo)nent
? < D 1 ?? D
2 ? D 2
D ? ? ? D + 1
2
D + 1 ? ? < ? ?1 0
??D
? ? 0 if D? 2 or V <?? 0
? otherwise.
Table 3.2: Easiness and hardness timescale exponents when ? = 1.
because the model maps to that of free fermions, or equivalently, matchgate circuits, which
are easy to simulate at all times [50; 51].
The results for the regime ?=1, or m=?(n), can be experimentally relevant and
are shown in Table 3.2. In this case, the separation between clusters is a constant, and when
??D + 1, the algorithm works only for time t=O(n?1/(??D)), which asymptotes to 0 as
n??. For ???, this is (almost) matched by the hardness timescale being thard=O(1),
since the leading exponents in the timescales are zero in both cases. We now outline the
proofs of our results, whose details may be found in Sections 3.6, 3.7, 3.9 and 3.10.
3.2 Easy-sampling timescale
To derive teasy, we give an efficient sampling algorithm. The algorithm performs time
evolution on eac(h cluster )Ci separately. This takes polynomial time in the number of basis
states, which is |Ci|+bi?1 =O(|C |bii ) and hence polynomial in n when bi =O(1). Thisbi
product-state approximation of the exact time-evolved state |?(t)?=Ut |?(0)? is achieved by
decomposing the propagator Ut via a spatial decomposition scheme for quantum simulation
[44; 171] that we call the HHKL decomposition for the authors of Ref. [44]. We complete
the derivation of the easiness timescale by showing that the approximation is good for times
49
t<O(teasy).
We briefly present the HHKL decomposition, which is powerful but remarkably sim-
ple. LetHR be the sum over all terms in the Hamiltonian supported completely in region R
and implicitly let XY =X ?Y represent the union of regions. The decomposition scheme
approximates the time evolution unitary acting on region XY Z (where Y separates regions
X and Z) by forward evolution on Y Z, backward evolution on Y , and forward evolution
on XY :(UXY Z ?UXY (U )
?
Y UY Z . The op)erator norm error made by this approximation is
[171] O (evt ? 1)?(X)(???+D+1 + e??) , where v > 0 is a characteristic velocity, ?(X)
is the area of the boundary of X , and ? is the minimum distance between any pair of sites
in X and Z. The error is small for times t shorter than the time it takes for information to
propagate from X to Z.
The velocity v of information propagation is also known as a Lieb-Robinson velocity
and is determined by the operator norm of terms in the Hamiltonian which couple different
sites [168]. Since bosonic operators have unbounded operator norm, this could result in an
unbounded velocity [84]. However, because of boson number conservation under the Hamil-
tonian, the dynamics is fully contained in the n-boson subspace, within which the operator
norm of each term is O(n). While free bosons (V =0) behave as in the single-particle
subspace, implying the Lieb-Robinson velocity is O(1), in the interacting case, an O(n)
Lieb-Robinson velocity would cause the asymptotic easiness timescale to vanish (teasy? 0).
Nevertheless, we can derive an easiness timescale independent of V for a clustered
initial state. Intuitively, at short times each boson is well-localized within its original cluster.
Therefore, the relevant subspace has at most b bosons in each cluster Ci. Truncating the
Hilbert space to allow only b + 1 bosons per cluster is therefore a good approximation at
50
short times [174], and the truncation error vanishes in the asymptotic limit. The modified
Hamiltonian H ? after truncation has terms with norm only O(b), giving an effective Lieb-
Robinson velocity v=O(b)=O(1) for states close to the initial state6. For this modified
Hamiltonian, we apply the HHKL decomposition to bound the error caused by simulating
each cluster separately. Once the error has been calculated, the timescale immediately follows
by solving ?(t)=O(1) for t= teasy, which is a lower bound on the transition timescale t?. In
Section 3.7, we give the full dependence of teasy on various system parameters, including the
filling fraction of bosons.
3.3 Sampling hardness timescale
To derive thard, we give protocols to simulate quantum circuits by setting the time
dependent parameters Jij(t) of the long-range bosonic Hamiltonian. This implies sampling
is worst-case hard after time thard. Specifically, if a general sampling algorithm exists for
times t? thard, we prove this algorithm can also simulate hard instances of boson sampling
[13] when interactions are weak, and quantum circuits that are hard to simulate [24] when
interactions are strong.
In the interacting case, our reduction from universal quantum computation to a long-
range Hamiltonian hinges on implementing a universal gate set. Using a dual-rail encoding
to encode a qubit in two modes of each cluster Ci, we show in Section 3.9 how to implement
arbitrary single-qubit operations in O(1) time and controlled-phase gates [175] between
adjacent clusters in a time that depends on their spacing L. The two-qubit gate uses free
particle state-transfer as a subroutine [172; 173] to bring adjacent logical qubits near each
6In other words, this is a state-dependent Lieb-Robinson velocity, or a butterfly velocity.
51
other. We implement the constant-depth circuit of Ref. [24], which consists only of nearest-
neighbor gates between qubits in a 2D grid. The total time for hardness under this scheme
takes timeO(min [L ,L??D]) when ?>D andO(1) when ?? [D/2, D]. In 1D, simulating
a 2D circuit introduces extra overhead. Nevertheless, we can recover the same timescale up
to an infinitesimal ? > 0 in the exponent by only encoding n? logical qubits. For hardcore
bosons, the above scheme mentioned does not work and the entangling gate is constructed
differently, and features an easiness result for the 1D nearest-neighbor case. Lastly, when
?<D / 2, state transfer takes time o(1) but the time for an entangling gate is O(1). We can
still achieve coarse hardness for time o(1) by mapping the system onto free bosons, which
we now come to.
In the noninteracting case, we implement the boson sampling scheme of Ref. [13],
which showed that a Haar-random unitary applied to m sites containing n bosons gives a
hard-to-sample state. It also gave an O(n logm)-depth decomposition of a linear-optical
unitary in the circuit model without spatial locality. In Section 3.10, we give a faster
implementation for the continuous-time Hamiltonian model, which can include simulta-
neous noncommuting terms but imposes spatial locality, a result of independent interest.
Specifically, we show that most linear-optical states of n bosons on m sites can be con-
structed in time min [O(nm1/D) , O?(nm?/D?1/2]), which is faster than the circuit model
when ?<D / 2. This result also uses free-particle state transfer as a subroutine. As in the
1D interacting case, we can implement the reduction on a polynomially growing number of
bosons n?, resulting in the timescale of Table 3.1 for free bosons. This result resolves an im-
portant conceptual question posed by Ref. [96] for the noninteracting, nearest-neighbor case
by closing the gap between teasy and thard. In this limit, the transition timescale is at?(L/v),
52
both with and without interactions, showing that the algorithm of Ref. [96] is optimal and
that the presence of interactions does not change the phase diagram.
3.4 Sharp and coarse transitions
In the nearest-neighbor limit ???, the exponents on our hardness and easiness
timescales match up to an infinitesimal (?), so we can make precise statements about the
nature of the transition. In the presence of interactions and in two dimensions and above,
the bounds on the timescale in the nearest-neighbor limit coincide up to a multiplicative
constant at teasy= thard=?(L), proving the transition is sharp. However, in 1D, the hardness
timescale only matches up to an infinitesimal ? > 0 in the exponent chard = ceasy + ?. In 1D,
we show that chard cannot be improved any further, proving that this is a coarse transition.
To understand the physics behind the two kinds of transitions, it is illuminating to
study the approach to the transition point from both sides. On the easiness side, the impor-
tant quantity is the many-body entanglement. Recall that at short times, the wavefunction
is approximately separable, implying easiness of classical simulation. The separable state is
computed using an HHKL decomposition, whose errors grow in time until they become
O(1) at the transition timescale. These errors upper bound the amount of entanglement
present across any cut, so the easy phase corresponds to states with no entanglement, and
the complexity transition occurs as the entanglement grows from zero entanglement to area-
law entanglement.
However, sampling complexity in one dimension is special because area-law entangle-
ment is classically simulable using matrix product states [176; 177]. Specifically in 1D, we
53
prove an extended easiness timescale of teasy = cL for any constant c. Sampling is easy for all
times O(L), implying that the ? in the hardness exponent cannot be removed, and the tran-
sition is coarse. We leave open the question of entanglement scaling between the area-law
and volume-law regimes. However, if D? 2, the argument based on entanglement breaks
down because tensor-network contraction takes time exponential in the system size in both
the worst case and average case [178; 179], and there are known examples of constant-depth
2D circuits that are hard to simulate [24].
On the hardness side, many-body entanglement is necessary but not sufficient for sam-
pling hardness [50; 51; 180]. Since our hardness results in the interacting case rely on map-
ping bosons to qubits via a dual-rail encoding, we understand the transition by counting the
number of encoded logical qubits. For coarse transitions, as the evolution time approaches
the transition timescale t?, the number of encoded logical qubits shrinks as n?, where ?? 0
as t? t?. This illustrates that while the problem is still asymptotically hard as n??, one
needs to go to higher boson numbers n to achieve the same computational complexity. For
sharp transitions on the other hand, the number of encoded logical qubits seems to suddenly
jump to O(nc) for some constant c. In Section 3.11, we elaborate on how to use number of
effective logical qubits as an order parameter for the phase transition. Whether or not such
an order parameter is a universal way to characterize complexity phase transitions deserves
closer attention.
Finally, in the noninteracting case, we cannot definitively prove that the transition is
coarse in D ? 2. However, the infinitesimal in the hardness timescale is suggestive of a
coarse transition. More precisely, our work implies that either the transition is coarse, or
there exists a constant-depth boson sampling circuit on a nearest-neighbor architecture for
54
which approximate sampling is classically hard. It has been proved by Brod [94] that exact
sampling of constant-depth boson sampling is classically hard. Nevertheless, Brod points
out that it is very unclear if this exact sampling hardness can be extended to the approximate
sampling hardness we consider in this Chapter. As such, in D ? 2 without interactions, the
type of transition is an open problem.
3.5 Related Models
We first show that our results can be easily adapted to a wide range of experimentally
and theoretically interesting Hamiltonians. First, in the hardcore limit, spin Hamiltoni-
ans naturally map onto our model. Fermionic systems with nearest-neighbor interactions
can also be incorporated by performing the mapping described in Ref. [181]. Our model is
also relevant to cold atom experiments that have been proposed as candidates for observing
quantum computational supremacy [24; 95; 159; 160], especially in the nearest-neighbor
limit.
The power-law hopping 1/r? can be engineered to directly implement the classes of
Hamiltonians we study. This can be done by virtually coupling the band of interest to another
with a quadratic band edge to implement exponentially decaying hopping [165; 182; 183].
Doing this simultaneously with multiple detunings approximates a power-law with high
accuracy as a sum of exponentials [184].
In the hardcore limit, the long-range hops translate to long-range interactions between
spins, which model quantum-computing platforms such as Rydberg atoms and trapped ions
[141; 161; 185?187]. Therefore, the Hamiltonian we study models various physically inter-
55
esting situations, both in the several limiting cases (???, V ? 0, V ??) as well as in
the general case of finite nonzero ? and V .
Our methods are fairly general for lattice models with long-range power law decay-
ing interactions. However, we do require number-conservation so the local Hilbert space
dimension can be bounded at short times. As an example of how to extend our results to
different models, we can straightforwardly incorporate long-range density-density interac-
tions Kij(t)ninj . The only effect on the easiness times is to modify the Lieb-Robinson
velocity to v=O(b2). This is an overall constant that does not affect the exponent.
Our model can also be used to describe a distributed modular quantum network when
the Hubbard interaction V can vary spatially. Specifically, a module of qubits can be rep-
resented by hardcore bosons (V ??), while photonic communication channels linking
distant modules can be represented by sites with V =0 separating the modules. As in quan-
tum networks, our hardness times in the nearest-neighbor regime are dominated by gates
between nodes, while operations within a single node are free.
3.6 Approximation error under HHKL decomposition
We first argue why it is possible to apply the HHKL decomposition lemma toH ? with
a Lieb-Robinson velocity of order O(1). As mentioned in Section 3.2, H ? is a Hamiltonian
that lives in the truncated Hilbert space of at most b + 1 bosons per cluster. Let Q be
a projector onto this subspace. Then H ? = QHQ. T[ime-evolutio]n under this modified
Hamiltonian H ? keeps a state within the subspace since e?iQHQt, Q = 0.
The Lieb-Robinson velocity only depends on the norm of terms in the Hamiltonian
56
Figure 3.2: (a) Decomposition of the first two steps of the unitary evolution followed by
(b) pushing the commuting terms past A?i (the product of all initial creation operators in a
cluster i) to the vacuum. Red boxes represent forward evolution and blue boxes backward
evolution in time.
which couple lattice sites. On-site terms do not contribute, which can be seen by moving to
an interaction picture [168; 170]. Therefore, since no state has more than b +? 1 bosons?on
any site within the image ofQ, the maximum norm of coupling terms inH ? is ?? ?Qa?iajQ? ?
b + 1. Therefore, the Lieb-Robinson velocity is at most O(b) instead of O(n), and we can
apply the HHKL decomposition to the evolution generated by the truncated Hamiltonian
H ?. We now prove that the error made by decomposing the evolution due to H ? is small.
Lemma 7 (Decomposition error for H ?). For all V and ?>D + 1, the error incurred (in
2-norm) by decomposing the evolution due to H ? for time t is
( )
N??1
?(t) ? O K(evt1 ? 1)(???+D+1 + e??) (r0 + j?)D?1 , (3.1)
j=0
57
where N = t/t1 and ??L/N can be chosen to minimize the error, and r0 is the radius of the
smallest sphere containing the initially occupied bosons in a cluster.
The sketch of the proof is as follows: recall that within each cluster Ci, there is a
group of bosons initially separated from the edge of the cluster by a region of width Li.
Naive application of the HHKL decomposition for the long-range case results in a timescale
teasy? log(L), because of the exponential factor (evt ? 1) in the error. To counter this, we
apply the HHKL decomposition in small time-steps t1. Thus, within each time-step, the
exponential factor can be approximated as evt1?1? vt1, turning this exponential dependence
into a polynomial one at the cost of an increased number of time-steps.
The first two time-steps are depicted pictorially in Fig. 3.2, and illustrate the main
ideas. The full propagator acting on the entire lattice is decomposed by applying the HHKL
decomposition K times, such that two of every three forward and reverse time-evolution
operators commute with all previous operators by virtue of being spatially disjoint, allowing
them to be pushed through and act identically on the vacuum. The remaining forward
evolution operator effectively spreads out the bosonic operators by distance ?. The error per
time-step is polynomially suppressed by O(???+D+1 + e???).
While it reduces the exponential factor to a polynomial one, using time-slices comes
a?t the cost of extra polynomial factors, originating from the sum over boundary termsN?1(r D?1j=0 0 + j?) .
Proof of Lemma 7. Let the initial positions of the bosons be denoted by (in1 , . . . , inn). The
initial state is |?(0)? = a?in . . . a
?
in |0? As before, the first two time-steps are illustrated1 n
in Fig. 3.2. Within each cluster Ci, there is a group of bosons initially separated from
58
?
the edge of the cluster by a region of width Li. Let A?i (0) =
?
inj?C a be the cre-i inj
ation operator for the group?of bosons in the cluster Ci, so that the initial state can also
be expressed as |(?(0)?? =
K ?
i=1 A)i (0) |0?. The forward-time propagator on a region R
is R tU 1t0,t = T exp ?i H (s)ds . When evolved for short times, each creation opera-1 t R0
tor a?in (t) is mostly supported over a small region around its initial position. Therefore, asi
long as these regions do not overlap, each operator approximately commutes, and the state
is approximately separable.
Let Ai be the smallest ball upon which A?i (0) is supported. Let Bi0 = Ai and denote
its radius ri0, and define r0 = max ri0. Bik is a ball of radius ri0+k? containingAi, where ? will
be chosen to minimize the error. Sik is the shell Bi \Bik k?1 (see Fig. 3.2). The complement
of a set X is denoted as Xc. We divide the evolution into N time steps between t0 = 0
and tN = t, and first show that the evolution is well-controlled for one time step from 0 to
t1 = t/N . We apply this decomposition K times, once for each cluster, letting X = Bi0,
Y = Si1 and Z be everything else:
B11 S
1 (B1)c
U0,t1 ? U0,t (U 1 ? 01 0,t ) U1 0,t (3.2)1
B1 S1 ? B2? 1 1 1 S
2 1 2 c
U (U ) U (U 1 )?
(B B )
0,t 0,t 0,t 0,t U
0 0 (3.3)
1 1 1 1 0,t1
1 1 K K 1 K c
? B1 S1 ? B1 S1 ? (B0 ...B )U 00,t (U1 0,t ) . . . U1 0,t (U1 0,t ) U1 0,t . (3.4)1
(? ) ( )
The total error isO K (evt1i=1 ? 1)?(Bi ??+D+1 ???0)(? + e ) =O K(evt1 ? 1)rD?1(???+D+1 + e???0 ) .
Applying the decomposed unitary to the initial state and pushing commuting terms through
59
to the vacuum state, we get
(
1 K ? )K i
U0,t1 |?(0)? ?
B
U 1 ?
B
0,t A1 . . . U
1 ?
0,t AK |0?
B
= U 1 A?
1 1 0,t1 i
|0? .
i=1
We can repeat the procedure for the unitary Ut1,t2 , where t2 = 2t1. Now, the separating
region Y will be Si2, so that Si2 ? Bi1 = ?. Each such region still has width ?, but now the
boundary of the interior is ?(Bi1) = O((r + ?)D?10 ). We get
(? )K
Bi i 1 K c
U 2
S2 ? (B1 ...B1 )
t1,t2 ? Ut1,t (U2 t1,t ) Ut ,t , (3.5)2 1 2
i=1
with error O(K(evt1 ? 1)(r + l)D?1(???+D+1 + e???0 )). The unitaries supported on Si2
and (B11 . . . BK c1 ) commute with all the creation operators supported on sites Bi1, giving
B1 B1 BK| B
K
?(t2)? ? U 2 U 1 2 1t ,t 0,t . . . Ut ,t U0,t |?(0)?. By applying this procedure a total of N times,1 2 1 1 2 1
once for each time step, we get the approximation
B1 1 K K
U N
B1 BN B1
0,t |?(0)? ? Ut ? ,t . . . U0,t . . . Ut ? ,t . . . U0,t |?(0)?. The total error in the stateN N 1 N 1 N 1 N 1
(in 2-norm) is
( )
N??1
? ? O K(evt1 ? 1)(???+D+1 + e???) (r0 + j?)D?1 (3.6)
( j=0 )
= O n(evt1 ? 1)(???+D+1 + e???)NLD?1 , (3.7)
proving Lemma 7. The last inequality comes from the fact that K ? n and that r0 + (N ?
1)? ? minLi = L. The latter condition ensures that the decomposition of the full unitary
is separable on the clusters.
60
In the regime ?> 2D +D/(? ? 1), teasy is optimized by choosing a fixed time-step
size t1 =O(1). Then, the number of steps N scales as the evolution time N = t/t1. By the
last few time-steps, the bosonic operators have spread out and have a boundary of size LD?1,
so the boundary terms contribute O(NLD?1) in total. In the regime D + 1<?? 2D +
D/(??1), the boundary contribution outweighs the suppression ???+D+1. Instead, we use
a single time-step in this regime, resulting in teasy = ?(logn) when ? > 1.
3.7 Closeness of evolution under H and H ?.
Next, we show that the states evolving due toH andH ? are close, owing to the way the
truncation works. This will enable us to prove that the easiness timescale for H is the same
as that of H ?. Suppose that an initial state |?(0)? evolves under two different Hamiltonians
H(t) and H ?(t) for time t, giving the states |?(t)? = Ut |?(0)? and |??(t)? = U ?t |?(0)?,
respectively. Define |?(t)? = |?(t)? ? |??(t)? and switch to the rotating frame, |?r(t)? =
U ? ? ?t |?(t)? = |?(0)? ? Ut Ut |?(0)?. Now taking the derivative,
i? |?r(t)? = 0 + U ?t tH(t)U ?t |?(0)? ? U
?H ?t (t)U
?
t |?(0)? (3.8)
= U ? ?t (H(t)?H (t)) |??(t)? . (3.9)
The first line comes about because i? ? ? ? ? ?tUt = H (t)Ut and i?tUt = ?UtH(t), owing to the
time-ordered form of Ut.
61
Now, we can bound the norm of the distance, ?(t) := ?|?(t)?? = ?|?r(t)??.
? t
?(t) ? ?(0) + d??(H(?)?H ?? (?)) |?
?(?)?? (3.10)
0
t
= d??(H(?)?H ?(?)) |??(?)??, (3.11)
0
since ?(0) = 0.
The next step is to bound the norm of (H?H ?) |??(?)? (we suppress the time label ?
in the argument ofH andH ? here and below). We use the HHKL decomposition: |??(?)? =
|?(?)?+ |?(?)?, where the state |?(?)? is a product state over clusters, and |?(?)? is the error
induced by the decomposition. We first show that (H ?H ?) |?(?)? = 0. Since |?(?)? is a
product state of clusters, each of which is time-evolved separately, boson number is conserved
within each cluster. Therefore, each cluster has at most b bosons, and Q |?(?)? = |?(?)?.
Furthermore, only the hopping terms inH can change the boson number distribution among
the different clusters, and these terms move single bosons. This implies thatH |?(?)? has at
most b+1 bosons per cluster, and remains within the image ofQ, denoted imQ. Combining
these observations, we get H ? |?(?)? = QHQ |?(?)? = H |?(?)?. This enables us to say
that (H ?H ?) |?(?)? = (H ?QHQ) |?(?)? = 0. Equation (3.11) gives us
? t
?(t) ? ? d??(H(?)?H
?(?))(|?(?)?+ |?(?)?)? (3.12)
0
t
= d??(H(?)?H ?(?)) |?(?)??,
0 ? (3.13)t
? max ?(H(?)?H ?(?)) |??? d??|?(?)??. (3.14)
?,|???im Q 0
In the last inequality, we have upper bounded ?(H(?)?H ?(?)) |?(?)?? bymax ?|???im Q ?(H ?H ) |????
62
?(?), where ?(?) := ?|?(?)??. The quantity max|???im Q ?(H ?H ?) |??? can be thought of
as an operator norm of H ? H ?, restricted to the image of Q. It is enough to consider a
maximization over states |?? in the image of Q because we know that the error term |?(?)?
also belongs to this subspace, as |??(?)? belongs to this subspace. Further, we give a uniform
(time-independent) bound on this operator norm, which accounts for the maximization over
times ? .
?? ?
Lemma 8. max ? ? ?|?? (H ?QHQ) |??? ? ? ? D??i?C ,j?C Jijaiaj? ? O(bL ).k l
Proof. Notice that for each term Hi in the Hamiltonian, the operator H ? QHQ contains
H?i ?QHiQ, where the rightmost Q can be neglected since Q |?? = |??. The on-site terms
i J
?
iiaiai+V ni(ni? 1)/2 do not change the boson number. Therefore, they cannot take
|?? outside the image ofQ, and do not contribute to (H?QHQ) |??. The only contribution
comes from hopping terms that change boson number, which we bound by
????? ? ?
?
J a?
?
ij iaj??, (3.15)
i?Ck,j?Cl
where the sum is over sites i and j in distinct clusters Ck and Cl, respectively. This is because
only hopping terms that connect different clusters can bring |?? outside the image ofQ, since
hopping terms within a single cluster maintain the number of bosons per cluster.
For illustration, let us focus on terms that couple two clustersC1 and C2. The distance
between these two clusters is denoted L12. For any coupling Jij with i ? C1 and j ? C2,
63
3 3 3 3 3 3 3 3 3
3 2 2 2 2 2 2 2 3
3 2 1 1 1 1 1 2 3
3 2 1 0 0 0 1 2 3
3 2 1 0 0 1 2 3
3 2 1 0 0 0 1 2 3
3 2 1 1 1 1 1 2 3
3 2 2 2 2 2 2 2 3
3 3 3 3 3 3 3 3 3
(a) (b)
Figure 3.3: (a) Cluster distances between the blue cluster in the center and nearby clusters.
The total number of clusters at cluster distance l = 2 (pink background) is given by (2l +
3)2? (2l+1)2 = 24. (b) Non-overlapping pairing between clusters separated by a diagonal.
The distance between these clusters is l = 0, since they share a boundary and contain adjacent
sites.
we can bound |Jij| ? L??12 by assumption. Let
hop ?
H12 = Jija
?
iaj + h.c. (3.16)
i?C1 j?C2
? ?
denote the sum over all such?pairs of sites. Then, we can bound
?? hop ?H12 |??? ? O(b). To
see this, diagonalize hopH ?12 = i wibibi. Since
hop
H12 only acts on two clusters, each normal
mode contains up to 2b bosons. The maximum eigenvalue of hopH12 is bounded by 2bmaxi wi,
where wi is the maximum normal mode frequency, given by the eigenvalue of the matrix
Jij : i ? C1, j ? C2. We now apply the Gershgorin circle?theorem, which states that the
maximum eigenvalue of J is bounded by the quantity maxi( j |Jij|) ? LDL??12 .
Taking advantage of the fact that the clusters form a cubic lattice in D dimensions,
we can group pairs of clusters by their relative distances. If we label clusters i by their D-
dimensional coordinate i1, i2, ..., iD, then we can define the cluster distance l between i
64
and j as l + 1 = maxd |id ? jd|. Cluster distance l corresponds to a minimum separation
l ? L between sites in different clusters. With this definition, there are ((2l + 3)D ? (2l +
1)D) ? 2DDlD?1 clusters at a cluster distance l from any given cluster (Fig. 3.3(a)), and
K ? 2DDlD?1 total pairs of clusters at cluster distance l. Notice that for a given separa-
tion vector, K/2 pairs of clusters (K total) can be simultaneously coupled without overlap
(Fig. 3.3(b)). Therefore, there are approximately 2D+1DlD?1 non-overlapping groupings
per distance l. The sum over these non-overlapping Hamiltonians hop hopHa + ... +H1b1 aK/2bK/2
for each grouping is block diagonal. Therefore, the spectral norm (maximum eigenvalue)
of the total Hamiltonian is equal to the maximum of the spectral norm over all irreducible
blocks. Putting all this together, as long as D ? 1? ? < ?1 the bound becomes
?lmax
max ?(H ?QHQ) |??? ? O(2D+1DlD?1)(2b)LD(lL)?? = O(bLD??). (3.17)
|???im Q
l=0
We are now in a position to prove our main easiness result.
Theorem 9 (Easiness result). For ?>D+1, and for all V , including V = o(1) and V = ?(1),
we have teasy=?(n?easy), with
? ? 1 ? ?? 2D 1?easy = ? , (3.18)
D ??D ??D
and teasy=?(logn) if ?easy< 0.
Proof. There are two error contributions, ? and ?, to the total error. The HHKL error ? is
given by evaluation of Eq. (3.7), which is minimized by either choosing N = 1 or N = t/t1
65
with t1 a small fixed constant. This leads to three regimes with errors
?
?????????nevt?L, ? ? ?
? ? O(1)????? nt??D D? (3.19)?L??2D , 2D + ? < ? < ?? ? 1??n(evt?1)
??D?1 , D + 1 < ? ? 2D + D .L ??1
The truncation error, arising from using H ? rather than H in the first step, is given
by
? t
?(t) ? O(bLD??) d??(?). (3.20)
0
Therefore, we can upper bound ?(t) by ? times an additional factor. This factor is bLD??t,
?(t) = bLD??t?, when ?(?) = poly(?) (? < ?), and it is LD??, ?(t) = LD??? when
?(?) = exp(v?) (? ? ?). Our easiness results only hold for ? > D + 1, so the L-
dependent factor serves to suppress the truncation error in the asymptotic limit. Although
the additional factor of t could cause ?(t) > ? at late times, by this time, ? > ?(1) and
we are no longer in the easy regime. Therefore, the errors presented in Eq. (3.19) can be
immediately applied to calculate the timescales in Section 3.2.
The resulting timescales are summarized in Table 3.3, which highlights the scaling of
the timescale with respect to different physical parameters. We also consider the scaling of
the easiness timescales when the density of the bosons increases by a factor ?. In our setting,
we implement this by scaling the number of bosons by ? while keeping the number of lattice
sites and the number of clusters (and their size) fixed. The effect of this is to increase the
66
Regime Error teasy(n, L) teasy(?)/teasy(? = 1)
? ? ? nevt?L L 1/?
2D + D
??D ?1 ??2D ?2
< ? < ? nvt? n??DL ??D ???D??1 L? 2D
D + 1 < ? ? 2D + D n(e
vt?1)
? ??D?1 (??D ? 1) logL? logn 1/?? 1 L
Table 3.3: Summary of easiness timescales in the different regimes. Timescales follow from
the error and are presented first as a function of n and L, which are the relevant physical
scales of the problem. We study the effect of the density by performing the scaling n ?
?n, L ? L, b ? ?b,m ? m. The last column shows the timescale as a function of ? in
terms of the timescale when ? = 1, namely teasy(? = 1).
Lieb-Robinson velocity: v ? v?. For all three cases, the net effect of increasing the density
by a factor ? is to decrease the easiness timescale.
3.8 Extended Easiness timescale for 1D
In this section, we prove that 1D systems with nearest-neighbor interactions (? ? ?)
can be simulated for longer times by using matrix product states (MPS). Specifically, we can
show that approximate sampling up to time t = cL for any constant c is easy. This shows
sampling is easy for any timescale t = O(L). When combined with our hardness result for
timescales t = O(L1+?), this proves the transition is coarse.
The first step is to show how time evolution up to time t = cL can be approximated
with low error using the HHKL decomposition. Earlier, we proved that for t = L/v, we
can self-consistently truncate the local Hilbert space dimension while incurring little error.
Then, the sampling algorithm relies on the separability of the approximate wavefunction.
For c > 1, this separability no longer holds. Nevertheless, we can use a similar ar-
gument to self-consistently truncate the local Hilbert space. However, due to the lack of
67
Figure 3.4: Schematic of the HHKL decomposition used to extend the classical simulation
algorithm to longer times. Red and orange blocks denote forward time evolution, and blue
blocks denote backwards time evolution. Two timesteps are depicted, k and k + 1. The
past causal lightcone of one the blocks is outlined in black. The lightcone dimensions and
HHKL block size double during subsequent timesteps, leading to an exponentially large
causal region.
separability, we only have an efficient simulation algorithm in D = 1, where tensor network
representations of low-entanglement (area law) states can be efficiently contracted.
We divide the time evolution into time intervals, where we have already proved sam-
pling is easy in the first interval up to time t = t1 = L/v0. Recall that during this first
time interval, we applied the HHKL decomposition with a block size L. For the next time
interval, we decompose the unitary Ut2,t1 choosing an HHKL block size 2L. By looking at
the past causal lightcone of a single site, it is clear that the boson number per site is bounded
by 4b. Therefore, the Lieb-Robinson velocity during this time interval is f(our times large)r
that the first time interval, v v1(t2?t1)?2L1 = 4v0. Setting the approximation error O ne
to be o(1), we can determine that for t L2 ? t1 = O( ) the approximation errors are well2v0
controlled. Next, we need to generalize the argument to k time evolution steps.
We prove that we can define a self-consistent Hilbert space truncation scheme and
an HHKL decomposition of the time-evolution unitary. During the k-th time interval,
68
we choose the HHKL block size to be L kk = 2 L. To estimate the maximum local boson
number, we look at the spatial extent of the past causal lightcone. Let Ck + 2Lk denote the
spatial extent of the past causal lightcone. Here Ck measures how far sideways the lightcone
extends (see Fig. 3.4). Looking at the lightcone, we can write down a recurrence relation for
Ck.
2Lk+1 + Ck+1 = 4Lk+1 + Ck (3.21)
Ck+1 = 2Lk+1 + C = 2
k+2
k L+ Ck (3.22)
=? Ck = 4(2k ? 1)L, (3.23)
where we have applied the initial condition C0 = 0. Combined with the boson density
? = 1 , the maximum boson number after step k is proportional to the size of the lightcone:
2L
Nk = ?(Ck + 2Lk) (3.24)
= ?L(6? 2k ? 4) = 3? 2k ? 2 (3.25)
This tells us we can choose timesteps tk ? tk?1 = O(L/3v0) for the error to be o(1).
To complete the proof, we need to argue that the HHKL time-evolution can be simu-
lated efficiently. The simplest way to do this is to fix the evolution time t = cL/3v0. Then,
we can think of the HHKL decomposed unitary as a finite-depth unitary circuit, which
generates constant (system-size independent) entanglement across any cut. Therefore, stan-
dard MPS algorithms can be applied to with cost polynomial in the system size, the local
Hilbert space dimension (physical index dimension), and the entanglement across any cut
69
(virtual index dimension) [176]. Although both the physical and virtual dimensions scale
exponentially or faster with c, since we have fixed c to be system-size independent, we have
an efficient sampling algorithm. Furthermore, this algorithm works for any fixed c, proving
that sampling cannot be hard for evolution times t = O(L) and that the transition is coarse.
3.9 Hardness timescale for interacting bosons
In this section, we provide more details about how to achieve the timescales in our
hardness results, which we state in more detail first.
Theorem 10 (Hardness result). When ??D/2, V =?(1), andD? 2, the hardness timescale
is It ?hard=O(n hard), where
???????1? min[1, ??D], ? > DD?Ihard = ??? (3.26)0, ? ? [D , D].
2
In all other cases, i.e. when at least one of the following cases hold: ?<D / 2, V = o(1), or
D=1, the timescale is IIthard=O(n?hard), where
??????? [ ]?????1 min 1 +
O(log(V+1))
D log , ??D , ? > Dn
?IIhard = ? +??????0, ? ? [
D , D] (3.27)
??? ( )
2
? ?? D , ? < D/2
D 2
for an arbitrarily small ? > 0.
Almost any bosonic interaction is universal for BQP [166] and hence these results are
70
applicable to general on-site interactions f(ni). Reference [188] also answers the questions
of what additional gates or Hamiltonians can make linear optics universal. We first describe
how a bosonic system wi?th fully controllable local fields Jii(t), hoppings Jij(t), and a fixed
Hubbard interaction V i n?i(n?i ? 1) can implement a universal quantum gate set. To2
simulate quantum circuits, which act on two-state spins, we use a dual-rail encoding. Using
2n bosonic modes, and n bosons, n logical qubits are defined by partitioning the lattice
into pairs of adjacent modes, and a boson is placed in each pair. Each logical qubit spans a
subspace of the two-mode Hilbert space. Specifically, |0?L = |10? , |1?L = |01?. We can
implement any single qubit (2-mode) unitary by turning on a hopping between the two sites
(X-rotations) or by applying a local on-site field (Z-rotations). To complete a universal gate
set, we need a two-qubit entangling gate. This can be done, say, by applying a hopping term
between two sites that belong to different logical qubits [175]. All these gates are achievable
in O(1) time when V = ?(1). In the limit of large Hubbard interaction V ? ?, the
entangling power of the gate decreases as 1/V [175] and one needs O(V ) repetitions of the
gate in order to implement a standard entangling gate such as the CNOT.
For hardness proofs that employ postselection gadgets, we must ensure that the gate
set we work with comes equipped with a Solovay-Kitaev theorem. This is the case if the gate
set is closed under inverse, or contains an irreducible representation of a non-Abelian group
[189]. In our case, the gate set contains single-qubit Paulis and hence has a Solovay-Kitaev
theorem, which is important for the postselection gadgets to work as intended.
We will specifically deal with the scheme proposed in Ref. [24]. It applies a constant-
? ?
depth circuit on a grid of n ? n qubits in order to implement a random IQP circuit
?
[12; 19] on n effective qubits. This comes about because the cluster state, which is a
71
(a) (b)
(c) (d)
Figure 3.5: A protocol that implements the logical circuit of Ref. [24]. Each subfigure shows
the location of the site that previously encoded the |1? state in gray. The current site that
encodes the |1? state is in black. The site that encodes |0? is not shown but moves similarly
as the |1? state. The distance traversed by each qubit is L+ L+ 2L+ 2L = 6L.
72
universal resource for measurement-based quantum computation, can be made with constant
depth on a two-dimensional grid.
For short-range hops (? ? ?), we implement the scheme in four steps as shown in
Fig. 3.5. In each step, we move the logical qubits to bring them near each other and make
them interact in order to effect an entangling gate. For short-range hopping, the time taken
to move a boson to a far-off site distance L away dominates the time taken for an entangling
gate. The total time for an entangling gate is thus O(L) +O(1) = O(L).
For long-range hopping, we use the same scheme as in Fig. 3.5, but we use the long-
range hopping to speed up the movement of the logical qubits. This is precisely the question
of state transfer using long-range interactions/hops [172; 173; 190]. In the following we
give an overview of the best known protocol for state transfer, but first we should clarify the
assumptions in the model. The Hamiltonian is a sum of O(m2) terms, each of which has
norm bounded by at most 1/d(i, j)?. Since we assume we can apply any Hamiltonian subject
to these constraints, in particular, we may choose to apply hopping terms across all possible
edges. This model makes it possible to go faster than the circuit model if we compare the
time in the Hamiltonian model with depth in the circuit model. This power comes about
because of the possibility of allowing simultaneous noncommuting terms to be applied in the
Hamiltonian model.
The state transfer protocols in Ref. [172; 173] show such a s?peedup for state transfer.
The broad idea in both protocols is to apply a map |1?1 ? |1?A := 1j? ?A |1? , followed|A| j
by the steps |1?A ? |1?B and |1?B ? |1?2, where A and B are regions of the lattice to be
specified. In the protocol of Ref. [172], which is faster than that of Ref. [173] for ? ? D/2,
?
A = B = {j : j =? 1, 2} and each step takes time O(L?/ N ? 2), where N ? 2 is the
73
number of ancillas used and L is the distance between the two furthest sites. In the protocol
of Ref. [173], which is faster for ? ? (D/2, D + 1], A and B are large regions around the
initial and final sites, respectively. This protocol takes time O(1) when ? < D, O(logL)
when ? = D, and O(L??D) when ? > D.
In our setting, we use the state transfer protocols to move the logical qubit faster than
timeO(L) in each step of the scheme depicted in Fig. 3.5. If ? < D/2, we use all the ancillas
in the entire system, giving a state transfer time of O(m?/D?1/2 ? 1) = O(n?( ? )D 2 ). If ? >
D/2, we only use the empty sites in a cluster as ancillas in the protocol of Ref. [173], giving
the state transfer time mentioned above. This time is faster thanO(L), the time it would take
for the nearest-neighbor case, when ? < D+1. Therefore, for 2D or higher and ? ? D/2,
the total time it takes to implement a hard-to-simulate circuit is min[L,L??D logL]+O(1),
proving Theorem 10 for interacting bosons. When ? < D/2, the limiting step is dominated
by the entangling gate, which takes time O(1). Therefore for this case we only get fast
hardness through boson sampling, which is discussed in Section IV. Note that when t = o(1)
and interaction strength is V = ?(1), the effect of the interaction is governed by V t = o(1),
which justifies treating the problem for short times as a free-boson problem.
3.9.1 One dimension
In 1D with nearest-neighbor hopping, we cannot hope to get a hardness result for
simulating constant depth circuits, which is related to the fact that one cannot have universal
measurement-based quantum computing in one dimension. We change our strategy here.
The overall goal in 1D is to still be able to simulate the scheme in Ref. [24] since it provides
74
a faster hardness time (at the cost of an overhead in the qubits). The way this is done is to
either (i) implement O(n) SWAPs in 1D in order to implement an IQP circuit [12], or (ii)
use the long-range hops to directly implement gates between logical qubits at a distance L
away.
For the first method, we use state transfer to implement a SWAP by moving each
boson within a cluster a distance ?(L). This takes time O(ts(L)), where ts(L) is the time
taken for state transfer over a distance L and is given by
?
??????
????L, ? > 2
??
???????L??1, ? ? (1, 2]
ts(L) = c???????logL, ? = 1
(3.28)
?????????1, ? ? [
1 , 1)
? 2???L??1/2, ? < 1 .
2
( )
Wewrite this su(ccinctly asO min[L,L
??1 logL+ 1, L??1/2) ] . The total time for n SWAPs
is therefore O n?min[L,L??1 logL+ 1, L??1/2] .
The second method relies on the observation that when ? ? 0, the distinction be-
tween 1D and 2D becomes less clear, since at ? = 0, the connectivity is described by a
complete graph and all hopping strengths are equal. Let us give some intuition for the
? ? 0 case. One would directly ?sculpt? a 2D grid from the available graph, which is a com-
plete graph on n vertices (one for every logical qubit) with weights wij given by d(i, j)??.
If we want to arrange qubits on a 1D path, we can assign an indexing to qubits in the 2D
75
Figure 3.6: A snaking scheme to assign indices to qubits in 2D for a n1/k ? n1?1/k grid,
which is used in mapping to 1D.
grid and place them in the 1D path in increasing order of their index. One may, in particu-
lar, choose a ?snake-like indexing? depicted in Fig. 3.6. This ensures that nearest-neighbor
gates along one axis of the 2D grid map to nearest-neighbor gates in 1D. Gates along the
other axis, however, correspond to nonlocal gates in 1D. Suppose that the equivalent grid in
2D is of size n1/k ? n1?1/k. The distance between two qubits that have to participate in a
gate is now marginally larger (O(Ln1/k) instead of O(L)), but the depth is greatly reduced:
it is now O(n1/k) instead of O(n). We again use state transfer to move close to a far-off
qubit and then perform a nearest-neighbor (entangling gate. This time is set by the) state
transfer protocol, and is now t 1/k 1/k ??1s(n L) = O( n ?min[L,L logL+ 1, L
??1/2
) ] . For
large k = ?(1), this gives us the bound O min[L1+?, L??1+? + L?(?), L??1/2+?] for any
? > 0, giving a coarse transition. Notice, however, that faster hardness in 1D comes at
a high cost? the effective number of qubits on which we implement a hard circuit is only
?(n1/k) = n?(?), which approaches a constant as ? ? 0.
This example of 1D is very instructive? it exhibits one particular way in which the
76
(Figure 3.7: (a) A hopping between s)ites 2 and 3 that implements the mode unitary
cos(|J |t) ?i sin(|J |t)J/|J | ?it(Re{J}X+Im{J}Y )
? sin ? | | cos | | = e . When |J |t = ?, this isi (Jt)J / J ( J t)
a SWAP between two modes with phases (?iJ/|J |,?iJ?/|J |) that depends on arg J , the
argument of J . (b) A ?physical? SWAP between sites 2 and 3 by using ancilla sites available
whenever the system is not nearest-neighbor in 1D. The colors are used to label the modes
and how they move, and do not mean that both sites are occupied. The total hopping phase
incurred when performing the physical SWAP can be set to be (+i,?i), which cannot be
achieved with just the hopping term shown in (a).
complexity phase transition can happen. As we take higher and higher values of k, the
hardness time would decrease, coming at the cost of a decreased number of effective qubits.
This smoothly morphs into the easiness regime when ? ? ? since in this regime both
transitions happen at t = ?(L).
If the definition of hardness is more stringent (in order to link it to fine-grained com-
plexity measures such as explicit quantitative lower bound conjectures), then the above men-
tioned overhead is undesirable. In this case we would adopt the first strategy to implement
SWAPs and directly implement a random IQP circuit on all the n qubits. This would increase
the hardness time by a factor n.
77
3.9.2 Hardcore limit
In the hardcore limit V ? ?, the strategy is modified. Let us consider a physical
qubit to represent the presence (|1?) or absence (|0?) of a boson at a site. A nearest-neighbor
hop translates to a term in the Hamiltonian that can be written in terms of the Pauli operators
asXX+Y Y . Further, an on-site field J ?iiaiai translates to a term ? Z. There are no other
terms available, in particular single-qubit rotations about other axesX or Y . This is because
the?total boson number is conserved, which in the spin basis corresponds to the conservation
of i Zi. This operator indeed commutes with both the allowedHamiltonian terms specified
above.
Let us now discuss the computational power of this model. When the physical qubits
are constrained to have nearest-neighbor interactions in 1D, this model is nonuniversal and
classically simulable. This can be interpreted due to the fact that this model is equivalent to
matchgates on a path (i.e. a 1D nearest-neighbor graph), which is nonuniversal for quantum
computing without access to a SWAP gate. Alternatively, one can apply the Jordan-Wigner
transformation to map the spin model onto free fermions. One may then use the fact that
fermion sampling is simulable on a classical computer [51].
When the connectivity of the qubit interactions is different, the model is computa-
tionally universal for BQP. In the matchgate picture, this result follows from Ref. [191],
which shows that matchgates on any graph apart from the path or the cycle are universal for
BQP in an encoded sense. In the fermion picture, the Jordan-Wigner transformation on
any graph other than a path graph would typically result in nonlocal interacting terms that
are not quadratic in general. Thus, the model cannot be mapped to free, quadratic fermions
78
and the simulability proof from Ref. [51] breaks down.
Alternatively, a constructive way of seeing how we can recover universality is as follows.
Consider again the dual rail encoding and two logical qubits placed next to each other as in
Fig. 3.7. Apply a coupling J(a?2a3 + a
?
3a2) on the modes 2 and 3 for time t = ? . This2J
effects the transition |10?23 ? ?i |01?23 and |01?23 ? ?i |10?23, while leaving the state
|11?23 the same. Now we swap the modes 2 and 3 using an ancilla mode that is available by
virtue of having either long-range hopping or having D > 1. This returns the system back
to the logical subspace of exactly one boson in modes 1 & 2, and one boson in modes 3 &
4, and effects the unitary diag{1, 1, 1,?1} in the (logical) computational basis. This is an
entangling gate that can be implemented in O(1) time and thus the hardness timescale for
hardcore interactions is the same as that of Hubbard interactions with V = ?(1).
We finally discuss the case when V is polynomially large. Using the dual-rail encoding
and implementing the same protocol as the non-h(ardcore cas)e now takes the state |11?23?
to | |20? +|02? 2 2? 11? + ? 23? 23 , with ? ? ? J23 2 2 sin t 8J +V . When |?| ?= 0, we get an2 8J +V 2
error because the state is outside the logical subspace. The probability with which this action
happens is suppressed by 1/V 2, however, which is polynomially small when V = poly(n).
However, one can do better: by carefully tuning the hopping strength J ? [0, 1]
and the evolution time t, one[can alw]ays achieve the goal of getting ? = 0 exactly and
im?plementing an operation exp ?i?X in the |10?23 , |01?23 subspace. This requires setting2
2
t 2J2 + V = m? and t = 2? for integer m. This can be solved as follows: set m =
4 J
? ?
? 8 + V 2?, and J = ? V (which is ? 1 since m ? 8 + V 22? ). The time is set bym 8
the condition t = 2? , which is ?(1). This effects a logical CPHASE[?] gate with angle
J
? = ??V /J .
79
Finally, the above parameters that set ? exactly to zero work even for exponentially large
V = ?(exp(n)), but this requires exponentially precise control of the parameters J and t,
which may not be physically feasible. In this case, we simply observe that |?|2, the probability
of going outside the logical subspace and hence making an error, is O(1/V 2), which is
exponentially small in n. Therefore, in this limit, the gate we implement is exponentially
close to perfect, and the complete circuit has a very small infidelity as well.
3.10 Hardness timescale for free bosons
In this section, we review Aaronson and Arkhipov?s method of creating a linear optical
sta(te that is har)d to sample from [13]. We then give a way to construct such states in time
O? nm?/D?1/2 with high probability in the Hamiltonian model, and prove Theorem 10 for
free bosons.
For free bosons, in order to get a state that is hard to sample from, we need to apply a
Haar-random linear-optical unitary on m modes to the state |1, 1, . . . 1, 0, 0, . . . 0?. Aaron-
son and Arkhipov gave a method of preparing the resulting state in O(n logm) depth in the
circuit model. Their method involves the use of ancillas and can be thought of as implement-
ing each column of the Haar-ran?dom unitary separately in O(logm)-depth. Here we mean
that we apply the map |1?j ? i?? Uij |1?i to ?implement? the column i of the linear-
optical unitary U . In the Hamiltonian model, we can apply simultaneous, non-commuting
terms of a Hamiltonian involving a common site. The only constraint is that each term of
the Hamiltonian should have a bounded norm of 1/d(i, j)?. In this model, when ? is small,
it is possible to implement each unitary in a time much smaller than O(logm)? indeed, we
80
show the following:
Lemma 11. Let U be a Haar-random unitary on m modes. Then with probability 1 ?
1
pol(y over t)he Haar measure, each of the first n columns of U can be implemented in time(m)?
logm
O
m1/2??/D
.
To prove this, we will need an algorithm that implements columns of the unitary. For
convenience, let us first consider the case ? = 0. The algorithm involves two subroutines,
which we call the single-shot and state-transfer protocols. Both protocols depend on the
following observation. If we implement a Hamiltonian that couples a site i to all other sites
j =? i through coupling?strengths Jij , then the e?ffe?ctive dynamics is that of two coupled
modes a?i and b? = 1
?
j ?=i Jijaj , where ? = j=? i J2ij . The effective speed of the?
dynamics is given by ?? for instance, the time period of the system is 2?? .?
The single-shot protocol implements a?map a
? ? ? ? ?i iai + j ?=i ?jaj . This is done
by simply applying the Hamiltonian H ? a?i ( j ?=i ?jaj) + h.c. for time t = 1 cos?1 |?? i|.
In the case ? = 0, we can set the proportionality factor equal to 1/max|?j|. This choice
means that the coupling strength between i and the site k with maximum |?k| is set to 1
(the maximum), and all other couplings are equal to | ?j |.
?k
The other subroutine, the state-transfer protocol is also an application of the above
observation and appears in Ref. [172]. It achieves the map a? ? ? a?+ ? a?i i i j j via two rounds
of the previous protocol. This is done by first mapping site i to the uniform superposition
over all sites except i and j, a(nd then coupli)ng this uniform superposition mode to site j.?
The time taken for this is 1 ? + cos?1 |?i| . Since ? = m(? 2 )(all m ? 2 modes are? 2
coupled with equal strength to modes i or j), this takes time O ?1 .
m
81
These subroutines form part of Algorithm 2. It can be seen that Algorithm 2 imple-
Algorithm 2: Algorithm for implementing one column of a unitary
Input: Unitary U , column index j
1 Reassign the mode labels for modes i ?= j in nonincreasing order of |Uij|.
2 Implement the?state-transfer protocol to map the state a?j |vac? to
U a? |vac?+ 1? |U |2a?jj j jj 1 |vac?. Skip this step if |Ujj| ? |Uj1| already.
3 Use the single-shot pr?otocol between site 1 and the rest (i ?= 1, j) to map
a?1 ? ?
U1j Ua?1 + ? ij
?
1?|U |2 i=? 1,j
a .
1?|U |2 ijj jj
?
ments a map a?j ? Ujja
?
j+ i ?=j Uija
?
i , as desired. To prove Lemma 11 we need to examine
the runtime of the algorithm when U is drawn from a Haar-random distribution.
Proof of Lemma 11. First, notice that since the Haar measure is invariant under the action of
any unitary, we can in particular apply a permutation map to argue that the elements of the
i?th column are drawn from the same distribution as the first column. Next, recall that one
may generate a Haar-random unitary by first generating m uniform random vectors in Cm
and then performing a Gram-Schmidt orthogonalization. In particular, this means that the
first column of a Haar-random unitary may be generated by generating a uniform random
vector with unit norm. This implies that the marginal distribution over any column of a
unitary drawn from the Haar measure is simply the uniform distribution over unit vectors,
since we argued above that all columns are drawn from the same distribution.
N(ow,)let us examine the runtime. The first step (line 2 of the algorith(m) requires tim)e
t = O ?1 irrespective ofUjj because the total time for state-transfer is 1 ? + cos?1 Um ? 2 jj ?
82
( )
? = ? ?? . Next, the second step takes time t =
1 cos?1 ? U1j = O( 1 ). Now,
? m 2 ? 1?|U21j | ?
?
|U |23j /(1? |Ujj|2) |U 22 4j|? =?1 + + + . . . (3.29)? |U 2 2 22j| /(1? |U?jj| ) |U2j|m |U |2i=2,i ?=j ij 1? |U 21j| ? |U 2jj|
= = (3.30)
|U 2 22j| |U2j|
Now in cases where |Ujj| ? |U1j| (where |U1j| is the maximum absolute value of the column
entry among all other modes i ?= j), which happens with probability 1 ? 1 , we will have
m
1?2|U |2
?2 ? 1j| |2 . In the other case when |Ujj| ? |U1j|, meaning that the maximum absoluteU2j
2
value among all entries of column j is in row j itself, we again have 2 ? 1?2|U |? jj|U 2 . Both these2j |
2
cases can be written together as 2 ? 1?2|U1j |? | |2 , where we now denote U1j as the entry withU2j
maximum absolute value among all elements of column j. The analysis co(mpletely hin)ges
on the typical ? we have, which in turn depends on |U1j|. We will show Pr ?2 ? cmlog ?m
1? 1poly , which will prove the claim for ? = 0.(m)
( ) ( )
Pr cm?2 ? log ? Pr 1? 2|U |
2
1j ? c 21 & |U2j| ?
c1 logm (3.31)
m cm
since the two events on the right hand side suffice for the first event to hold. Further,
( ) ( )
Pr 1? 2|U |21i ? c1 & | 2
c1 logm 2 c1 logmU2j| ? ? Pr |U1j| ? (3.32)
cm cm
for large enough m with some fixed c1 = 0.99 (say), since |U2j|2 ? |U |21j and 1 ?
1.98 logm/m ? 0.99 for large enough m.
To this end, we refer to the literature on order statistics of uniform random unit
83
vectors (z1, z2, . . . zm) ? Cm [192]. This Chapter gives an explicit formula for F (x,m),
the probability that all |zj|2 ? x. We are interested in this quantity at x = c1 logm/(cm),
since this gives us the probability of the desired event (?2 ? cm/ logm). We have
( )
1 1 ?k ?
? ?
m?
Pr ? x ? = ?? ?? (?1)l(1? lx)m?1. (3.33)k + 1 k
l=0 l
It is also argued in Ref. [192] that the terms of the series successively underestimate or
overestimate the desired probability. Therefore we can expand the series and terminate it at
the first two terms, giving us an inequality:
( )
1 1 2Pr ? x ? = 1? mm(1? x)m?1 + (1? 2x)m?1 ? . . . (3.34)
k + 1 k 2
? 1?m(1? x)m?1. (3.35)
Choosing c = c1/4 = 0.2475, we are interested in the quantity when k = ? m4 log ?:m
Pr(x ? 4 logm/m) ?[1?m(1? 4(logm/m)m?1)?
1
]1? [? , since (3].36)m3 4/m
(1? 4 log 4 logm logmm/m)m?1 = exp (m? 1) log 1? ? exp ?4(m? 1) = m?4(1?1/m).
m m
(3.37)
?
This implies that the time for the single-shot protocol is also t = O( 1 ) = O( logm) for a
? m
single column. Notice that we can make the polynomial appearing in Pr(?2 ? cm/ logm) ?
1?1/poly(m) as small as possible by suitably reducing c. To extend the proof to all columns,
84
we use the union bound. In the following, let tj denote the time to implement column j.
( ? ) ( ? )
Pr ? logm
?
j : tj > ? Pr
logm
tj > (3.38)
cm cm
j
? ? 1 1m poly = poly (3.39)(m) (m)
when the degree in the polynomial is larger than 1, just as we have chosen by setting c =
0.2475. This implies
( ? ) ( ? )
Pr ? logm logm 1j : tj ? = 1? Pr ?j : tj > ? 1? poly . (3.40)cm cm (m)
This completes the proof in the case ? = 0. When ? =? 0, we can in the worst-case set each
coupling constant to a maximum of O(m??/D), which is the maximum coupling strength
of the furthest two sites separated by a distance O(m1/D). This factor appears in the total
time for both the state-tran(sf?er [172] and sin)gle-shot(protocols,)and simply multiplies the?
required time, making it logm ? ?/D logmO m = O 1/2??/D . Finally, if there are anym m
phase shifts that need to be applied, they can be achieved through an on-site term J ?iiaiai,
whose strength is unbounded by assumption and can thus take arbitrarily short ti(me.? )
( The total)time for implementing boson sampling on n bosons is therefore
logm
O n
m1/2??/D
=
? 1
O? n1+?( ? )D 2 , since we should implement n columns in total.
85
3.10.1 Optimizing hardness time
We can optimize the hardness time by implementing boson sampling not on n bosons,
but on n? of them, for any ? ? (0, 1]. The explicit lower bounds on running time of classical
algorithms we woul[d get as]suming fine-grained complexity-theoretic conjectures is again
something like exp npoly(?) for any ? ? (0, 1]. This grows very slowly with n, but it
still qualifies as subexponential, which is not polynomial or quasipolynomial (and, by our
definition, would fall in the category ?hard?). This choice of parameters allows us to achieve
a smaller hardness timescale at the cost of getting a coarse (type-II) transition. We analyze
this idea in three cases: ? ? D/2, ? ? (D , D] and ? > D.
2
When ? ? D/2, we perform boson sampling on the nearest set of n? bosons with the
rest of the empty sites in the lattice as target sites. In terms of the linear optical unitary, the
unitary acts on m ? n? = ?(m) sites in the lattice, although only the n? columns corre-
sponding to initially occupied sites are relevant. Using the protocol in Lemma 11, the total
time to implement n? columns of anm?m linear optical unitary isO(n?m?/D?1/2 logn) =
?
O?(n?n (??D/2)D ).
When ? ? (D , D], the strategy is modified. We first move the nearest set of n? bosons
2
into a contiguous set of sites within a single cluster. This takes time O(n?), since each boson
may be transferred in time O(1). We now perform boson sampling on these n? bosons
with the surrounding n2? sites as targets, meaning that the effective number of total sites is
m = O(n2?eff ), as required for the hardness of boson sampling. Applying Lemma 11, the
time required to perform hard instances of boson sampling is nowO(n?n2?(?/D?1/2) logn) =
nO(?) for arbitrarily small ? > 0.
86
Lastly, when ? > D, we use the same protocol as above. The time taken for the
state transfer is now n? ? min[L,L??D]. Once state transfer has been achieved, we use
nearest-neighbor hops instead of Lemma 11 to create an instance of boson sampling in time
O(n2?/D). Since state transfer is the limiting step, the total time is n? ? min[L,L??D].
The hardness timescale is obtained by taking the optimum strategy in each case, giving the
hardness timescale IIthard = O?(n?hard), where
????????????1 min[1, ??D] ? > DD
?IIhard = ? +?????????
0 ? ? (D , D] (3.41)
( ) 2
? ?? D ? < D
D 2 2
for an arbitrarily small ? > 0. This proves Theorem 10 for free bosons and for interacting
bosons in the case? < D/2. When we compare with Ref. [96], which states a hardness result
for ? ? ?, we see that we have almost removed a factor of n from the timescale coming
from implementing n columns of the linear optical unitary. Our result here gives a coarse
hardness timescale of?(L) that matches the easiness timescale of L. More importantly, this
makes the noninteracting hardness timescale the same as the interacting one.
3.10.2 Almost free bosons, V = o(1)
When the interaction strength satisfies V = o(1) or when the bosons are almost free,
we can treat the evolution as being close to that of free bosons. The total variation distance
error between the actual distribution and the distribution modeled by free bosons can be
87
upper bounded by
?
? ??| ? ? ?? ???? ?(t) ???(t) ? =: ?(t), (3.42)
2
? ?
where we take H and |?? to be the actual Hamiltonian and state and H? and ???? to be
their respective free-bosonic approximations. Therefore, by the same logic leading up to
Eq. (3.11), we have the same expression here for ?(t):
? t ???( ) ?? ????(t) ? ? d?0 ? H(?)? H?(?) ???(?) ? (3.43)t
? d ???? ? ?? ??
??
? V f(ni) ???(?) ??. (3.44)0 i
Just as before, we use the fact that at short times, the boson number in each cluster
(and hence on each site) is bounded. Specifying to the case of Bose-Hubbard interactions,
we have
? ??t ? ? ???
? ? d V ? ? ?? ?? ni(ni ? 1) ???(?) ??. (3.45)0 2 i
3.11 Types of transitions
In this section, we study in more detail the number of encoded logical qubits in the sys-
tem inherent in the hardness proof, which we define more carefully below. Our definition is
valid for any family of Hamiltonians and the definition for circuit architectures is analogous.
Given constraints on the evolution time t (taken to be the depth in case of circuits), we would
88
Exponent of Easy
h(t, n) in n Type-I hard
Type-II hard
All D
D ? 2
D = 1
Easy
??1 ??1/2 ? Exponent of
D D D t in n
Figure 3.8: Schematic of types-I and II complexity phase transitions. The points on the
X-axis, (? ? 1)/D and ?/D, are different for different dimensions and therefore the lines
do not really intersect in the way depicted here.
like to reduce the problem of simulating arbitrary circuits of nearest-neighbor gates in one
dimension acting onm qubits for depthm to the problem of simulating postselected Hamil-
tonian evolution for time t under the given Hamiltonian. We call this reduction exploiting
postselection an ?embedding?. For a given reduction, we define the quantity h(t, n) to be the
number of qubits m that we can embed into a Hamiltonian evolution on n qubits/particles
evolving for time t. The motivation for defining this quantity is that if the function h(t, n)
is at least some (possibly sublinear) polynomial in n, then simulating time evolution for time
t under the Hamiltonian is hard modulo some conjectures.
3.11.1 Defining phase transitions
The function h(t, n) with respect to a certain reduction is an increasing function of the
time t for a fixed n, since given more time, the number of qubits that can be embedded cannot
decrease. At the transition timescale t?, the quantity h transitions from being subpolynomial
89
in n to polynomial in n as the system transitions from being easy to simulate to being hard
to simulate. The 2D and 1D phase diagrams in Fig. 3.1 can also be viewed in terms of the
function h of the corresponding family of Hamiltonians.
Figure 3.8 shows the behavior of h(t, n) under our hardness reductions for the two
types of transitions. We can see from the figure that if these lines are correct, sharp transi-
tions are akin to a I-order phase transition, and coarse transitions like II-order transitions.
Specifically, if the exponent of n in h(t, n) has a discontinuity with respect to the exponent
of n in the evolution time t, we have a I-order transition. For II-order transitions, on the
other hand, there is no discontinuity in the exponent of n in h(t, n). The exponent may be
defined as limn??(logh(t, n)/ logn). Therefore, the exponent and hence the transition is
only well-defined in the thermodynamic limit n ? ?, just as in regular phase transitions
where non-analyticities are only visible in the thermodynamic limit.
Let us look at how we obtained the lines. First, the region before exponent ??1
D
corresponding to time o(L) is easy in all dimensions from our easiness results, meaning
h necessarily scales smaller than any polynomial in n in the blue region. Also, we know
from our hardness results that near ?/D (i.e. time t = O(Ln1/D)) we have enough time
to ?touch? all n qubits, giving h(t, n) = n for all dimensions. This can be extended to
hardness for h(t, n) = n? for all ? ? (0, 1] in all dimensions, with the corresponding time
Ln?/D = n(??1+?)/D. For D ? 2 on the other hand, we can do better: we can always
?
encode n qubits even if the time is ?(L). This strategy is better than the first strategy for
time t < n(??1/2)/D. It is not known if these curves for the function h are optimal, since
for optimality we should rule out other reductions that might achieve a better scaling.
To sum up, considering the h-index gives a complementary way of looking at complex-
90
ity phase transitions of the sort we study. This is a fine-grained view of looking at complexity
phase transitions, since we care not just about whether the h-index is polylog (easy) or poly
(hard), but how exactly it scales with n as a function of the evolution time.
3.12 Outlook
We have mapped out the complexity of the long-range Bose-Hubbard model as a
function of the particle density ?, the degree of locality ?, the dimensionality D, and
the evolution time t. A particularly interesting open question concerns the regions of the
phase diagram without definitive easiness/hardness results. These gaps are closely related
to open problems in other areas of many-body physics and quantum computing. In the
nearest-neighbor limit, there is no gap between teasy and thard. When ? is finite, closing the
gap is closely tied to finding state-transfer protocols which saturate Lieb-Robinson bounds.
Stronger Lieb-Robinson bounds can increase teasy, and faster state-transfer can reduce thard,
as evidenced by the improvement over the original version of the manuscript [93] due to
results from Ref. [173]. These observations show that studying complexity phase transitions
provides a nice testbed for, and gives an alternative perspective on results pertaining to the
locality of quantum systems.
Our results can be easily adapted to a wide range of Hamiltonians in atomic physics
and quantum information. Several interesting Hamiltonians are special cases of the one that
we study, or straightforward extensions. For example, some experimental platforms such as
cold atoms, neutral atoms, and trapped ions can be naturally mapped onto our model. This
model is also ideal for studying interesting phenomena in quantum information such as dy-
91
namics in long-range interacting systems, quantum computational supremacy, entanglement
phase transitions, and models of modular networks. We elaborate on these connections in
Section 3.5.
If the qualitative features of the phase diagram we have derived for the Bose-Hubbard
model hold more generally, our results may hint at a notion of universality present in tran-
sitions between complexity phases. In 1D, we have proved that the transition is always
coarse. However in 2D and higher, when there are interactions, the transition is sharp. In
contrast, in 2D and higher for non-interacting transitions, it is unknown whether the tran-
sition is coarse or sharp, and the classification depends on approximate sampling hardness
of constant-depth boson sampling. This dependence on the dimension and possible de-
pendence on whether the system is interacting or noninteracting suggests the possibility of
classifying complexity phases of matter, and the transitions between them, based on generic
features such as connectivity, dimensionality, and kinds of interactions.
Along this line, it would be interesting to study whether similar features occur for
different kinds of complexity phase transitions. In our work, the transition occurs in the
dynamics of a many-body Hamiltonian. However, different approaches are possible. For
example, one could consider open quantum systems, where decoherence might drive dynam-
ical transitions from hard to easy. A particularly rich class of open quantum systems are
random quantum circuits with interspersed measurements [193?195]. These have been used
recently to study new non-equilibrium phases, entanglement phase transitions, and could be
a promising platform to study complexity phase transitions.
92
Chapter 4: Complexity of Fermionic Dissipative Interactions and Applica-
tions to Quantum Computing
Understanding whether a particular quantum system is easy or hard to simulate from
the perspective of classical computation is a crucial task serving several goals. The first goal,
as a primary step of many numerical studies, is to find efficient classical algorithms describ-
ing the desired quantum phenomena. Another goal arises in quantum computing, where
finding many-body systems lacking an effective classical description may be worthwhile for
constructing quantum computation [196] and simulation [3; 197] devices. The versatility
of the classical simulability problem can be illustrated by considering the sampling problem
for noninteracting and interacting fermions [14; 50; 51; 198]. There are efficient classical
algorithms to simulate fermions described by a quadratic Hamiltonian: the amplitudes of
time-evolved many-body configurations are expressed by an efficiently-computable analytical
formula [51; 199]. The existence of an efficient algorithm makes the free-fermion approxi-
mation a numerically accessible and valuable method with applications to condensed-matter
systems. At the same time, simulating interacting fermions is believed to be classically in-
tractable. Indeed, simulating general interacting fermions is as hard as simulating the output
of a universal quantum computer (see, for example, Ref. [166]). A similar practical differen-
tiation between easy and hard problems can be applied to other systems [18; 19; 66; 95; 96].
93
?r
H(t), Ak(t)
?r'
Figure 4.1: Classical simulability. We look for the existence of an efficient algorithm running
on a classical computer and producing (sampling) the many-body configurations with the
probability distribution close to the physical system after measurement in some basis. We
show that, for fermionic systems with Hamiltonian H(t) an with dissipation described by
quadratic Lindblad jump operators Ak(t), such an algorithm exists for at least a restricted
number of problems, while the worst-case scenario requires a quantum computer in order
to be solved efficiently. The three optical lattices illustrate the state of the system at initial,
intermediate, and final times.
In this Chapter, we study the fate of classical simulability of fermionic systems in the
presence of dissipation, both for computing local observables and for sampling from the
many-body output distribution (to be defined shortly). To obtain a classification of the com-
plexity of simulating free fermions with dissipation, we consider a general class of Markovian
processes, i.e. dynamics that depends only on the instantaneous system state and is inde-
pendent of the preceding evolution [200]. In previous studies, it was shown that Marko-
vian single-fermion loss or gain terms keep the noninteracting system classically tractable
[201; 202]. As a step forward, we consider a much wider class of quadratic-linear Lindblad
jump operators. Using the method of stochastic trajectories [203; 204], we establish a wide
subclass of problems that can be simulated classically. At the same time, we demonstrate
that, in general, quadratic Lindblad jump operators are at least as hard to simulate as most
elastically interacting systems. More precisely, we establish a connection between dissipative
94
interactions and fault-tolerant universal quantum computation exploiting the quantum Zeno
effect [205?209]. Therefore, simulating evolution under quadratic Lindblad jump operators
is equivalent in power to quantum computation. The tractability and intractability re-
sults together show that simulation of quartic dissipative Liouvillian operators is a problem
whose complexity can be changed from hard to easy by varying one or more parameters in
the system [96].
One motivation behind this Chapter is the existence of a variety of accessible fermionic
physical systems involving inelastic processes. Examples of dissipative processes described by
quadratic Lindblad jump operators include two-body loss in trapped alkali atoms [210?212],
alkaline-earth atoms [213?218], and cold molecules [164; 219]. As we will show, Feshbach
resonances [220; 221] can be used to significantly suppress coherent interactions between
cold atoms, simultaneously increasing the rate of atom-pair trap losses. More general types
of dissipation can be created by adding a source of atoms [222?224] or inelastic photon scat-
tering [225?227]. In condensed matter physics, an example of a process that can potentially
be described by a Lindblad equation in the Markovian approximation is Cooper-pair loss
[228; 229] and dephasing [230]. Recent progress in the control of dissipative electronic sys-
tems has brought them into focus in condensed matter physics. Some of the novel effects
in noninteracting and mean-field fermionic systems include dissipation-induced magnetism
[231?233], dissipative superfluids and superconductors [234; 235], dissipative Kondo effect
[236; 237], non-Hermitian topological phases [238?245], and non-Hermitian localization
[246?248].
We provide a classification of dissipative fermionic processes into easy (efficiently simu-
lable) and hard (not efficiently simulable) classes according to their worst-case computational
95
complexity. The classical simulability problem may be phrased in two ways, either in terms
of evaluation of few-body observables or sampling from the full probability distribution on
many-body outcomes. In the first task (few-body observables), a classical computer is required
to output the expectation value of an observable supported on k sites, where k does not
grow with the system size. In the second task (sampling), a classical computer is tasked with
producing samples from the same distribution as the one obtained by measuring the time-
evolved state in some canonical basis (see Fig. 4.1). Both tasks allow for the computer to
make a small error ?, measured appropriately in each case1. The task of sampling is com-
putationally harder; an algorithm producing samples in some product-state basis can also be
used to obtain expectation values of few-body observables in the same basis. Therefore, in
this Chapter, we focus mainly on the easiness of sampling in arbitrary product-state bases
as a criterion for overall easiness and on the hardness of computing few-body observables as
a criterion for overall hardness. This choice gives the stronger of the two results for both
easiness and hardness.
We note that a limited version of classical simulability for some models below was also
studied in previous works [249?251]. In particular, it was shown that two-point correlation
functions in such models can be evaluated via solving a closed set of equations. This result
establishes classical simulability of local observables and can be used in various problems
such as dissipative transport or optical response. However, this result alone is not sufficient
to establish the simulability of sampling. In contrast to local observables, simulated sampling
requires the full knowledge of the many-body output probabilities, therefore the sampling
1To be precise, for the first task (estimating few-body observables), the error is measured by the maximum
difference in the estimated and ideal expectation values of a unit-norm observable O. For the second task
(sampling), the error is measured through the maximum variation distance between the ideal and the sampled
probability distributions. The definitions are elaborated upon in Appendix A.
96
complexity of systems with simulable low-order correlations remains unclear. As we revisit
below, Gaussian systems are the exception that allow reducing these output probabilities to
two-point correlation functions via Wick?s theorem; other systems we study below do not
have such simple reduction. To overcome this problem, we develop the easiness proof that
does not require applicability of Wick?s theorem. In conclusion, sampling is a stronger notion
of simulation compared to two-point correlators in a sense that any local observables can be
efficiently obtained using an oracle producing sampling outcomes
Let us emphasize the importance of the provided complexity analysis. While estab-
lished easy dissipation classes are limited to certain fine-tuned processes, such limited simu-
lable models have an important application for quantum computing. For example, classical
models can be used in calibration of quantum computers, simulation of the impact of noise
on sampling, and analysis of fermionic quantum error-correcting codes [252]. More funda-
mentally, identifying easy classes is an important first step to analyze easy-hard transitions
in open fermionic systems, as we analyze below. At the same time, the hardness result we
obtain in this Chapter establishes utilizing dissipative interactions as an alternative path to-
ward building a universal quantum computer. This conclusion is surprising, since dissipative
interactions generally produce mixed states. Therefore, dissipative interactions should be
used only in a manner utilizing a blockade mechanism induced by the quantum Zeno effect,
as we show below. In cold atomic systems, controlling dissipative interactions differs from
photonic systems studied before [207; 208] and can be achieved using an atomic Feshbach
resonance. In Ref. [253], we analyze in detail a scheme for universal quantum computation
with 40K atoms and illustrate that, with realistic experimental parameters, an entangling
gate with low error-rate of roughly 10?4 is possible. The existence of both easy and hard
97
classes for two-body dissipation establishes it as a valuable model for physical analysis of noisy
intermediate-scale quantum devices.
4.1 Model
We consider dynamics generated by the Lindblad master equation [200; 254]
d? ?kA
= ? 1i[H(t), ?] + Ak(t)?A?k(t)? {A
?
k(t)Ak(t), ?}, (4.1)dt 2
k=1
where {X,Y } ? XY +Y X is the anticommutator, ?(t) is the density matrix of the system,
H(t) is a noninteracting Hamiltonian, and Ak(t) ? A(t) form a set of kA Lindblad jump
operators. We set h? = 1 throughout the Chapter unless specified otherwise. Both the
Hamiltonian and the Lindblad jump operators may depend explicitly on the time but not on
the state itself. The correspondingmap ?(t1) = V(t1, t2)?(t2) between arbitrary times t1 and
t2 ? t1 satisfies V(t2, t1) = V(t2, ?)V(?, t1) for any t2 ? ? ? t1. This divisibility condition
is commonly referred to as the most general definition of Markovian dynamics [255]. The
master equation in Eq. (4.1) is invariant under certain transformations of the set of Lindblad
jump operators A(t), such as operator permutations, multiplying any Lindblad operator by
a phase factor, or splitting/merging of identical operators as Ak ? {
? ?
pAk, 1? pAk},
0 ? p ? 1.
As a physical system of interest, we consider a fermionic many-body problem where
N ? L spinless fermions initially occupy L available levels (modes). Such systems are com-
monly described by the second quantization method, which expresses any operator, including
the Hamiltonian and Lindblad jump operators, in terms of fermionic Fock operators c?n and
98
Type Examples of Ak Complexity
Dephasing c?1c1
Particle shuffle c?1c
?
2 & c2c1
Classical fluctuations ? & Easy (EC1)c1 c1
Classical pair fluctuations c?1c
?
2 & c1c2
Mixing unitaries 2c?1c1 ? 1 + i(c
?
2 + c2) Easy (EC2)
Single-particle loss/gain c1 OR c?1 Easy (EC3)
Incoherent hopping c?1c2
Pair loss/gain ? ? Hardc1c1 OR c1c1
Table 4.1: Comparison between different types of noninteracting fermion dynamics with
additional dissipation. For simplicity, we provide examples for two modes out of L, denoted
by numbers 1 and 2. The symbol & means that both operators are present in the set A(t)
with factors equal in absolute value. Abbreviations EC1, EC2, EC3 stand for Easy Class 1,
2, and 3 described in the text.
cn, n ? {0, 1, . . . L? 1}. Fock operators create and annihilate a single fermion in a partic-
ular mode and satisfy the canonical commutation relations {cn, c ?m} = 0, {cn, cm} = ?nm.
Though the conventional fermion operators are suitable in most physical problems, in the
absence of fermion number conservation it is convenient to use the 2L Hermitian Majorana
fermion operators ?2n = cn+ c?n and ?2n+1 = ?i(cn? c?n), due to their simple anticommu-
tation relations {?i, ?j} = 2?ij , i, j ? {0, 1, . . . , 2L ? 1}. We consider the most general
form of a noninteracting Hamiltonian [256]
2?L?1 2?L?1i
H(t) = ?ij(t)?i?j + ?i(t)?i, (4.2)
2
i,j=0 i=0
where ?(t) is a real-valued antisymmetric 2L? 2L matrix and ?(t) is a real 2L vector. We
assume that the magnitude of all entries of ?(t) and ?(t) and their time derivatives scale at
most polynomially with system size.
In this Chapter, we focus on the classical resources needed to approximately sample
99
from the fermion distribution at time t,
P (r|r?) = ??r|?(t)|?r?, ?(0) = |?r????r? |, (4.3)
where r? and r denote the positions of occupied modes in the initial and final (measured)
product-state configurations, respectively, and |?r? is a product state defined as |?r? =
c?r . . . c
?
r |0? = ?2r1 . . . ?2r |0?. Importantly, because the dynamics may not conserve the1 N N
total fermion number, the final number of fermions N? can, in general, be different from the
initial number: N =? N? .
We establish the sufficiency of polynomial resources for classically simulating dynamics
due to arbitrary noninteracting Hamiltonians in Eq. (4.2) and a limited set of Lindblad jump
operators Ak(t) ? A(t) in the worst case. In order to prove polynomial-time simulability
(also called easiness) for limited classes of dissipative dynamics, we reduce the problem to
that of simulating unitary noninteracting fermionic evolution, an easy problem for a clas-
sical computer. In order to prove hardness for more general Lindblad jump operators, we
exploit the ability of dissipative dynamics to perform arbitrary quantum computation (i.e.
we prove that simulating universal quantum computation reduces to simulating Lindbladian
dynamics).
The results of this Chapter are briefly illustrated in Table 4.1. First of all, we define
three classically tractable classes of Lindblad jump operators (defined as Easy Classes 1, 2, and
3). All of these cases allow for polynomial-time sampling of any Hamiltonian and Lindblad
jump operators from the given class on a classical computer, with error scaling inverse-
polynomially withL. Easy Class 1 (EC1) allows for simulation of self-adjoint sets of quadratic
100
Lindblad jump operators: all Lindblad jump operators in the set A(t) come with their
Hermitian conjugate. This class includes such widely used examples as dephasing, incoherent
particle shuffle, and classical fluctuations of the number of fermions and of the number of
fermion pairs. Easy Class 2 (EC2) works with unitary quadratic Lindblad jump operators.
Finally, Easy Class 3 (EC3) describes the loss or gain of a single particle in the system
and can be used in combination with EC1 and/or EC2. At the same time, there exists a
class of Lindblad jump operators with a nonzero measure that is hard to classically simulate.
Examples from this class include pair loss/gain and incoherent fermion hopping. Below we
explore each class separately.
We focus on quadratic-linear Lindblad jump operators of the form
2
i ?L?1 2?L?1
A (t) = aijk (t)?i?j + b
i (t)?i + dk(t), (4.4)
2 k k
i,j=0 i=0
where ak(t) and bk(t) are arbitrary complex-valued 2L?2Lmatrices and 2L-vectors respec-
tively, and dk(t) is a number. In this problem, we assume that the number kA of nontrivial
Lindblad jump operators from this class is at most L(L+1). In fact, any instance where A
has a larger number of operators can be reduced to a smaller set through a linear transforma-
tion [200]. Also, as with the Hamiltonian, we assume that the magnitude of the entries of
ak(t), bk(t), and dk(t) and their time derivatives grow at most polynomially with the system
size.
This Chapter is organized as follows. In Section 4.2, we provide a brief introduction to
free-fermion sampling, recalling established results in the literature and connecting them to
the most general form of quadratic-linear Hamiltonians. In Section 4.3, we derive three new
101
algorithms allowing us to solve distinct classes of fermionic problems involving quadratic
Lindblad jump operators and prove that these algorithms run in time that is polynomial
in both L and the inverse of the distance from the exact distribution. In Section 4.4, we
establish generic class of systems that belong to the hard class and show their robustness to
the presence of minor imperfections.
4.2 Free-fermion sampling
In this Section, we discuss the noninteracting fermion problem in the absence of dis-
sipation. We recap the work of Terhal and DiVincenzo [51], which shows that all output
probabilities P (r|r?) in Eq. (4.3) and the marginal probabilities can be obtained using a clas-
sically tractable analytical formula. Before referring to this result, we need to incorporate
the linear terms present in Eq. (4.2) into effective quadratic dynamics. In order to do so, we
consider a slightly larger system containing an extra ancilla (L + 1)th mode [256], labeled
as n = L. Next, we choose new effective dynamics such that the ancilla mode remains in
?
the state |+? ? (|0? + |1?)/ 2 during the entire evolution, including the initial and final
times, i.e.
|?r?? ? |?r?? ? |+?, |?r? ? |?r? ? |+?. (4.5)
To construct such dynamics, we consider a newHamiltonian by replacing ?i ? i?i?2L, where
?2L and ?2L+1 are Majorana operators acting on the ancilla mode. It is straightforward to
check that such a transformation results in a new purely quadratic Hamiltonian (without any
linear terms) that keeps the state of the ancilla stationary and does not modify the dynamics
102
of the original Hamiltonian. The new coefficients in Eq. (4.2) are
?ij ? ??ij = ?ij + ?i2L+2?j ? ?j2L+2?i, (4.6)
where we use by default ?2L = ?2L+1 = 0. Given that the modified initial and final condi-
tions for the system and the ancilla are {r?} ? {r?, s?}, {r} ? {r, s}, s, s? ? {0, 1}, the
probability P (r|r?) of obtaining outcome r for the original system can be computed from the
probability P ({r, s}|{r?, s?}) for the system with the ancilla as follows:
1 ?
P (r|r?) = P ({r, s}|{r?, s?}). (4.7)
2
s,s??{0,1}
Summarizing, this method ensures that the dynamics of a linear-quadratic Hamiltonian can
always be reduced to the dynamics of a quadratic one by expanding the system size by one
mode. Therefore, we henceforth consider only quadratic Hamiltonians.
Let us derive the formula for the sampling probability. We start from(a (b?ackwards) )
time-evolvedMajorana fermion operator ? (t) = U ? U ?i t i t , whereUt = T exp ?
t
i H(t?)dt? .
0
Here T exp is the standard time-ordered exponential. Given the qu?adratic structure of
the Hamilto(nian,?this evolu)tion is a linear transformation ?i(t) = i Rij(t)?j , where
R = T exp ? t2 ?(t?)dt? is a unitary 2L ? 2L matrix. One can use this expression
0
to derive the time evolution of a fermion operator as
1 ?
U ?tcnUt = Ut(?2n + i?2n+1)U
? = Tnj?j, (4.8)
2 t
j
where Tnj ? R2n,j + iR2n+1,j are elements of a L? 2L transformation matrix T . Labeling
103
the initially empty sites as l?i and recalling that the initial fermion positions are r?i and that
the final positions are ri, the linearity allows to write the output probability in Eq. (4.3) at
any time as
P (r|r?) = ??r|Ut |?r????r? |U ?t |?r? (4.9)
= ?? ?r|Ut?cr? cr? . . . c c
? ?
l? ? U |?r? (4.10)
1 1 L?N lL? tN
= T ?r? n Tr?m . . . T
?
?
1 1 1 lL?Nm
T
1 L l
?
L?Nn
? (4.11)
L
n1,...nL;m1,...mL?N
?0|?2r . . . ?2r ?n1?m1 . . . ?m ?n ?2r . . . ?2r |0?. (4.12)N 1 L L 1 N
This expression can be computed efficiently using Wick?s theorem and written in a
compact form. Let I be a subset of indices with increasing order and A[I] be the matrix
whose elements satisfy A[I]ij ? AI ,I . Consider the set I = {r?i, L+ l?j, 2L+2rk}, wherei j
i ? {1, 2, . . . N}, j ? {1, 2, . . . L ? N}, and k ? {1, 2, . . . N?} take all possible values.
Then the result can be written as [51]
P (r|r?) = PfM [I], (4.13)
where Pf is the Pfaffian, and M is a 4L? 4L matrix
? ?
??? T?T T T?T ? T?
M = ?
?
? ?? ??T ??T T T ??T ? T ?????? , (4.14)
?T T ?T ? I
104
where, in turn, the 2L? 2L matrix ? is
? ?
??? 1 i?? = I ? ? ?L L ? . (4.15)
?i 1
The expression in Eq. (4.13) can be efficiently evaluated on a classical computer using existing
polynomial-time algorithms for computing Pfaffians [199]. Similarly, marginal probabilities
can be efficiently computed conditioning on the output of a given fraction of sites, as in
Ref. [51], which is enough to efficiently sample from the output probability distribution.
4.3 Easy Classes
Here we present three algorithms that allow simulating specific fermionic problems
involving quadratic Hamiltonians and quadratic-linear Lindblad jump operators. All meth-
ods are based on stochastic unraveling, i.e., replacing dissipative dynamics by a stochastic
free-fermion Hamiltonian without changing the outcome distribution (see also Ref. [204]).
Since each stochastic realization can be simulated efficiently by a classical computer, as es-
tablished in the previous section, a classical computer can serve as a black box sampler that
reproduces measured outcomes. In this Section, we demonstrate that the classes of problems
belonging to the aforementioned Easy Classes 1?3 are efficiently simulable. In particular, we
show that these problems require computation resources C (number of operations) bounded
as C ? poly (L, t2/?) to sample from a distribution that is ?-close to the target distribu-
tion. Therefore, we establish efficient classical algorithms for approximate dissipative fermion
sampling in the presence of certain classes of quadratic-linear Lindblad jump operators.
105
4.3.1 Efficient classical algorithms
Let us define Easy Class 1 (EC1) as problems that involve quadratic-linear self-adjoint
Lindblad sets A(t) defined as follows. We assume that at any time one can divide the set as
a union of two equal-size subsets, A = A1 ? A2, where the Hermitian conjugate of every
Lindblad operator in A1 returns an operator in A2 (and vice versa). Under this division, any
Hermitian Lindblad operator must be included in both subsetsA1 andA2 with normalization
?
factor 1/ 2. The latter splitting can be seen as a transformation that keeps the Lindblad
equation invariant, as defined earlier below Eq. (4.1). Examples from EC1 include several
important physical models such as dephasing and classical fluctuations (see examples of sets
in lines 1?4 in Table 4.1).
In previous works, it was shown that such systems have two-point correlation functions
that are classically simulable by solving a closed set of linear equations [249?251]. This
is indeed a strong indication that the system can be simulable in the broader context of
sampling complexity. However, as we noted previously, Wick?s theorem is not applicable to
non-Gaussian states. This means that the scheme we utilized to obtain Eq. (4.13) does not
work any more. We now show an alternative scheme using stochastic unraveling that leads
us to the easiness result.
To efficiently simulate dynamics from EC1, we consider a stochastic Hamiltonian
?
H ?(t) = H(t) + ?k(t)Ak(t) + ?
? ?
k(t)Ak(t), (4.16)
Ak?A1
?
where ?k(t) is a complex random variable taking constant values ?k(t) = ?nk/ ?? during
106
short time intervals t ? [n??, (n + 1)?? ]. The discrete complex Gaussian variables ?nk
satisfy E?nk = 0, E??nk?n?k? = ?nn??kk??ab, where E denotes the expectation value taken over
the random variables. Then, given a stochastic Hamiltonian of the form in Eq. (4.16), the
original dynamical map V(t2, t1) generated by the Lindblad equation can be approximated
as
V(t , t ) = EV (t , t ) + ?V(t , t )?? +O(?? 22 1 st 2 1 2 1 ), (4.17)
where ?V(t2, t1)?? is the lowest-order correction (to be explicitly derived below) and Vst is
a stochastic unitary map
Vst(t2, t1)? = U(t2, t ?1)?U (t2, t1). (4.18)
( ? )
In the above, U(t2, t1) = T exp ? ti 2 dt?H ?(t?) encodes the time-evolution due toH ?(t)t1
in Eq. (4.16). The average E in Eq. (4.17) is taken over the stochastic fields ?k(t). The
resulting output probabilities satisfy
P (r|r?) = EP ?st(r|r ) +O(??), (4.19)
where P ?st(r|r ) is the output probability for the unitary dynamics in Eq. (4.18) obtained via
the formula in Eq. (4.13). Therefore, a computer programmed to sample from the distribu-
tion for a randomly chosen set of unitary trajectories will produce outcomes with the same
probabilities as the physical process following Lindbladian evolution, up to O(??) error.
The cost of suppression of this error in terms of computational resources will be discussed
107
later in this section. Here we just specify that the correction to the dynamical map, which
we treat as an error, can be expressed as
? t2
?V(t2, t ) = E dt?1 Vst(t2, t?)D(t?)V ?st(t , t1), (4.20)
t1
where D(t) is a time-local superoperator
?
D(t)? = D(1)(t)?D(2)? ? (t). (4.21)
?
Here, the operators (i)D? (t) = poly4(H(t), Ak(t)) can be expressed as polynomials of degree
four in terms of the Hamiltonian and Lindblad jump operators at time . Therefore, (i)t D? (t)
can always be presented as a sum of terms, each being a product of no more than eight
Majorana operators. The specific form of these operators and the derivation of Eq. (4.17)
can be found in Section 4.5.
Although the proposed unraveling scheme represents dynamics in terms of stochastic
trajectories for Gaussian pure states, the resulting averaged mixed state is non-Gaussian, in
contrast to previously studied problems [201; 202]. Therefore, the overall dynamics of EC1
represents dissipative interactions of fermions, while the proposed method can be seen as a
good choice of time-dependent density matrix decomposition.
Let us consider another class of problem?s, Easy Class 2 (EC2), that include uni-
tary quadratic Lindblad jump operators Ak = ?k(t)Yk(t), where ?k(t) ? 0 are time-
dependent rates and Yk(t) = exp(?iGk(t)) are unitary operators generated by quadratic-
linear Hamiltonians Gk(t) of the form in Eq. (4.2). To classically simulate dynamics under
108
EC2, we also consider discretized time evol?ution with sufficiently small timesteps ?? and
set the unitary transformation U(t1, t2) = n2n=n Un, where the timestep unitaries Un are1
generated randomly according to the rule
??????Yk(tn), pk = ?k(tn)??,U = U0n n ???? ? (4.22)I, p0 = 1? k ?k(tn)??.
Here pk are (proba?bilities that are u)sed to generate the respective outcomes,
0 (n+1)??Un = T exp ?i H(t)dt , and tn ? [n??, (n+1)?? ] are random times generatedn??
from the uniform distribution.
Notwithstanding the slightly different stochastic unraveling, the procedure for approx-
imating EC2 is the same as for EC1. In particular, the system dynamics is described by the
expression in Eq. (4.17) leading to the distribution in Eq. (4.19), with the average taken
over stochastic unitary realizations. The correction term has the form in Eq. (4.21), but the
operators (i)D? (t) here are degree-two polynomials in the Hamiltonian and Lindblad jump
operators. The detailed form of the operators along with the derivation can be found in
Section 4.6.
Fin?ally, let us consider Easy Class 3 described by generic linear Lindblad jump operators
A ik(t) = i bk?i + dkIL?L, which can be obtained by setting ak = 0 in Eq. (4.4) without
assuming any additional restrictions on the setA(t). Previous works had already shown that
linear jump operators can be simulated classically [201; 202]. However, this proof applies
only to Gaussian states and cannot be extended to, for example, Lindblad equations that also
contain easy quadratic jump operators. Below we propose a different way of simulating linear
109
jump operators similar to one we used for EC1. This technique would allow us to combine
EC1, EC2, and EC3 into a single easy class of Lindblad equations, including both quadratic
and linear processes.
Now let us show that simulation of linear jump operators is equivalent to simulating
a unitary system extended by a number of ancilla modes. In particular, we require La =
t/?? ancilla fermion modes equal to the number of time steps after discretization. Let us
enumerate the ancilla modes described by Majorana fermion operators ?2n and ?2n+1 using
indices n = L, . . . , L + La ? 1. We also assume that the ancilla modes are initialized
in the vacuum state and traced out after performing the evolution. The dynamics of both
the system and the ancillas can be described as unitary evolution with the Hamiltonian in
Eq. (4.16), with one important difference. Now, the quantities ?k(t) in the time interval
t ? [n??, (n+ 1)?? ] are operators instead of numbers, and are given by
?k(t) = ? ??
?1/2
nk (?2(L+n?1) + i?2(L+n)?1). (4.23)
The random variables ?nk are the same as in EC1. The idea is that we pair a loss (gain) term on
the system with a gain (loss) term on the ancilla to make the overall Hamiltonian quadratic.
After discarding the ancilla modes, the evolution becomes equivalent to the target dissipative
dynamics, up to a discretization error that originates from the approximation in Eq. (4.17)
and Eq. (4.21), with (i)D? (t) expressed as degree-four polynomials in the Hamiltonian and
Lindblad jump operators, as shown in Section 4.7.
110
4.3.2 Performance of the classical algorithms
Let us quantify the error of the method of quantum trajectories used for Easy Classes
1?3, and then show that the sampled distribution can be made arbitrarily close to the exact
one with an appropriate choice of the timestep ?? . In order to characterize the approxi-
mation error ? associated with sampling from a distribution P? (r|r?) different from the ideal
distribution P (r|r?), we utilize the total variation distance
1 ?? ?
? = max ?P? (r|r?)? P (r|r?)?, (4.24)
2 r? r
where the maximization is over all possible initial product-state configurations r?.
Using convexity of the absolute value and the expression for the correction in Eqs. (4.20)?
(4.21), the error can be bounded as (see Section 4.8),
?
? ??
? t
? max dt?C? ? 2r? (t, t ) +O(?? ), (4.25)2 r?
? 0
where
?
? ?? ?C (t, t?) = E ??r|D(1)(t, t? ?r? ? )?r?(t)D(2)(t, t?? )|r??. (4.26)
r
Here (i) (i)D? (t, t?) = Vst(t, t?)D? (t?) and ?r?(t) = Vst(t, 0)?r? are operators transformed ac-
cording to unitary evolution for a single stochastic trajectory, and the average E is taken over
all trajectories. We now use the following lemma to further bound this expression.
111
Lemma 12. Consider two sparse operators O1 and O2 whose matrix elements satisfy
?r|O |r?? ? = 0 if dH(r, r?) ? k?, ? ? {1, 2}, (4.27)
where dH is the Hamming distance, and r, r? are binary strings of length L representing compu-
tational basis states. Let ? be a normalized positive semidefinite operator, ? ? 0, Tr ? = 1,
then
? k1+k2
|?r|O1?O2|r?| ?
L ?O1?max?O2?max, (4.28)
r k1!k2!
where ?O??max is the max-norm.
The proof of the lemma can be found in Section 4.8. The result of the Lemma allows
us to simplify Eq. (4.26) as
k1?+k2?
C?r? (t, t
?) ? L E?D(1)? (t, t?)?max?D(2)(t, t?? )?max, (4.29)k1?!k2?!
where ki? is the locality of the operator (i)D ?? (t, t ), i.e. the maximum number of Majorana
operators in its decomposition. Because Vst is a map describing free-fermion evolution, the
locality ki? of the operator (i)D? (t, t?) is equal to the locality of (i)D? (t?). At the same time, as
analyzed in the previous section, the localities of operators (i)D? (t?) satisfy k?? ? km, where
km = 8 for EC1/EC3, and km = 4 for EC2. We can also bound the max-norm by the
112
(spectral) operator norm
?D(i)? (t, t?)?max ? ?D(i)? (t, t?)? = ?D(i)? (t?)?. (4.30)
As a result, the error bound is given by
?? L2k ??m t
? ? E dt??D(1)(t?? )??D(2)? (t?)?. (4.31)2 (km!)2 ? 0
Since the matrices (i)D? = poly(H,Ak) are generated by a quadratic-linear Hamiltonian H
and set of Lindbladians Ak, we can always find a polynomially large bound for the norm
? (i)D? (t?)? ? poly(L). Thus, there always exists a discretization step
?
?? ?
t poly (4.32)(L)
that keeps the error in Eq. (4.31) arbitrarily small, suppressed at least polynomially with the
number of modes L.
Let us now estimate the amount of computational resources required to perform the
above sampling procedure. For each sample, the algorithm randomly chooses a unitary trajec-
tory according to the given prescription for each class EC1?EC3 and, according to the Terhal-
DiVincenzo algorithm, samples outputs from the free-fermion distribution in Eq. (4.13). In
particular, it samples the output at site i conditioned upon the outcomes sampled at sites
j < i, for which the marginal probabilities should also be computed. Consider cases of EC1
and EC2 that do not require ancillas. Once the matrix T is obtained in Eq. (4.8), the number
of steps to compute the distribution is equal to C ? = L?O(L3) = O(L4), where the factor
113
O(L3) is the upper bound on the time it takes to compute a Pfaffian of an O(L) ? O(L)
matrix. Further, the runtime for obtaining the matrix T is proportional to t/?? ?M(2L),
where M(n) ? O(n3) is the time for n? n matrix multiplication. In sum, the total bound
on the runtime for each trajectory is bounded as C ? O(L4) + O (L3t/??) . Choosing
?? = ?/(t? poly(L)), the runtime is
( )
2
C ? poly tL, . (4.33)
?
For EC3, the derivation is the same up to adjusting the system size to include the ancilla
modes, L ? L+ t/?? . This case also has a similar polynomial upper bound on the classical
runtime in the form of Eq. (4.33) as long as the evolution time t is polynomial.
Finally, let us analyze the case when the conditions of Class 1?3 are violated. Strictly
speaking, then the stochastic method fails as it generally maps the problem to a non-quadratic
fermionic evolution, which is not believed to be simulable for arbitrarily long time. However,
we can still efficiently simulate the system after this mapping if the product of evolution time
t and the correction rate ?? (of processes violating easiness conditions) remains small. In
particular, if the product is bounded as ??t < c/L2 for some constant c, the dynamics
remains classically easy. To obtain this result, we consider a more general form of stochastic
unraveling in Eq. (4.18) with nonunitary unraveling. This formula can be Taylor-expanded as
Vst ? V ?st = Vst+ ???K1+(???)2K2+ . . . , whereKn are local correction superoperators
and ? is the evolution time. Therefore, we can update the bound for the operator norms in
Eq. (4.30) as ? (i) ? ? ? ? (i) ? ? ? (i) ? ? 2? (i)D? (t, t ) D? (t ) + ??? K1D? (t ) + (???) K2D (t?? )?+ . . . .
Since the operator (i)D? involves at most km fermion Fock operators and the action of Kn
114
involves at most four fermion operators, we see that ? (i)K D (t)? ? O(Lkm+2nn ? ). Therefore,
the norm ? (i)D? (t, t?)? is always bounded by a poly(L) value if ?? < c/(L2?), where ? =
t ? t? ? t. This result leads to Eq. (4.33). As a result, if the dissipation is close enough
to the symmetric point, the evolution remains classically easy. This result may be helpful
for analyzing the precision needed for implementing this dynamics in intermediate-scale
quantum devices.
4.4 Hard class
We have so far demonstrated cases when the probability distribution generated by
the Lindblad equation is efficiently simulable on a classical computer. Can we extend these
proofs to the most general case of quadratic Ak ?s? Since quadratic operators Ak correspond
to single-fermion jumps in many cases, one may expect that the problem can be solved in the
single-particle sector, similar to unitary free-fermion dynamics. However, such an intuition
is incomplete. A simple explanation can be obtained using the Fermi exclusion principle that
requires the transition between two modes to depend on the occupation of the target mode;
thus a quadratic Lindbladian jump operator can induce many-body correlations in the system
that quickly become classically intractable.
4.4.1 Reduction from a generic quantum circuit
We now provide a rigorous argument for worst-case hardness based on the equivalence
of dynamics under classesH(t) andAk(t) in Eq. (4.1) on the one hand and universal quantum
computing on the other. Let us start with the simplest map utilizing quadratic Hamiltonians.
115
We distribute all modes into L/2 pairs, each pair corresponding to a logical qubit in the state
|0?L = |01? or |1?L = |10?. Then, utilizing only quadratic Hamiltonians and Lindblad
jump operators, we can implement any quantum circuit with arbitrary precision. Thus,
by showing the equivalence of the dynamics to universal quantum computation, we obtain
hardness results for both estimating time-evolved local observables ?O(t)? and sampling
from the time-evolved state in any local basis. The obtained hardness result is therefore on
par with the best complexity-theoretic evidence that simulating quantum circuits (in both
senses) is hard.
First, using single-fermion hopping between the two sites of a qubit, we can reproduce
arbitrary single-qubit operations [175]. Second, to approximately generate a desired two-
qubit gate, we can use hopping combined with a quadratic Lindblad operator. In particular,
assigning the two-qubit logical states |00?L = |0101?, |01?L = |0110?, |10?L = |1001?,
and |11?L = |1010?, the control-Z gate can be implemented by simultaneously applying
the hopping Hamiltonian H = J(c?2c3 + c
?
3c2) and pair-loss operator A = ?c3c4 for time
t = ?/J , in the limit ? ? J . This type of dynamics can be analyzed as follows. The logical
states |01?L and |10?L remain invariant in the course of the evolution. At the same time, in
the limit ? ? ?/J ? ?, due to the quantum Zeno effect, the Lindblad operator?s action
disallows any coherent transition involving states where qubits 3 and 4 are both occupied (i.e.
| ? ?11?). As a result, the logical state |00?L is unaffected by the evolution. Therefore, the
only evolving logical state is |11?L, which acquires a phase factor exp(i?) = ?1 after time
t = ?/J . As a result, the effective transformation on the two logical qubits is the control-Z
116
gate
? ?
1 0 0 0
???? ????0 1 0 0? ?
???
|?? ? U?|??, U? =
???
?? . (4.34)
0 0 1 0 ????
0 0 0 ?1
Together with arbitrary single-qubit operations, the control-Z gate is enough to obtain dy-
namics universal for quantum computing and hence hard to approximately sample from,
assuming standard conjectures in complexity theory [14; 30].
Importantly, the performance of the dissipative gate relies on the Zeno-effect blockade
effective for ? ? ?. In the limit of large but finite ?, the two-qubit system has the
probability ? = 2?/? +O(??2) of ending up in states |0011? or |0000?, which could result
in an error in the gate. To avoid computational error, we can choose the ratio ? to be
arbitrarily large by taking vanishing J ? poly?1(L) for any given ? < 0. Therefore, we can
keep the error below any given threshold at the cost of increased overall computation time,
which remains polynomial in system size.
The proposed architecture is not unique and allows for modified/generalized realiza-
tions of logical qubits and gates. For example, if the pair decay is always present on any two
neighboring modes, one may introduce an empty ancilla mode between two logical qubits
in order to ensure that logical states don?t decay. As another example, if the control Hamil-
tonian is linear in terms of Majorana operators, a logical qubit can be encoded using just a
single mode. Moreover, for a reader focused on applications, we discuss below a practical
117
modification of qubit encoding implementable in cold atoms.
Now let us show that pair loss is not the unique dissipation present in the hard class.
In fact, this class also includes any quadratic dissipation connected to pair loss by a time-
dependent linear Bogoliubov transformation, A?(t) = ?Y (t)c ?1c2Y (t), where Y (t) =
exp(?iG(t)) is a free-fermion unitary transformation and G(t) is a Hermitian operator
from the quadratic-linear class in Eq. (4.2). To demonstrate this equivalence, we consider
the pair loss scheme described above but simultaneously replace all pair losses A with A?(t),
the Hamiltonian H(t) with H ?(t) = Y ?(t)H(t)Y (t), and instead of the initial and final
states, choose states transformed by Y (0)? and Y (t), respectively. The resulting process
has the same probability distribution; thus its complexity would be the same. As a result,
Lindbladians such as incoherent transitions A = ?c?1c2 or pair gains A = ?c
? ?
1c2 are also
classically hard in combination with free-fermion dynamics (see Table 4.1).
4.4.2 Robustness of the hardness result
The error associated with imperfect Zeno blockade cannot be arbitrarily suppressed
by slowing down the computation if there are small generic corrections to the dissipative
dynamics. These corrections can be viewed as the presence of additional Lindblad jump
operators with total rate ??. Such terms generate additional transitions with the probabil-
ity ?? ? ???/J , where ?? is the combined rate of added operators A? and/or other errors.
In contrast to the imperfect-Zeno-blockad?e error, this type of error diverges for small J .
Therefore, there is an optimal value J ? ??/? that minimizes the overall gate error to
? + ?? ? O(1), including, besides standard errors, leakage into states outside of the logi-
118
Hard Easy Hard
p0
p20 /8?2 1 8?2/p2 0
Pair loss-to-gain ratio ?/? 
Figure 4.2: Complexity phase diagram for a fermionic system with simultaneous pair
losses and gain. The plot illustrates the connection between complexity of simulation and
the hypothetical dissipative control-Z gate error ? + ?? (both axes use log scale). When the
error is smaller than the best-known two-qubit error-correction threshold p0, the worst-
case system dynamics is equivalent to that of a fault-tolerant quantum computer (blue shaded
regions) and, according to existing complexity conjectures, is classically computationally hard
to simulate. In contrast, when the rates of gain and loss are exactly equal, the problem belongs
to EC1 (vertical red line) with the effective classical algorithm provided in the text. The
result for the unshaded region remains inconclusive. The dashed line represents qualitative
extrapolation.
cal Hilbert space. For fixed ?, there always exists a choice of ?? ? O(1) that keeps the
error below any provided threshold, ? + ?? < p0, where p0 < 0. According to the leak-
age threshold theorem in Ref. [257], which is a generalization of earlier standard threshold
results [258?260], a universal set of such gates can be used to implement fault-tolerant quan-
tum computing. Therefore, there are instances of Lindblad evolutions that remain hard to
simulate for arbitrarily long times.
One particular example of a dissipative correction to ideal dynamics is the presence of
pair gain A? ? ? ?ij = ? cicj tha?t acts on exactly the same sites as pair loss Aij . In this case, the
minimum error is ?+?? = 8?2??/? and the problem remains hard for a classical computer
if ?? ? p2 20/8? ?. Since the entangling gate is also implementable using pair gain instead
of loss, this inequality also works after replacing ? by ??. Thus, the problem of simulating
119
Error ?+?' 
the evolution in the regions ??/? ? p20/8?2 and ??/? ? 8?2/p20 is classically hard. The
complexity for the rest of parameter space remains an open problem. Notably, there exists at
least one point in this range, ? = ??, that is easily simulable by a classical computer since it is
in EC1. Therefore, by changing the ratio ??/?, we can potentially induce a complexity phase
transition. Figure 4.2 illustrates the connection between gate error and sampling complexity.
Summarizing, we established quantum computational universality of quadratic dissi-
pation combined with free fermion dynamics, where dissipation replaces the unitary interac-
tions between fermions. This result open a possibility of using simple dissipation processes
as a resource for quantum computing. In the following section, we illustrate the feasibility
of this proposal by considering a system of cold atoms.
4.5 Easy Class 1
In this section, we analyze the convergence of the average unitary stochastic evolution
to the exact Lindblad dynamics in the case of Easy Class 1 (EC1). First, we set the initial
time to be zero and consider the final time t being an integer multiple of timestep ?? .
This assumption holds without loss of generality since ?? may be adjusted appropriately to
capture any particular final time. Then the overall evolution of unitary can be written as a
product
t?/??
U(t) = Un, (4.35)
n=0
120
where the timestep unitary Un is expressed in terms of a time-ordered exponential
( ? (n+1)?? )
Un = T exp ?i dtH ?(t) (4.36)
n??
generated by the stochastic Hamiltonian H ?(t) in Eq. (4.16),
? ?R(t)H (t) = H(t) + . (4.37)
??
?
HereH(t) is the original time-dependent Hamiltonian, andR(t) = ? ? ?k nkAk(t)+?nkAk(t)
is the normalized stochastic part, where ?nk are independent complex Gaussian variables de-
fined for times n?? ? t ? (n+ 1)?? .
Let us consider the ordered exponential expansion of the timestep unitary in Eq. (4.36):
( )
U = I ? i?? 1/2 ? 1n ( Rn ?? iH)n + R
2
2 n
? i?? 3/2 PHnRn ? R3
6 n
1 ( ) (4.38)? ?? 2 H2 ? iP 1H R R ? R4n n n n2 3 12 n
+O(?? 5/2),
where we denote the discretized value of an operator On and permutation sum respectively
as
? ?
1 1 (n+1)??
On = dtO(t) ? dtO(t),
?? n ? ?? n?? (4.39)
PO1 . . . Om = O?(1) . . . O?(m).
??Sm
121
The average over the stochastic field can be taken for each timestep independently. Therefore,
the effect of the timestep unitary in Eq. (4.38) is
( 1 )
EUn?U ?n = I + Ln?? + L2n?? 2 ?2 (4.40)
+Dn??? 2 +O(?? 3).
In the equation above, Ln is the generator of the original Lindblad equation, lim???0 Ln =
L(n??), expressed as
?( )
L ? = ?i[H , ?] + A ?A? ? 1 ?n n kn kn {AknAkn, ?}
?k ( 21 ) (4.41)
+ A? ?A ? {A A?kn kn kn kn, ?} ,2
k
and Dn represents the lowest-order correction occurring due to the timestep being nonzero:
D 1
?(
? = A? ?n k?nAkn?[Ak?n, A
?
k] + Ak?nAkn?[A
?
k?n, A
4 kn
]
kk? )
+ A A??( k?n kn?[A
?
k?n, Ak] + Ak?nAkn?[A
? , A?k?n kn] ) (4.42)
+ Akn?V + V ?A + V
?
kn kn kn kn?A
?
kn + A
? ?
kn?Vkn
k
+Wn?+ ?W
?
n.
122
Here we have used the notation
? 1
V ? ?
1
kn = {Akn, {Ak?nAk?n}} ? PA
? ?
knAk?nAk?n,
? 4 6k
? i
?([ ] )
Wn = [Hn, A ], A
?
kn kn +{Hn, A
?
knAkn} (4.43)
1(
6?k )
? 2? { } 1
?
A ,A + PA? A ?kn kn kn knAk?nAk?n.8 48
k kk?
The overall expression in Eq. (4.42) can be written in a compact form,
?
D ? = D(1)n ?n?D(2)?n , (4.44)
?
where (i)D?n = poly(Hn, Akn) are polynomials of degree less than four.
The averaged stochastic map in Eq. (4.40) can be rewritten as a continuous evolution
and then decomposed using Dyson series for the small parameter ?? ,
(? t2 )
EVst(t2, t1) = T ?exp dt
?(L(t?) +D(t?)??) +O(?? 2)
t1 (4.45)
t2
= V(t , t ) + dt?2 1 V(t2, t?)D(t?)V(t?, t1)?? +O(?? 2),
t1
where the generators L(t) and D(t) are continuous versions of the operators in Eq. (4.41)
and Eq. (4.42), in which the ??-averaged operators Akn and Hn are replaced by the cor-
responding instantaneous values at time t, i.e. A(t) and H(t), respectively. To obtain the
expression in Eq. (4.20), we recursively replace V(t , t?2 ) and V(t?, t1) on the right-hand side
by their stochastic average and collect all O(?? 2) terms.
123
4.6 Easy Class 2
In this section, we analyze the convergence of the average stochastic unitary evolution
to the Lindblad dynamics in the case of Easy Class 2 (EC2). The single timestep evolution
averaged over stochastic unitaries in Eq. (4.22) is equivalent to the map
( ? ? ( ))
EU ?U ?n n = U0n ?+ dt ?k(t) Y
?
k(t)?Yk (t)? ? U
0?
n
( n k) (4.46)1
= I + L + L2n ?+Dn??? 2 +O(?? 3),
2 n
where the target Liouville operator is
?
Ln? = ?i[Hn, ?] + A ?kn?Akn ? ?kn?. (4.47)
k
The correction now takes the form
?( )
D ? = A ?C + C? ?n kn kn kn?Akn
k ? (4.48)
? 1 ? ? 1A 2knAk?n?Ak?nAkn ? ?n?,2 2
kk?
? ?
denoting C ? i ?kn = ?nAkn+ [Akn, Hn] and ? = ???1n dt k ?k(t). This expression has2 n
the form of Eq. (4.44) with operators (i)D?n being a sum of products of at most four Majorana
fermion operators.
124
4.7 Easy Class 3
In this section, we analyze Easy Class 3 (EC3) and show the convergence of the system-
ancilla stochastic evolution under the Hamiltonian in Eq. (4.16) using the stochastic oper-
ators in Eq. (4.23) to the dissipative dynamics with linear Lindblad jump operators. Let us
start from a many-body pure state of the fermions occupying L modes of the system and La
ancilla modes at time t = n?? , denoting it as |?n?. At the nth timestep, the evolution acts
on the system and the nth ancilla mode only. Thus, the state at time t = n?? is a product
state of subsystem states: (1) correlated state of L system modes together with the first n
ancilla modes and (2) the product states of the remaining La ? n ancilla modes, i.e.
|?n? = |?n?L+n ? |0?La?n. (4.49)
The evolution is governed by the Hamiltonian
1 ( )
H ?(t) = H(t)? IA + ? K(t) +K?(t) , (4.50)
??
where the stochastic terms are
?
K(t) = fnkAk(t)(?2(L+n?1) + i?2(L+n)?1) (4.51)
k
125
at times n?? ? t ? (n+ 1)?? , and fnk are independent real Gaussian variables. Then, at
the (n+ 1)th step, the system-ancilla state |?n? = Un|?n? is
|? 1/2n+1?(= |?n? ? i?? )Kn|?n?|1?|0?T?n?1
? 1?? iH(n + K
?
nKn |?n?|0?La?n2 )
??? 3/2 1{ iH ,K } ? K K?( n n n nKn |?n?|1?|0?La?n?12 6 (4.52)
? 1?? 2 H2 in ? {Hn, K?Kn}2 3 n
? i ? ? 1
)
R H K (K? 2n n n nKn) |?n?|0?La?n3 12
+O(?? 5/2),
where we used the discrete-time operator values Hn and Kn obtained as in Eq. (4.38).
The interpolated continuous-time evolution for the density matrix of the system can
be presented in the form
d 1 ( )??
? = ( E TrA |?n+)1???n+1| ? |?n???n| ?dt ?? n=?t/??? (4.53)1
= I + L 2 2 3n + Ln ?+Dn??? +O(?? ),2
where ?x? is the floor function. The target Liouville operator is
?
Ln? = ?
1
i[H , ?] + A ?A? ? {A?n kn kn knAkn, ?} (4.54)2
k
126
and the correction is
?(
D 1 1
)
n? = A
?
kAk??A
?
k?Ak ? AkA ?A
? A?k?
4 2 k
? k
?kk?( ) (4.55)
+ Ak?Qk +Q
?
k?A
?
k +M?+ ?M
?,
k
where
1
Q = {A?k ?(k, A
?
k?Ak?}12
i 1 )
M = A?HA ? {H,A?k k kAk}6 (4.56)
k
1 ?(
2 )
? A? ? 1 ? ??Ak?A Ak ? A ?AkA Ak? .
12 k k 2 k k
kk?
As is the case for EC1 and EC2, the correction is described by Eq. (4.44) with operators (i)D?n
being a sum of products of at most eight Majorana fermion operators.
4.8 Error analysis
In this section, we first derive Eq. (4.25) and then provide the proof of Lemma 12 in
Section 4.3.1. The error can be formally expressed in terms of evolution superoperators as
1 ? ?
? = max |?r|EVst(t, 0)?r?(0)? V(t, 0)?r?(0)|r??, (4.57)
2 r? r
where V(t2, t1) is the Markovian map generated by Eq. (4.1) in Section 4.1, Vst(?, t2, t1) is
a unitary trajectory map depending on either a realization of the discrete stochastic field ?kn
(EC1 and EC3) or a random choice of unitaries (EC2). We use the Dyson-like expansion in
127
Eq. (4.20) and the convexity of the absolute value to upper bound the error as
?
?? t ?? ?
? ? Emax dt? ???r|V ? ? ? ?st(t, t )D(t )Vst(t , 0)?r? |r??
2 r? 0 r (4.58)
+O(?? 2).
Using the fact that Vst is a unitary map, we can rewrite
V (t?, 0) = V?1 ?st st (t , t)Vst(t, 0), (4.59)
where the inverse of a unitary map is well-defined through the inverse unitary transforma-
tions. This expression leads directly to Eq. (4.25), taking into account that
?
V?1st (t?, t)D(t)?V?1st (t?, t) = D(1) ?? (t, t )?D(2)? (t, t?), (4.60)
?
where (i) (i)D? (t, t?) = Vst(t, t?)D? (t?).
128
4.8.1 Proof of the Lemma
? Let us rewrite the left-hand side of Eq. (4.28) using the spectral decomposition ? =
? p?|??????| and triangle inequality as
? ??? ?
|?r| ?O ?O |r?| = ? p ????? ?1 2 ? r r ?r|O|r1 2 1??r2|O|r??
r ? ?? r ?,r1r2
? p |??? r ||??r ||?r|O|r1?||?r2|O|r?|1 2 (4.61)
? r r1r2 ? ? ?
? ?O1?max?O ? ? ?2 max p?|?r ||?1 r |,2
?,r r1?D(k1,r) r2?D(k2,r)
where we denote ?O?max = maxij |Oij| to be the max-norm of the matrix O, and Dk(r) is
a sphere with radius k with respect to Hamming distance. Using the inequality
| 1
( )
?? ||??r r | ? |?? 2r | + |??r |2 (4.62)1 2 2 1 2
( )
and the property that the sphere D(k, r) contains L ? Lk/k! states, we obtain
k
?
|?r|O ?O |r?| ? 1 ?O ? ?O ? Lk1+k21 2 1 max 2 max , (4.63)
r k1!k2!
where we use the fact that the density matrix is properly normalized, Tr ? = 1.
129
4.9 Discussion
In this Chapter, we have demonstrated how simple forms of dissipation affect the com-
plexity of simulation of noninteracting fermions. In particular, focusing on linear-quadratic
Lindblad jump operators, we have shown the existence of two complementary complexity
classes of Lindblad jump operators, easy and hard for simulation on a classical computer.
Using the error-correction formalism, we showed that the hard class has a finite volume in
the parameter space and tolerates the presence of small arbitrary corrections. At the same
time, the easy classes may have small measure and could become hard even as a result of
arbitrarily small corrections to the master equation.
We have expanded the region of classical simulability of free-fermions in the presence
of Markovian errors from single-qubit loss/gain to more general quadratic-linear Lindblad
jump operators. The algorithms we devise for EC1?EC3 based on the stochastic unraveling
approach provably work in polynomial time. This shows that a large class of dissipation
processes such as dephasing or single-fermion decay can be treated with the help of efficient
classical algorithms.
At the same time, more complex processes are BQP-complete, which we show by ex-
plicitly constructing an entangling gate and showing the equivalence of the problem with
universal quantum computation. We thus place limitations on the extent to which the sim-
ulability result may be extended, since we believe quantum computation is strictly more
powerful than classical computation. Our detailed analysis shows that it is within the range
of experimental feasibility to implement with cold atoms a quantum computer with purely
dissipative atom-atom interactions, an exciting possibility for experiments in quantum com-
130
puting. For example, dissipative quantum systems such as alkaline-earth atoms may serve in
the next generation of quantum supremacy experiments. Also, our result suggests that sim-
ulating fermion dynamics may be hard for quantum particles experiencing dissipation, for
example, quasiparticles in solid-state systems. Future work can explore the hardness of sim-
ulation of electronic systems with quasiparticle dynamics approximated with quadratic-linear
Lindblad jump operators that include the effects of electron-electron, electron-phonon, and
electron-impurity scattering processes. Alternatively, physical systems following such dy-
namics with high accuracy may be a future platform for quantum computing experiments.
It may be interesting to explore the connection of our results with the theory of
matchgate (free-fermion) computations and the role played by non-Gaussianity. Quadratic
fermionic Hamiltonians and single-fermion loss give rise to Gaussian operations and are
hence easily simulable [202]. It is known [261] that any non-Gaussian fermionic state is
a resource for fermionic computation, boosting the computational power of free fermions
from being classically simulable to being universal for quantum computation. Our results
suggest that quadratic-linear Lindblad jump operators are non-Gaussian in general. There-
fore, it would be interesting to quantify the amount of non-Gaussianity (or ?magic?) for the
Lindblad operations we study here.
Along the same lines, one can quantify a different resource for non-classicality, such
as a suitable measure of entanglement for open fermionic systems. Efficient sampling from
the full output distribution in arbitrary bases can allow for efficient computation of certain
measures of entanglement such as R?nyi entropies [262]. Relatedly, Ref. [263] has studied
the logarithmic negativity for free fermions with gain/loss Lindblad terms.
Further, one may also consider how the complexity of simulating dynamics under
131
quadratic Lindbladians changes with time. Since the system starts off in a Fock state that is
easy to sample from, and dynamics under quadratic Hamiltonians with quadratic Lindblad
jump operators can generate states that are hard to sample from, one can see a dynamical
transition in sampling complexity [93; 96]. It is worthwhile to investigate whether these
transitions are sharp or coarse (as defined in Ref. [93]) since this can identify what ?uni-
versality class? free fermions with noise belong to. Techniques such as Lieb-Robinson-like
bounds for the evolution of free particles with dissipation [264; 265] would be relevant here.
Another exciting direction is the study of worst-to-average-case equivalence in com-
plexity, which seeks to understand the complexity of typical instances as opposed to worst-
case instances [30; 31]. It would be interesting to see if the Cayley path technique of Ref. [31]
can be adapted to argue for average-case hardness of dissipative fermionic dynamics.
132
Chapter 5: QuantumComputational Supremacy via High-Dimensional Gaus-
sian Boson Sampling
5.1 Introduction
We are arriving at an exciting era for quantum computing in which quantum experi-
ments are pushing the limits of what is efficiently computable by the most powerful classical
supercomputers. The first major goal for this era is the demonstration of a scalable quan-
tum advantage or quantum computational supremacy (QCS) over classical computers. QCS is
important as a probe of the foundations of computer science, where it can be seen as an ex-
perimental violation of the extended Church-Turing thesis, and it also serves as an important
benchmarking tool for comparing near-term experiments on different platforms in a fair and
consistent manner. The recent groundbreaking demonstrations of QCS [33; 34] constitute
the first significant experimental evidence against the extended Church-Turing thesis.
Notwithstanding, multiple potential loopholes have been pointed out [266?270]. In-
deed, QCS will not be marked by a single isolated experiment but rather will be established
by gradually improving and scaling up ?high complexity? experiments run over the course of
many years, which improving classical algorithms will try to simulate. Our confidence that
we have arrived in this new era will grow as multiple experiments, performed in different
133
physical architectures, independently reach this conclusion in a comparable fashion. In this
way, the goal may be seen as being analogous to Bell inequality violations, which were orig-
inally conducted in landmark experiments starting in the 1970s [271; 272] performed on a
variety of different platforms but only much later were loopholes closed in a recent series of
impressive experiments [273?276].
Among different approaches to demonstrating QCS [14; 33; 34; 36; 277; 278], pho-
tonics provides a promising path as it enables room-temperature operation, fast gate speeds
and remarkable potential for scalability [279; 280]. Arguably, the most feasible approach to
demonstrating QCS with photonics is to perform the Gaussian boson sampling (GBS) pro-
tocol [53; 54; 281]. Indeed, this protocol is at the heart of the recent QCS demonstration
performed by a team from USTC [34], which employed a GBS device with 100 modes and
an average of around 45 photons. However, GBS has several important limitations. On the
experimental side, current implementations of GBS either lack programmability [34] or have
high loss rates, which could render the system classically simulable [177; 282]. Also, from
a theoretical standpoint, there is a comparative lack of complexity-theoretic evidence for the
hardness of GBS [268?270].
In this Chapter, we overcome these dual challenges. We close important theoreti-
cal loopholes in the hardness argument for GBS and provide evidence for the hardness of
classically simulating GBS even in the presence of loss.
We first address the open theoretical questions about GBS, namely, hardness in the
regime with little overall noise in the form of optical loss (Section 5.2). More specifically,
to provide complexity-theoretic evidence for the hardness of approximately simulating GBS,
we prove average-case hardness of computing output probabilities in the noise-free case as
134
well as the so-called ?hiding property? [14]. These results bring GBS to the level of evidence
shared by other QCS proposals such as random circuit sampling (RCS) and conventional boson
sampling (see e.g., Refs. [14; 30; 36]). We then show that average-case hardness of computing
output probabilities still holds in a regime of high loss rates, building on recent results [32],
and discuss the implications of this result on the noise-regimes in which one may still expect
GBS to be hard to simulate on a classical computer. These results bolster the evidence for
QCS in the USTC experiment and also any future GBS experiments.
5.2 Hardness of approximate GBS
We begin by reviewing and significantly strengthening the hardness argument for the
task of simulating GBS as introduced in Refs. [53; 54]. The structure of this section is as
follows. We first introduce the model of Gaussian boson sampling and then examine the
evidence for the hardness of approximate boson sampling. In order to bring the hardness
evidence for GBS to the same standard as that of boson sampling, we identify two properties
required for establishing hardness using the standard QCS arguments, namely hiding and
average-case hardness of approximating probabilities, and we provide strong evidence for these
properties in GBS. Specifically, we reduce the hiding property to a highly plausible con-
jecture in random matrix theory, for which we provide analytical and numerical evidence.
Additionally, we provide evidence for approximate average-case hardness by proving approx-
imate worst-case hardness and near-exact average-case hardness of computing the output
probabilities. We then extend the latter results to the case of computing output probabilities
of noisy GBS, which can be well-motivated when the noise model describing the experimen-
135
tal data is trusted. These results show that the evidence of a quantum ?signal? remains in
the output distribution even in the presence of noise. Finally, we discuss the implications of
these results on the complexity of simulating GBS in the presence of noise.
5.2.1 Recap: Gaussian boson sampling
GBS is the computational task of sampling the photon number statistics of a Gaussian
state. Obtaining a sample from a typical GBS experiment involves the following steps. First,
M single-mode squeezed vacuum states are prepared1. These states are then interfered on an
M-mode linear/optical interferometer containing beam-splitters and phase shifters. Finally,
the Gaussian state at the output of the interferometer is impinged on M photon-number-
resolving (PNR) detectors. The resulting pattern of photon number outcomes from the de-
tectors is the required sample. Because single-mode squeezed states can be generated and
interfered deterministically and at room temperatures and with high rates, GBS is experi-
mentally feasible on large scales already today, as evidenced by the recent experiment from
USTC [34].
In more detail, a typical GBS experiment involves interferingM single-mode squeezed
vacuum states with squeezing parameters {r Mi}i=1 at an interferometer specified by anM?M
linear-optical unitary matrix U . Note that some of the modes can be optionally prepared in
the vacuum state, and these can be specified by setting their squeezing parameter to zero.
The probability of detecting n1 photons in the first mode, n2 in the second, and so
1In the general setup, a GBS experiment can have a general Gaussian state at the input.
136
on, denoted by n = (n1, . . . , nM), is
Pr n ?|Haf(An,n)|2( ) = . (5.1)M
j=1 nj! cosh rj
( )
Here,A = AT = U ?Mi=1 tanh(ri) UT is the so-called adjacency matrix of the (pu?re, zero-
displacement) Gaussian state [53], and An,n is the symmetric matrix of size N = Mi=1 ni
(i.e the total photon number) obtained by repeating the ith column and row of A a total of
ni times. In particular, if ni = 0 then the corresponding row and column is deleted. Finally,
the Hafnian Haf(?) of a symmetric N ?N matrix B is given by
? ?
Haf(B) = Bi,j, (5.2)
??PMP(N) (i,j)??
where PMP(N) is the set of perfect matching permutations ofN elements, i.e., permutations
? : [N ] ? [N ] satisfying ?(2k ? 1) < ?(2k), ?(2k ? 1) < ?(2k + 1). The Hafnian of a
0? 0 matrix is defined to be 1 and the one of an odd-size matrix is defined to be 0, which is
a manifestation of the fact that squeezed states are supported on even photon number states
only. By allowing arbitrary linear-optical unitaries and arbitrary squeezing parameters of each
squeezer, an arbitrary symmetric matrix A can be encoded (up to scaling pre-factors) into a
Gaussian state. For generic instances, the best-known algorithms to calculate Hafnians scale
like N32N/2 where N is the size of the matrix [283].
137
5.2.2 Recap: Approximate sampling hardness of boson sampling
Before we state our technical results, we review the main steps of the hardness ar-
gument for conventional boson sampling as given by Aaronson and Arkhipov [14]. These
steps provide context for the hardness results of GBS that we present in the remainder of
this section.
In a standard boson sampling experiment, instead of interfering single-mode squeezed
states at an interferometer as done in Gaussian boson sampling, anN-photonM-mode Fock
state is prepared and evolved under a linear-optical unitary and then measured in the photon-
number basis. The boson sampling task is to, given a linear-optical unitary as an input, output
samples from the output distribution of a corresponding boson sampling experiment.
Aaronson and Arkhipov showed that it is not possible for a classical computer to ef-
ficiently do this task unless certain complexity-theoretic conjectures are false. In particu-
lar, they reduce the task of approximating the probabilities of outputs to sampling using
an efficient classical algorithm, making use of an approximate counting algorithm due to
Stockmeyer [15]. This probability estimation can in turn be related to approximating the
permanent of a certain sub-matrix of the linear-optical unitary, which is is provably hard for
a class known as #P. While the Stockmeyer reduction is not efficient, the existence of a clas-
sical efficient sampling algorithm would imply that #P-hard problems could be solved using
fewer computational resources than expected, amounting to an argument by contradiction.
The main difficulty in the hardness argument for boson sampling arises when ex-
tending it to the setting of approximate sampling. Here, the task is to sample from from
any distribution that is within a constant-size total-variation distance from a given ideal bo-
138
son sampling distribution. This additional constraint takes into account that actual devices
are bound to achieve only some finite and typically additive precision. In this setting, one
may therefore argue for a separation of computational power between quantum and classical
devices.
Given this constraint, the hardness argument for the task of approximate sampling
must take into account that the constant error budget on the distribution can be distributed
arbitrarily across all outcome probabilities. In particular, this means that any specific outcome
probability of the actually sampled distribution might have a large (constant-size) error when
compared to the ideal distribution, which would imply that the sampler cannot be used to
estimate the true outcome probabilities. To get around this issue, the argument is extended
to random problem instances: via a property of the distribution over problem instances
called hiding, one can then translate typical outcomes of fixed instances to fixed outcomes
of random instances. This enforces that with high probability, the overall constant error
budget for the entire distribution is manifest in small errors on the indivi(dua)l probabilities
that are proportional to the inverse size of the sample space, that is, ? 1/ M . Technically,
N
in standard boson sampling, showing the hiding property boils down to showing that the
distribution of any small enough sub-matrix of a Haar-random unitary is approximately (in
total-variation distance) an entry-wise complex normal distribution. In particular, Aaronson
and Arkhipov show that when M ? ?(N5), we can ?hide? a random Gaussian matrix in
a small enough sub-matrix of the large Haar-random unitary by an appropriate procedure
[14, Lemma 5.7] because all of these sub-matrices are indistinguishable random Gaussian
matrices.
For the approximate sampling task to remain computationally intractable, it remains
139
to show that estimating the outcome probabilities up to inverse-exponentially small error
is #P-hard for any fraction of the problem instances on which the obtained error bound
holds?a property called approximate average-case hardness. More precisely, given a random
problem instance, estimating the probability of a given outcome must be #P-hard with high
probability. As evidence toward this property, it has been shown that exactly computing
those output probabilities is in fact #P-hard on average (and this was a motivation for boson
sampling in the first place), and it is known that estimating them to the required robust-
ness level is worst-case hard. However, the hardness of computing those probabilities to a
sufficiently large robustness level on average is still unknown.
5.2.3 Average-case hardness of computing GBS probabilities
As outlined in Section 5.2.2, the question of hardness of approximate sampling boils
down to whether it is #P-hard to approximate most output probabilities. In this subsection,
we show the average-case hardness of this task when the allowed additive approximation error
is exponentially small2. ? ?
The output probabilities of GBS are given in terms of ?Haf 2((UI UT )n n)?K , . In Ref. [284],
we show that the distribution over the N ? N matrices (UI UTK )n,n for Haar random U
is well approximated by complex, symmetric Gaussian matrices XXT . Hence, to show the
average-case hardness of computing output probabilities of GBS, it suffices to consider the
following problem:
2Note, however, that this is a different exponential from what is desired in order to show a polynomial
hierarchy collapse.
140
(?, ?)-SQUARED-HAFNIANS-OF-GAUSSIANS
Input A matrix XXT with X ? GN,K(0, 1/M).
? ?
Output ?Haf (XXT 2)? to additive error ?, with probability ? ? over the distribution
GN,K(0, 1/M).
To complete the argument that an efficient classical approximate sampling algorithm
for GBS cannot exist, it remains to prove the #P-hardness of (?, ?)-SQUARED-HAFNIANS-OF-
GAUSSIANS as formalized by the following approximate average-case hardness conjecture.
Co(?ecture 13. The (?, ?)-SQUARE)D-HAFNIANS-OF-GAUSSIANS problem is #P-hard for any ? =
O N ! tanhN(r)/(coshK(r)MN) and any constant ? > 3/4.
As in all other known proposals for demonstrating QCS, this approximate average-
case hardness conjecture remains open. Nonetheless, just like in other proposals, it turns
out that one can give evidence for Conjecture 13. Namely, we can prove a weaker version of
the conjecture with a smaller robustness level ? = O (exp[?6N logN ? ?(N)]) as opposed
to ? = O (exp[?N logN ? ?(N)]) in Conjecture 13.
Theorem 14. The (?, ?)-SQUARED-HAFNIANS-OF-GAUSSIANS problem is #P-hard under PH
reductions for any ? ? O (exp[?6N logN ? ?(N)]) and any constant ? > 3/4.
We provide a detailed proof of Theorem 14 in Section 5.3. The technique we employ
in the proof is a worst-to-average-case reduction [14; 285]. That is, by assuming access to an
oracle for the (?, ?)-SQUARED-HAFNIANS-OF-GAUSSIANS problem, we show that one in fact
approximate Haf(XXT ) for any matrix X ? CN?K . This latter task is #P-hard in the
141
worst-case as we show i?n Section 5.3?.1. At a high level, the worst-to-average-case reduction
relies on the fact that ?Haf (XXT )?2 is a low degree (of degree 2N ) polynomial over the
entries of the matrixX . This allows us to use the oracle to perform polynomial interpolation.
Therefore, by combining this observation with the techniques of Refs. [14; 32; 285; 286],
we obtain a worst-to-average case reduction for exactly computing the output probabilities.
Together, our results on the hiding property and the approximate average-case con-
jecture in GBS, significantly strengthen the evidence for the hardness of approximately sim-
ulating GBS in terms of the total-variation distance to the ideal output distribution. Given
our results, GBS is now on par with the other leading QCS proposals in terms of complexity-
theoretic evidence for approximate sampling hardness [14; 30?32; 36; 277; 286], up to a plau-
sible conjecture in random matrix theory?for which we provided theoretical and numerical
evidence in Ref. [284]. To achieve a demonstration covered by those complexity-theoretic
results, however, the loss rate at every element of the linear-optical circuit, must scale in-
versely with the total number of such elements?a daunting challenge from an experimental
perspective.
5.2.4 Hardness of computation of output probabilities for noisy GBS
In the remainder of this section, we will go one step further and assess how the
complexity-theoretic argument for sampling hardness is affected by more realistic noise lev-
els, in particular, in terms of photon loss. In terms of scaling, any constant loss rate of the
individual optical elements will lead to an exponential suppression of quantum effects in the
output distribution. We now show that, nonetheless and surprisingly, an evidence of a quan-
142
tum signal remains even in the presence of such significant loss. We then discuss to what
extent and in which regimes such a quantum signal might lead to the hardness of simulating
a lossy GBS experiment.
Our main result in this section is the average-case hardness of computing the noisy
output probability of a random GBS instance, which we obtain by using similar arguments
to recent work of Bouland et al. [32], but now extended to the GBS setting. Our results are
valid for any noise model that is local, stochastic, and is error-detectable using linear optics.
More specifically, we consider a setting where the noise acts locally after every gate, and is of
the form
Ni[?] = (1? ?i)?+ ?iEi[?], (5.3)
where stochasticity requires Ei to itself be a valid channel (i.e. a completely positive trace
preserving map) with no identity component.
Consider the following problem.
143
(?, ?)-NOISYGBS-PROBABILITY
Input A noisy GBS instance, consisting of the linear-optical unitary U on M modes
chosen from the Haar measure H, the squeezing parameters at the input, and a
description of the noise channels with parameters ?i. Let ? = maxi ?i.
Output With probability ? over instances, an estimate of the quantity Pr(n) to addi-
tive error ?, where Pr(n) is the probability of obtaining an outcome n that is
collision-free and has a total number of photons N scaling as N = poly(M).
With probability 1? ?, an arbitrary output.
In the above definition, we take ? = 1 to mean the worst-case problem. We prove the
following statement of average-case hardness of computing noisy probabilities.
Theorem 15. There exists a noise threshold ?? and a sufficiently large polynomial such that the
problem (?, ?)-NOISYGBS-PROBABILITY is #P-hard under PH reductions for any constant ? >
3/4, ? ? ??, and ? ? 2?poly.
There are two parts to the proof. The first part is a proof of worst-case hardness of the
problem (when ? = 1), and the second a worst-to-average-case equivalence. For worst-case
hardness, it turns out that it suffices for the noise to be stochastic and to be able to error-
detect it [287]. These conditions are both met for optical loss. Optical loss is stochastic
by virtue of being a convex combination of the channels corresponding to no photon loss,
single-photon loss, and so on [288]. Moreover, optical loss can also be detected and corrected
using only linear-optical operations and photo-detection with high thresholds [280].
These two properties, namely stochasticity and error-detectability, allows us to apply
144
a result of Fujii [287] to prove worst-case hardness of estimating the noisy probabilities.
For the worst-to-average-case equivalence, all we need is for the polynomial structure in the
problem to be preserved. This can be satisfied for any local noise model. Preserving the
polynomial structure of the output probability enables us to continue to use the same proof
techniques as in the previous Section 5.2.3, as we elaborate in Section 5.4.
We close out this section by reminding the reader again that, in this section, we
considered the hardness of computing output probabilities. While these are not tasks that
are feasible for any realistic quantum device, our results nevertheless indicate that there is a
computationally intractable (but exponentially small) ?quantum signal? present in the system.
5.2.5 The complexity of noisy and approximate GBS
We now discuss the implications of the hardness result for computing noisy GBS
probabilities on the complexity of sampling from the output distribution of noisy GBS. An
immediate implication of this result is that it is classically hard to exactly sample from the
noisy distribution of a worst-case GBS experiment. This is because the quantum signal is still
present in the distribution, so the argument based on Stockmeyer?s algorithm (and sketched
in Section 5.2.2) is valid. Thus, in the idealized situation in which loss is the only source
of noise of an experimental system and the exact loss rate is known, simulating a worst-case
GBS experiment is classically intractable. Note that loss rates can be inferred from standard
optical tomography procedures such as that of Ref. [289]. Given that this result links the
hardness of simulating the noisy experiment to an exponentially small quantum signal in the
form of output probabilities, it is crucial that the noise model accurately captures the working
145
of the device.
We now discuss the more realistic situation in which loss is the predominant, but not
the sole, source of noise in a photonic experimental system. What can we say about the
hardness of approximate sampling in such a situation? To begin with, let us draw on some
intuition from RCS schemes acting on n qubits. Here, the additive error incurred in esti-
mating output probabilities using the Stockmeyer algorithm isO(2?n) with high probability
(since this is the size of a typical output probability in an RCS experiment). In the presence
of uncorrected noise, an error of O(2?n) in the noisy output probability can be too large for
hardness. For example, there is evidence that with gate-wise depolarizing noise, the proba-
bilities will deviate from uniform by merely O(2?m), where m is the total number of gates3
[290]. This means that approximate-sampling hardness cannot be shown using these tech-
niques, since it is not hard to approximate the noisy probabilities any more. Indeed, in this
regime, the noisy distribution is exponentially close in total-variation distance to the uniform
distribution, rendering the approximate sampling task for the noisy distribution classically
simulable.
In the case of noisy GBS, the dominant noise model, namely loss, leads to the vacuum
state for a sufficiently deep network, which is again a distribution that is easy to classically
sample from (similar to the uniform distribution in qubit RCS schemes). However, if we
post-select on a certain minimal number of photons surviving, the distribution need not
be easy to simulate. This post-selection is efficient when the depth of the circuit scales
poly-logarithmically in the number of modes. In this case, the quantum signal will be large
enough so that even with an inverse exponential error, deviations from the easy distribution
3Here we are considering the size m to satisfy m = ?(n).
146
can be detected.
This excludes the simulation algorithm that samples from the trivial distribution4 and
is a necessary condition for approximate average-case hardness to hold. In summary, our
results indicate that there might be ?room in the middle? in terms of gate depth and noise
rates, where hardness of sampling might hold. In fact, this intuition lies at the heart of
the high-dimensional architecture presented in Ref. [284]. This architecture is designed in
such a way that only as few gate applications as necessary for hardness are executed, so that
the leeway for noise to ruin the hardness of sampling is minimized. We stress, however,
that at the moment, existing proof techniques do not suffice to make a claim of this nature.
In fact, in certain regimes of noisy GBS, approximate sampling is known to be classically
efficient [282].
5.2.6 Hardness for computing noisy probabilities in high-dimensional GBS
In this subsection, we now argue for the hardness of computing output probabilities
for the noisy, high-dimensional GBS setup. In particular, we show that hardness is present
even in shallow depth noisy high-dimensional GBS architectures. This is in contrast to the
results in Section 5.2.4, where no restriction is made on the depth.
To do this, we simply observe that the previous argument for worst-case hardness,
which depends on the noise being local, stochastic, and error-detectable, continues to hold
for the limited-depth setup. For average-case hardness of computing noisy probabilities, we
again use a worst-to-average-case reduction. However, the polynomial interpolation in this
4This distribution is uniform on every photon number sector and every such sector is sampled according to
the ideal photon number distribution.
147
case is different, since a random instance is not Haar distributed any more but rather according
to U , the distribution over random instances of high-dimensional GBS. To explain further,
consider the usual interpolation X(t) = (1? t)X + tY , where X(0) = X is drawn from U
andX(1) = Y is the matrix corresponding to a worst-case high-dimensional GBS instance.
In this case, there is no guarantee that the interpolated matrices X(t) also correspond to
high-dimensional GBS instances of small depth. We get around this issue by choosing a
gate-wise interpolation that is similar to that seen in RCS [30; 31].
We first define the problem of computing output probabilities of a restricted-depth
high-dimensional GBS architecture.
(?, ?)-HIGHDIMENSIONAL-NOISYGBS-PROBABILITY
Input A noisy GBS instance drawn from U that can be implemented in D dimensions
with a constant number of cycles C = O(1) with noise parameter ?.
Output With probability ? over instances, an estimate of Pr(n), a collision-free outcome
with N = poly(M) photons, to additive error ?.
With probability 1? ?, an arbitrary output.
Similar to the previous sections, we can again obtain an average-case hardness result,
that we state here and prove in Section 5.5:
Theorem 16. There exists a noise threshold ?? and a sufficiently large polynomial such that the
problem (?, ?)-HIGHDIMENSIONAL-NOISYGBS-PROBABILITY is #P-hard under PH reductions
for any constant ? > 3/4, ? ? ? , and ? ? 2?poly? .
148
5.3 Average-case hardness of computing GBS output probabilities
In this section, we show average-case hardness of computing GBS output probabilities.
As explained in Section 5.2.3, this amounts to showing that the following problem is #P-
hard.
(?, ?)-SQUARED-HAFNIANS-OF-GAUSSIANS
Input A matrix XXT with X ? GN,K(0, 1/M).
? ?
Output ?Haf (XXT )?2 to additive error ?, with probability ? ? over the distribution
GN,K(0, 1/M).
The proof will proceed in two steps: First, we will show that an?oracle for th?e (?, ?)-
SQUARED-HAFNIANS-OF-GAUSSIANS problem allows one to approximate ?Haf 2(Y Y T )? for ar-
bitrary Y ? C2N?2K . This first part of the proof c?onstitutes t?he worst-to-average-case re-
duction. Second, we will show that approximating ?Haf(Y Y T ?2) for arbitrary Y ? C2N?2K
is actually #P-hard in the worst-case. We show this by reducing the task of approximat-
i?ng the perm? anent of an arbitrary complex N ? N matrix to the task of approximating?Haf(Y Y T 2)? .
5.3.1 Worst-case hardness
Consider the following problem:
149
?-SQUARED-HAFNIANS
Input A matrix Y Y T with Y ? CN?K for K ? N, N ? 2N such that the entries of
?
Y are of the form (x + iy)/ M for |x|, |y| some O(1)-bounded integers and
additive-error toler?ance ? > 0.? ? ?
Output An estimate s.t. ?h ?h? ?Haf(Y Y T 2?)? ? ? ?.
We prove the following Lemma.
Lemma 17. The problem ?-SQUARED-HAFNIANS is worst-case #P-hard for any additive error
? ? 1/(2MN).
Proof. Without loss of generality, we restrict to N ? K. We begin the proof by noting that
the permanent of any square matrix G can be expressed as the Hafnian of a corresponding
block matrix twice the size of G [54],
???? ??
Per(G) = Haf???? 0 G?????? . (5.4)
GT 0
Hence, computing the squared permanent of any complexN/2?N/2matrixG ? CN/2?N/2
reduces to computing the squared Hafnian of a corresponding block matrix
?? ?
B(G) = ?? 0 G??? . (5.5)
GT 0
Computing the squared permanent exactly is known to be worst-case #P-hard even over
0/1-matrices [14; 291].
150
Next we note that any matrix B(G) for G ? CN/2?N/2 can be decomposed as XXT
in terms of some complex matrix X ? CN?K . Indeed the block matrix B(G) is a complex,
symmetric matrix, so we can decompose it using the Takagi decomposition as WDW T ,
where W ? U(N) is a unitary matrix and D ? RN?N is a nonnegative diagonal matrix.
We now define X ? = (WD1/2) and X by appending (K ? N) all-0-columns to X ?. This
gives rise to a decomposition of B(G) = XXT with X ? CN?K . Hence it is #P-hard to
exactly compute the Hafnian of matrices of the form XXT in the worst case. Additionally,
since the Hafnian is a continuous function, we can compute Haf(XXT ) to an arbitrary level
of precision by considering Haf(Y Y T ) with the entries of Y being of the form x+ iy, with
x and y integers (by suitably rescaling the entries of the matrix). Finally, we note that by
?
normalization we can assume that the entries of the matrix Y are of the form (x+ iy)/ M
with x and y O(1) bounded integers. Then the squared Hafnian of Y Y T is an integer
multiple of 1/MN . Therefore, computing the Hafnian of Y Y T up to additive error of
1/(2MN) serves to compute the squared Hafnian of B(G) exactly, which is #P-hard. This
concludes the proof.
The proof holds equally for N ? poly(K): in this case we embed a square matrix in
CK?K and append 0 rows instead of columns.
5.3.2 Worst-to-average equivalence
In this section, we prove the average-case hardness of computing GBS output proba-
bilities. That is, we prove the following Lemma:
Theorem 18 (Theorem 14 restated). The (?, ?)-SQUARED-HAFNIANS-OF-GAUSSIANS problem
151
is #P-hard under PH reductions for any ? ? O (exp[?6N logN ? ?(N)]) and any constant
? > 3/4.
We first sketch the proof idea and elaborate on the technique used. The overall idea
is to give a worst-to-average-case reduction from the problem ?-SQUARED-HAFNIANS to the
problem (?, ?)-SQUARED-HAFNIANS-OF-GAUSSIANS. The worst-case #P-hardness of problem
?-SQUARED-HAFNIANS has been established in the previous section.
We use the same technique as Refs. [14; 32] to establish this reduction. Assume that
we are given an oracle O that solves (?, ?)-SQUARED-HAFNIANS-OF-GAUSSIANS, meaning that
with probability at least ? over the inputX , it outputs a squared Hafnian ofXXT to additive
error ?. The rest of the time, it may output an incorrect value, with no guarantees whatsoever
on how close the output is to the desired output. In the following, we will show how to use
the oracle O to obtain the squared Hafnian of an arbitrary worst-case matrix Y Y T with high
probability (this latter probability is over the choice of the random variables instantiated in
the algorithm). ?
The key idea is that forX ? CN?K , the quantity ? ?Haf 2(XXT )? is a degree 2N poly-
nomial over the entries of the matrix X . This allows for the use of polynomial interpolation
to recover the squared Hafnian of an arbitrary worst-case matrix Y Y T . An important tech-
nique we use in this proof is the robust Berlekamp-Welch algorithm due to Ref. [32], which
is important for polynomial interpolation over R as opposed to a finite field. Polynomial in-
terpolation over the reals is a technique often used for the problem of average-case hardness
of computing output probabilities of random quantum circuits [30; 31]. The Berlekamp-
Welch algorithm cannot be used as is for the reals, and therefore, recent works [30; 31]
152
use techniques like Lagrange interpolation. The new robust Berlekamp-Welch algorithm of
Ref. [32] allows for improved robustness of the worst-to-average-case reduction.
As an example, in the context of random quantum circuits over n qubits and m gates,
Lagrange interpolation can only give average-case #P-hardness of computing output prob-
abilities to error ?O(m32 ) rather than the O(2?n) that suffices for proving the hardness of
approximate sampling (see [14; 31]). The modified Berlekamp-Welch algorithm of Ref. [32],
which is boosted with an NP oracle, can sidestep the need for Lagrange interpolation and
obtain average-case #P-hardness with 2?O(m logm) error (see also, the recent work of Kondo
et al. [286] which also obtains this robustness error).
Theorem 19 (Robust Berlekamp-Welch algorithm [32]). Let p be a univariate polynomial of
degree d over the reals. Suppose that we have k ? 100d2 points (xi, yi), with {xi} uniformly
spaced in the interval [0, ?] and obeying the promise
Pr 1[|yi ? p(xi)| ? ?] ? ? < . (5.6)
4
Then there is a PNP algorithm that can estimate p(1) to additive error? exp[d log??1 +O(d)]
with probability at least 2/3.
Proof of Theorem 14. The polynomial interpolation procedure is as follows. Let X(t) be the
matrix obtained by drawing a random X ? GN,K(0, 1/M) and setting
X(t) := (1? t)X + tY, (5.7)
153
where Y is the matrix corresponding to the worst-case instance. Now, the quantity
?? ?Haf T ?2p(t) := (X(t)X (t)) (5.8)
is a polynomial of degree 2N over the entries of X(t), and consequently, over t itself. For t
close to 0, X(t) is close to Gaussian distributed, while when t is close to 1, the distribution
is close to being deterministic. We select k points in the range [0, ?] and query the oracle O
for the value of p(t) for these points. By the promise, the oracle outputs the correct value
of p(t) for most values of t with high probability. Conditioned on this event, the robust
Berlekamp-Welch algorithm stated in Theorem 19 allows one to reconstruct the polynomial
in the second level of the polynomial hierarchy. The polynomial can then be evaluated at the
point t = 1 to obtain an estimate of the squared Hafnian of the worst-case matrix Y Y T .
We now check that the conditions of Theorem 19 are met. We say that a call to
the oracle O is successful if it outputs the squared Hafnian of a matrix to additive error ?.
By assumption, for X drawn at random from GN,K(0, 1/M), the oracle is successful with
probability at least ?. Note however that the matrix X(t) = (1 ? t)X + tY is not exactly
distributed according to GN,K(0, 1/M). Instead, for small t, due to the rescaling by (1? t)
and the shift by tY , X(t) is distributed according to a slightly different distribution G ?. If
we query the oracle for the value of p(t) with matrices drawn from this different distribution
G ?, the probability of success can, in the worst case, decrease. By definition, the success
probability can decrease at most by the variation distance between the two distributions
GN,K(0, 1/M) and G ?, which is O(tmax(N,K)2). Therefore, for K ? N , the probability
of success is at least ? ? O(?K2). We choose ? to be O(c/K2) with some small enough c
154
so that the success probability is at least ? ? O(c) > 3/4. This ensures that the conditions
of the theorem are met.
We finally conclude by examining the additive error to which we can compute, using
the BPPNP reduction, the squared Hafnian of the worst-case matrix Y Y T . If the additive
error for successful queries to the oracle is at most ?, Theorem 19 implies that the error in
computing p(1) is ? exp[d log??1 +O(d)]. Plugging in d = 2N and ? = c/N2, we get the
total additive error in estimating p(1) to be ? exp[4N logN +O(N)]. Finally, we note that
the squared Hafnian is shown to be worst-case hard for additive errorO(1/MN). Therefore,
we make the choice ( )
1
? exp[4N logN +O(N)] = O , (5.9)
MN
or
? =O (exp[?4N logN ? ?(N)? 2N logN ]) (5.10)
=O (exp[?6N logN ? ?(N)]) ,
where we have assumed M = ?(N2). This choice ensures that we can, with probability at
least 2/3, compute the squared Hafnian of an arbitrary matrix with bounded entries of the
form Y Y T to additive error O(1/MN). As shown in Lemma 17, this task is #P-hard. This
completes our proof.
155
5.4 Average-case hardness of computing noisy GBS output probabilities
In this section, we argue that computing the output probabilities for a noisy random
GBS experiment is #P-hard on average. That is, we show the following lemma.
Lemma 20. There exists a polynomial p(N) and a loss threshold ?? such that (?, ?)-NOISYGBS-
PROBABILITY with ? ? ??, ? > 3/4, and ? ? 2?p(N) is #P-hard under PH reductions.
Proof. For worst-case hardness despite the presence of noise, we follow the proof technique
in Refs. [32; 287]. At a high level, the worst-case hardness follows from the error-detection
property of the system. In particular, the error-detection property implies that as long as
the noise ? is smaller than a certain threshold ??, there is a fixed outcome on a subset of the
modes, say m, such that conditioned on this outcome, the probability distribution on the
rest of the modes is exponentially close to the target noiseless distribution. In other words,
we have
???? ??Pr [n|m]? Pr [n]?? ? 2?poly(N) (5.11)noisy ideal
for any desired polynomial on the right hand side. Since Prideal[n] is #P-hard to approximate
in the worst case by virtue of Lemma 17, so is computing the conditional probability
Prnoisy[n,m]Pr [n|m] =
noisy Prnoisy[m
. (5.12)
]
The denominator here is the probability of seeing the outcome m, which flags the no-error
156
event. The probability of this can be exponentially small, and satisfies [32; 287]
???? ??Pr [m]? (1? ?)O(Md)? ? Pr [m]2?poly(N), (5.13)noisy ? noisy
where ? is the maximum noise parameter as in Eq. (5.3). In other words, for an error-
detected circuit, the probability that the outcome on the subset of heralding modes is in the
state m is exponentially close to the probability that no error occurred, which is given by
(1? ?)O(Md).
Therefore, approximating Prnoisy[n,m] is also #P-hard:
???? ??Pr [n,m]? Pr [n|m](1? ?)O(Md)? ??? ? Pr [n,m]2
?poly(N) (5.14)
nois?y noisy ? noisy
? ?? Pr [n,m]? Pr [n](1? ?)O(Md)?? ? Pr [n,m]2?poly(N)noisy ideal noisy
+2?poly(N). (5.15)
Since computing Pr [n] to additive error ?O(2?poly(N)ideal ) is #P-hard, so is computing
Pr [n,m] to additive error O(2?poly(N)(1??)O(Md)noisy . A similar analysis in Ref. [32] shows
that it is coC=P-hard to compute a noisy probability in the worst case to additive error
2?O(m logm) in the context of RCS. This proves the worst-case hardness.
For the worst-to-average-case reduction, we again use the technique of polynomial
interpolation in conjunction with a robust Berlekamp-Welch algorithm. We observe that
any noisy output probability for a local noise model can still be written as a polynomial
in the gate entries of the circuit, using the Feynman sum-over-paths idea. As before, we
perform interpolation from a random instance from the ensemble to the worst-case-hard
157
instance. This is achieved now using the Cayley path interpolation technique of Ref. [31]
instead of the direct interpolation between two matrices5. The full interpolation involves
interpolating every gate of a circuit implementation from the average-case instance Ai to the
worst-case instance Wi along the Cayley path
(
C (t) = t1+ (2? t)A W?
) ( )
1 ?1
i i i (2? t)1+ tA W?1i i ?Wi, (5.16)
which satisfies Ci(0) = Ai and Ci(1) = Wi. Using this interpolation and the fact that any
local noise can be ?purified? gate-wise by introducing ancillary systems of finite dimension,
we can again write the noisy probability Prnoisy[n,m][t] as a polynomial in t. The rest of the
proof follows from before.
5.5 Average-case hardness of computing noisy probabilities in high-dimensional
GBS
For the worst-case hardness of computing noisy probabilities of the high-dimensional
GBS architecture, we mainly use the previous results on error-detection of noise. The ad-
ditional ingredient used is the fact that a constant-depth linear-optical architecture in two
dimensions (and higher) has been shown by Brod [94] to be hard to exactly sample from.
The proof of Ref. [94] uses post-selection to argue for exact sampling hardness. Note
that the post-selection result does not, by itself, imply the #P-hardness of computing output
probabilities6. However, we note that the post-selection proof can often be ?opened up? in
5This is because the noisy output probability is no longer a simple function of only the linear-optical unitary
(like the Hafnian), but is also a function of the circuit implementation.
6It implies the PP-hardness of strong simulation, which involves computing both the output probabilities
158
order to directly argue about the hardness of computing output probabilities. This is done by
giving an amplitude-preserving reduction from aBQP circuit to the circuit family in question
(here, high-dimensional GBS). Since computing output amplitudes of BQP circuits is #P-
hard, so is computing that of the circuit family in question. Using the results in the previous
section, so is computing the noisy output probability in the worst case for an error-detected
circuit as long as the noise level is smaller than some (constant) threshold ??.
The average case hardness again essentially follows by observing that there is a poly-
nomial structure in the output probability, to prove Theorem 16. We again use the Cayley
technique of Ref. [31] to set up the polynomial interpolation in this case, and use results
from Ref. [32] to strengthen it, such as using a variable rescaling and applying a robust
version of the Berlekamp-Welch algorithm (Theorem 19).
5.6 Open problems and discussions
5.6.1 Open problems
In this Chapter, we have provided strong evidence for the hardness of Gaussian boson
sampling, putting it on eye level with other known schemes to show QCS such as random
circuit sampling and have substantially advanced the theory of GBS. Still, some theoretical
questions are outstanding.
1. In Ref. [284], we have been able to show that two plausible conjectures in random ma-
trix theory allow us to obtain the hiding property for a noiseless GBS set up, without
restrictions on the number of active modes. Can we obtain a similar hiding property
and the marginals.
159
for the high-dimensional GBS set-up introduced in Ref. [284]? Is this also possible
in the presence of noise? Answering these questions is crucial for extending the hard-
ness of computing output probabilities to the hardness of approximate sampling from
experimentally realizable distributions.
2. Informally, the anti-concentration conjecture for boson sampling (or GBS) states that
the output probability of a random instance is unlikely to be very small. If this con-
jecture was true, then now-standard arguments can show that the output probability
corresponding to an approximate sampler is, with high probability, a good multiplica-
tive estimate to the ideal output probability. Proving this conjecture true, in either
the case of boson sampling or GBS, would give increased evidence to support the goal
of proving QCS via photonics. A proof of such a conjecture is challenged due to
the fact that tools of unitary designs [155] are presumably unavailable in the bosonic
setting [14; 29].
3. Notwithstanding, it would be insightful to compute the secondmomentsE |Haf(XXT )|4X?G(0,1/M)
for the distribution we have found to characterize GBS problem instances. These mo-
ments thus characterize the so-called collision probability of seeing the same outcome
twice in an experiment, which in turn can be related not only to anticoncentration but
also the verifiability of approximate GBS from samples [10; 292], thus shedding some
light on the structure of the GBS output distribution.
4. An important task in demonstrating QCS is to verify that the performed experiment
indeed contains a non-trivial quantum signal that cannot be efficiently spoofed. The
Google QCS demonstration relied on linear cross-entropy benchmarking fidelity, and
160
the USTC experiment used a heavy-output generation (HOG) ratio test as an alterna-
tive path to verifiable hardness. Whether the HOG-ratio test can be spoofed efficiently
by a classical adversary such as the algorithms considered in Refs. [267; 268] is an open
problem.
5. The recent result in Ref. [293] presents a classical algorithm for the simulation of
high-dimensional boson sampling experiments in certain regimes. As described, this
algorithm is not applicable to the architecture as described in Ref. [284]. Extending
the algorithm to be relevant to the present architecture is an open problem.
6. With current optical technology, loss is the dominant source of noise in any GBS
experiment. Consequently we were motivated to obtain hardness results for computing
the output probabilities of a GBS experiment in the presence of significant photon loss.
It is natural to investigate if similar hardness results can be obtained in the presence of
other possible sources of experimental noise such as such as mode mismatch, multiple
Schmidt modes, interferometer phase drift and detector dark counts.
7. It is a challenge to the community, after all, to relate boson sampling closer to prac-
tically important computational tasks, following up on the applications reviewed in
Ref. [281] and to identify new applications.
5.6.2 Conclusion and outlook
In summary, this Chapter brings the convincing demonstration of QCS on a pro-
grammable photonics device substantially closer to reality. It overcomes previously out-
standing major theoretical challenges in the field by providing stronger evidence for the
161
hardness of GBS. Crucially, we have presented a novel architecture for high-dimensional
GBS using optical delay lines that promises low levels of noise without compromising on
its programmability. We benchmarked this architecture against the best available classical
simulation algorithms and found that already experiments involving a moderate number of
modes are far beyond reach for those algorithms.
We close by briefly commenting on the experimental prospects of realizing high-
dimensional GBS. Since high-dimensional GBS can be implemented in the time domain
according to the scheme presented in Ref. [284], only a single squeezer and a single de-
tector are required. If multiple detectors are available, these can be de-multiplexed using
optical switches in order to increase the effective repetition rate of the experiment and re-
duce the length of the delay lines. Especially promising is the case of D = 3, a = 6,
C = 1, which can be implemented with only three optical delay lines and three each of
re-programmable beam-splitters and phase shifters. Assuming reasonable values of squeezer
out-coupling losses, free-space to fiber coupling loss and detector efficiency, we estimate
that such a setup can be built using current optical technology with around 40% transmis-
sion, higher than that enabled by the ultra-low non-programmable loss interferometer in
the USTC experiment. Such a setup would enable the largest demonstration of QCS yet
with a mean detected photon number of 80 in a programmable device with 216 total modes.
162
Chapter 6: The importance of the spectral gap in estimating ground-state
energies
6.1 Introduction
Several exotic phenomena in our world occur only at very low temperatures, most
notably, those occurring in condensed matter such as superconductivity, superfluidity and
the fractional and integer quantum Hall effects. Beyond these examples in condensed-matter
physics, the low-energy physics of systems of several interacting particles is of interest in
several fields such as particle physics, atomic, molecular, and optical physics, chemistry, and
quantum computing. Accordingly, finding effective descriptions of ground states of many-
body Hamiltonians is a very natural and important task in physics.
Given the prevalence and importance of this task, a natural question is that of the
computational difficulty of solving this task in naturally occurring situations. Questions
such as the hardness of solving a computational task belong to the domain of computational
complexity theory. A good proxy for the difficulty of obtaining ground-state descriptions is
the difficulty of solving a weaker problem, namely that of computing ground-state energies
of many-body Hamiltonians. This question is studied in the domain known as ?Hamiltonian
complexity? (see e.g. Ref. [294]), an area of research at the intersection of quantum many-
163
body physics and computational complexity theory.
This area of research originated from Kitaev?s result that the LOCALHAMILTONIAN
problem, which is the problem of computing the ground-state energy of a local Hamiltonian,
is QMA-complete [59] (we refer a reader unfamiliar with complexity-theoretic language to
Appendix B). The complexity classQMA is the quantum generalization ofNP. Kitaev?s result
may be viewed as an analogue of the seminal Cook-Levin theorem [295; 296] in computer
science, generalized to the setting of quantum constraint satisfaction problems. Despite the
tremendous amount of progress in understanding the power of local Hamiltonians, many
important questions remain, such as whether the task remains hard under less-demanding
notions of approximation error [297?299] and whether there exist short classical descriptions
of ground states of local Hamiltonians (see e.g., Refs. [297; 300; 301]), among others.
One important question about LOCALHAMILTONIAN is the role played by the spectral
gap. The spectral gap is a traditionally important quantity in the context of ground-state
properties of any physical system and is defined as the difference between the smallest two
eigenvalues of the Hamiltonian. Many important families of Hamiltonians in physics have
the ?gap property?, meaning that the spectral gap in the limit of large system size n ? ?
is lower-bounded by a constant. Important conjectures in physics are concerned with the
existence of the gap property for certain Hamiltonians [302; 303], a problem that is known to
be undecidable in general [304]. Furthermore, the existence of a spectral gap implies various
tractability results for the ground states of Hamiltonians. For instance, in one dimension,
the gap property significantly restricts the entanglement structure of ground states through
the area law of entanglement, implying efficient classical representations of the same [305],
and further, classically efficient algorithms to compute the ground-state energy [306; 307].
164
It is not known whether these properties hold for higher dimensions.
Despite the physical importance of the spectral gap, its role in the context of the
LOCALHAMILTONIAN problem itself is much less clear. In particular, it is not known whether
LOCALHAMILTONIAN is QMA-complete in the presence of nontrivial lower bounds on the
spectral gaps, even when the lower bound is ?(1/poly(n)) [308; 309]. Meanwhile, if the
spectral gap is promised to be lower bounded by a constant, there are no-go results [310; 311]
that rule out any QMA-hardness proof that proceeds by generalizing the clock construction
technique. This technique underlies all known QMA-completeness results, in analogy with
the theory of NP-completeness, where the Cook-Levin theorem plays a foundational role.
Therefore, Hamiltonians with any nontrivial lower bounds on the spectral gap can be less
complex than the general case.
In this Chapter, we take an initial step towards answering the question of the role
played by the spectral gap in the LOCALHAMILTONIAN problem. To do so, we study QMA
in the precise setting, i.e. the class PreciseQMA. In the precise setting, the completeness
(the minimum probability of accepting a correct statement) and soundness (the maximum
probability of accepting an incorrect statement) of the protocol are separated by a quantity
called the promise gap that scales inverse-exponentially in the size of the input. For the
LOCALHAMILTONIAN problem, this translates to computing the ground-state energy to within
inverse-exponential precision in the system size.
Computing ground-state energies to inverse-exponential precision is not an artificial
task. This task corresponds to computing polynomially many digits of the answer, which
is very desirable in some cases [312]. Algorithms whose runtimes scale as polylog(1/?)
for additive error ? can compute quantities to inverse-exponential precision in polynomial
165
time, and such algorithms have been found for Hamiltonian simulation and linear systems
[40; 313; 314]. There are also situations where precise knowledge of the ground-state energy
of a Hamiltonian is essential. For example, in quantum chemistry, chemical reactivity rates
depend on the Born-Oppenheimer potential-energy surface for the nuclei. Each point on
this surface is an electronic ground-state energy for a particular arrangement of the nuclei.
Small uncertainties in the ground-state energy can exponentially influence the calculated rate
k via Arrhenius?s law k ? exp[???E], where ?E is an energy barrier and ? the inverse
temperature (see, e.g., Ref. [315]). Another example is in condensed-matter physics, where
algorithms such as the density matrix renormalization group (DMRG) routinely compute
several digits of the ground-state energy (see, e.g., Ref. [316]). Precise knowledge of the
ground-state energy can enable one to identify the locations of quantum phase transitions
by identifying non-analyticities [317]. Interestingly, the class of Hamiltonians for which the
energy can be precisely measured correspond to Hamiltonians that can be fast-forwarded
[318].
Fefferman and Lin [319] studied the complexity of the class PreciseQMA, and showed
the mysterious result that it equals PSPACE. This is surprising sinceQMA ? PP [320?322]
(also see Fig. B.1 in Appendix B for reference), and an alternative characterization of the class
PP is PreciseBQP, the precise analogue of BQP (the class of problems efficiently solvable
on quantum computers). Since PreciseBQP can handle inverse-exponentially small promise
gaps and contains QMA, one might have expected that adding the modifier Precise? to
QMA would not have changed the power of the class by much.
We provide an explanation for this seemingly unexpected boost in complexity from
166
QMA, which is a subset of PP, to PreciseQMA, which equals PSPACE1. Specifically, we
find that in order for the precise version of LOCALHAMILTONIAN, i.e. PRECISELOCALHAMIL-
TONIAN, to be PSPACE-hard, the spectral gap of the Hamiltonian must necessarily shrink
superpolynomially with the size of the system n (measured by the number of qudits in the
system). We give strong evidence that if the spectral gap shrinks no faster than a polynomial
in the system size, i.e. if the spectral gap is bounded by ?(1/poly), the complexity of the
problem is strictly less powerful. In particular, we show that this problem characterizes the
complexity class PP, which is a subset of PSPACE and is widely believed to be distinct from
PSPACE. If the problem were PSPACE-hard, the so-called counting hierarchy, defined as
CH = PP ? PPPP ? . . . [323], would collapse, which is considered an unlikely possibil-
ity. Our results therefore bring out the importance of the spectral gap, a quantity not well
understood so far in Hamiltonian complexity.
Another main result of ours concerns the existence of polynomial-size quantum circuits
to prepare ground states of local Hamiltonians. This is an important question that has
implications in circuit-complexity of ground states of natural Hamiltonians and is directly
related to whether natural Hamiltonians can be efficiently cooled down to zero temperature.
In complexity-theoretic language, the question may be phrased in terms of the power of
classical versus quantum witnesses in Merlin-Arthur proof systems, or more formally, the
so-called QMA vs. QCMA question. The (in)equivalence of these classes is an important
open question in quantum complexity theory and many-body physics, which has remained
unsettled despite recent progress in the oracle setting (see e.g., Refs. [300; 301]). The precise
version of QCMA, or PreciseQCMA, is known to be equal to NPPP (see e.g., Refs. [324;
1The class PSPACE contains PP, and is believed to be unequal to and much larger than PP: see Fig. B.1.
167
325]), indicating a separation between PreciseQCMA and PreciseQMA (= PSPACE) unless
the counting hierarchy collapses. Interestingly, we show strong equivalence results for the
PreciseQMA vs. PreciseQCMA question in the presence of spectral gaps.
Our results and the proof techniques we develop here also have consequences for other
areas of complexity theory and many-body physics. Our second main result mentioned ear-
lier roughly says that in the precise regime, the promise of an inverse-polynomial lower
bound on the spectral gap is equivalent to the promise that there exists a polynomial-size
circuit to prepare the ground state. We make an interesting conjecture in this regard in Sec-
tion 6.1.3, which could have a bearing on the performance of near-term quantum algorithms
for quantum chemistry and on the circuit complexity of various low-energy states, which
is an important question in gravitational and high-energy physics [326; 327]. Furthermore,
our results can shed light on an attempt to give a quantum-inspired reproof [328; 329] of the
celebrated IP = PSPACE result [330] via interactive protocols for the class PreciseQMA.
Our results also allow us to rule out sufficiently strong error-reduction techniques for the
class postQMA.
This Chapter is structured as follows. In the rest of Section 6.1, we state the main
results, give a high-level overview of the proof techniques and their implications, and dis-
cuss the relation of our results to other work in the literature. In Section 6.2, we give the
definitions of some other complexity classes and define some new classes that appear in this
Chapter. We also define natural problems complete for these classes. We then formally state
the results pertaining to the class PP in Section 6.3 and PSPACE in Section 6.4. We also
consider the complexity of related classes in Section 6.5, after which the Appendices have
detailed proofs of our claims.
168
6.1.1 Results
We describe a general problem we study here, called (?,?)-LOCALHAMILTONIAN. In-
formally, it is the problem of estimating the ground-state energy of a given k-local Hamilto-
nian acting on n qudits2 to additive error at most ?, when promised that the spectral gap is at
least ? (see precise definitions in Section 6.2). In the absence of any bound on the spectral
gap (i.e.? = 0), the problem (1/poly(n), 0)-LOCALHAMILTONIAN is, by definition, the same
as k-LOCALHAMILTONIAN, which is complete for QMA for k ? 2 [59; 331; 332]. Mean-
while, (1/ exp(n), 0)-LOCALHAMILTONIAN is, by definition, PRECISE-k-LOCALHAMILTONIAN
[319], which is complete for PreciseQMA. We henceforth suppress the dependence on the
number of qudits n in the notation exp and poly for the rest of the Chapter.
To our knowledge, Aharonov et al. [308] were the first to study the k-LOCALHAMILTONIAN
problem in the presence of a spectral gap. Specifically, they considered (1/poly, 1/poly)-
LOCALHAMILTONIAN and showed it to be complete for the class PGQMA (Polynomially
Gapped QMA). The definition of PGQMA, which is given in Section 6.2, depends on a
notion of a spectral gap for proof systems, distinct from that for Hamiltonians. For complex-
ity classes associated with proof systems such as QMA, QCMA and the variants we study
in this Chapter, the spectral gap corresponds to the gap in the highest and second-highest
accept probabilities of the optimal witness and the next-optimal orthogonal witness. A pri-
ori, the two notions of a spectral gap have no relation with each other. We show that the
two notions are equivalent for various cases (? and ? each behaving as 1/poly or 1/ exp),
by showing that (?,?)-LOCALHAMILTONIAN is complete for the appropriate spectral-gapped
2The results in this Chapter are applicable generally to qudits of any dimension d ? 2, but we will often
work with qubits in our proofs.
169
QMA class.
To understand the relation between the gapped QMA classes and the regular versions
without a spectral gap, we focus on the precise regime, so that ? = 1/ exp henceforth for
the rest of this section. By specifying the spectral gap to be ?(1/poly), we get the prob-
lem (1/ exp, 1/poly)-LOCALHAMILTONIAN. We show in Lemma 31 that this problem is
in a class we call PrecisePGQMA (Precise Polynomially Gapped QMA), which is the pre-
cise analogue of PGQMA. We also show (Lemma 32) that PrecisePGQMA ? PP, im-
plying that PrecisePGQMA is likely different from PreciseQMA, which equals PSPACE.
Specifically, assuming that PP =? PSPACE, there is a separation between PrecisePGQMA
and PreciseQMA. The PP upper bound on PrecisePGQMA is optimal: we show that
(1/ exp, 1/poly)-LOCALHAMILTONIAN is PP-hard (Lemma 34). Thus, we tightly character-
ize the complexity of the class by showingPrecisePGQMA = PP and prove that (1/ exp, 1/poly)-
LOCALHAMILTONIAN is its associated complete problem.
The results in the previous paragraph show that the PSPACE-hardness result of
Ref. [319] relies on the fact that the spectral gaps of the associated Hamiltonians can decay
rapidly with the system size. This raises the question of the maximum scaling of the spectral
gap required in order to retain PSPACE-hardness. This is an important question since if the
PSPACE-hardness results only apply when there is no promise whatsoever on the spectral
gap, it would indicate that PSPACE-hardness of PRECISE-k-LOCALHAMILTONIAN is artifi-
cial. We rule out this possibility by showing that if the spectral gap is bounded below by
1/ exp, i.e. if we consider the problem (1/ exp, 1/ exp)-LOCALHAMILTONIAN, the problem
remains PSPACE-hard. Specifically, we show in Theorem 23 that this problem is complete
for a class called PreciseEGQMA (Precise Exponentially Gapped QMA). Next, we show that
170
Spectral (?,?)-GS-DESCRIPTION-LOCALHAMILTONIAN (?,?)-LOCALHAMILTONIAN
gap (?)
? = 1/poly ? = 1/ exp ? = 1/poly ? = 1/ exp
1/poly PGQCMA {26} (=R QCMA {44}) PrecisePGQCMA {28} (= PP {30}) PGQMA PrecisePGQMA {22} (= PP {29})
1/ exp EGQCMA (=R QCMA {44}) PreciseEGQCMA {27} (= NPPP {45}) EGQMA(?) PreciseEGQMA {23} (= PSPACE {38})
0 QCMA {24} PreciseQCMA {25} (= NPPP) QMA PreciseQMA (= PSPACE)
Table 6.1: Complexity of variants of the LOCALHAMILTONIAN problem as a function of the
parameters ?, the promise gap, and?, the spectral gap. The problem is complete for the class
mentioned in each cell. For reference, we mention in curly brackets the theorem number
corresponding to the results proved in this Chapter. The question mark corresponding to
the entry EGQMA indicates that the result is a conjecture and the notation =R denotes
equivalence under randomized reductions (defined in Section 6.5.3).
PreciseEGQMA equals PSPACE (Theorem 38), implying that instances with ?(1/ exp)
spectral gaps are no less complex than the general case.
Lastly, we consider the analogues of these classes when the witness is classical, which
gives us the classesQCMA (QuantumClassicalMerlin Arthur), PreciseQCMA, PrecisePGQCMA
(Precise Polynomially GappedQCMA) and PreciseEGQCMA (Precise Exponentially Gapped
QCMA). The complete problems for these classes are the appropriate versions of the LOCAL-
HAMILTONIAN problem under the additional promise that there is an efficient classical de-
scription of a circuit to prepare a low-energy state, as we show in Theorems 24 to 27. We de-
fine this problem in Section 6.2.1 and denote it (?,?)-GS-DESCRIPTION-LOCALHAMILTONIAN,
which is the problem of computing the ground-state energy to additive error ?, given the
promise that there exists a polynomial-size circuit to prepare a low-energy state and promised
that the spectral gap of the Hamiltonian is at least ?. As stated in Corollary 37, we show
that PrecisePGQCMA has the same complexity as PrecisePGQMA, implying that in the
precise setting, once there is a ?(1/poly) promise on the spectral gap, a further promise
that there exists an efficient circuit to prepare a low-energy state is redundant. We comment
more on this result in Section 6.1.3.
171
In Table 6.1, we give an overview of the parameter dependence of the complexity of
two main problems studied in this Chapter, namely (?,?)-LOCALHAMILTONIAN and (?,?)-
GS-DESCRIPTION-LOCALHAMILTONIAN. The problems are completely characterized by the
appropriately gapped versions of QMA or QCMA, or their precise variants. The complexity
class in any cell in the table is a subset of all the classes below it in the same column, since these
classes correspond to weaker promises on the spectral gap. Similarly, the complexity class
associated with (?,?)-GS-DESCRIPTION-LOCALHAMILTONIAN is a subset of that associated
with (?,?)-LOCALHAMILTONIAN, because the former problem is associated with an extra
promise. While we have given evidence that PrecisePGQMA ?= PreciseQMA, it is unknown
whether the same holds for the question PGQMA ?= QMA. Similarly, while we have proved
PreciseEGQMA = PreciseQMA, it would be interesting to see if a similar result holds for
EGQMA.
6.1.2 Techniques
Here, we give an overview of the primary techniques used in proving our results.
Imaginary-time evolution and the powermethod.?To show the containmentPrecisePGQMA ?
PP, we use a technique called the ?power method? [333]. The broad idea behind the al-
gorithm is that if a matrix A is promised to have a spectral gap between the largest two
eigenvalues, the behavior of Ad( fo)r large d is dominated by the largest eigenvalue. We give a
PP algorithm to compute Tr Ad for an exponentially large matrix A and d = poly(n) for a
wide class of matrices A. This wide class includes sparse matrices and matrices representing
local observables as special cases. The PP algorithm uses the Feynman sum-over-paths idea
172
[334] to express the trace as a sum over 2poly many terms, each of which is a product over
quantities of the form ?x|R |y? for some matrix R whose entries are efficiently computable.
A PP algorithm can decide whether the sum over 2poly many terms, each term computable
in polynomial time, is above or below a threshold.
The power method is closely related to another technique called the ?cooling algo-
rithm?, inspired by a brief discussion by Schuch et al. [178]. The idea is that letting a sys-
tem evolve in imaginary time can produce an unnormalized state close to the ground state.
Imaginary-time evolution is a linear, albeit nonunitary, operation and produces an unnor-
malized state ?? in general. Schuch et al. relied on a quantum characterization of PP, namely
postBQP. The class postBQP [335] is the class of problems solvable in polynomial time on
a quantum computer with access to the resource of postselection, which is the ability to con-
dition on exponentially unlikely events. Aaronson [335] showed that any linear operation,
even nonunitary ones, may be simulated in postBQP. Schuch et al.?s algorithm [178] works
by decomposing the imaginary-time evolution operation exp[??H] into a series of local op-
erations exp[??Hi] using Trotterization, and implementing each local operation using the
resource of postselection. Unfortunately, the state-of-the-art error bounds for Trotterization
of imaginary-time evolution [48] give, at best, a multiplicative error that is exponential in n
(see also Refs. [44; 171]), and hence this technique does not work in the precise regime. Us-
ing a modification of this idea, we prove a more general statement about precise computation
of ground-state local observables for Hamiltonians with a spectral gap. Specifically, we use
the idea of imaginary time evolution and give a PPP algorithm that provably works not just
for 1/poly precision, but also 1/ exp precision in computing local observables in addition to
the Hamiltonian. Our technique is closely related to the power method, since the core of
173
the algorithm is to compute expectation values of powers of the Hamiltonian.
Small-penalty clock construction.? Our second major technical contribution is a
modification of the clock construction that we call the small-penalty clock construction.
One of the ways this technique is useful is as follows. As mentioned earlier and as will
be described in detail in Section 6.2, it is possible to consider spectral-gapped versions of
both the LOCALHAMILTONIAN problem and the class QMA and their variants. We have
already discussed the (natural) notion of a spectral gap for Hamiltonians. For QMA and
related classes, the spectral gap is related to the difference in accept probabilities between the
optimal and next-optimal witness. Our technique allows us to bridge the notion of spectral
gap in both cases by constructing spectral-gap-preserving reductions. In other words, the
small-penalty clock construction allows us to prove that the Hamiltonians resulting from
the construction inherit a spectral gap related to the gap in accept probabilities in the circuit,
for several variants of QMA. This ability is used in the proofs of Theorems 23 to 27. An
interesting feature of the modified clock construction is that it also allows us to show that,
when there is a classical witness (i.e. a QCMA computation), the resulting Hamiltonian has a
classical description for a state with energy close to the ground-state energy. Another related
application of the small-penalty clock construction is that it also allows us to show complexity
lower bounds like in Lemmas 34 and 36. In these cases, we directly reduce from PP to the
appropriate gapped version of the LOCALHAMILTONIAN problem instead of a reduction from
the corresponding ?QMA class.
We now spell out what enables the small-penalty clock construction to show the above
results. As mentioned before, the clock construction and its variants encompass all current
proofs of hardness for QMA and related classes. Typically, this consists of mapping a circuit
174
to a Hamiltonian H = Hinput + Hprop + Hclock + Houtput. Roughly speaking, each term
locally enforces that the computation is a valid step of a QMA protocol by adding energy
penalties to undesirable states. The ?witness register?, where a quantum prover may input
any quantum state, is left unpenalized and the Hamiltonian therefore has no terms acting on
the witness register. The role of Houtput is to ensure that witnesses and computations that
lead to a low accept probability at the output get a high energy penalty. In the absence of the
penalty term at the output, the ground-state space of the Hamiltonian is well-known and
is given by the subspace of the so-called ?history states?, each with the same energy. The
output penalty term Houtput is what breaks the degeneracy and helps create a promise gap,
and we will henceforth refer to this as simply the penalty term without qualification.
However, the addition of the penalty term makes the eigenstates of the Hamiltonian
difficult to analyze, since the magnitude of the penalty can be large, i.e. ?(1) in strength. In
this Chapter, we often choose the output penalty terms to have small strength. This might
seem like a strange choice to make since one is typically interested in making the promise
gap as large as possible. However, since we are dealing with instances where the promise gap
is already exponentially small, our choice is not too costly. The advantage this gives us is
that the ground-state energy tracks the effect of the output penalty more faithfully. More
concretely, the smallness of the penalty term allows us to use tools like the Schrieffer-Wolff
transformation [336; 337], which can be viewed as a rigorous formulation of degenerate
perturbation theory. We review the Schrieffer-Wolff transformation in Section 6.6.
Spectral gap in adjacency matrix.? For the proof of Theorem 38, we show a reduc-
tion3 from a natural PSPACE-complete graph problem to an instance of a problem known as
3This reduction is inspired by unpublished work by one of us and Cedric Lin [338], and we supplement it
175
(1/ exp, 1/ exp)-SPARSEHAMILTONIAN4. This problem is a generalization of (1/ exp, 1/ exp)-
k-LOCALHAMILTONIAN, allowing for the Hamiltonian to be any sparse Hamiltonian with a
spectral gap? 1/ exp. Sparse Hamiltonians are Hermitian matrices that can be exponentially
large, with at most poly(n) nonzero entries per row in some basis and an efficient algorithm
for computing any entry of the matrix. They are a generalization of local Hamiltonians.
The PSPACE-complete graph problem may be described as SUCCINCTGRAPHREACH-
ABILITY, which is to decide if there is a path from one vertex to another in a succinctly
described graph of exponential size (also see Ref. [318]). We show that one can always con-
struct a PSPACE-bounded Turing machine such that the resulting Hamiltonian after the
reduction always has a spectral gap that is at least 1/ exp(n). We do this through an explicit
analysis of the eigenvalues of the Hamiltonian, which are related to the lengths of cycles and
paths of the graph constructed from the Turing machine. Next, we give a PreciseEGQMA
upper bound to (1/ exp, 1/ exp)-SPARSEHAMILTONIAN, i.e. the problem in the presence of a
spectral gap, establishing that PSPACE ? PreciseEGQMA.
6.1.3 Discussion
Our first main result was that the addition of even an inverse-polynomially small
spectral gap takes the complexity of precisely estimating the ground-state energy of a lo-
cal Hamiltonian from PreciseQMA = PSPACE to PrecisePGQMA = PP. Note that this
result also implies a difference between the case of no spectral gap and a constant spectral
gap. Therefore, we have given a provable setting where the difference in complexity between
with a technique to create spectral gaps.
4We actually show a reduction to the complement of the problem (where YES and NO instances are re-
versed), but this turns out not to matter because PSPACE is closed under complement.
176
two problems is attributable entirely to the spectral gap.
Our second main result concerned a modification of the same problem of precisely
estimating the ground-state energy of a local Hamiltonian promised to have an inverse-
polynomial spectral gap. When additionally promised that there exists a classical description
of a circuit to prepare a state whose energy is exponentially close to the ground-state energy,
our results show that the complexity of the problem does not get weaker. Specifically, we
show that the class PrecisePGQCMA is equivalent to PrecisePGQMA.
The above equivalence result is in sharp contrast with the belief PreciseQCMA ?=
PreciseQMA in the non-spectral-gapped case. This inequality follows from the conjecture
that NPPP ?= PSPACE, which, if false, would lead to a collapse of the counting hierar-
chy. The inequality PreciseQCMA ?= PreciseQMA rules out the possibility of there being
polynomial-size circuits to prepare ground states of local Hamiltonians to exponential pre-
cision, since otherwise the prover could simply supply a description of such a circuit. Our
equivalence result that PrecisePGQMA = PrecisePGQCMA is consistent with the following
intriguing conjecture about the circuit-complexity of ground states of low-energy Hamilto-
nians, although it does not imply the conjecture.
Co?ecture 21. Consider any Hamiltonian H on n qubits with ground-state energy E1 and
a 1/poly spectral gap. Then there exists a low-energy state |?? satisfying ??|H |?? ? E1 +
2?poly(n) that can be prepared by an efficient quantum circuit, namely a circuit of the form |?? =
U |0?m, where m and the size of U are both polynomials in n.
Note that Conjecture 21 implies both PrecisePGQMA = PrecisePGQCMA = PP
and PGQMA = PGQCMA(=R QCMA); however the reverse direction need not hold. Our
177
result does not imply Conjecture 21 because the reduction does not imply anything about
how the PrecisePGQCMA verification protocol looks like. We also note that the quantum
circuits referred to in Conjecture 21 may be hard to find?the conjecture is only concerned
with the existence of such circuits, and not with whether these circuits can be obtained by
an efficient algorithm. In complexity-theoretic language, these circuits may be nonuniform.
This is why Conjecture 21 is not in contradiction with Ref. [339], which argues that finding
efficient matrix-product-state representations of Hamiltonians with a?(1/poly) spectral gap
can be hard.
If Conjecture 21 were true, it would also explain the observed success of quantum
algorithms such as the variational quantum eigensolver (VQE) [64; 340], which seek to
solve a much simpler problem of preparing low-energy states of translation-invariant many-
body Hamiltonians with energy 1/poly-close to the ground-state energy. A large class of
translation-invariant Hamiltonians have a spectral gap that is either a constant,?(1) (gapped
phases), or vanishing in the system size as?(1/n1/D) (gapless phases described by conformal
field theories in D-dimensions). Therefore, Conjecture 21 applies to both these cases and
would imply the existence of polynomial-size circuits to prepare states with high overlap with
the ground state. Such circuits are generally found in the VQE algorithm if one optimizes
over sufficiently many parameters. This behavior is in line with other instances where a
lower bound on the spectral gap implies tractability of the ground state in various senses
[305; 307; 341; 342].
Coming to the case of exponentially small spectral gaps, we have shown thatPreciseEGQMA =
PreciseQMA. This implies that PreciseEGQMA ?= PreciseEGQCMA unless the counting
hierarchy collapses. Therefore, we have given a class of local Hamiltonians (in the proof of
178
Lemma 46) with exponentially small spectral gaps, whose ground states have exponentially
large circuit complexity. This is a result of independent interest, and it might be interesting
to study the whether these Hamiltonians can be classified as quantum spin glasses, which
are believed to be hard to cool down to zero temperature [343].
In another intriguing line of work, Aharonov and Green [328] and Green, Kindler, and
Liu [329] have given interactive protocols for precise quantum complexity classes with a com-
putationally bounded prover P and a computationally bounded verifier V , denoted IP[P ,V ].
A goal of this line of work is to give a quantum-inspired proof of the result IP = PSPACE
[330] by giving an interactive protocol for PreciseQMA [329] (which equals PSPACE) with
a BPP verifier. This has been successful so far with PreciseBQP and PreciseQCMA (which
equals NPPP) but not yet with PreciseQMA. From the result of Ref. [328] and our result
that PrecisePGQMA = PP, there is an IP[PreciseBQP,BPP] protocol for PrecisePGQMA.
Our results indicate that the spectral gap might play an important role in extending such an
interactive protocol to PSPACE. Namely, such an extension would need to be able to work
with inverse-exponentially small spectral gaps.
In addition, the class postQMA [325; 344] is the class where there is a quantum prover
and a postBQP verifier, where one may condition (postselect) on exponentially unlikely
outcomes. This class has been shown to be equal to PreciseQMA [325], so an alternative
approach mentioned by Green et al. [329] to reprove the result IP = PSPACE is to exhibit
an IP[postQMA,BPP] protocol for postQMA. To complete such a proof, it would suffice to
prove a witness-preserving amplification technique like in QMA [322; 345] that additionally
handles postselection. Witness-preserving amplification is a technique for improving the
promise gap of an interactive protocol by modifying the verifier?s strategy while keeping
179
the witness fixed. We show in Lemma 41 that, assuming PP ?= PSPACE, the soundness
of a postQMA protocol cannot be reduced beyond a particular point without requiring the
witness to grow larger or requiring the postselection success probability to shrink. Therefore,
we obtain evidence that a witness-preserving amplification technique for postQMA should
differ significantly from the technique of Marriott and Watrous [322], since in the latter,
repeating the verifier?s circuit suffices to get any soundness parameter s ? 2?poly.
So far, we have considered the spectral-gap promise to be applicable to both YES and
NO instances of the problems defined. We can also define asymmetric problems where only
the YES instances are promised to have a spectral gap. The motivation for considering such
asymmetric promises is that they are related to complexity classes where the accepting witness
is promised to be unique, such as the class UQMA [308]. The problems with asymmetric
promises can only be harder than their symmetric analogues, since the promise is weaker. We
show that for both ?(1/poly) and ?(1/ exp) spectral gaps in the precise setting, there is no
difference between symmetric and asymmetric promises on the spectral gaps. Specifically, we
show in Theorem 42 that the classes with asymmetric promises are of the same complexity
as those with symmetric promises.
We remark here that the promise of a spectral gap above a unique ground state is dis-
tinct from assuming that we have a UQMA instance. The reason is that for LOCALHAMILTO-
NIAN, the presence of a spectral gap does not imply that there is a unique accepting witness,
it only implies a unique ground state. In case the ground-state subspace is polynomially
degenerate, the PP algorithm continues to work to produce estimates of the ground-state
energy.
Lastly, we add that results shown in the precise regime do not always imply analogous
180
results in the non-precise regime. For example, our work gives evidence thatPrecisePGQCMA ?=
PreciseQCMA, but in the non-precise regime we can show PGQCMA =R QCMA. In this
respect, inequivalence results in the high-precision regime resemble oracle separation re-
sults in complexity theory, which is a mature area of research with several important results
[346?348]. While oracle separations do not constitute strong evidence for the inequiva-
lence of two complexity classes, they are useful in ruling out proof techniques that work
relative to oracles, or ?relativize?. Similarly, inequivalence results in the precise regime can
rule out proof techniques from extending to the precise regime. For example, a purported
proof that QCMA = QMA must not work in the precise regime, otherwise we would obtain
PreciseQCMA = PreciseQMA, or PSPACE = PP, which is believed to be unlikely.
6.1.4 Related work
The study of Hamiltonian complexity [60?63; 294; 331; 332; 349; 350] has given rise
to many techniques and important results applicable in quantum many-body physics, such
as [304; 339; 351?357]. The clock construction has also been analyzed in detail recently
[358?360].
The study of exponentially small promise gaps in the context of quantum classes can
be traced to Watrous [361], who defined PQP and showed its equivalence with postBQP,
which equals PP [335]. In the precise setting, one can sometimes give far stronger evidence
for the (in)equivalence of complexity classes than in the analogous bounded error setting,
as is the case for precise versions of the questions of QCMA vs. QMA [319] and QMA(2)
vs. QMA [319; 362?364]. There has been work on quantum interactive proof systems with
181
exponentially small promise gaps, such as in the context of QMA(2) [364], or with even
smaller gaps, such as in Refs. [365?367]. Fefferman and Lin [319; 338] studied the precise
regime of QMA, showing it to equal PSPACE, leading to other works concerning precise
classes [325; 368]. Gharibian et al. [324] considered quantum generalizations of the poly-
nomial hierarchy, where precise classes and spectral gaps are relevant to the definitions and
proof techniques.
Aharonov et al. [308] were the first to consider the complexity of the LOCALHAMILTO-
NIAN problem in the presence of spectral gaps, motivated by the question of uniqueness [369]
for randomized and quantum classes. They showed the equivalence of UQCMA and QCMA,
and that of UQMA and PGQMA, using similar techniques as Valiant and Vazirani [369] in
their proof of equivalence of UNP and NP. Jain et al. [309] defined the class FewQMA
and showed that it is contained in PUQMA, giving a technique to reduce the dimension of
accepting witnesses.
More recently, Gonz?lez-Guill?n and Cubitt [310] studied the spectral gap of a large
class of Hamiltonians that encode history states in their ground state and showed that the
spectral gap is upper bounded by O(1/poly). A similar result was obtained by Crosson
and Bowen [311] using different techniques. These works are mainly concerned with the
existence of a ?(1) spectral gap, whereas our results distinguish between 1/poly and 1/ exp
spectral gaps.
Finally, Ambainis [370] studied the problem of estimating spectral gaps and local
observables and gave a PQMA[log] upper bound for these problems, while also giving PQMA[log]-
hardness results (also see Ref. [371]). The class PQMA[log] is the class of problems solvable
in polynomial time by making logarithmically many (adaptive) queries to a QMA oracle.
182
Gharibian and Yirka [371] showed that PQMA[log] ? PP and extended previous hardness
results to more natural Hamiltonians. Gharibian, Piddock, and Yirka [372] also gave a very
natural complete problem for the class PQMA[log] in the context of computing local observables
in ground states. Novo et al. [373] have recently studied the closely-related problem of
sampling from the distribution obtained by making energy measurements and obtain various
interesting hardness results, under different notions of error.
6.2 Definitions and complete problems
We have seen the definition of BQP in terms of the class BQP[c, s] with general
parameters c and s. The Precise- version of BQP can be defined similarly.
Definition 2. PreciseBQP = ?c?s?1/ expBQP[c, s].
This class is known to be equal to PP (see, e.g., Ref. [324]).
We now give an equivalent definition ofQMA in terms of the eigenvalues of an operator
called the accept operator. We will then define a very general class called Gapped QMA,
GQMA[c, s, g1, g2], which has several parameters. By specifying these parameters, we can
define the major complexity classes in this Chapter. The complexity classes corresponding
to classical witnesses (QCMA and its derivatives) are defined analogously.
The alternative definition of QMA is in terms of the ?accept operator? Q(Ux) =
?0|?m U ? ?mx?outUx |0? on the witness register, where ?out is the projector on to the ac-
cept state (|1?o). For any state |?? provided as a witness, the quantity ??|Qx |?? is the
accept probability of the circuit. We will henceforth suppress the dependence of Q on the
unitary Ux and the instance x. The eigenvalues of Q, ?1(Q) ? ?2(Q) ? . . . are important
183
quantities to consider since the accept probability of any input proof state is a convex com-
bination of these eigenvalues. The alternative definition of QMA in terms of the operator Q
is as follows:
Definition 3 (Alternative definition of QMA[c, s]). A = (Ayes, Ano) is a QMA[c, s] problem
iff for every instance x there exists a uniformly generated circuit Ux of size poly(n) acting
on m+ w = poly(n) qubits, with the property that
If x ? Ayes: ?1(Q) ? c
If x ? Ano: ?1(Q) ? s,
where Q = Q(Ux) is as above.
Note that we are typically interested in the behavior of the maximum accept probability,
which equals the largest eigenvalue of Q. We are also interested in the lowest eigenvalue
of a Hamiltonian H for the LOCALHAMILTONIAN problem and its variants. Therefore, we
order eigenvalues in nonincreasing order for accept operators and in nondecreasing order for
Hamiltonians. For the same reason, we define the spectral gap differently for accept operators
and Hamiltonians. For a Hamiltonian, we define the spectral gap to be the difference in the
smallest two eigenvalues E2 ? E1. For accept operators, the spectral gap is the difference
between the highest two eigenvalues ?1(Q) ? ?2(Q). This is equal to the difference in the
accept probabilities of the optimal witness and the next-optimal witness orthogonal to it. It
will usually be clear from context which spectral gap we are referring to.
Now let us define the class GQMA[c, s, g1, g2]. It corresponds to a promise on the
operator Q having a spectral gap of at least g1 in the YES case, and at least g2 in the NO
case:
Definition 4 (Gapped QMA). GQMA[c, s, g1, g2] is the class of promise problems A =
184
(Ayes, Ano) such that for every instance x, there exists a polynomial size verifier circuit Ux
acting on poly(n) qubits and its associated accept operator Q such that
If x ? Ayes: ?1(Q) ? c and ?1(Q)? ?2(Q) ? g1
If x ? Ano: ?1(Q) ? s and ?1(Q)? ?2(Q) ? g2.
This definition is a generalization of the class PGQMA (Polynomially Gapped QMA) defined
by Aharonov et al. in Ref. [308]:
Definition 5. PGQMA=?c?s,g1,g2?1/polyGQMA[c, s, g1, g2].
To see the relation of this class with QMA, notice that by setting g1 = g2 = 0, the
promise on spectral gaps becomes vacuous, since ?1(Q) ? ?2(Q) by definition. Therefore,
we get the equality GQMA[c, s, 0, 0] = QMA[c, s]. We also define
Definition 6 (Exponentially Gapped QMA). EGQMA = ?c?s?1/polyGQMA[c, s, g1, g2].
g1,g2?1/ exp
We now come to precise versions of these classes, where the completeness?soundness
gap c ? s can be exponentially small, giving us more powerful classes. The first of these is
PreciseQMA, which was defined in Ref. [319] and shown to be equal to PSPACE.
Definition 7. PreciseQMA = ?c?s?1/ expQMA[c, s].
This definition should be compared to the precise version of GQMA, which comes
in two varieties: the spectral gaps can either be polynomially small (PrecisePGQMA) or
exponentially small (PreciseEGQMA).
Definition 8 (PrecisePGQMA). PrecisePGQMA, short for Precise Polynomially Gapped
QMA, is the class with exponentially small promise gaps and polynomially small spectral
gaps:
PrecisePGQMA = ? c?s?1/ exp GQMA[c, s, g1, g2].
g1,g2?1/poly
185
Definition 9 (PreciseEGQMA). PreciseEGQMA, short for Precise Exponentially Gapped
QMA, has both the promise gap and spectral gap exponentially small:
PreciseEGQMA = ? c?s?1/ exp GQMA[c, s, g1, g2].
g1,g2?1/ exp
We now come to complexity classes in which the prover sends a classical witness but
the verifier remains quantum. The classicality of the witness can be enforced by measur-
ing the qubits sent by the prover in the computational basis and interpreting qubits in the
computational basis as classical bits. If the verifier is only allowed to make measurements at
the end, we use the standard protocol for deferring measurements: we apply a ?copy opera-
tion? Uc that has CNOTs from the qubits in the witness register to an ancilla register in the
state |0?w. We leave the qubits in the witness state unmeasured. This modified circuit has
the property that it preserves the accept probabilities of input witness states that are in the
computational basis. Further, the eigenstates of the modified accept operator acting on the
register can be taken to be computational basis states. This allows us to define QCMA and
its derivatives in terms of the accept operator and also allows us to consider a gapped version
of QCMA:
Definition 10 (GQCMA[c, s, g1, g2]). A = (Ayes, Ano) is a GQCMA[c, s] problem iff for
every instance x there exists a uniformly generated circuit Ux of size poly(n) acting on
m+ w = poly(n) qubits, with the property that
If x ? Ayes: ?1(Q) ? c and ?1(Q)? ?2(Q) ? g1
If x ? Ano: ?1(Q) ? s, and ?1(Q)? ?2(Q) ? g2,
where Q = Q(UxUc) is the accept operator of the modified circuit with the copy operation
Uc described above.
Definition 11. The derived classes of GQCMA are given by
186
? QCMA[c, s] = GQCMA[c, s, 0, 0].
? QCMA = ?c?s>1/polyQCMA[c, s].
? PreciseQCMA = ?c?s>1/ expQCMA[c, s].
? Polynomially Gapped QCMA: PGQCMA = ? c?s>1/poly GQCMA[c, s, g1, g2].
g1,g2>1/poly
? Precise Polynomially Gapped QCMA:
PrecisePGQCMA = ? c?s>1/ exp GQCMA[c, s, g1, g2].
g1,g2>1/poly
? Exponentially Gapped QCMA: EGQCMA = ?c?s>1/polyGQCMA[c, s, g1, g2].
g1,g2>1/ exp
? Precise Exponentially Gapped QCMA:
PreciseEGQCMA = ? c?s>1/ exp GQCMA[c, s, g1, g2].
g1,g2>1/ exp
6.2.1 Complete problems
We now come to the definitions of problems that are complete for these classes. The
classic problem complete for the class QMA is the LOCALHAMILTONIAN problem [59; 331;
332]. We define a k-local observable to be a Hermitian operat?or A that can be written as a
sum over operators Ai supported on k qudits at most: A = poly(n)i Ai. We assume that
each term has bounded operator norm ?Ai? ? poly(n). The task in the LOCALHAMILTONIAN
problem is to estimate the ground-state energy of a local Hamiltonian. The decision version
of the problem is as follows:
187
k-LOCALHAMILTONIAN[a, b]
?
Input A description of a k-local Hamiltonian H = i hi on n qubits with hi ? 0,
two numbers a and b with b > a.
Output YES if the ground-state energy E1 ? a,
NO if E1 ? b, promised that one of them is the case.
Henceforth we omit the phrase ?promised that one of them is the case? because we will
be exclusively considering promise problems unless otherwise specified. Kitaev [59] showed
that 5-LOCALHAMILTONIAN[a, b] with b ? a = ?(1/poly) is QMA-complete, which was
improved to k = 3 and then k = 2 in Refs. [331; 332]. The parameter ? := b ? a, the
promise gap, is a measure of the accuracy to which the solution is desired. We define the
problem in terms of ? only, as follows:
Definition 12. ?-k-LOCALHAMILTONIAN := ?b?a?? k-LOCALHAMILTONIAN[a, b].
We now come to the gapped and precise versions of the problem, which turn out to be
complete for their respective ?QMA variants. We also suppress the notation k in the name
of the problem, though there is formally a dependence on k. In this Chapter, our hardness
results hold for k ? 3 and it may be possible to improve our results to hold for k = 2.
188
LOCALHAMILTONIAN[a, b, g1, g2]
?
Input Description of a k-local Hamiltonian H = i hi with hi ? 0, numbers a, b,
g1, and g2 with b > a.
Output YES if the ground-state energy E1 ? a and any state orthogonal to the ground
state has energy ? E1 + g1,
NO ifE1 ? b and any state orthogonal to the ground state has energy? E1+g2.
In both the YES and NO cases above, we see that the Hamiltonian has a unique ground state
and a spectral gap of at least g1 in the YES case and g2 in the NO case. The above problem
with promise gap ? = b? a and spectral gap ? = min[g1, g2] is defined to be:
Definition 13. (?,?)-LOCALHAMILTONIAN := ? b?a?? LOCALHAMILTONIAN[a, b, g1, g2].
g1,g2>?
In the non-precise regime, the problem (1/poly, 1/poly)-LOCALHAMILTONIAN was shown
to be complete for PGQMA for k ? 2 [308].
We now focus on the precise regime, i.e. ? = ?(1/ exp). From the results of Ref. [319],
we know that (1/ exp, 0)-LOCALHAMILTONIAN is PreciseQMA-complete for k ? 3. We
show that:
Theorem 22. (1/ exp, 1/poly)-LOCALHAMILTONIAN is PrecisePGQMA-complete.
Theorem 23. (1/ exp, 1/ exp)-LOCALHAMILTONIAN is PreciseEGQMA-complete.
By virtue of these theorems, we can talk about the complexity of the classes PrecisePGQMA
and PreciseEGQMA interchangeably with their complete problems. The proofs of these
theorems are given in Sections 6.7 and 6.8. The hardness results rely on the small-penalty
clock construction, where the size of the penalty term is either?(1/poly) or?(1/ exp). The
189
upper bounds are shown in Lemmas 52 and 53 and rely on a modification of the standard
phase-estimation protocol used to show k-LOCALHAMILTONIAN is in QMA. Specifically, we
consider the modified protocol of Ref. [319] used for PRECISE-k-LOCALHAMILTONIAN and
observe that the spectral gaps in the energies translate to separations in the accept probabil-
ities.
Finally, we turn to complete problems for QCMA and its derivatives. The first prob-
lem, GS-DESCRIPTION-LOCALHAMILTONIAN, concerns finding the ground-state energy of a
k-local Hamiltonian when there is a polynomial-size circuit to prepare a state close to the
ground state (which constitutes a classical description of the ground state).
GS-DESCRIPTION-LOCALHAMILTONIAN[a, b, g1, g2]
?
Input Description of a k-local Hamiltonian H = i hi, numbers a, b ? a + ?,
polynomials T (n),m(n), together with the promise that there exists a circuit
V of size T such that V |0m? = |?? satisfies ??|H |?? ? E1 + ?3/f(n)2 for
some polynomial f(n) ? ?H?.
Output YES if the ground-state energy of H satisfies E1 ? a and the spectral gap of H
is at least g1,
NO if E1 ? b and the spectral gap of H is at least g2.
Definition 14. (?,?)-GS-DESCRIPTION-LOCALHAMILTONIAN := ?b?a??,g1,g2?? GS-DESCRIPTION-
LOCALHAMILTONIAN[a, b, g1, g2]
As in the case of (?,?)-LOCALHAMILTONIAN, if we take ? = 0, we get a version
without any promise on the spectral gap. This is a close relative of the following problem
190
proved to be QCMA-complete for ? = ?(1/poly) [374].
?-LOWCOMPLEXITY-LOWENERGYSTATES
?
Input Description of a k-local HamiltonianH = i hi, numbers a, b and polynomials
T (n), m(n), with b ? a+ ?.
Output YES if there exists a circuit of size ? T (n) that acts on |0m? to prepare a state
|?? with energy ??|H |?? ? a,
NO if any state |?? obtained by applying a circuit of size T (n) on |0m? has
energy ??|H |?? ? b.
This latter problem has a weaker promise than (?, 0)-GS-DESCRIPTION-LOCALHAMILTONIAN.
This is because a NO instance of ?-LOWCOMPLEXITY-LOWENERGYSTATES is automatically a
NO instance of (?, 0)-GS-DESCRIPTION-LOCALHAMILTONIAN, since any state necessarily has
energy ? b. Meanwhile, a NO instance of (?, 0)-GS-DESCRIPTION-LOCALHAMILTONIAN
need not be a NO instance of ?-LOWCOMPLEXITY-LOWENERGYSTATES, since for the latter
there is no guarantee of a circuit to prepare a state with energy close to the ground-state
energy.
Despite having a stronger promise on (?, 0)-GS-DESCRIPTION-LOCALHAMILTONIAN
(which only makes the problem less complex), our small-penalty clock construction allows
us to prove the same hardness result for both ? = 1/poly and ? = 1/ exp:
Theorem 24. (1/poly, 0)-GS-DESCRIPTION-LOCALHAMILTONIAN is QCMA-complete.
Theorem 25. (1/ exp, 0)-GS-DESCRIPTION-LOCALHAMILTONIAN is PreciseQCMA-complete.
For the latter theorem in the precise regime, we use the small-penalty clock construc-
tion with an exponentially small energy penalty. Lastly, when we add the promise of spectral
191
gaps, we have the following results:
Theorem 26. (1/poly, 1/poly)-GS-DESCRIPTION-LOCALHAMILTONIAN isPGQCMA-complete.
Theorem 27. (1/ exp, 1/ exp)-GS-DESCRIPTION-LOCALHAMILTONIAN is PreciseEGQCMA-
complete.
Theorem 28. (1/ exp, 1/poly)-GS-DESCRIPTION-LOCALHAMILTONIAN is PrecisePGQCMA-
complete.
The upper bounds in Theorems 24 to 28 follow from a precise version of phase esti-
mation, together with the promise that there is a classical description of a circuit to prepare
a low-energy state. The lower bounds either follow directly through a small-penalty clock
construction or through a reduction from a class that contains the relevant class.
6.3 Problems characterized by PP
In this section, we discuss the complexity of the classesPrecisePGQMA andPrecisePGQCMA,
both of which turn out to equal PP.
Theorem 29. PrecisePGQMA = PP.
Theorem 30. PrecisePGQCMA = PP.
We describe here the overall strategy for proving these results. First, we adapt the
one-bit phase estimation circuit in Ref. [319] to show that it is possible to compute ground-
state energies of sparse Hamiltonians with a spectral gap in the corresponding GQMA class.
In particular, we have
192
Lemma 31. (1/ exp, 1/poly)-LOCALHAMILTONIAN ? PrecisePGQMA.
Next, we use the ?power method? [333] to give a PP algorithm for any problem in
PrecisePGQMA.
Lemma 32 (One half of Theorem 29). PrecisePGQMA ? PP.
Proof. Suppose we have a GQMA[c, s, g1, g2] instance. Then we should give a PP algorithm
to precisely compute the maximum eigenvalue ?1 of the accept operator Q associated with
the instance, under the promise that the spectral gap of Q is bounded below by an inverse
polynomial. In particular, the spectral gap of the accept operator, given by ?1 ? ?2, is at
least min[g1, g2] =: ?. Consider the power method to compute the maximum eigenvalue
and eigenvector of a positive semidefinite operator Q. This method relies on the observation
that upon taking positive powers of the operator Q and estimating its trace, the quantity is
dominated by the maximum eigenvalue of Q. In the following, we suppress the dependence
of ?i on Q:
?
Tr(Qq) = ?qi (6.1)
i ( ( )q )
?2
= ?q1 1 + +( . . . ) (6.2)?1 q
? ?q + ?q(2w ?1 1 ? 1) 1? . (6.3)?1
On the other hand, we have Tr(Qq) ? ?q1. Therefore, in the YES case, we have
Tr(Qq) ? cq, (6.4)
193
while in the NO case,
( )q
Tr(Qq ? ?) sq + sq(2w ? 1)(1? (6.5)?1)q
? sq + sq ?(2w ? 1) 1? , (6.6)
s
where w is the size of the witness register and we assume s ? ? ? 1/poly, since otherwise
the promise cannot be satisfied. The difference in the two cases is
( )q
cq ? ?sq ? sq(2w ? 1) 1? [ ( )] (6.7)s
= cq ? sq ? sq(2w ? 1[) exp ]q log
?
1? (6.8)
s
? cq ? q q w q?((s ? s 2 e)xp ?q )s [ ] (6.9)
c? s
= sq 1 + ? 1 ? sq2w q?( [ ])exp ? (6.10)s s
? q q(c? s)s ? 2w( e[xp ?
q?
s ]s) . (6.11)
? q c? ss ? 2w exp ?q? . (6.12)
s s
? ( )?
If we pick q = s log c?sw+1 = O(poly), we can ensure that the difference in Tr(Qq)? 2 s
between the YES and NO cases is at least
c? s
sq = ?(2?poly). (6.13)
2s
This observation suggests that a PP algorithm can decide between the YES and NO
cases by computing Tr(Qq) for some large enough polynomial q. This is possible because a
194
PP algorithm can compute a sum of 2poly terms, where every term is efficiently computable
in polynomial time. We justify this more rigorously in Section 6.10 (Lemma 60).
The above result implies that, since (1/ exp, 1/poly)-LOCALHAMILTONIAN is inPrecisePGQMA,
a PP algorithm can precisely compute ground-state energies of local Hamiltonians with a
?(1/poly) spectral gap. A similar technique can also be used to show a slightly more general
result:
Lemma 33. Given a local Hamiltonian H and a local observable A, along with a promise that
?A? = O(poly) and the spectral gap ofH is lower-bounded by ?(1/poly), a PPP algorithm can
decide if the ground-state local observable ?E1|A|E1? is either? a or? b, for b?a = ?(2?poly),
where |E1? is the ground state of H .
This lemma is proved in Section 6.10. Note that both of these results include the case
? = ?(1), the important case of constant spectral gaps.
We complete the characterization of the power of PrecisePGQMA with the following
result.
Lemma 34. (1/ exp, 1/poly)-LOCALHAMILTONIAN is PP-hard.
For this proof, we use the small-penalty clock construction, albeit one for the class
PreciseBQP as opposed to the class PreciseQMA. In this aspect, it resembles the clock
construction of Aharonov et al. [351], where it was used to show BQP-universality of the
model of adiabatic quantum computing. We use the technique of applying ?(1/poly) small
penalties at the output so as to preserve the lower bound of ?(1/poly) on the spectral gap
shown in Ref. [351]. In sum, Lemmas 31, 32 and 34 together imply Theorems 22 and 29.
195
We now come to the class PreciseQCMA and its complete problem, (1/ exp, 0)-GS-
DESCRIPTION-LOCALHAMILTONIAN, where we are promised that there is an efficient circuit
to prepare a low-energy state. We know that PreciseQCMA = NPPP [325], which lies in
the second level of the counting hierarchy. Since PrecisePGQMA is characterized by PP,
the promise of having a spectral gap is only slightly stronger than the promise of an efficient
circuit to prepare the ground state.
Consider now the gapped version of the problem, (1/ exp, 1/poly)-GS-DESCRIPTION-
LOCALHAMILTONIAN, where there is a 1/poly spectral gap in addition to the promise of an
efficient circuit to prepare the ground state. This characterizes the class PrecisePGQCMA,
for which the proof technique is similar to PrecisePGQMA.
We first show that the gapped version of GS-DESCRIPTION-LOCALHAMILTONIAN is in
the corresponding GQCMA class, and in particular,
Lemma 35. (1/ exp, 1/poly)-GS-DESCRIPTION-LOCALHAMILTONIAN ? PrecisePGQCMA.
PP-hardness of the problem follows by the same argument as the proof of Lemma 34:
Lemma 36. (1/ exp, 1/poly)-GS-DESCRIPTION-LOCALHAMILTONIAN is PP-hard.
We give a unified proof of Lemmas 34 and 36 in Section 6.7. Since PrecisePGQCMA ?
PrecisePGQMA = PP, this implies:
Corollary 37. PrecisePGQMA = PrecisePGQCMA = PP.
6.4 Problems characterized by PSPACE
In this section, we discuss the complexity of the class PreciseEGQMA, which turns
out to equal PSPACE.
196
Theorem 38. PreciseEGQMA = PreciseQMA (= PSPACE).
Proof. The containmentPreciseEGQMA ? PreciseQMA follows trivially since anyPreciseEGQMA
instance is automatically aPreciseQMA instance. We show the other direction, PreciseEGQMA ?
PreciseQMA, in two steps. Our proof relies on the complexity of the following problem:
SPARSEHAMILTONIAN[a, b, g1, g2]
Input A succinct description of a Hermitian matrix of size 2poly(n) ? 2poly(n), with at
most d = poly(n) many entries in each row and two numbers a and b, with
b > a. The magnitude of each entry is bounded by k = poly(n).
Output YES if the smallest eigenvalue E1 ? a and the spectral gap of the matrix is at
least g1,
NO if E1 ? b, and the spectral gap of the matrix is at least g2.
We define (?,?)-SPARSEHAMILTONIAN to be ? b?a?? SPARSEHAMILTONIAN[a, b, g1, g2] and
g1,g2??
consider the problem with parameters ?,? = ?(1/ exp). First, in Lemma 39, we prove that
(1/ exp, 1/ exp)-SPARSEHAMILTONIAN is PSPACE-hard, or equivalently PreciseQMA-hard.
Next, we show in Lemma 40 that (1/ exp, 1/ exp)-SPARSEHAMILTONIAN may be solved in
PreciseEGQMA. The theorem then follows.
Lemma 39. (1/ exp, 1/ exp)-SPARSEHAMILTONIAN is PSPACE-hard.
The reduction is from any problem in PSPACE to an instance of co-(1/ exp, 1/ exp)-
GAPPED-
SPARSEHAMILTONIAN, which is the complement of the problem, in the sense that the YES
and NO instances are reversed. Since PSPACE is closed under complement, this still gives
197
YES case, original graph YES case, modified graph
NO case, original graph NO case, modified graph
Figure 6.1: Schematic of the original and modified graphs for both YES and NO cases.
The original graph in both YES and NO cases consists of vertices with in-degree and
out-degree at most 1, due to the fact that the Turing machine is reversible. The start vertex
sx is marked in blue, the accept vertex tx in green, and the reject vertex in orange. The
modified graphs have self-loops on all vertices except the start and the accept vertices. They
have additional vertices 1, 2, . . . t(n) without self-loops. All modifications are in maroon.
198
the desired hardness result. The broad idea is to represent a PSPACE computation as an
exponentially large, but sparse, graph. The smallest eigenvalue of the adjacency matrix of
this graph encodes information about whether the computation accepts or rejects.
Proof. We use a proof technique adapted from an unpublished manuscript by Fefferman and
Lin [338]. First, we use the fact that PSPACE with reversible operations in every step still
equals PSPACE: revPSPACE = PSPACE [375]. Indeed, it is known that SPACE[s(n)] =
revSPACE[s(n)] [376] with an overhead in time that is exponential in the space, s(n). Let
t(n) be this upper bound on the running time of the Turing machine, so that we can restrict
our attention to the class revSPACE[s(n)]?TIME[t(n)] = SPACE[s(n)]. Any computation
on a reversible Turing machine may be viewed as traversing a directed configuration graph,
where each vertex of the graph is determined by the state of the head and the list of symbols
on the input and work tapes (Fig. 6.1). When such a Turing machine is restricted to use
space polynomial in the input length n, the number of vertices in the graph is upper bounded
by an exponential, 2poly(n). Consider the adjacency matrix of the graph, Ax. The description
of this exponentially large matrix is succinct because it only requires specifying the input x
and the rules of the Turing machine.
We modify the configuration graph G ? G?x x so that the smallest eigenvalue of the
matrix A? ?x A?x is 0 in the NO case and bounded away by an exponentially small amount in the
YES case. We do this modification in a way that ensures the matrix has a spectral gap lower
bound of at least ?(1/ exp). This is done as follows. First, we modify the configuration
graph of the Turing machine by adding self-loops to all vertices except for the start and
accept configurations sx and tx. We then add a sequence of vertices {1, 2, . . . t(n)} from
199
the accept configuration tx, with the directed edges tx ? 1 ? 2 ? . . . ? t(n) ? sx, as
shown in Fig. 6.1. The adjacency matrix of this modified directed graph G? is A?x x, and we
are interested in the eigenvalues and spectral gap of ? ?Ax A?x, which is Hermitian and sparse,
and also has a succinct representation.
We now analyze this construction. The proof relies on an explicit computation of the
eigenvalues for the various subgraphs of the modified configuration graph. In the NO case,
the graph G?x has a path of vertices ending in the reject state (Fig. 6.1). This path contains
the starting configuration sx. Let ? be the graph distance between sx and the reject state.
Since we have added the edges tx ? 1 ? . . . t(n) ? sx, these vertices and the vertices
leading to the accept state are also part of the path (the Turing machine does not explore
these vertices in practice). All vertices in this path except for tx, sx, and i : i ? [t(n)]
have self-loops on them. As we show in Lemma 63, there is a zero eigenvalue in the NO
case, with a spectral gap above the zero eigenvalue. The spectral gap is lower bounded by
?(1/?2 ?polymax) = ?(2 ), where ?max is the number of vertices in the longest subgraph.
In the YES case, the subgraph containing the starting vertex is a cycle, with self-loops
on all vertices except for tx, sx, and the intermediate verti(ces i. In )each case, the eigenvalues
for any subgraph are given by 2 ? 2 cos (2k?1)? = 4 sin2 (2k?1)? , k ? [?] [338], where ?
2?+1 4?+2
is the number of vertices in the subgraph. The smallest eigenvalue is therefore given by the
longest subgraph and this eigenvalue is nondegenerate if no two subgraphs have the same
number of vertices. This is why we have added the sequence of edges tx ? 1 ? . . . t(n). The
role played by these vertices is to elongate the length of the subgraph containing the start and
accept configurations by t(n). This ensures that no other subgraph has a length equal to the
longest subgraph (since t(n) is the upper bound on the total number of vertices in the graph
200
( )
before elongation). Therefore, the smallest two eigenvalues are given by 4 sin2 (2k?1)? ,
4?+2
which are separated by ?(t(n)?2) = ?(2?poly).
To summarize, in the YES case we have E ? 2?poly1 and E ? E ?poly2 1 ? 2 . In the
NO case, we have E1 = 0 and E2 ? 2?poly. Therefore, we have a promise gap of 2?poly
and spectral gap 2?poly in both the YES and NO instances. Furthermore, the matrix A? ?x A?x
has entries of magnitude at most 2, and is 3-sparse because of the bounded degree of the
configuration graph. Since the minimum eigenvalue is small in the NO case and large in the
YES case, we have a reduction to co-(1/ exp, 1/ exp)-SPARSEHAMILTONIAN. Due to the fact
that PSPACE is closed under complement, we get PSPACE-hardness of (1/ exp, 1/ exp)-
SPARSEHAMILTONIAN.
Lemma 40. (1/ exp, 1/ exp)-SPARSEHAMILTONIAN ? PreciseEGQMA.
The proof of this is mostly the same as the proof of containment of (1/ exp, 1/ exp)-
LOCALHAMILTONIAN in PreciseEGQMA and is also given in Section 6.8. The only difference
is that we have a sparse Hamiltonian instead of a local Hamiltonian. This distinction turns
out not to matter, however, because of quantum algorithms for Hamiltonian evolution that
work well with sparse Hamiltonians [40].
6.5 Other related classes
In this section, we discuss implications of our proof techniques for other complexity
classes. The first concerns a technique for amplifying the promise gap in QMA and related
classes, called in-place amplification, due toMarriott andWatrous [322]. The second is about
the complexity of related classes when the spectral gap promise only applies to one kind of in-
201
stance (YES instances, for example). We also complete a discussion of the results in Table 6.1
by characterizing the complexity classes PGQCMA, EGQCMA, and PreciseEGQCMA.
6.5.1 Amplification for postQMA
We first define the class postQMA:
Definition 15 (postQMA). postQMA[c, s] is the class of promise problems A = (Ayes, Ano)
that can be decided in the following way: Apply a uniformly generated quantum circuit
U of size poly(n) on a state |x? encoding the input, together with a proof state of size
w(n) supplied by an arbitrarily powerful prover. Postselect the first l = poly(n) qubits
at the output onto the |0?l state, and measure the first qubit of the remaining register at
the output, called the decision qubit (o). The postselection probability is ?(2?f(n)) for a
polynomial f(n).
If x ? Ayes: ? |?? such that Pr(o = 1) ? c
If x ? Ano: ? |??, Pr(o = 1) ? s.
Morimae and Nishimura [325] defined this class and showed that postQMA := postQMA[1 , 2 ] =
3 3
PreciseQMA = PSPACE. This result is similar to the result postBQP = PreciseBQP(=
PP). They raised the question of whether one can do a Marriott-Watrous type in-place
amplification for this class, which, for instance, means boosting the parameters c and s to
be c = 1? 2?poly, s = 2?poly without changing the size of the witness. If one is allowed to
change the witness size, one can simply ask for polynomially many copies of the witness and
run the verification in parallel to get the required parameters. The benefit of in-place ampli-
fication is that it allows for good completeness and soundness parameters without blowing
up the witness size, which turns out to be useful in the proof of QMA ? PP. In-place
202
amplification for postQMA would also be useful to show IP = PSPACE [328; 329]. Here
we give a negative result for a sufficiently strong in-place amplification for postQMA.
Lemma 41 (Upper bound for in-place amplified postQMA). If f(n) = O(w(n)), then
postQMA[1? 2?t(n), 2?u(n)] ? PP for u(n) > w(n) + 1 and for any polynomial t(n) > 1.
Proof. Consider a postQMA[1 ? 2?t(n), 2?u(n)] language. Replace the witness state in the
amplified protocol by a maximally mixed state 1w . Now, since the overlap of any witness state2
with the maximally mixed state is 2?w, we have that the postselection success probability is
at least ?(2?f(n)?w(n)). Further, in the YES case, the probability of accepting the string x
(conditioned on success) is
Pr(o = 1) ? 2?w(n) ? (1? 2?t(n)). (6.14)
In the NO case, we have that no matter what state is in the witness register, the accept
probability is
Pr(o = 1) ? 2?u(n). (6.15)
In PreciseBQP = PP, we can distinguish between these two cases if 2?w ? 2?t?w > 2?u,
i.e. if 1? 2?t > 2w?u, for which it suffices to have u(n) > w(n) + 1 and t > 1.
This result implies that the completeness-soundness gap for postQMA cannot be
boosted beyond a point without incurring a blowup in the size of the witness or by reducing
the success probability of postselection.
203
6.5.2 Asymmetric promises on spectral gap and uniqueness
Motivated by a possible connection to the study of unique witnesses for quantum
complexity classes, we consider the complexity class GQMA[c, s, g1, 0]. Here, there is no
promise on the spectral gap for NO instances. In the YES case, we have ?1(Q) ? c and
?2 ? ?1? g1 ? 1? g1. If we choose the spectral gap g1 to be larger than 1? s, we see that
?2 ? s, ensuring that in the YES case, there is exactly one accepting witness5. The existence
of one accepting witness is exactly the promise that defines the class UQMA:
Definition 16 (Unique QMA [308]). UQMA[c, s] is the class of promise problems A =
(Ayes, Ano) such that for every instance x, there exists a polynomial-size verifier circuit Ux
acting on m = poly(n) qubits and an input quantum proof on w = poly(n) qubits and the
associated accept operator Q has properties
If x ? Ayes: ?1(Q) ? c and ?2(Q) ? s
If x ? Ano: ?1(Q) ? s.
Definition 17. UQMA := ?c?s?1/polyUQMA[c, s].
The earlier statement can be rephrased as ?an instance of GQMA[c, s, 1 ? s, 0] is a
UQMA[c, s] instance?. In the reverse direction, we can see that a UQMA[c, s] instance
necessarily has a spectral gap ?1??2 ? c?s, and therefore is an instance of GQMA[c, s, c?
s, 0]. This hints at, but does not prove, an equivalence between the promise of uniqueness
and that of an asymmetric spectral gap of?(1/poly). Aharonov et al. [308] proved a stronger
result by showing that the class UQMA is equivalent to the class PGQMA under randomized
reductions (defined below), where PGQMA is the class with spectral gaps for both the YES
5In the sense that any witness orthogonal to the accepting witness rejects with probability at least 1? s.
204
and the NO cases.
In the precise regime, we show the following results for the asymmetric variants of
PrecisePGQMA and PreciseEGQMA.
Theorem 42. PrecisePGQMA = ?c?s?1/ exp,GQMA[c, s, g1, 0].
g1?1/poly
PreciseEGQMA = ?c?s?1/ exp,GQMA[c, s, g1, 0].
g1?1/ exp
The proofs are given in Section 6.12 and hinge on the problem of computing ground-
state energies when there is a spectral gap only for the YES case, i.e. LOCALHAMILTONIAN[a, b, g1, 0].
Since the problem with an asymmetric gap can only be more complex than the symmetric
case, the nontrivial part of this lemma is to show that this problem has the same PP upper
bound as the symmetric case. This is not straightforward since the power method we de-
scribed before does not necessarily work for the NO case, since there is no spectral gap. We
work around this by making use of Ambainis?s technique [370] of identifying spectral gaps,
which is possible in PP [371].
6.5.3 Complexity of PGQCMA, EGQCMA, and PreciseEGQCMA
In this subsection we show that the classes PGQCMA and EGQCMA are both equiv-
alent to QCMA under randomized reductions, which we now define.
We say a problem A is random reducible to problem X if every instance a of A can
be mapped to a random set of polynomially instances xi of X , such that
If a ? Ayes: Pri(xi ? Xyes) ? 1/poly
If a ? Ano: Pri(xi ? Xyes) = 0.
A class Y is random reducible to another class Z if every problem in Y is random reducible
to some problem in Z (and vice versa), and is denoted =R.
205
To show PGQCMA =R QCMA and EGQCMA =R QCMA, we make use of the class
UQCMA (Unique QCMA), which has been defined in Ref. [308], and was shown to be equal
to QCMA under randomized reductions.
Definition 18 (UQCMA[c, s] [308]). UQMA[c, s] is the class of promise problems A =
(Ayes, Ano) such that for every instance x, there exists a polynomial-size verifier circuit Ux
acting on m = poly(n) qubits and an input classical proof on w = poly(n) qubits, whose
associated accept operator Q has properties
If x ? Ayes: ?1(Q) ? c and ?2(Q) ? s
If x ? Ano: ?1(Q) ? s.
Definition 19. UQCMA := ?c?s?1/polyUQCMA[c, s].
Aharonov et al. [308] showed that UQCMA =R QCMA using generalizations of tech-
niques in Ref. [369] to complexity classes with randomness. In order to show PGQCMA =R
QCMA and EGQCMA =R QCMA, we show
Lemma 43. PGQCMA =R UQCMA.
Since PGQCMA ? EGQCMA ? QCMA, the equivalence of EGQCMA with QCMA
follows.
To show Lemma 43, we observe that the proof of PGQMA =R UQMA in Ref. [308]
works for classical witnesses. For completeness, we give a self-contained proof here.
Proof of Lemma 43. First, we show the direction UQCMA ? PGQCMA. We observe that
in a YES instance of UQCMA[c, s], ?1 ? c and ?2 ? s. Thus, a YES instance already has a
spectral gap of g1 ? c? s and is a YES instance of PGQCMA. In the NO case, we modify
the verifier?s strategy so that it creates a spectral gap. The verifier expects an additional qubit
206
we call the ?flag qubit? from the prover, which is measured in the beginning just like the
other qubits of any QCMA proof. The associated accept operator now has twice as many
eigenvalues because it acts on a space with one larger qubit.
The verifier?s protocol is as follows. If the state of the flag qubit is |0?, the verifier
continues with the original protocol. This gives the same eigenvalues for the accept operator
as the original protocol. If the state of the flag qubit is |1?, the verifier accepts with probability
s+(c?s)/poly if the state of the rest of the witness qubits is |1??w. If the state of the rest of
the witness register is anything else, the verifier rejects. In the latter case (when the state of
the flag qubit is |1?), the accept operator has one eigenvalue at s+ (c? s)/poly and 2w ? 1
eigenvalues with eigenvalue 0, each case corresponding to some state in the witness. The
modified verifier is a PGQCMA instance with completeness c, soundness s + (c ? s)/poly
and spectral gaps g1 ? c?s and g2 ? (c?s)(1?1/poly). ThereforeUQCMA ? PGQCMA.
For the other direction, we give a randomized reduction PGQCMA ?R UQCMA.
Consider a YES instance of PGQCMA[c, s, g1, g2]. We know ?1 ? c and ?2 ? ?1 ? g1, but
we do not know if ?2 ? s, as is required for the instance to be a UQCMA instance. The idea
in Ref. [308] is to make a query to a UQCMA[cj, sj] oracle with completeness cj = c+(j+
1)g1/2 and soundness sj = c + jg1/2, for j chosen randomly from {0, 1, . . . ? 2(1? ?}.c)g1
In the NO case, all the queries are valid queries to a UQCMA oracle and return the correct
answer (NO). In the YES case, since the completeness and soundness in each query differ by
g1/2, there is at least one j where ?1 ? cj and ?2 ? s 6j . Therefore, this is a randomized
reduction to UQCMA.
6In the YES case, there could be some queries that are not valid UQCMA instances, and the oracle can
answer arbitrarily for such ill-formed queries. This does not, however, hamper the proof, since at single valid
query is enough to give a nonzero probability of saying YES.
207
Therefore, we obtain
Corollary 44. PGQCMA =R QCMA.
EGQCMA =R QCMA.
Our final result concerns the classPreciseEGQCMA. Just like we havePreciseEGQMA =
PreciseQMA, we can show that exponentially small spectral gaps are no less complex in the
case of classical witnesses. We show
Lemma 45. PreciseEGQCMA = PreciseQCMA.
Proof. The direction PreciseEGQCMA ? PreciseQCMA is trivial. For the other direction,
we take a PreciseQCMA[c, s] instance and give a PreciseEGQCMA[c, s, g1, g2] instance with
an exponentially small spectral gap. This is done by modifying the verifier so that no two
witnesses yi and yj are accepted with the same probability. First, we choose the verifier?s gate
set so that the accept probability of any witness y is given by kx,y/2l(n), for k ? [2l(n)x,y ],
where l(n) is the size of the verifier?s circuit [377]. The modified verifier rejects the instance
straightaway with probability y /2polyb , where yb is a number in [2w ? 1] when interpreting
the witness y in binary and the polynomial is at least l(n) + w(n) + log 12( ? ). If thec s
verifier does not reject at this step, they run the original(verificati)on protocol. The overall
accept probability when given y is given by p = kx,y 1? yby w poly . Since the polynomial2 2
satisfies poly ? l(n) + w(n) + log2( 1? ), the completeness and soundness are given byc s
c? ? c? 2?w(n)(c? s) and s? = s, which are still separated by 2?poly.
We now claim that the resulting accept probabilities are distinct for distinct witnesses,
and hence separated by an amount ?(2?poly). This is easily seen for two distinct yi and yj
208
such that kx,y = k . If ki x,yj x,y =? kx,y , then for p = p , we needi j yi yj
2w
kx,y ? kx,y = poly (yj ? yi ), (6.16)i j 2l+w+ b b
which cannot be satisfied by integers yj and y lb i in [2 ].b
The same technique also works to give a more direct proof of EGQCMA = QCMA.
6.6 The Schrieffer-Wolff transformation
In this section, we give a brief introduction to the Schrieffer-Wolff transformation
[336], which is an important tool in some of our subsequent proofs. We follow the exposition
in Ref. [337], specialized to our context.
In the context relevant for us, we usually have an ?unperturbed? Hamiltonian H0 and
a ?perturbation? H1, together forming the full Hamiltonian H = H0 +H1. The (possibly
degenerate) ground-state subspace of H0, denoted S0, has energy ?0 and is separated from
the rest of the spectrum by a gap ?. We are interested in cases when the Hamiltonian H1
has small strength relative to the gap ?, in the sense ?H1? =: ? < ?/2. This ensures that
all eigenvalues of H0 are shifted by an amount smaller than ?/2 under the perturbation.
Therefore, the low-energy subspace of H , given by
{ [ ]}
S | ? ?= ?? : ??|H |?? ? ?0 ? , ?0 + , (6.17)
2 2
has the same dimension as that of H0. We denote the the projectors on to S0 and S by P0
and P , respectively. As long as ? < ?/2, we have ?P ? P0? < 1, which captures the fact
209
that the dimension of the two subspaces is the same.
Since the dimension of the two subspaces is the same, there exists a unitary U that
maps the subspace S0 to S:
UPU ? =?P0, with (6.18)
U = (2P0 ? 1)(2P ? 1). (6.19)
We are interested in the effective Hamiltonian in the subspace S0, given by
H ?eff = P0U(H0 +H1)U P0. (6.20)
The Schrieffer-Wolff transformation allows one to express the generator V = log(U), and
consequently, Heff, as a convergent series in the perturbation H1. We first write H1 as
Hd1 + H
o
1 , where Hd1 is block-diagonal in the subspace S and Ho0 1 is block-off-diagonal.
Let the eigenstates of H0 be given by {|i?}, with corresponding energies {Ei}. We denote
I0 = {i : Ei = ?0}, which is the set of indices corresponding to the ground-state space.
The first few terms of the Schrieffer-Wolff expansion are given by
Heff =?H0P0(+ P0H1P0 + )1 ?i|H1 |j? | ?? | ?j|H1 |i?P0 i j H + H |j??i| P
2 Ei ?
1 1 0
E E
?I ?I j i
? Ej
i 0,j / 0
+O(?H ?31 ). (6.21)
In our work, we use the first-order expansion of the Schrieffer-Wolff series. The series
210
converges absolutely as long as ?H1? ? ?/16 [337]. We can upper bound the error caused
by truncating the formal series to first order:
???Heff ??H0P0(? P0H1P0? ? O(1)????
) ?
?i|H1 |j? | ?? | ?j|H1 |i?
??
P0 ? i j H1 + H |j??i| P? E ?
1 0?
E E ? E ?
i?I0,?j?/I0?? ?
i j i j
? 1O(1) ? (?i|H1 |j? ?j|H1 |k? |i??k| +Ei ? Eji?I0,j?/I0,k?I0 ?
?j|H1 |i? ?k?|H1 |j? |
?
k??i|)?? ? (6.22)( )
1 ?? ? ( )?? ?O ?? ?i|H21 |k? |i??k|+ ?k|H21 |i? |k??i| ?? (6.23)(?)?i?I0,k?I01 ?
= O( )?2P0H
2P ?1 0 (6.24)?
? ?
2
O , (6.25)
?
where we have used |Ei ? Ej| > ? for states i ? I0, j ?/ I0.
6.7 Modified clock constructions with spectral gaps
In this section, we present the small-penalty clock construction and use it to prove
the main hardness results in this Chapter. We first illustrate the technique by proving the
following lemma.
Lemma 46. (1/ exp, 1/ exp)-LOCALHAMILTONIAN is PreciseEGQMA-hard.
Proof. Consider a GQMA[c, s, g1, g2] instance x, where the verifier?s circuit Ux acts on m =
poly(n) qubits apart from the proof state. We assume that the circuit has T = poly(n) gates.
211
The idea behind the technique is valid generally, but for concreteness we focus on the clock
construction of Kempe et al. [331], which proves QMA-hardness of k-LOCALHAMILTONIAN
for k ? 3. The clock Hamiltonian takes the form
H = Hinput +Hprop +Houtput +Hclock. (6.26)
The first term Hinput ensures that the ground state of Hinput coincides with input state to the
circuit. The term on the proof register is identity, allowing for any witness state given by the
prover to be input into the verifier?s circuit. It is given by
?m
Hinput = |1??1|i ? 1proof ?Hclockinit. (6.27)
i=1
In the above, the term Hclockinit ensures that the clock is properly initialized to the |1?clock
state. Next, Hprop is a Hamiltonian that ensures the ground state is ?propagated? correctly
with each gate applied by the verifier:
?T
Hprop = ?Ui+1 ? |i+ 1??i| ?clock ? Ui+1 ? |i??i+ 1|clock
i=0
+ 1? (|i??i|clock + |i+ 1??i+ 1|clock). (6.28)
The ground-state subspace of Hprop contains valid ?partial? computations until step i ? T ,
namely Ui . . . U2U1 |?0? on any initial state |?0? ? i. The term Houtput penalizes states that
212
have any nonzero probability of saying ?NO? at the output qubit o of the circuit:
Houtput = ? |0??0|o ? |T ??T |clock . (6.29)
Lastly, Hclock ensures that states in the clock register that do not encode a valid time step are
penalized. The HamiltoniansHclock andHclockinit both depend on the details of the particular
clock construction. Our analysis does not depend on these details is largely independent of the
way the clock register encodes the time. We refer the reader to Ref. [331] for an explanation
of their construction.
First consider just the Hamiltonian H0 = Hinput +Hprop +Hclock, which is the clock
Hamiltonian without a penalty term at the output. The ground-state space of H0 is exactly
given by the subspace S0 of history states:
S0 = span{|?h? : |?? arbitrary},where?T
|?h?
1
:= ? Ui . . . U0 |0m? ? |?? ? |i?
T + 1 clock
. (6.30)
i=0
where U0 = 1. Any state having zero support on S0 has an energy at least ?(1/T 3) [351],
implying that the gap above the zero energy subspace is ??= ?(1/?T
3).
Now, let us add in the term H1 = Houtput, with ?Houtput? = ?. We choose ? <
?/16, unlike the regular clock construction where ? is usually taken to be ?(1). As long as
? < ?/2, we can restrict our attention to the zero energy space of H0, since H1 can change
eigenvalues by at most ?. We use the tool of Schrieffer-Wolff transformation as described
in Section 6.6 to obtain a description of the Hamiltonian in the low-energy subspace. The
213
subspace S0 is the ground-state space of states with energy 0. Since ?H1? = ?, the associated
low-energy subspace of H = H0 +H1 is
S = span{|?? : ??|H |?? ? [??, ?]}, (6.31)
the subspace with energies in [??, ?]. In our caseH0P0 = 0 in the ground subspace spanned
by history states |?h?, and the matrix elements of P0H1P0 are given by
??h|P0H1(P0 |?h? = ??? h
|H1 |?h? ) (6.32)
T
1
= ?0|m( ? ??| ? ?i|
?
clock)U0 . . . U
? H1?
?T + 1
i
i=0
T
Uj .(. . U0 |0
m? ? |?? ? |j?clock ) (6.33)j=0
T
1 ?
= ?0|m ? ??| ? ?i| ? ?
T + 1 ( clock U0 . . . Ui ?i=0 ? )T
? |0??0|o ? |T ??T |clock Uj . . . U
m
0 |0 ? ? |?? ? |j?clock (6.34)
j=0
1
= ?0|m ? ??| ? ?T |U ?? |0??0| ?
T + 1 o
|T ??T |clock U |0
m? ? |?? |T ?clock (6.35)
?
= ?0|m ? ??|U ? |0??0|o U |0
m? ? |?? (6.36)
T + 1
?
= ?0|m ? ??|U ?(1? ? mout)U |0 ? ? |?? , (6.37)
T + 1
where ?out is the projector onto the accepting state |1?o. Continuing, we have
? ??h|P0H1P0 |?h? = (??|?? ? ??|Q |??), (6.38)
T + 1
214
meaning that the first order correction P0H1P0 is simply related to the accept operator
Q, which was defined as Q(U) = ?0|?m U ??outU |0??m. Let the eigenstates of Q be
|?1? , |?2? , . . . |?2w? with eigenvalues ?1 ? ?2 ? . . . ?2w . We use the associated history
states |?i ? as a basis for the subspace S0. In this basis, the first order correction Ph 0H1P0 is
diagonal:
? ?
P0H1P0 = (1? ?i) |?i ???i | . (6.39)
T + 1 h h
i
We conclude that in the ground space of the original HamiltonianH0, the full Hamil-
tonianH has eigenvalues ?(1??i)/(T +1)?O(?2/?), where the quantity ?i is the accept
probability of the verifier?s circuit given |?i? as witness. This is the same conclusion we would
obtain by applying degenerate perturbation theory, except that the error bound is rigorous.
We now analyze the YES and NO cases to obtain a lower bound on the promise gap. In each
case, we also lower bound the spectral gaps in the resulting Hamiltonian.
In the YES case the ground-state energy is E1 ? ?(1 ? c)/(T + 1), as can be seen
from the fact that the history state |?h? corresponding to an accepting witness |?? would have
energy ?(1???|Q |??)/(T + 1) ? ?(1?c)/(T +1). Our small-penalty clock construction
and the Schrieffer-Wolff transformation comes in handy for the NO case. We see in the
NO case that the ground-state energy is at least E1 ? ?(1 ? s)/(T + 1) ? O(?2/?).
Therefore, the promise gap is at least ?(c? s)/(T + 1)?O(?2/?) = ?(1/ exp) as long as
?/? = o((c? s)/(T + 1)).
In the above, if we had chosen ? = ?(1) instead of ? < ?/16, the NO case would
have given us a bound E1 ? ?(1? s)/T 3. This would mean that one would have to amplify
215
the completeness and soundness c, s to near unity in order to get a nontrivial promise gap.
However, such an amplification can, in general, shrink the spectral gap of the accept operator.
Independently, a large penalty term ? = ?(1) could also reorder some eigenvalues, meaning
that the spectral properties of the resulting clock Hamiltonian would not faithfully track
those of the original accept operator.
The spectral gap in the YES/NO case is ? ? ? ? 2E2 E1 (?1(Q) ?2(Q))? O( ? ).T+1 ?
We take ? = o(?(c? s)/(T + 1)) = o((c? s)/T 4), which is exponentially small if c? s
is. As long as ?/? = o(min[g1, g2]/(T + 1)), both the YES and NO cases will have an
exponentially small spectral gap. In summary the choice
min [g1, g2, (c? s)]
? = = ?(1/ exp) (6.40)
nT 4
suffices to have a promise gap and spectral gaps bounded below by ?(1/ exp). This proves
PreciseEGQMA-hardness of (1/ exp, 1/ exp)-LOCALHAMILTONIAN and one half of Theo-
rem 23.
We generalize the above proof technique to the case of GQCMA-hardness of GS-
DESCRIPTION-LOCALHAMILTONIAN. In addition to showing a promise gap and a spectral gap,
we should show that the resulting Hamiltonian has a classical description of a circuit to
prepare a low-energy state. We show the following general lemma.
Lemma 47. (?,?)-GS-DESCRIPTION-LOCALHAMILTONIAN is GQCMA[c, s, g1, g2]-hard for
any ?,? satisfying both the following conditions.
i. ? = O((c? s)2/poly(n)) for some polynomial.
216
ii. If c? s = o(min[g1, g2]), then any ? satisfying ? = O((c? s)min[g1, g2]/poly(n)).
Else, ? = 0.
Proof. To prove GQCMA-hardness, we give a reduction from GQCMA[c, s, g1, g2] to GS-
DESCRIPTION-LOCALHAMILTONIAN[a, b, g? , g?1 2]. We are promised that the input witnesses are
computational basis states (this can be assumed without loss of generality), corresponding to
the classical witness sent by the prover. We would like to show that there exists a circuit V
to prepare a state ?-close in energy to the ground state of the clock Hamiltonian both the
YES and NO cases.
Consider again the small-penalty clock co?nstructi?on, with the clock Hamiltonian
Eq. (6.26). Let the norm of the penalty term be ?Houtput? = ?. When ? = 0, the ground-
state space is given by valid history state computations corresponding to computational basis
witness states. The spectral gap above this subspace is at least ?(1/T 3). The addition of
the penalty term changes the energies to ? (1 ? ?k) + O(?2T 3), where ?k is the acceptT+1
probability upon input computational basis state |yk? as witness. Consider the history state
associated with witness |yk?:
?T
|yk ? := ?
1
U . . . U mi 0 |0 ? ? |yk? ? |i?h
T + 1 clock
. (6.41)
i=0
This state has energy ?yk |H |yk ? = ? (1??k) and is t(hereforh h T+1 )eO(?2T 3)-close in energy
to the true ground state. Therefore, as long as 3?2T 3 < O (b?a)2 , a classical description off(n)
a circuit that prepares |yk ? is a valid ground-state description. The circuit may be describedh
by specifying yk and a circuit that prepares the histo?ry state |?h? upon any quantum input
|??. This latter circuit first prepares the state ? 1 Ti=0 |0m? |i?T+1 clock and then applies the
217
unitaries Uj . . . U0 controlled on the clock being in time-step j [374].
The same promise gap and spectral gap analyses as in the proof of Lemma 46 hold.
In the YES case, the Hamiltonian has ground-state energy ? ? (1 ? ? ) ? ? (1 ? c).
T+1 1 T+1
In the NO case, the ground-state energy is at least ? (1 ? ?1) ? O(?2T 3) ? ? (1 ?T+1 T+1
s)?O(?2T 3). The promise gap between the ground-state energy for YES and NO cases is
? ? ? (c? s)?O(?2T 3). We make the choice ? = ?( c?s) to ensure the promise gap is
T+1 T 4
?((c? s)2/T 5). This choice is consistent with the choice 3?2T 3 ? O( (b?a)
f(n)2
) made above.
Let us now analyze the spectral gap of the resulting Hamiltonian. Using the Schrieffer-
Wolff expansion to obtain the eigenvalues of the Hamiltonian for small ?, we have ? ?
? (?1 ? ?2) ? O(?2T 3). The spectral gap is at least ? min[g1, g2] as long as ?2T 3 =T+1 T+1
o( ? min[g1, g2]). Using the choice of ? above, this means the spectral gap is ?((c ?T+1
s)/T 5min[g1, g2]) as long as c ? s = o(min[g1, g2]). Otherwise, the best bound on the
spectral gap is ? ? 0. Observing that T = poly(n) by assumption, we obtain the lemma.
The lemma allows us to show the following:
Corollary 48 (Second half of Theorems 24 to 27). (1/poly, 0)-GS-DESCRIPTION-LOCALHAMILTONIAN
is QCMA-hard.
(1/ exp, 0)-GS-DESCRIPTION-LOCALHAMILTONIAN is PreciseQCMA-hard.
(1/poly, 1/poly)-GS-DESCRIPTION-LOCALHAMILTONIAN is PGQCMA-hard.
(1/ exp, 1/ exp)-GS-DESCRIPTION-LOCALHAMILTONIAN is PreciseEGQCMA-hard.
For the problem with ? = 1/ exp,? = 1/poly, we do not give a direct reduction from
a PrecisePGQCMA instance. Instead, we show PP-hardness through the characterization of
218
PP in terms of the class PreciseBQP. From the PP upper bound to PrecisePGQCMA, we
obtain PrecisePGQCMA-completeness of the problem (1/ exp, 1/poly)-GS-DESCRIPTION-
LOCALHAMILTONIAN. The argument is similar forPrecisePGQMA-hardness of (1/ exp, 1/poly)-
LOCALHAMILTONIAN.
Lemma 49 (Lemmas 34 and 36 restated). .
(1/ exp, 1/poly)-GS-DESCRIPTION-LOCALHAMILTONIAN is PP-hard.
(1/ exp, 1/poly)-LOCALHAMILTONIAN is PP-hard.
Proof. We give a reduction from any problem inPreciseBQP to (1/ exp, 1/poly)-GS-DESCRIPTION-
LOCALHAMILTONIAN, which is also an instance of (1/ exp, 1/poly)-LOCALHAMILTONIAN.
Since PreciseBQP is the class of problems that can be decided by quantum circuits with
a promise gap c ? s = ?(1/ exp), it can also be thought of as ?PreciseQMA without an
input witness?. The Hamiltonian is constructed out of the PreciseBQP computation as
H = Hinput +Hprop +Houtput +Hclock, where the terms are now
?m
Hinput = |0??0|i ?Hclockinit, (6.42)
?i=1T
Hprop = ?Ui+1 ? |i+ 1??i|clock ? U
?
i+1 ? |i??i+ 1|clock
i=0
+ 1? (|i??i|clock + |i+ 1??i+ 1|clock), and (6.43)
Houtput = ? |0??0|o ? |T ??T |clock . (6.44)
The only difference from Eqs. (6.27) to (6.29) is that Hinput does not have support on an
unpenalized proof register, since PreciseBQP does not rely on a proof state given as input.
This is analogous to the clock construction of Ref. [351], which was instrumental in the
219
proof that adiabatic quantum computation is universal for BQP.
We again let the Hamiltonian H0 be Hinput +Hprop +Hclock and H1 = Houtput. The
ground state of H0 is now nondegenerate (unique) and given by the history state
?T
|0h? := ?
1
U mi . . . U0 |0 ? ? |i?clock . (6.45)T + 1
i=0
Let us denote the ground-state space of H0 and the projector onto it by ?0. As for H1, the
ground space ?1 is spanned by states belonging to subspaces L and L?, with
L = |1?o ? |T ?clock (6.46)
L? = span {|??} ? span {|0?clock , |1?clock , . . . |T ? 1?clock}, (6.47)
with |?? arbitrary.
We observe that when ? = 0, the Hamiltonian exactly corresponds to Aharonov et
al.?s Hfinal [351]. Aharonov et al. [351] showed that this Hamiltonian H0 has a spectral gap
of ? = ?(1/T 3) in the full Hilbert space. Further, the ground state of H0 corresponds to
the history state of the BQP computation (PreciseBQP in this case), which starts off in a
fixed, known state |0m?.
In the YES case, the ground-state energy of H = H0 + H1 can be bounded above
by ? (1 ? c). For the NO case, we again use the expression for the perturbed energies in
T+1
the ground-state space coming from the Schrieffer-Wolff transformation. Specifically, in the
NO case, we have E1 ? ?(1 ? s)/(T + 1) ? O(?2/?), where ? is the spectral gap above
220
the ground state, just as in the proof of Lemma 46. The promise gap is lower-bounded by
( )
1? s 2
? ? 1? c? ? ?O . (6.48)
T + 1 T + 1 ?
Therefore, as long as ?/? = o((c ? s)/(T + 1)) and ? = ?(2?poly), the promise gap is at
least ?(?(c? s)/(T + 1)) = ?(2?poly). The spectral gap for the unperturbed Hamiltonian
H0, which is the same as the final Hamiltonian in Ref. [308], is at least ?(1/T 3). Therefore,
we pick ? = (c? s)/(nT 4), which ensures that the conditions above are satisfied.
Coming to the spectral gap of the full Hamiltonian, we observe that since the orig-
inal Hamiltonian had a spectral gap of ?(1/T 3) and the perturbation H1 is exponentially
small, the eigenvalues can change at most by ?H1? = ?, preserving the spectral gap. So
far, we have a reduction from any PreciseBQP instance to an instance of (1/ exp, 1/poly)-
LOCALHAMILTONIAN.
It remains for us to see that there is an efficient circuit that can prepare a state close
in energy to the ground state. By the justification in the proof of Lemma 47, we know that
choosing the output penalty term to be exponentially small causes the history state of the
computation |0h? to be exponentially close to the ground state in energy. We have also seen
the existence of a polynomial size circuit that prepares the history state given a description
of the input (which here is |0m? for PreciseBQP). Note that when ? = 0, the ground state
is unique and has a ?(1/poly) spectral gap above and therefore taking ? exponentially small
does not pose a problem with spectral gaps.
The difference between the proof of Lemma 49 and the proof of Lemma 46 is that it
is the perturbation ? that creates the spectral gap in the proof of Lemma 49, while in the
221
proof of Lemma 46, the spectral gap already exists in the unperturbed Hamiltonian. This is
why we can afford to take ? exponentially small here, which is needed to obtain an instance
with a promise gap.
Thus, we have seenPP-hardness of (1/ exp, 1/poly)-LOCALHAMILTONIAN. PrecisePGQMA-
hardness of the problem follows from the fact that PrecisePGQMA ? PP (Lemma 32).
Corollary 50. (1/ exp, 1/poly)-LOCALHAMILTONIAN is PrecisePGQMA-hard.
Similarly, thePP-hardness of (1/ exp, 1/poly)-GS-DESCRIPTION-LOCALHAMILTONIAN
and the result PrecisePGQCMA ? PrecisePGQMA = PP together imply the following re-
sult.
Corollary 51. (1/ exp, 1/poly)-GS-DESCRIPTION-LOCALHAMILTONIAN is PrecisePGQCMA-
hard.
Lastly, the remaining case is (1/poly, 1/ exp)-GS-DESCRIPTION-LOCALHAMILTONIAN
with ? = 1/poly, ? = 1/ exp, for which we argue that an instance with spectral gap
? = ?(1/poly) is also an instance with ? = ?(1/ exp). Therefore, (1/poly, 1/ exp)-GS-
DESCRIPTION-LOCALHAMILTONIAN is PGQCMA-hard, and, since PGQCMA =R EGQCMA,
EGQCMA-hard under randomized reductions. For the case of (1/poly, 1/ exp)-LOCALHAMILTONIAN,
we do not currently have a hardness result. This is because, in performing a reduction from
EGQMA, we get an instance of (1/poly, 0)-LOCALHAMILTONIAN and do not get any promise
on the spectral gap that results.
222
6.8 Precise phase estimation of gapped Hamiltonians
In this section, we show that the (1/ exp,?)-LOCALHAMILTONIAN problems with
either 1/poly or 1/ exp spectral gaps defined in Section 6.2.1 are in the corresponding
PreciseGQMA class. Together with the results of the previous section, this proves The-
orems 22 and 23.
Lemma 52. (1/ exp, 1/poly)-LOCALHAMILTONIAN ? PrecisePGQMA.
Lemma 53. (1/ exp, 1/ exp)-LOCALHAMILTONIAN ? PreciseEGQMA.
The proof relies on phase estimation to infer energies of a local Hamiltonian. The
standard phase estimation circuit requires exp(n)many gates in order to infer the eigenvalues
to 1/ exp precision. However, since we want to show containment in a Precise- class, we can
use the power of being able to distinguish between two cases with exponentially close accept
probabilities. It turns out that phase estimation with a single ancillary qubit is enough to
distinguish between the YES and NO cases, as shown in Ref. [319]. Moreover, we show
that the circuit preserves spectral gaps of the Hamiltonian: if two eigenstates have energies
separated by some amount, then the phase estimation circuit also has a gap in the accept
probabilities corresponding to these input states.
Below we give a unified proof of Lemmas 40, 52 and 53. Specifically, we show
PrecisePGQMA (PreciseEGQMA) containment of the problem (1/ exp,?)-GAPPEDSPARSEHAMILTONIAN
with ? = 1/poly (? = 1/ exp).
Lemma 54. GAPPEDSPARSEHAMILTONIAN[a, b, g1, g2] has a GQMA[c, s, g?1, g?2] protocol with
spectral gaps g?1 = ?(g21/poly) and g?2 = ?(g22/poly) and promise gap c? s = (b? a)2/poly.
223
|?? e?iHt |??
|0? 
H ? H 

Figure 6.2: One-qubit phase-estimation circuit. The symbol H on the bottom line denotes
the Hadamard gate.
Proof. The strategy is to ask the prover for the ground state of the sparse Hamiltonian. The
verifier then performs phase estimation on the witness state with a single ancillary qubit
and uses the power to decide between two cases with exponentially close accept probabilities.
This power effectively enables computation of the phase of e?iHt to exponential precision,
despite having a single ancilla qubit in the phase estimation circuit (see Ref. [319] for more
details). The phase estimation circuit is shown in Fig. 6.2. If t ? ?? ? , all eigenstates of2 H
H would correspond to a unique phase and a unique accept probability for the circuit. We
know an upper bound dk on ?H? through the Gershgorin circle theorem because we are
assured that the magnitude of the entries is ? k and the sparsity is d. Therefore, it suffices
to choose t ? ? .
2dk
In order to perform phase estimation to exponentially small error, we need to apply a
controlled-e?iHt rotation to error ? = 1/ exp. This is possible due to Hamiltoni(an)simu-
lation algorithms for sparse Hamiltonians, whose circuit size scales as poly(n) log 1 [40],
?
which is polynomial in n, as desired. The accept probability of the circuit upon input an
eigenstate |Ei? of the Hamiltonian is 1+cos(Eit) . The promise gap can be lower bounded by2
an inverse exponential, as has been analyzed previously [319].
We can also show a spectral gap in the accept operator, or equivalently, a gap in the
accept probabilities of the circuit for the optimal state and any state orthogonal to it. Since
224
the phase estimation circuit does not apply the exact controlled-e?iHt unitary but a unitary
Ux exponentially close to it, the eigenstates of Q =? ?0|?inU
?
x??outUx?in |0? are not exactly
the eigenstates of e?iHt (or of H). However, since ?e?iHt ? U ?x ? ?, the eigenvalues of Q
are exponentially close to the accept probabilities of the eigenstates |Ei? ofH . The difference
in accept probabilities can be bounded by ?.
The difference in the ideal accept probabilities of the ground state and the first excited
state is cos(E0t)?cos(E1t) . Applying Taylor?s theorem to cos(E1t) around the point E0t, we get2
cos(E1t) = cos(E0t)? sin(E0t)t(E1 ? E0)?
2 2 3
cos t (E1 ? E0) h(E0t) + sin(E0t) (6.49)
2 6
for some h ? [0, (E1 ? E0)t]. Therefore,
cos(E0t)? cos(E1t) = sin(E0t)t(E1 ? E0)+
2 2 3
cos t (E1 ? E0)(E0t) ? sin
h
(E0t) (6.50)
2 6
3 3
? t2 ? t (E1 ? E0)(E1 E0)2/2? . (6.51)
6
? ?(t2(E1 ? E0)2), (6.52)
where in the second line we use the fact that E t,E t < ?/2 and (E ?E )3t30 1 1 0 = O(t(E1?
E0)) in the third. Therefore, the ideal accept probabilities also have a gap of ?((E1 ?
E )2/?H?20 ) = ?(?2/poly) as long as ? ? O(t2?2/n) = O(?2/poly). Now, when the
applied unitary differs from the ideal one by ? in operator norm distance, the gap in the
accept probabilities differs from the ideal accept probabilities by 2?. We therefore choose ?
225
sufficiently small, i.e. we choose, say, ? = ?(t2(E1 ? E0)2/2n), which is still ?(2?poly), as
needed.
To see the existence of a promise gap, notice that E0 ? a in the YES case and E1 ? b
in the NO case, giving c ? s = ?(t2(b ? a)2 ? 2?) = ?((b ? a)2/poly). This proves the
lemma.
As corollaries, we obtain Lemmas 40, 52 and 53, since a local Hamiltonian is also a
sparse Hamiltonian.
6.9 Phase estimation in the presence of efficient circuit descriptions
In this section, we show the problem (?,?)-GS-DESCRIPTION-LOCALHAMILTONIAN is
in GQCMA with appropriate bounds on the promise and spectral gaps (Theorems 24 to 28).
We first deal with the case of zero spectral gap:
Lemma 55 (One half of Theorems 24 and 25). (1/poly, 0)-GS-DESCRIPTION-LOCALHAMILTONIAN
? QCMA.
(1/ exp, 0)-GS-DESCRIPTION-LOCALHAMILTONIAN ? PreciseQCMA.
Proof. For the upper bound, we describe a QCMA or PreciseQCMA protocol. We are
promised that in both the YES and NO cases, there exists a classical description of a circuit
V of polynomial size that will create a state with energy close to the ground-state energy.
Specifically, the energy of this state is ?-close to the ground-state energy, for ? < (b?a)3 for a
f(n)2
polynomial f(n) ? ?H?. ForQCMA, we have b?a ? ?(1/poly), while for PreciseQCMA,
b? a ? ?(1/ exp). The verifier asks the prover to give this description (which is promised
to exist). The verifier then creates a state |?? with low energy by applying V to |0m?. The
226
verifier measures the energy of this state via the one-bit phase-estimation protocol outlined
in Section 6.8, which involves applying a controlled-e?iHt for time t ? ?? ? .2 H
The proof that this verification protocol works is slightly more involved than the
QMA[c, s] case. This is because, in the case of QMA a verifier can assume without loss
of generality that the prover sends the optimal eigenstate as a witness. However, in the case
of GS-DESCRIPTION-LOCALHAMILTONIAN, we are only promised the existence of an efficient
circuit to prepare a state close in energy to the ground state, and not the ground state itself7.
Despite this complication, we can still show that a state close in energy to the ground state
behaves similarly with respect to the accept probabilities of the QCMA[c, s] verifier.
In the YES case, there is a description V that produces a state |?? with energy close to
the ground-state energy (i.e. with energy ? E + ? < a+ (b?a)31 poly ). We show in Lemma 56(n)
that the accept probabilit(y o)f the verifier upon performing one-bit phase estimation on the
state |?? is at least cos2 bt + ?((b ? a)2/poly). In the NO case, the optimal strategy
2
for the prover is to send the description of a circuit that makes a state as close as possible
to the ground state, since the accept probabilities are monotonic in energy and there exists
no other state with smaller energy, by definition. Even if the prover sends the verifier a
circuit that exactly prepares the ground state |E1?, its energy in the NO case is already
? b. This means that the verifier will accept with probability at most (1 + cosE1t)/2 ?
(1 + cos bt)/2. Therefore there is a separation in the accept probabilities in the YES and
NO cases of c ? s = ?((b ? a)2/poly), which is ?(1/poly) for b ? a = ?(1/poly) and
?(1/ exp) for b? a = ?(1/ exp).
Lemma 56. 3If a state |?? has energy ??|H|?? = ?E? ? E + 5(b?a)1 for some polynomial24f(n)2
7The weaker promise is more natural since it is more robust.
227
f(n) ? ?H?, then in the YES ca(se,)the accept probability(of the sta)te upon phase estimation with
one bit of precision is ?p? ? cos2 bt + ?, where ? = ? 5(b?a)2 .
2 24f(n)2
Proof. We are given a state |?? with energy ?E?. Let pj = |?E 2j|??| be the weight of the
energy eigenstate Ej . Then we know p1E1 + p2E2 + . . . p2nE2n = ?E?. The pro(babi)lity
of accep(ting)|?? in the one(-bit ph)ase estimation circuit is given by ?p? = p cos
2 E1t
1 +2
p cos2 E2t + . . . p cos2 ?H?t2 2n , where ?H(? = )E2n . Given the co(nstrai)nt on the energy2 2
?E?, we show in Lemma 57 that ?p? ? cos2 E1t (1? x) + cos2 ?H?t( ) x, where x :=2 2?E??E1
? ?? . Now in order to have ?p? ? cos2
bt + ?, it suffices to have
H E1 2
( ) ( )
cos2 (E1t )? cos2x ? 2 (bt ? ?2 ) (6.53)
cos2 E1t ? cos2 ?H?t
2 2
cos (E1t)? cos (bt)? 2?
= cos . (6.54)(E1t)? cos (?H?t)
It is therefore sufficient if
( )
?(b? a)t b
3t3
x (bt? )? ?, since (6.55)2 6
(b? a)t b3? t
3
? ? (b? a)t sin(bt)? 2?bt ? (6.56)
2 6 2
? (b? a)t sin(bt)? 2?cos (6.57)(at)? cos(?H?t)
? cos(at)? cos(bt)? 2?cos(at)? cos(?H?t)
? cos(at)? cos(bt)? 2?cos , (6.58)(E1t)? cos(?H?t)
where we use the inequalities sin 3 3(bt) ? bt ? b t , E1 ? a, cos(at) ? cos(bt) ? (b ?6
a)t sin(bt), and 2 ? cos(at)? cos(?H?t). We now require ? ? (b? a)t sin(bt)/4 , so that
228
( )
the condition Eq. (6.55) translates to 3x ? (b?a)t bt? b t3 .
4 6
Let us choose t = min[1/f(n), 1/b] = 1/f(n), since otherwise b ? f(n) ? ?H?
and the instance is trivial. We thus know ?H?t ? 1 ? ?/2, t ? 1/f(n), and bt < 1. We
also assume that in the YES case, ?H??E1 ? b?a. This is because otherwise a verifier can
compute Tr(H)n efficiently given the Hamiltonian and accept straightaway if Tr(H)2 2n ? b. This
works since E ? Tr(H)1 n , and by the promise, E1 ? b =? E1 ? a. Therefore, without2
loss of generality, one can assume that the nontrivial instances satisfy b ? Tr(H)
2n
? ?H?, or
?H? ? E1 ? b? a.
Therefore, since 3?E? ? E 5(b?a)1 + , we have24f(n)2
2
?E? ? 5(b? a) (?H? ? E1)E1 + (6.59)
24f(n)2 ( )
=? ? (b? a)
2 5 (b? a)2 1
x =
4f(n)2 6( 4f(n) 1? (6.60))2 6
(b? a)2 b2t2? 1?( ) (6.61)4f(n)2 6
? (b? a) b b
2t2
4f(n) f( 1? ) (6.62)(n) 6
? (b? a)t ? b
3t3
bt , (6.63)
4 6
as required. To sum up, we have shown that 3?E? ? E + 5(b?a)1 2 implies ? ? (b ?24f(n)
2 2 2
a)t sin(bt)/4 ? (b?a) (1?b t /6) 22 ? 5(b?a) .4f(n) 24f(n)2
?
Lemma 57. For probabilities pj : j ? [2n] satisfying ?j pjEj ?( ?E)? and numbers E1 ?
? E tE2 ( . . ). E2n satisfying Ej(t ? [)0, ?/2], the quantity j pj cos
2 j is bounded below by
2
cos2 E1t (1? x) + cos2 E2n t x, where x is given by ?E??E1 .
2 2 E2n?E1
229
Proof. Since the function f(x) = ? cos2(xt/2) is convex for xt/2 ? [0, ?/2), we have
f(E1)(E2n ? Ej) + f(E2n)(Ej ? E1) ? f(Ej). (6.64)
E2n ? E1
Therefore,
f(E1)(E2n ? Ej) + f(E2n)(Ej ? E1)
pj ? p? ?j
f(Ej) (6.65)
E2n E1
=? f(E1)(E2n ? ?E?) + f(E2n)(?E? ? E1) ? pjf(Ej) (6.66)
? ( E)2n ? E1 ( ) j
=? cos2 Ejt ? cos2 E1t E2n ? ?E?pj +
2 2 E
j ( ) 2n ? E1
cos2 E2nt ?E? ? E1 , (6.67)
2 E2n ? E1
which completes the proof.
We now turn to the cases where in addition to the promise of an efficient circuit to
prepare a low-energy state, the Hamiltonian is promised to have a spectral gap ?. For this
case, we can show the following:
Lemma 58(. GS-D)ESCRIPTION-LOCALHAMILTONIAN[a, b, g1, g2] ? GQCMA[c, s, g? , g?1 2] for
2
c? 2s = ? (b?a) and min[g? , g? ] ? 5?2 1 2 , where f(n) is a polynomial upper bound tof(n) 36f(n)
?H?, and ? = min[g1, g2] ? (b? a)3/f(n)2.
Proof. We analyze the same algorithm as the non-gapped case and show that the verification
protocol, with slight modifications, preserves the spectral gap. In particular, in the first
step of the original protocol, the verifier straightaway accepts if Tr(H)/2n ? b or if the
230
upper bound to the norm of the Hamiltonian, f(n) satisfies f(n) ? b. We modify this to
requiring the verifier to accept only if, in addition to the previous conditions, measurement of
the witness register yields the all zeroes string 0w (where w is the size of the witness register).
This has the effect of creating a spectral gap, since in this case only the all-zeroes state is
accepted and all other computational-basis states are rejected.
If the first step does not cause the verifier to accept, the verifier assumes that the witness
state is a description of the circuit V to prepare a low-energy state |??. The verifier then
proceeds to prepare this state and measure its energy using the one-bit phase estimation
pro(tocol. )As shown in the proof of Lemma 55, the protocol has a promise gap c ? s =
? (b?a)
2 .
f(n)2
We now analyze the spectral gap. Let us denote by y the quantity ?E??E1? and by x theE2 E1
quantity ?E??E1? ? y. Any state with energy
3
?E? := ??|H|?? ? E + (b?a)1 2 ? EE n E f(n) 1 + ?2 1
has a large overlap with the ground state:
|? | ?|2 ? ? ?E? ? E1? E1 1 = 1? y. (6.68)
E2 ? E1
Therefore, any state |?? orthogonal to |?? must have an overlap with the ground state that
satisfies |??|E1?|2 ? y. This means that the accept probability for any witness orthogonal
to the one corresponding to the ground-state description is
? ( )
?p?? = pj cos2
Ejt (6.69)
j ( )2 ( )
? y cos2 E1t ? cos2 E2t+ (1 y) . (6.70)
2 2
231
On the other hand, the accept probability of the optimal witness is at least (Lemma 56)
( ) ( )
? 2 E1t 2 E2tp?? ? (1? x) cos + x cos . (6.71)
2 2
The difference in these two is a lower bound for the spectral gap of the accept operator:
( )
E1t
g1, g2 ??p 2??(? ?p?)? ? cos (1? x? y)+2
cos2 E2t (x+ y ? 1) (6.72)
2
(1? x? y)
= (cos(E1t)? cos(E2t)) (6.73)
2
? (1? 2y)(E2 ? E1) sin(E2t). (6.74)
2
Now, we know from the promise that y = ?E??E1 ? (b?a)3 1? 2 ? , and E2 ? E1 +? ? ?.E2 E1 f(n) ? 3
Also, we have chosen t ? 1/f(n) for a polynomial f(n) ? ?H?. Therefore,
min ?[g?1, g?2] ? s(in(?t)6 ) (6.75)
? ?
3 3
(?t?
? t
) (6.76)6 6
? ? ?
3
( ?
? (6.77)
6 f(n) 6f(n)3)
?2 ?2
= 1? (6.78)
6f(n) 6f(n)2
5?2? , (6.79)
36f(n)
since ? ? ?H? ? f(n).
This proves the following results:
232
Corollary 59 (One half of Theorems 26 to 28). .
(1/poly, 1/poly)-GS-DESCRIPTION-LOCALHAMILTONIAN ? PGQCMA.
(1/ exp, 1/ exp)-GS-DESCRIPTION-LOCALHAMILTONIAN ? PreciseEGQCMA.
(1/ exp, 1/poly)-GS-DESCRIPTION-LOCALHAMILTONIAN ? PrecisePGQCMA.
6.10 Details of PP algorithm
In this section we complete the proof of Lemma 32 by expanding upon the PP algo-
rithm. We also prove Lemma 33 by giving a PPP algorithm to precisely compute ground-state
local observables of ?(1/poly)-spectral-gapped Hamiltonians.
Lemma 60. A PP algorithm can decide whether Tr[QqA] ? a? or ? b? when input thresholds a?
and b?, for matrices Q and A of size 2poly(n) ? 2poly(n) satisfying the following properties (we use
the symbol R to denote both matrices Q and A in the following):
1. The norm of the matrix R is upper bounded by a polynomial in n.
2. The matrix R may be written as a polynomial of degree d = poly(n) in terms of matrices
Ri, i ? [m] in the computational basis for m = poly(n), such that:
(a) The matrix elements of each matrix Ri are computable to precision ? in time poly-
nomial in n and log(1/?).
Proof. The quantity Tr(QqA) may be expressed as
? ? ?
?x|QqA|x? = ?x|Q |x1??x1|Q |x2? . . .
x x x1,x2,...xq
?xq?1|Q |xq??xq|A |x? . (6.80)
233
If Q is a polynomial of degree d in terms of matrices R1, . . . Rm for m = poly(n), we
can write it as
?
Q = p Ri1Ri2 imi1i2...im 1 2 . . . Rm , (6.81)
i1,i2,...im?[d]
i1+i2+...im?d
where each tuple (i1, . . . im) specifies a monomial. The number of terms in the polynomial
is bounded above by (d + 1)m = exp[m log(d+ 1)] = O(exp[poly(n)]). We write a term
of Eq. (6.92), ?xj|Q |xj+1?, as
?
?xj|Q |x i1j+1? = pi1i2...im ?xj|R1 |zj,1??
i1,i2,...im?[d]
i1+i2+...im?d
?zj,1|Ri22 |zj,2??zj,2| . . . ?z imj,m?1|Rm |xj+1? . (6.82)
We can further insert resolutions of the identity in Eq. (6.82) to get a sum over yet more
terms. Each term in the resulting sum is a product over polynomially many quantities of
the form ?w1|Rs |w2? for some computational basis states |w1? , |w2? and an index s ? [m].
Each of these can be computed in polynomial time. The number of terms in the final sum
of the form in Eq. (6.92) is still bounded above by 2poly.
From the assumption, the matrix elements of the matrices Ri can be computed to
additive error 2?g(n) in time scaling as O(g(n)). We therefore choose g(n) to be such
that the total additive error resulting from the 2poly many paths in Eq. (6.92) is negligible
compared to (b? ? a?) ? 2poly, where the second term (2poly) corresponds to the number of
terms in the sum. This can be ensured by taking g(n) to be a sufficiently large polynomial.
234
Equation (6.92) is a sum over T = O(2poly) many terms fi, each of which may be
computed in polynomial time. Each term of Eq. (6.92) may b?e interpreted as a path in a
Turing machine. Therefore, a PP machine can decide whether Ti=1 fi is ? a? or ? b? for
some thresholds a?, b? ? a? + ?(2?poly) input to the PP machine. This is seen as follows.
Each term fi is an efficiently computable real-valued function of the trajectory xi , xi , . . . xi0 1 K .
Let amax be an upper bound to the norm of A. The PP machine selects a uniformly random
trajectory and computes fi. It accepts with probability 1? fi? n+1 > 0 and rejects otherwise.2 2 amax
The overall acceptance probability is 1 (1 fii ? n+1 ). In the YES case, this is at leastT 2 2 amax
1 ? a?n+1 , while in the NO case, it is at most 1 ? b
?
n+1 . Since we at least have a2 2 Tamax 2 2 Tamax
separation of 2?n?1/T ? ?(b? ? a?) = ?(2?poly(n)) between the YES and NO instances,
this is a valid PP algorithm.
Lemma 60 applies to the proof of Lemma 32 because the accept operator Q in that
proof is a degree 2T + 3-polynomial in matrices with efficiently computable entries.
For the proof of Lemma 33, we show in Lemma 61 that beginning from the maximally
mixed initial state, imaginary time evolution for ?time? ?i? produces a thermal state with
high enough overlap with the ground state for a suitable ?. Computing local observables in
the obtained thermal state then suffices to get exponentially good estimates of ground-state
local observables for gapped systems. We make the choice of a maximally mixed initial state
in the above because it is guaranteed to have at least overlap 2?n with the ground state.
Lemma 61. For a Hamiltonian H with spectral gap at least ?, let ?? be the thermal state at
temperature 1/(2?). Also let |E1? be the ground state of H and let A be any local observable
satisfying ?A? ? poly(n). Then for ? = ?(n??1), the thermal expectation value satisfies
235
|Tr[??A]? ?E1|A |E1?| ? 2?poly.
Proof. Let the eigenstates of the Hamiltonian be given by |Ei?, i ? [2n], with the eigenvalues
Ei arranged in nondecreasing order. Consider the initial state ? = 1/2n and apply the linear
operation exp(??H), which performs imaginary time evolution for ?time? ?i?:
? ? ?? = exp(??H)? exp(??H), (6.83)
?
up to normalization. The maximally mixed initial state ? = 1 = 1n i n |Ei??Ei| transforms2 2
to the state ??, given by
?? 1 ? 1
?? = = e
??H |E ??E | e??Hi i (6.84)N N 2n
i
1 ?
= e?2?Ei |Ei??Ei| . (6.85)
2nN
i
This state is the same as the thermal state e?2?H at tem?perature 1/(2?) up to normaliza-
tion. The normalization factor N = Tr ?? is given by i e?2?Ei/2n. The overlap of the
normalized state with the ground state is thus
?2?E1
Tr e[?? |E1??E1|] = (6.86)
2nN
?e?2?E1= ( e? (6.87)2?E?ii )?1
= 1 + e?2?(Ei?E1) . (6.88)
i ?=1
Since Ei ? E1 = [? = ?(1]/n
c), if ? is taken to be ?(nd) with d ? c + 1,[we have that ]
e?2?(Ei?E1) ? exp ?2nd?c . This means that the overlap is at least 1/(1+exp n log 2? 2nd?c ) ?
236
[ ]
1 ? exp n log 2? 2nd?c[ ], which means that the trace distance between the normalized
states is ? = O(exp ?nd?c ). Therefore, the choice ? = ?(n??1) suffices to ensure that
the resulting (normalized) state ?? is exponentially close to the ground state. Therefore,
the thermal expectation value of any local observable A with polynomially bounded spectral
norm is also exponentially close to the ground-state expectation value ?E1|A |E1?.
We now show the following lemma about computing unnormalized thermal expecta-
tion values, which is the core subroutine of our PPP algorithm.
[ ]
Lemma 62. A PP algorithm can decide whether Tr e?2?HA ? a? or ? b? when input a sparse
Hamiltonian H , a number ? ? 0, a local observable A, and thresholds a? and b?.
Proof. We express the unnormalized thermal expectation value as a sum over several paths
as follows:
[
A ?
]
2?H
??= Tr e A (6.89)
= ?x| e?2?H |y??y|A |x? (6.90)
?x,y ( )K
? ? | (?2?H)x 1? 2?H + 2(?H)2 + . . . |y??
K!
x,y
?y|A |x? =: A? (6.91)
?K 1 ?
?
= ?x0| ? 2?H |x1??x1| ? 2?H |x2? . . .
k!
k=0 x0,x1,...xk
?x
?k?1
| ? 2?H |xk??xk|A |x0? . (6.92)
T
= fi. (6.93)
i=1
This expression is reminiscent of a Euclidean path integral, although there are some differ-
237
?
ences. In a Euclidean path integral, one Trotterizes themap exp(??H) ? ( i exp(??Hi/r))
r
and use the fact that each term of the Hamiltonian Hi is local in order to compute terms
in the series. In contrast, here we have used the Taylor expansion for exp(??H) and have
inserted resolutions of the identity in order to compute the terms ?x|Hk |y?. Using the
Taylor series allows us to get exponentially small additive error, which is not guaranteed by
Trotterization.
Before we move on, let us analyze the additive error in Eq. (6.91). It is given by:
? (2??H?)
K+1
? ?A? ?O(1). (6.94)
(K + 1)!
By choosing K > 2?e?H?+ f(n) for some polynomial f(n) = O(??H?/n) and f(n) =
?(n), we can ensure that the error is bounded above by ?A? exp[?f(n)]:
K + 1 ? 2?e?H?+ f(n) (6.95)
=? (K + 1) log((K + 1) ? (K +) 1) log(2?e?H?)+
log f(n)(K + 1) 1 + (6.96)
2?e?H?
? f(n)(K + 1() log(2?e?H)?) + (K + 1) ?2?e?H?2
K + 1 f(n) (6.97)
2 2?e?H?
?(K + 1) log(2?e?H?) + ?(f(n)), (6.98)
where we have used the fact that log 2(1 + x) ? x? x for small x and that f(n) = o(??H?).
2
238
Therefore,
( )
log (2??H?)
k+1
? ??(f(n)), giving (6.99)
(K + 1)!
? ? O (?A? exp[?f(n)]) . (6.100)
Proof of Lemma 33. From Lemma 61, we know that the normalized state is exponentially
close to the true ground state. Therefore, deciding whether the ground state has Tr[|????|A] ?
a or ? b is equivalent to deciding whether the unnormalized state has expectation value
Tr[??A] ? a? = N ?est(a+ ?A??) or ? b = Nest(b??A??), whereNest is an estimate of the
normalization of the state and ? the trace distance between the ground state and the thermal
state. To maintain a gap between the YES and NO cases, we need ? < 2?u(n)/?A? for some
polynomial u, which can be satisfied by taking nd?c in Lemma 61 to be ? u(n) + log ?A?.
The norm of A is bounded above by a polynomial in n and therefore is a subleading term.
Since the thresholds a? and b? depend on the normalization, we should compute the
normalization N beforehand. Since the normalization is a special case of Eq. (6.89) with
A = 1, we can use the PP procedure to decide if N ? a1 or N ? a2 for some a1, a2 with
a2 ? a1 = ?(1/ exp). Performing binary search over the interval (0, 1] with polynomially
many queries to the PP oracle, we can estimate the normalization to exponentially small
additive error, giving an estimate Nest.
Therefore, we have shown that a PPP machine can do all the above: compute the
normalization and then compute the thermal expectation value for a low-temperature state.
239
(a) (b)
Figure 6.3: (a) Graph G?x with adjacency matrix A?x, adapted from Fig. 6.1. (b) Graph with
(weighted, directed) adjacency matrix ? ?Ax Ax. Vertices with two self-loops can be thought of
as a single self-loop with weight 2.
Since we have also shown that setting ? = (n/?) suffices to get exponentially small error,
we have shown that the problem is in PPP.
This technique is also applicable to Hamiltonians or Hermitian operators that are not
necessarily local, or even sparse. For example, it can apply to Hermitian operators of the kind
in Lemma 60.
6.11 Turing machine construction for PSPACE-hardness
In this section, we complete the proof of Lemma 39.
Lemma 63 (Lower bound on spectral gap for PSPACE-hard construction). In the NO case,
the construction in the proof of Lemma 39 has a spectral gap of ?(??2max), where ?max is the number
of vertices in the largest subgraph of G?x.
Proof. Recall the form of the graph G?x in the NO case, reproduced here in Fig. 6.3(a).
We first restrict our attention to the subgraph of G?x containing the start and accept con-
240
figurations. The matrix A? ? ?x Ax, when restricted to this subspace, is further composed of
three subspaces, each corresponding to a subgraph, as shown in Fig. 6.3(b). We write
? ?A A?x x = G1 ? G2 ? G3. The block G1 corresponds to the vertices leading to tx (not
including tx). The block G2 corresponds to the vertices {tx} ? {1, . . . t(n)}. Lastly, G3 is
the block with the vertices starting from sx and leading to the reject state, which are the
configurations visited by the Turing machine. We have
? ?
???2 1??? ?
??
?1 2 1 ???
G ? .1 = ??? 1 2 . .?? ?
?
??? , G2 = 1?2??2 , and. .
?? . . . . 1 ????
? 1 2 ??1??1
???1 1??
?
? ????
1 2 1 ????
? .? 1 2 . .? ??
??
G3 = ? ??? . (6.101)? .? . .
. . . 1
??? ?
?
? ?1 2 1???
1 1
?3??3
It may be seen that there is a zero eigenvector (0, 0, . . . , 0, 1,?1, 1, . . . (?1)?3)T , with the
zeros corresponding to the subspaces G1 and G2. We now lower-bound the next-smallest
241
eigenvalue. Let
rn(?) := det?[G3 ? ?1n]? ???1? ? 1? ????
?
??
?
1 2? ? 1 ???
? . ?? 1 2? ? . .? ??
?
= det? ??? (6.102)??
. . . . . . 1 ??
???? ? ???1 2 ? 1 ?
? 1 1? ? ?n?n
???2? ? 1 ??? ???? 1 2? ? 1 ?? ?? ?? . ?? 1 2? ? . .? ?
?
pn(?) := det
?? . . . . . . 1?? ?
??? . (6.103)?
??? ?
?
1 2? ? 1 ???
1 1? ?
n?n
The polynomial pn(?) can be computed exactly [338], and is given by pn(2 ? 2 cos ?) =
sin 1((n+1)?)?sin(n?) cos((n+ )?2 )
sin = cos ? . We can obtain rn(?) in terms of pn(?): rn(?) = (1 ?? ( 2)
?)pn?1(?)? pn?2(?), giving us
rn(?) = fn(?) ( ) ( )
cos (n? 1)? cos (n? 3)?
= (2 cos ? ? 1) ( )2 ? ( )2 , (6.104)
cos ? cos ?
2 2
242
where ? = cos?1(1? ?), or ? = 2? 2 cos ?. The eigenvalues of G3 are related to the roots2
of the characteristic polynomial fn(?) = 0. We can see that ? = 0 is always a root of the
polynomial, giving us the zero eigenvalue (? = 2? 2 cos ? = 0) for the NO case.
Now, it remains to be shown that the next smallest eigenvalue is bounded away from
zero. First consider G1, whose eigenvalues are the roots of the characteristic equation
det[G1 ? ?1?1 ]. The eige(nvalues o)f G1 can be computed in a similar fashion to those of
G and are given by 4 sin2 k?3 ( ) , k ? [n]. The smallest eigenvalue of G1 is therefore at2(?1+1)
least ? 1/? 21 . It is also easily seen that G2 ? 0.
We now come to G3. As we have seen, G3 has a zero eigenvalue. In order to show a
spectral gap for G3, we show that the next root of the polynomial f?3(?) must occur at least
a distance ?(? ?23 ) away. The roots of G3 are given by [378]
( )
?j
?j = 2 + 2 cos , j ? [?3]. (6.105)
?3
Setting j = ?3 gives the zero eigenvalue and j = ?3 ? 1 the first nonzero eigenvalue. The
spectral gap of G3 is therefore
( )
??3?1 = 2? 2(cos
?
) (6.106)?3
= 4 sin2 ?( (6.107)2?3 )
2
? ? ? ?
4
O (6.108)
?2 43( ) ?3
= ? 1/?23 . (6.109)
Finally, we consider other subgraphs that do not contain the start vertex. Just like the
243
analysis of the YES case, the eigenvalues for these are bounded away from 0 by ??2, where ?
is the number of vertices in the subgraph. We have therefore lower bounded the value of the
nonzero eigenvalue in each case, showing that the spectral gap is ?(??2max) = ?(2?poly).
6.12 Complexity of PrecisePGQMA and PreciseEGQMA with asymmetric
spectral gaps
We show here that the promise of asymmetric spectral gaps does not change the com-
plexity class for both PrecisePGQMA and PreciseEGQMA, proving Theorem 42.
Proof of Theorem 42. It is easy to see that GQMA[c, s, g1, g2] ? GQMA[c, s, g1, 0] simply by
ignoring the promise on the NO instance. It remains to show that the same upper bounds
as the symmetric case hold for the asymmetric case too. For the case of c? s = ?(1/ exp),
g2 = ?(1/ exp), we observe that one can also ignore the promise on the YES instance and
obtain containment in PreciseQMA = PSPACE, which equals PreciseEGQMA.
It remains to give an upper bound for the class ?c?s??(1/ exp)GQMA[c, s, g1, 0]. We
g1??(1/poly)
give a PP algorithm for any instance from this class, which implies equivalence of the two
classes.
We are given a description of a circuit, with the promise that the YES case has
?(1/poly) spectral gap for the accept operator Q. We want to decide if ?1(Q) is ? c
(YES) or ? s (NO). The overall PP algorithm is as follows.
1. Use the PQMA[log] algorithm of Ambainis [370] to determine whether an instance has
spectral gap ? ? g1 (YES) or ? g1/2 (NO), for g1 = ?(1/poly).
244
2. If the spectral gap is g1 or larger, run the algorithm in Lemma 60 with Hamiltonian
1?Q and accept or reject according to the answer returned by the algorithm.
3. Otherwise reject.
We claim that the algorithm of Ambainis works not just for local Hamiltonians, but also for
accept operators likeQ. This is because the QMA queries in Ambainis?s algorithm pertain to
whether the ground-state energy (or the minimum eigenvalue 1??1 in this case) is smaller
or larger than a threshold. AQMA verifier can compute the eigenvalue of the accept operator
given an eigenstate, using phase estimation. Therefore, all queries to the oracle about 1??1
are still valid QMA queries. Also, the final query in Ambainis?s algorithm is for the operator
(1?Q)?1+1? (1?Q) on two registers, restricted to the antisymmetric subspace. Since
a QMA verifier can also perform a projection onto the antisymmetric subspace, Ambainis?s
algorithm (i.e. the first step) works to estimate the spectral gap of Q in PQMA[log].
Now, since PQMA[log] ? PP [371], the overall algorithm is a valid PP algorithm, since
the two queries can be made in parallel. To see the correctness, we see that if the instance
has a YES answer, then it has a spectral gap of at least g1 by virtue of the promise. In this
case the spectral gap algorithm would return YES. This ensures that the PP algorithm in
Lemma 60 works correctly and returns the correct answer E1 ? a (YES) or E1 ? b (NO).
The algorithm outputs YES since the instance has low energy.
In the NO case, there may or may not be a spectral gap. If the spectral gap ? ? g1/2
is not large enough, the spectral gap algorithm returns NO. We reject in this case. If the
spectral gap algorithm returns YES, then the spectral gap is at least? ? g1/2 (this includes
the cases when the spectral gap is in the window [g1/2, g1], which is outside of the promise
245
in the spectral gap algorithm). This means that the algorithm in Section 6.10 will work, and
return the correct output (NO). Therefore, we see that ?c?s??(1/ exp)GQMA[c, s, g1, 0] =
g1??(1/poly)
PrecisePGQMA.
We remark that it can be seen that LOCALHAMILTONIAN[a, b, g1, 0] with b ? a =
?(1/ exp) isPrecisePGQMA-complete when the spectral gap g1 is 1/poly andPreciseEGQMA-
complete when g1 is 1/ exp.
246
Appendix A: Mathematical Preliminaries
In this Chapter, we define some notation and some standard terminology in use
throughout this dissertation.
A.1 Notation
Definition 20 (Big-Oh). We define O(g(n)) as the set of all functions f(n) that satisfy
?n ? n0, f(n) ? cg(n) for some constant c > 0 independent of n.
Definition 21 (Big-Omega). We define ?(g(n)) as the set of all functions f(n) such that
g(n) ? f(n), or equivalently, the set of all functions f(n) that satisfy ?n ? n0, g(n) ?
cf(n) for some constant c.
We have f(n) ? ?(g(n)) ? g(n) ? O(f(n)).
Definition 22 (Theta). We define ?(g(n)) as the set of all functions f(n) satisfying ?n ?
n0, c1f(n) ? g(n) ? c2f(n) for some constants c2 > c1 ? 0.
In other words, f(n) ? ?(g(n)) is equivalent to f(n) ? O(g(n)) and f(n) ?
?(g(n)). For all these definitions, we sometimes abuse notation and write f(n) = O(g(n))
instead of f(n) ? O(g(n)).
Lastly, we also have the o (little-oh) and ? (little-omega) notation.
247
Definition 23 (Little-oh). We define o(g(n)) as the set of all functions f(n) satisfying
lim f(n)n?? = 0.g(n)
Definition 24 (Little-omega). The set ?(g(n)) is the set of functions f(n) satisfying g(n) ?
o(f(n)), or equivalently, satisfying lim f(n)n?? ? ?.g(n)
A.2 Notions of error
Broadly speaking, we have two notions of error: additive and multiplicative. Below, we
take the example of computing a target function to some error, but the same notions extend
to that of sampling from a target distribution.
Definition 25 (Additive error). An algorithm computes a function f(x) to additive error ?
if its output g(x) satisfies g(x) ? [f(x)? ?, f(x) + ?].
Definition 26 (Multiplicative error). An algorithm computes a function f(x) to within a
multiplicative factor c > 1 if its output g(x) satisfies
f(x) ? g(x) ? cg(x). (A.1)
c
We are often interested in the setting c ? 1, where we write c = 1 + ?. In this case, we call
? the multiplicative error and the output satisfies
f(x) ? g(x) ? (1 + ?)f(x). (A.2)
1 + ?
=? (1? ?)f(x) ? g(x) ? (1 + ?)f(x). (A.3)
248
A.3 Notions of simulation
For an n-qudit quantum system and a polynomial-sized description of the quantum
state1, we define the following simulation tasks.
Definition 27 (Strong Simulation). In the task of strong simulation, we are given as input a
suitable description of the state |??. The task is to output the expectation value of any desired
observable in the state, namely |??|O |??|2. The output may be exact or approximate, and
either accurate to multiplicative or additive error.
This task includes computing quantities such as full output probabilities, marginals of
output probabilities, and local observables.
Definition 28 (Weak simulation). In weak simulation, the input is again a suitable descrip-
tion of the state |??. The task is to output a sample from the distribution D obtained by
measuring |?? in a suitable basis. The output may be exact or approximate up to multiplica-
tive error (measured in terms of the multiplicative error of each output probability).
This task includes the ability to compute local observables diagonal in the same basis
up to some small additive error.
Definition 29 (Additive-error approximate sampling). In additive-error approximate sam-
pling, the task is similar to the one above. The desired output is now allowed to be approx-
imate up to small additive error (measured in terms of the total variation distance between
the distributions).
1We do not mean an explicit description of the state, but one that can be implicit. For example, the state can
be defined as the output state after Hamiltonian evolution under an efficiently-specifiable Hamiltonian for time
t. Or it can be defined as an equilibrium state of such a Hamiltonian. Ultimately, we assume that physically
interesting states can be described efficiently in this sense.
249
Again, this includes the ability to compute local observables up to some small additive
error.
Definition 30 (Computing local observables). In this task, the input is the same, a suitable
description of the state |??. The desired output is the expectation value | ??|?O|??|
2 of a
k-local observable O, meaning an observable that can be decomposed as O = i Oi, each
term of which is supported on k qudits at most.
250
Appendix B: Complexity-theoretic basics
Here, we give a very brief introduction to the complexity-theoretic definitions and
terminology in this Chapter. The reader is referred to a textbook (e.g. Refs. [379; 380]) for a
more pedagogical exposition. We are generally concerned with decision problems, where the
answer is either ?YES? or ?NO?. These problems can be cast as follows: given an instance x,
the task is to decide if it belongs to the class of YES instances (x ? Ayes, also called ?accept?),
or to the class of NO instances (x ? Ano, also called ?reject?). In principle, there can be
problems where certain instances (for example, ill-defined ones) belong neither to Ayes or
Ano. In such cases, we either allow an algorithm to answer arbitrarily, or we supplant the
problem with a promise that such instances never occur. These are called promise problems.
In complexity theory, one is typically interested in the resources taken to solve various
classes of decision problems. Further, one is interested in how the resource cost scales with
the size of the problem to be solved, which is quantified in terms of the length of the input,
often denoted n. In this dissertation, we use the notation poly(n) to denote any function
that can be upper bounded by O(nc) for some constant c = ?(1). We also denote exp(n)
to be any function 2poly(n). We will omit the dependence on n, which in this dissertation is
usually taken to be the number of particles or qudits.
To characterize the complexity of a problem, we give ?upper? and ?lower? bounds on
251
the complexity of the problem. Upper bounds are statements of the form ?X ? Y?, which
means that the problem X can be solved with access to a solver for the complexity class
Y. For example, Shor [381] proved that FACTORING ? BQP, which means that quantum
computers can factor integers in polynomial time (since quantum computers may be viewed
as ?solvers for the class BQP?). Lower bounds are statements of the form ?X is Y-hard?.
This means that problem X is as hard as any problem in Y. Such statements are often shown
via reductions. One assumes the existence of an oracle, a black box that can solve any instance
of the problem X in one timestep. A reduction is a mapping from the complexity class Y
to the problem X with the property that any problem in Y can be solved by querying the
oracle for X. If such a reduction exists, it implies that the problem X is at least as hard as any
problem in the class Y. If a problem X is both in the class Y and is Y-hard, then it means
that the upper and lower bounds to the problem match. This means that the problem X is
the hardest in the class it belongs to, namely Y. In this case, we say ?X is Y-complete.? or
?X is complete for Y.?. We also denote by YZ the class of problems solvable by a Y machine
with access to an oracle for any problem in Z.
Definition 31 (P). The class P (Polynomial time) is defined to be the class of problems
efficiently solvable in polynomial time on a classical computer. More formally, it is the class
of problemsA = (Ayes, Ano = {0, 1}?\Ayes) such that there exists a polynomial-time Turing
machine M that on input a description x of the instance, outputs a bit M(x) satisfying:
x ? Ayes ? M(x) = 1, and
x ? Ano ? M(x; y) = 0.
We interpret the output M(x) = 1 as the machine accepting (answering YES), and
M(x) = 0 as the machine rejecting (answering NO).
252
We turn to the extremely important class in classical complexity theory, nondetermin-
istic polynomial time (NP). We imagine two parties, Merlin (the prover) and Arthur (the
verifier). The prover would like to convince the verifier that a certain problem instance x is
a YES instance. The prover, who is computationally unbounded, can supply any bit-string
y of length w = poly(n) to the verifier as a ?proof ? or ?witness?. The verifier performs a
polynomial-time computation that reads any bit in the bit-string y as many times as desired
and either accepts or rejects. NP is the class of problems such that a YES answer can be
reliably verified in this way and in case the answer is NO, no matter what string is sent by
the (possibly cheating) prover, the verifier rejects.
Definition 32 (NP). NP is the class of problems A = (Ayes, Ano) such that there is a
polynomial-time Turing machine M that takes as input the instance x and a string y with
|y| = poly(n) such that for every instance,
x ? Ayes ? ? y such that M(x; y) = 1, and
x ? Ano ? ? y, M(x; y) = 0.
One of the most important questions in computer science is the P vs. NP problem,
which asks whether P ?= NP. The widely believed answer to this question is P ?= NP.
Co?ecture 64 (P vs. NP). P ?= NP.
We can generalize the class P to allow a classical computer to use randomness and solve
problems merely with high probability rather than deterministically. This is the class BPP,
which stands for Bounded-error Probabilistic Polynomial time. The error here is measured
via the parameters c (minimum probability of saying ?YES? if the answer is YES) and s
(maximum probability of saying ?YES? if the answer is NO).
253
Definition 33 (BPP[c, s]). BPP[c, s] is the class of problemsA = (Ayes, Ano) such that there
is a polynomial-time Turing machine M that takes as input the instance x and a random
string y ? U from the uniform distribution over bit-strings of length |y| = poly(n) such
that for every instance,
x ? Ayes ? Pry?U [M(x; y) = 1] ? c, and
x ? Ano ? Pry?U [M(x; y) = 1] ? s.
The class BPP is simply defined to be BPP := ?c,s,:c?s?1/poly(n)BPP[c, s]. This turns
out to be the same as the class BPP[1 + 1 1 1poly , ? poly ], which in turn is the same as2 (n) 2 (n)
BPP[2/3, 1/3] and BPP[1 ? 2?r, 2?r] for some r = poly(n). The key to this fact is the
Chernoff bound, which allows us to repeat a computation with small promise gap c ? s and
take a majority vote to get exponentially small error c = 1? 2?poly(n), s = 2?poly(n).
If we remove the constraint that there be a small ?gap? between the completeness and
the soundness, we get a very different class, namely Probabilistic Polynomial time (PP)1.
Definition 34 (PP). PP is the class of problems A = (Ayes, Ano) for which there exists a
polynomial Turing machineM that takes as input the instance x and a random string y ? U
from the uniform distribution over bit-strings of length |y| = poly(n) and outputsM(x; y)
with the following properties:
x ? A 1yes ? Pry?U [M(x; y) = 1] > , and2
x ? Ano ? Pry?U [M(x; y) = 1] < 1 .2
In this definition, the ?1? can be replaced by any constant in (0, 1). The key here is
2
that the promise gap c?s can be exponentially small in n. By virtue of the weak requirement
on what constitutes solving a problem, PP is a large class, including, for instance, all of NP
1Note the absence of the modifier ?Bounded-error?
254
and coNP.
The largest class we deal with in this dissertation is PSPACE.
Definition 35 (PSPACE). PSPACE is the class of problems that can be deterministically
solved by a Turing machine that uses at most a polynomial amount of space.
The space used by a Turing machine is the maximum number of cells in use at any
given time step. In the above bound, we do not explicitly upper bound the amount of time
the Turing machine is allowed to take. Yet we can prove that no more than exponential time
is enough: PSPACE ? EXP, the class of problems solvable deterministically in exponential
time.
B.1 Quantum classes
We now define the class BQP (Bounded-error Quantum Polynomial time), which
is the class of problems solvable in polynomial time (in n) on a quantum computer with
bounded error. More formally,
Definition 36 (BQP[c, s]). BQP[c, s] is the class of promise problems A = (Ayes, Ano) such
that for every instance x, there is a uniformly generated circuit Ux of size poly(n) acting on
the state |0?m? for m = poly(n), with the property that upon measuring the first bit at the
output, o, also called the decision qubit, we have
If x ? Ayes: Pr(o = 1) ? c
If x ? Ano: Pr(o = 1) ? s.
255
Figure B.1: Major complexity classes featuring in this dissertation, and especially
Chapter 6. The classes PSPACE, NPPP, and PP can be defined purely in terms of quantum
computation, and are equal to PreciseQMA, PreciseQCMA, and PreciseBQP, respectively.
All inclusions except P ? BPP are believed to be strict.
In the above, we imagine that a quantum computer applies a circuit Ux that acts on
a standard initial state, measures the first bit at the output, and says YES (?accepts?) or NO
(?rejects?), depending on whether the bit is measured to be in state |1? or |0?. The choice of
the bit to measure at the output is arbitrary. The term uniformly generated circuit means that
given an instance x there is a polynomial-time classical algorithm to generate a description
of the circuit Ux to be applied.
Definition 37. BQP = ?c?s?1/polyBQP[c, s].
The class BQP is the quantum generalization of class BPP (Bounded-Error Proba-
bilistic Polynomial time), the class of problems solvable in polynomial time by a randomized
classical computer.
We now come to the class QMA (Quantum Merlin Arthur), which is a quantum
generalization of NP. Now the prover can send quantum states to the verifier, and the
256
verifier can perform quantum computation on top of it. QMA is the class of problems such
that a YES answer can be reliably verified in this way and in case the answer is NO, no matter
what state is sent by the (possibly cheating) prover, the verifier rejects with high probability.
Just like with BQP, QMA is defined with respect to parameters c and s, which are called
completeness and soundness, respectively.
Definition 38 (QMA[c, s]). QMA[c, s] is the class of problems A = (Ayes, Ano) with the
property that, for every instance x, there exists a uniformly generated circuit Ux with the
following properties: Ux is of size poly(n) and acts on an input state |0??m, together with
a proof (or witness) state |?? of size w supplied by an arbitrarily powerful prover. Both m
and w are bounded by polynomials in n. Upon measuring the decision qubit o of the output
register, the verifier accepts if o = 1, and rejects otherwise. We say A = (Ayes, Ano) is a
QMA[c, s] problem iff
If x ? Ayes: ? |?? such that Pr(o = 1) ? c
If x ? Ano: ? |??, Pr(o = 1) ? s.
QMA is defined as ?c?s?1/polyQMA[c, s].
Lastly, we depict the known inclusions between complexity classes in Fig. B.1. We
also describe here the classes not mentioned so far. The class QCMA is analogous to QMA,
except that the prover sends a classical witness instead of a quantum one. The class NPPP
is a subset of PPPP, since NP ? PP. These classes belong to the counting hierarchy (CH),
which is defined as CH = PP ? PPPP ? . . . [323]. All of these classes are in PSPACE.
257
Bibliography
[1] D. Deutsch, The Fabric of Reality (Penguin UK, 2011).
[2] S. Aaronson, ?Guest Column: NP-complete problems and physical reality,? ACM
SIGACT News 36, 30?52 (2005).
[3] R. P. Feynman, ?Simulating physics with computers,? Int. J. Theor. Phys. 21, 467?488
(1982).
[4] S. P. Jordan, K. S. M. Lee, and J. Preskill, ?Quantum Algorithms forQuantum Field
Theories,? Science 336, 1130?1133 (2012).
[5] D. Harlow and P. Hayden, ?Quantum Computation vs. Firewalls,? Journal of High
Energy Physics 2013, 85 (2013).
[6] S. Aaronson, ?The Complexity ofQuantum States and Transformations: FromQuan-
tum Money to Black Holes,? (2016), arXiv:1607.05256 [gr-qc, physics:quant-ph].
[7] A. Bouland, B. Fefferman, and U. Vazirani, ?Computational Pseudorandomness, the
Wormhole Growth Paradox, and Constraints on the AdS/CFT Duality (Abstract),?
in 11th Innovations in Theoretical Computer Science Conference (ITCS 2020), Leib-
niz International Proceedings in Informatics (LIPIcs), Vol. 151, edited by T. Vidick
(Schloss Dagstuhl?Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2020) pp.
63:1?63:2.
[8] I. H. Kim, E. Tang, and J. Preskill, ?The ghost in the radiation: Robust encodings
of the black hole interior,? Journal of High Energy Physics 2020, 31 (2020).
[9] U. Mahadev, ?Classical Verification of Quantum Computations,? in 2018 IEEE 59th
Annual Symposium on Foundations of Computer Science (FOCS) (2018) pp. 259?267.
[10] G. Valiant and P. Valiant, ?An Automatic Inequality Prover and Instance Optimal
Identity Testing,? SIAM Journal on Computing 46, 429?455 (2017).
[11] B. M. Terhal and D. P. DiVincenzo, ?Adaptive Quantum Computation, Constant
DepthQuantum Circuits and Arthur-Merlin Games,?Quantum Inf. Comput. 4, 134?
145 (2002).
[12] M. J. Bremner, R. Jozsa, and D. J. Shepherd, ?Classical simulation of commuting
quantum computations implies collapse of the polynomial hierarchy,? Proc. R. Soc.
Math. Phys. Eng. Sci. 467, 459?472 (2011).
258
[13] S. Aaronson and A. Arkhipov, ?The computational complexity of linear optics,? in
Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing (ACM
Press, New York, New York, USA, 2011) p. 333.
[14] S. Aaronson and A. Arkhipov, ?The computational complexity of linear optics,? The-
ory Comput. 9, 143?252 (2013).
[15] L. Stockmeyer, ?The complexity of approximate counting,? in Proceedings of the Fif-
teenth Annual ACM Symposium on Theory of Computing, STOC ?83 (Association for
Computing Machinery, New York, NY, USA, 1983) pp. 118?126.
[16] L. Stockmeyer, ?On Approximation Algorithms for # P,? SIAM J. Comput. 14, 849?
861 (1985).
[17] S. Toda, ?PP is as Hard as the Polynomial-Time Hierarchy,? SIAM J. Comput. 20,
865?877 (1991).
[18] B. Fefferman and C. Umans, ?On the Power of Quantum Fourier Sampling,? in 11th
Conference on the Theory of Quantum Computation, Communication and Cryptogra-
phy (TQC 2016), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 61
(Schloss Dagstuhl?Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2016) pp.
1:1?1:19.
[19] M. J. Bremner, A. Montanaro, and D. J. Shepherd, ?Average-case complexity versus
approximate simulation of commuting quantum computations,? Phys. Rev. Lett. 117,
080501 (2016).
[20] E. Farhi and A. W. Harrow, ?Quantum Supremacy through the Quantum Approxi-
mate Optimization Algorithm,? (2016), arXiv:1602.07674 [quant-ph].
[21] B. Fefferman, M. Foss-Feig, and A. V. Gorshkov, ?Exact sampling hardness of Ising
spin models,? Phys. Rev. A 96, 032324 (2017).
[22] X. Gao, S.-T. Wang, and L.-M. Duan, ?Quantum Supremacy for Simulating A
Translation-Invariant Ising Spin Model,? Phys. Rev. Lett. 118, 040502 (2017).
[23] J. Miller, S. Sanders, and A. Miyake, ?Quantum supremacy in constant-time
measurement-based computation: A unified architecture for sampling and verifica-
tion,? (2017), arXiv:1703.11002.
[24] J. Bermejo-Vega, D. Hangleiter, M. Schwarz, R. Raussendorf, and J. Eisert, ?Ar-
chitectures for Quantum Simulation Showing a Quantum Speedup,? Phys. Rev. X 8,
021010 (2018).
[25] T. Morimae, ?Hardness of classically sampling the one-clean-qubit model with con-
stant total variation distance error,? Phys. Rev. A 96, 040302 (2017).
[26] A. Harrow and S. Mehraban, ?Approximate unitary $t$-designs by short ran-
dom quantum circuits using nearest-neighbor and long-range gates,? (2018),
arXiv:1809.06957 [quant-ph].
259
[27] J. Haferkamp, D. Hangleiter, A. Bouland, B. Fefferman, J. Eisert, and J. Bermejo-
Vega, ?Closing Gaps of a Quantum Advantage with Short-Time Hamiltonian Dy-
namics,? Phys. Rev. Lett. 125, 250501 (2020).
[28] A. M. Dalzell, N. Hunter-Jones, and F. G. S. L. Brand?o, ?Random quantum circuits
anti-concentrate in log depth,? (2020), arXiv:2011.12277 [cond-mat, physics:quant-
ph].
[29] S. Nezami, ?Permanent of random matrices from representation theory: Moments,
numerics, concentration, and comments on hardness of boson-sampling,? (2021),
arXiv:2104.06423 [quant-ph].
[30] A. Bouland, B. Fefferman, C. Nirkhe, and U. Vazirani, ?On the complexity and veri-
fication of quantum random circuit sampling,? Nat. Phys. 15, 159?163 (2019).
[31] R. Movassagh, ?Quantum supremacy and random circuits,? (2019), arXiv:1909.06210
[cond-mat, physics:hep-th, physics:math-ph, physics:quant-ph].
[32] A. Bouland, B. Fefferman, Z. Landau, and Y. Liu, ?Noise and the frontier of quantum
supremacy,? (2021), arXiv:2102.01738 [quant-ph].
[33] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas,
S. Boixo, F. G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro,
R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gid-
ney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hart-
mann, A. Ho, M. Hoffmann, T. Huang, T. S. Humble, S. V. Isakov, E. Jeffrey,
Z. Jiang, D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov,
F. Kostritsa, D. Landhuis, M. Lindmark, E. Lucero, D. Lyakh, S. Mandr?, J. R.
McClean, M. McEwen, A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus,
O. Naaman, M. Neeley, C. Neill, M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt,
C. Quintana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank, K. J. Satzinger,
V. Smelyanskiy, K. J. Sung, M. D. Trevithick, A. Vainsencher, B. Villalonga, T. White,
Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Martinis, ?Quantum supremacy
using a programmable superconducting processor,? Nature 574, 505?510 (2019).
[34] H.-S. Zhong, H. Wang, Y.-H. Deng, M.-C. Chen, L.-C. Peng, Y.-H. Luo, J. Qin,
D. Wu, X. Ding, Y. Hu, P. Hu, X.-Y. Yang, W.-J. Zhang, H. Li, Y. Li, X. Jiang,
L. Gan, G. Yang, L. You, Z. Wang, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan,
?Quantum computational advantage using photons,? Science 370, 1460?1463 (2020).
[35] A. Arkhipov, ?Boson Sampling is Robust to Small Errors in the Network Matrix,?
Phys. Rev. A 92, 062326 (2015).
[36] S. Boixo, S. V. Isakov, V. N. Smelyanskiy, R. Babbush, N. Ding, Z. Jiang, M. J.
Bremner, J. M. Martinis, and H. Neven, ?Characterizing quantum supremacy in
near-term devices,? Nat. Phys. 14, 595?600 (2018).
260
[37] S. Aaronson and L. Chen, ?Complexity-theoretic Foundations of Quantum
Supremacy Experiments,? in Proceedings of the 32Nd Computational Complexity Con-
ference, CCC ?17 (Schloss Dagstuhl?Leibniz-Zentrum fuer Informatik, Germany,
2017) pp. 22:1?22:67.
[38] S. Aaronson and S. Gunn, ?On the classical hardness of spoofing linear cross-entropy
benchmarking,? Theory Comput. 16, 1?8 (2020).
[39] B. Barak, C.-N. Chou, and X. Gao, ?Spoofing Linear Cross-Entropy Benchmarking
in Shallow Quantum Circuits,? (2020), arXiv:2005.02421 [quant-ph].
[40] D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, ?Exponen-
tial improvement in precision for simulating sparse Hamiltonians,? Proc. 46th Annu.
ACM Symp. Theory Comput. - STOC 14 , 283?292 (2014).
[41] D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, ?Simulating
Hamiltonian Dynamics with a Truncated Taylor Series,? Phys. Rev. Lett. 114, 090502
(2015).
[42] D. W. Berry, A. M. Childs, and R. Kothari, ?Hamiltonian Simulation with Nearly
Optimal Dependence on all Parameters,? in 2015 IEEE 56th Annual Symposium on
Foundations of Computer Science (IEEE, Berkeley, CA, USA, 2015) pp. 792?809.
[43] G. H. Low and I. L. Chuang, ?Optimal Hamiltonian Simulation by Quantum Signal
Processing,? Physical Review Letters 118, 010501 (2017).
[44] J. Haah, M. B. Hastings, R. Kothari, and G. H. Low, ?Quantum algorithm for
simulating real time evolution of lattice Hamiltonians,? in 2018 IEEE 59th Annual
Symposium on Foundations of Computer Science (FOCS) (IEEE, Paris, 2018) pp. 350?
360.
[45] A. Gily?n, Y. Su, G. H. Low, and N. Wiebe, ?Quantum singular value transformation
and beyond: Exponential improvements for quantum matrix arithmetics,? (2018),
arXiv:1806.01838 [quant-ph].
[46] G. H. Low and I. L. Chuang, ?Hamiltonian Simulation by Qubitization,? Quantum
3, 163 (2019).
[47] A. M. Childs and Y. Su, ?Nearly Optimal Lattice Simulation by Product Formulas,?
Phys. Rev. Lett. 123, 050503 (2019).
[48] A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, ?Theory of Trotter Error
with Commutator Scaling,? Phys. Rev. X 11, 011020 (2021).
[49] P. W. Anderson, ?Absence of Diffusion in Certain Random Lattices,? Phys. Rev. 109,
1492?1505 (1958).
[50] L. Valiant, ?Quantum Circuits That Can Be Simulated Classically in Polynomial
Time,? SIAM J. Comput. 31, 1229?1254 (2002).
261
[51] B. M. Terhal and D. P. DiVincenzo, ?Classical simulation of noninteracting-fermion
quantum circuits,? Phys. Rev. A 65, 032325 (2002).
[52] R. Jozsa and A. Miyake, ?Matchgates and classical simulation of quantum circuits,?
Proc. R. Soc. Math. Phys. Eng. Sci. 464, 3089?3106 (2008).
[53] C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex,
?Gaussian Boson Sampling,? Phys. Rev. Lett. 119, 170501 (2017).
[54] R. Kruse, C. S. Hamilton, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, ?A
detailed study of Gaussian Boson Sampling,? Phys. Rev. A 100, 032326 (2019).
[55] D. Poulin and P. Wocjan, ?Sampling from the Thermal Quantum Gibbs State and
Evaluating Partition Functions with a Quantum Computer,? Phys. Rev. Lett. 103,
220502 (2009).
[56] C.-F. Chiang and P. Wocjan, ?Quantum Algorithm for Preparing Thermal Gibbs
States - Detailed Analysis,? NATO Sci. Peace Secur. Ser. Inf. Commun. Secur. , 138?
147 (2010).
[57] F. G. Brandao and K. M. Svore, ?Quantum Speed-Ups for Solving Semidefinite Pro-
grams,? in 2017 IEEE 58th Annual Symposium on Foundations of Computer Science
(FOCS) (2017) pp. 415?426.
[58] S. Arora, E. Hazan, and S. Kale, ?The Multiplicative Weights Update Method: A
Meta-Algorithm and Applications,? Theory of Computing 8, 121?164 (2012).
[59] A. Kitaev, A. Shen, and M. Vyalyi, Classical and Quantum Computation, Graduate
Studies in Mathematics, Vol. 47 (American Mathematical Society, Providence, Rhode
Island, 2002).
[60] R. Oliveira and B. M. Terhal, ?The complexity of quantum spin systems on a two-
dimensional square lattice,? Quantum Inf. Comput. 8, 0900?0924 (2008).
[61] D. Aharonov, D. Gottesman, S. Irani, and J. Kempe, ?The power of quantum systems
on a line,? Commun. Math. Phys. 287, 41?65 (2009).
[62] S. Bravyi, D. P. DiVincenzo, R. I. Oliveira, and B. M. Terhal, ?The Complexity
of Stoquastic Local Hamiltonian Problems,? Quantum Inf. Comput. 8, 0361?0385
(2008).
[63] D. Gottesman and S. Irani, ?TheQuantum and Classical Complexity of Translationally
Invariant Tiling and Hamiltonian Problems,? (2009), arXiv:0905.2419 [quant-ph].
[64] A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-
Guzik, and J. L. O?Brien, ?A variational eigenvalue solver on a photonic quantum
processor,? Nat. Commun. 5, 4213 (2014).
[65] J. Preskill, ?Quantum computing and the entanglement frontier,? (2012),
arXiv:1203.5813 [cond-mat, physics:quant-ph].
262
[66] R. Jozsa and M. V. den Nest, ?Classical simulation complexity of extended Clifford
circuits,? (2013), arXiv:1305.6190 [quant-ph].
[67] A. W. Harrow and A. Montanaro, ?Quantum computational supremacy,? Nature 549,
203?209 (2017).
[68] A. Neville, C. Sparrow, R. Clifford, E. Johnston, P. M. Birchall, A. Montanaro, and
A. Laing, ?Classical boson sampling algorithms with superior performance to near-
term experiments,? Nat. Phys. 13, 1153?1157 (2017).
[69] P. Clifford and R. Clifford, ?The Classical Complexity of Boson Sampling,? in Pro-
ceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms,
SODA ?18 (Society for Industrial and Applied Mathematics, Philadelphia, PA, USA,
2018) pp. 146?155.
[70] A. M. Dalzell, A. W. Harrow, D. E. Koh, and R. L. La Placa, ?How many qubits are
needed for quantum computational supremacy?? Quantum 4, 264 (2020).
[71] S. Kirkpatrick and B. Selman, ?Critical Behavior in the Satisfiability of Random
Boolean Expressions,? Science 264, 1297?1301 (1994).
[72] C. R. Laumann, R. Moessner, A. Scardicchio, and S. L. Sondhi, ?RandomQuantum
Satisfiabiilty,? Quantum Inf. Comput. 10, 1?15 (2010).
[73] K. P. Seshadreesan, J. P. Olson, K. R. Motes, P. P. Rohde, and J. P. Dowling, ?Boson
sampling with displaced single-photon Fock states versus single-photon-added coher-
ent states: The quantum-classical divide and computational-complexity transitions in
linear optics,? Phys. Rev. A 91, 022334 (2015).
[74] M. Heyl, A. Polkovnikov, and S. Kehrein, ?Dynamical Quantum Phase Transitions
in the Transverse-Field Ising Model,? Phys. Rev. Lett. 110, 135704 (2013).
[75] M. Heyl, ?Dynamical quantum phase transitions: A review,? (2017),
arXiv:1709.07461 [cond-mat, physics:quant-ph].
[76] E. B. Rozenbaum, S. Ganeshan, and V. Galitski, ?Lyapunov Exponent and Out-of-
Time-Ordered Correlator?s Growth Rate in a Chaotic System,? Phys. Rev. Lett. 118,
086801 (2017).
[77] E. H. Lieb and D. W. Robinson, ?The finite group velocity of quantum spin systems,?
Commun. Math. Phys. 28, 251?257 (1972).
[78] S. Aaronson, ?A linear-optical proof that the permanent is #P-hard,? Proc. R. Soc.
Math. Phys. Eng. Sci. 467, 3393?3405 (2011).
[79] M. Friesdorf, A. H. Werner, W. Brown, V. B. Scholz, and J. Eisert, ?Many-body
localisation implies that eigenvectors are matrix-product states,? Phys. Rev. Lett. 114,
170505 (2015).
263
[80] Y. Huang, ?Efficient simulation of many-body localized systems,? (2015),
arXiv:1508.04756 [cond-mat, physics:quant-ph].
[81] F. Pollmann, V. Khemani, J. I. Cirac, and S. L. Sondhi, ?Efficient variational di-
agonalization of fully many-body localized Hamiltonians,? Phys. Rev. B 94, 041116
(2016).
[82] X. Yu, D. Pekker, and B. K. Clark, ?Finding matrix product state representations
of highly-excited eigenstates of many-body localized Hamiltonians,? Phys. Rev. Lett.
118, 017201 (2017).
[83] A. Chapman and A. Miyake, ?Classical simulation of quantum circuits by dynami-
cal localization: Analytic results for Pauli-observable propagation in time-dependent
disorder,? (2017), arXiv:1704.04405 [cond-mat, physics:quant-ph].
[84] J. Eisert and D. Gross, ?SupersonicQuantum Communication,? Phys. Rev. Lett. 102,
240501 (2009).
[85] M. B. Hastings, ?Locality inQuantum Systems,? (2010), arXiv:1008.5137 [math-ph,
quant-ph].
[86] J. Fr?hlich, F. Martinelli, E. Scoppola, and T. Spencer, ?Constructive proof of lo-
calization in the Anderson tight binding model,? Commun. Math. Phys. 101, 21?46
(1985).
[87] E. Hamza, R. Sims, and G. Stolz, ?Dynamical Localization in Disordered Quantum
Spin Systems,? Commun. Math. Phys. 315, 215?239 (2012).
[88] M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, ?Experimental realization of
any discrete unitary operator,? Phys. Rev. Lett. 73, 58?61 (1994).
[89] W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A.
Walsmley, ?Optimal design for universal multiport interferometers,? Optica 3, 1460
(2016).
[90] M. Jerrum, A. Sinclair, and E. Vigoda, ?A Polynomial-time Approximation Algo-
rithm for the Permanent of a Matrix with Nonnegative Entries,? J. ACM 51, 671?697
(2004).
[91] F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert,
C.W. Clark, B. R. Miller, and B. V. Saunders, ?NIST Digital Library of Mathematical
Functions,? .
[92] A. Deshpande, B. Fefferman, M. C. Tran, M. Foss-Feig, and A. V. Gorshkov, un-
published .
[93] N. Maskara, A. Deshpande, M. C. Tran, A. Ehrenberg, B. Fefferman, and A. V.
Gorshkov, ?Complexity phase diagram for interacting and long-range bosonic Hamil-
tonians,? (2019), arXiv:1906.04178.
264
[94] D. J. Brod, ?Complexity of simulating constant-depth BosonSampling,? Phys. Rev. A
91, 042316 (2015).
[95] G. Muraleedharan, A. Miyake, and I. H. Deutsch, ?Quantum computational
supremacy in the sampling of bosonic random walkers on a one-dimensional lattice,?
New J. Phys. 21, 055003 (2018).
[96] A. Deshpande, B. Fefferman, M. C. Tran, M. Foss-Feig, and A. V. Gorshkov, ?Dy-
namical Phase Transitions in Sampling Complexity,? Phys. Rev. Lett. 121, 030501
(2018).
[97] S. Banerjee and E. Altman, ?Solvable model for a dynamical quantum phase transition
from fast to slow scrambling,? Phys. Rev. B 95, 134302 (2017).
[98] R.-Q. He and Z.-Y. Lu, ?Characterizing many-body localization by out-of-time-
ordered correlation,? Phys. Rev. B 95, 054201 (2017).
[99] S. V. Syzranov, A. V. Gorshkov, and V. M. Galitski, ?Interaction-induced transition
in the quantum chaotic dynamics of a disordered metal,? (2017), arXiv:1709.09296
[cond-mat, physics:quant-ph].
[100] P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, ?Chaos in quantum channels,? J.
High Energ. Phys. 2016, 4 (2016).
[101] D. A. Roberts and B. Yoshida, ?Chaos and complexity by design,? J. High Energ.
Phys. 2017, 121 (2017).
[102] Y. Huang, Y.-L. Zhang, and X. Chen, ?Out-of-time-ordered correlators in many-
body localized systems,? Ann. Phys. 529, 1600318 (2017).
[103] R. Fan, P. Zhang, H. Shen, and H. Zhai, ?Out-of-time-order correlation for many-
body localization,? Sci. Bull. 62, 707?711 (2017).
[104] Y. Chen, ?Universal Logarithmic Scrambling in Many Body Localization,? (2016),
arXiv:1608.02765 [cond-mat].
[105] B. Swingle and D. Chowdhury, ?Slow scrambling in disordered quantum systems,?
Phys. Rev. B 95, 060201 (2017).
[106] S. Sachdev and J. Ye, ?Gapless spin-fluid ground state in a random quantum Heisen-
berg magnet,? Phys. Rev. Lett. 70, 3339?3342 (1993).
[107] A. Kitaev, ?A simple model of quantum holography,? (2015).
[108] J. Maldacena and D. Stanford, ?Remarks on the Sachdev-Ye-Kitaev model,? Phys.
Rev. D 94, 106002 (2016).
[109] J. Maldacena, S. H. Shenker, and D. Stanford, ?A bound on chaos,? J. High Energy
Phys. 2016, 106 (2016).
265
[110] Y. Sekino and L. Susskind, ?Fast scramblers,? J. High Energy Phys. 2008, 065?065
(2008).
[111] N. Lashkari, D. Stanford, M. Hastings, T. Osborne, and P. Hayden, ?Towards the
fast scrambling conjecture,? J. High Energy Phys. 2013, 22 (2013).
[112] B. Swingle, ?Black holes and the limits of quantum information processing,? XRDS
Crossroads ACM Mag. Stud. 23, 52?56 (2016).
[113] A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y. Zhao, ?Holographic
Complexity Equals Bulk Action?? Phys. Rev. Lett. 116, 191301 (2016).
[114] A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y. Zhao, ?Complexity,
action, and black holes,? Phys. Rev. D 93, 086006 (2016).
[115] S. Aaronson and D. J. Brod, ?BosonSampling with Lost Photons,? Phys. Rev. A 93,
012335 (2016).
[116] A. R. Brown and L. Susskind, ?Second law of quantum complexity,? Phys. Rev. D 97,
086015 (2018).
[117] U. Marzolino and T. Prosen, ?Computational complexity of nonequilibrium steady
states of quantum spin chains,? Phys. Rev. A 93, 032306 (2016).
[118] L. Amico, R. Fazio, A. Osterloh, and V. Vedral, ?Entanglement in many-body sys-
tems,? Rev. Mod. Phys. 80, 517?576 (2008).
[119] B. Zeng, X. Chen, D.-L. Zhou, and X.-G. Wen, ?Quantum Information Meets
Quantum Matter ? From Quantum Entanglement to Topological Phase in Many-
Body Systems,? (2015), arXiv:1508.02595 [cond-mat, physics:quant-ph].
[120] J. H. Bardarson, F. Pollmann, and J. E. Moore, ?Unbounded Growth of Entanglement
in Models of Many-Body Localization,? Phys. Rev. Lett. 109, 017202 (2012).
[121] Z.-C. Yang, A. Hamma, S. M. Giampaolo, E. R. Mucciolo, and C. Chamon, ?En-
tanglement complexity in quantum many-body dynamics, thermalization, and local-
ization,? Phys. Rev. B 96, 020408 (2017).
[122] O. Morsch and M. Oberthaler, ?Dynamics of Bose-Einstein condensates in optical
lattices,? Rev. Mod. Phys. 78, 179?215 (2006).
[123] I. Bloch, J. Dalibard, and S. Nascimb?ne, ?Quantum simulations with ultracold quan-
tum gases,? Nat. Phys. 8, 267?276 (2012).
[124] I. Buluta and F. Nori, ?Quantum Simulators,? Science 326, 108?111 (2009).
[125] M. H. Devoret and R. J. Schoelkopf, ?Superconducting Circuits for Quantum Infor-
mation: An Outlook,? Science 339, 1169?1174 (2013).
266
[126] W. S. Bakr, J. I. Gillen, A. Peng, S. F?lling, and M. Greiner, ?A quantum gas mi-
croscope for detecting single atoms in a Hubbard-regime optical lattice,? Nature 462,
74 (2009).
[127] W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Foelling, L. Pollet,
and M. Greiner, ?Probing the Superfluid to Mott Insulator Transition at the Single
Atom Level,? Science 329, 547?550 (2010).
[128] J. F. Sherson, C.Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr, ?Single-
atom-resolved fluorescence imaging of an atomic Mott insulator,? Nature 467, 68
(2010).
[129] L. Mazza, D. Rossini, R. Fazio, and M. Endres, ?Detecting two-site spin-
entanglement in many-body systems with local particle-number fluctuations,? New
J. Phys. 17, 013015 (2015).
[130] M. Miranda, R. Inoue, Y. Okuyama, A. Nakamoto, and M. Kozuma, ?Site-resolved
imaging of ytterbium atoms in a two-dimensional optical lattice,? Phys. Rev. A 91,
063414 (2015).
[131] R. Yamamoto, J. Kobayashi, K. Kato, T. Kuno, Y. Sakura, and Y. Takahashi, ?Site-
resolved imaging of single atoms with a Faraday quantum gas microscope,? Phys. Rev.
A 96, 033610 (2017).
[132] C. Weitenberg, M. Endres, J. F. Sherson, M. Cheneau, P. Schau?, T. Fukuhara,
I. Bloch, and S. Kuhr, ?Single-spin addressing in an atomic Mott insulator,? Nature
471, 319 (2011).
[133] Y. Wang, X. Zhang, T. A. Corcovilos, A. Kumar, and D. S. Weiss, ?Coherent Ad-
dressing of Individual Neutral Atoms in a 3D Optical Lattice,? Phys. Rev. Lett. 115,
043003 (2015).
[134] T. Fukuhara, A. Kantian, M. Endres, M. Cheneau, P. Schau?, S. Hild, D. Bellem,
U. Schollw?ck, T. Giamarchi, C. Gross, I. Bloch, and S. Kuhr, ?Quantum dynamics
of a single, mobile spin impurity,? Nat. Phys. 9, 235?241 (2013).
[135] S. Paul, P. R. Johnson, and E. Tiesinga, ?Hubbard model for ultracold bosonic
atoms interacting via zero-point-energy\char21{}induced three-body interactions,?
Phys. Rev. A 93, 043616 (2016).
[136] K. Xu, T. Mukaiyama, J. R. Abo-Shaeer, J. K. Chin, D. E. Miller, and W. Ketterle,
?Formation ofQuantum-Degenerate SodiumMolecules,? Phys. Rev. Lett. 91, 210402
(2003).
[137] J. Herbig, ?Preparation of a Pure Molecular Quantum Gas,? Science 301, 1510?1513
(2003).
[138] S. D?rr, T. Volz, A. Marte, and G. Rempe, ?Observation of Molecules Produced from
a Bose-Einstein Condensate,? Phys. Rev. Lett. 92, 020406 (2004).
267
[139] A. M. Kaufman, B. J. Lester, C. M. Reynolds, M. L. Wall, M. Foss-Feig, K. R. A.
Hazzard, A. M. Rey, and C. A. Regal, ?Hong-Ou-Mandel atom interferometry in
tunnel-coupled optical tweezers,? Science 345, 306?309 (2014).
[140] M. Endres, H. Bernien, A. Keesling, H. Levine, E. R. Anschuetz, A. Krajenbrink,
C. Senko, V. Vuletic, M. Greiner, and M. D. Lukin, ?Atom-by-atom assembly of
defect-free one-dimensional cold atom arrays,? Science 354, 1024?1027 (2016).
[141] H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S.
Zibrov, M. Endres, M. Greiner, V. Vuleti?, and M. D. Lukin, ?Probing many-body
dynamics on a 51-atom quantum simulator,? Nature 551, 579?584 (2017).
[142] E. Flurin, V. V. Ramasesh, S. Hacohen-Gourgy, L. S. Martin, N. Y. Yao, and I. Sid-
diqi, ?Observing Topological Invariants Using Quantum Walk in Superconducting
Circuits,? Phys. Rev. X 7, 031023 (2017).
[143] S. Hacohen-Gourgy, V. V. Ramasesh, C. De Grandi, I. Siddiqi, and S. M. Girvin,
?Cooling and Autonomous Feedback in a Bose-Hubbard chain with Attractive Inter-
actions,? Phys. Rev. Lett. 115, 240501 (2015).
[144] X.-H. Deng, C.-Y. Lai, and C.-C. Chien, ?Superconducting circuit simulator of
Bose-Hubbard model with a flat band,? Phys. Rev. B 93, 054116 (2016).
[145] P. Roushan, C. Neill, J. Tangpanitanon, V. M. Bastidas, A. Megrant, R. Barends,
Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. Fowler, B. Foxen, M. Giustina, E. Jef-
frey, J. Kelly, E. Lucero, J. Mutus, M. Neeley, C.Quintana, D. Sank, A. Vainsencher,
J. Wenner, T. White, H. Neven, D. G. Angelakis, and J. Martinis, ?Spectral signa-
tures of many-body localization with interacting photons,? Science 358, 1175?1179
(2017).
[146] Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell,
Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J.
O?Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R.
Geller, A. N. Cleland, and J. M. Martinis, ?Qubit architecture with high coherence
and fast tunable coupling,? Phys. Rev. Lett. 113, 220502 (2014).
[147] S. Aaronson and D. Gottesman, ?Improved Simulation of Stabilizer Circuits,? Phys.
Rev. A 70, 052328 (2004).
[148] X. Ni and M. V. den Nest, ?Commuting quantum circuits: Efficient classical simula-
tions versus hardness results,? (2012), arXiv:1204.4570 [quant-ph].
[149] S. Lloyd, ?Almost Any Quantum Logic Gate is Universal,? Phys. Rev. Lett. 75, 346?
349 (1995).
[150] D. Deutsch, A. Barenco, and A. Ekert, ?Universality in Quantum Computation,?
Proc. R. Soc. Math. Phys. Eng. Sci. 449, 669?677 (1995).
268
[151] M. J. Bremner, C. M. Dawson, J. L. Dodd, A. Gilchrist, A. W. Harrow, D. Mortimer,
M. A. Nielsen, and T. J. Osborne, ?Practical Scheme forQuantum Computation with
Any Two-Qubit Entangling Gate,? Phys. Rev. Lett. 89, 247902 (2002).
[152] M. J. Bremner, A. Montanaro, and D. J. Shepherd, ?Achieving quantum supremacy
with sparse and noisy commuting quantum computations,? Quantum 1, 8 (2017).
[153] A. M. Childs, D. Leung, L. Man?inska, and M. Ozols, ?Characterization of universal
two-qubit Hamiltonians,? Quantum Inf. Comput. 11, pp0019?0039 (2011).
[154] A. Bouland, L. Man?inska, and X. Zhang, ?Complexity classification of two-qubit
commuting hamiltonians,? in 31st Conference on Computational Complexity (CCC
2016), Leibniz International Proceedings in Informatics (LIPIcs) (2016) pp. 28:1?
28:33.
[155] D. Hangleiter, J. Bermejo-Vega, M. Schwarz, and J. Eisert, ?Anticoncentration the-
orems for schemes showing a quantum speedup,? Quantum 2, 65 (2018).
[156] R. Raussendorf, C. Okay, D.-S. Wang, D. T. Stephen, and H. P. Nautrup, ?Compu-
tationally Universal Phase of Quantum Matter,? Phys. Rev. Lett. 122, 090501 (2019).
[157] E. Mossel, D. Weitz, and N. Wormald, ?On the hardness of sampling independent
sets beyond the tree threshold,? Probab. Theory Relat. Fields 143, 401?439 (2009).
[158] S. Vijay, ?Measurement-Driven Phase Transition within a Volume-Law Entangled
Phase,? (2020), arXiv:2005.03052 [cond-mat, physics:hep-th, physics:quant-ph].
[159] M. A. Norcia, A.W. Young, and A.M. Kaufman, ?Microscopic Control and Detection
of Ultracold Strontium in Optical-Tweezer Arrays,? Phys. Rev. X 8, 041054 (2018).
[160] C. Neill, P. Roushan, K. Kechedzhi, S. Boixo, S. V. Isakov, V. Smelyanskiy,
A. Megrant, B. Chiaro, A. Dunsworth, K. Arya, R. Barends, B. Burkett, Y. Chen,
Z. Chen, A. Fowler, B. Foxen, M. Giustina, R. Graff, E. Jeffrey, T. Huang, J. Kelly,
P. Klimov, E. Lucero, J. Mutus, M. Neeley, C. Quintana, D. Sank, A. Vainsencher,
J. Wenner, T. C. White, H. Neven, and J. M. Martinis, ?A blueprint for demonstrat-
ing quantum supremacy with superconducting qubits,? Science 360, 195?199 (2018).
[161] M. Saffman, T. G. Walker, and K. M?lmer, ?Quantum information with Rydberg
atoms,? Rev. Mod. Phys. 82, 2313?2363 (2010).
[162] J. W. Britton, B. C. Sawyer, A. C. Keith, C.-C. J. Wang, J. K. Freericks, H. Uys,
M. J. Biercuk, and J. J. Bollinger, ?Engineered two-dimensional Ising interactions
in a trapped-ion quantum simulator with hundreds of spins,? Nature 484, 489?492
(2012).
[163] N. Yao, L. Jiang, A. Gorshkov, P. Maurer, G. Giedke, J. Cirac, and M. Lukin, ?Scal-
able architecture for a room temperature solid-state quantum information processor,?
Nat. Commun. 3, 1?8 (2012).
269
[164] B. Yan, S. A. Moses, B. Gadway, J. P. Covey, K. R. A. Hazzard, A. M. Rey, D. S. Jin,
and J. Ye, ?Observation of dipolar spin-exchange interactions with lattice-confined
polar molecules,? Nature 501, 521?525 (2013).
[165] J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E.
Chang, ?Quantum many-body models with cold atoms coupled to photonic crystals,?
Nat. Photonics 9, 326?331 (2015).
[166] A. M. Childs, D. Gosset, and Z. Webb, ?Universal Computation by Multiparticle
Quantum Walk,? Science 339, 791?794 (2013).
[167] S. Janson, T. Luczak, and A. Ruci?ski, Random Graphs, Wiley-Interscience Series in
Discrete Mathematics and Optimization (John Wiley, New York, 2000).
[168] M. B. Hastings and T. Koma, ?Spectral Gap and Exponential Decay of Correlations,?
Commun. Math. Phys. 265, 781?804 (2006).
[169] Z.-X. Gong, M. Foss-Feig, S. Michalakis, and A. V. Gorshkov, ?Persistence of locality
in systems with power-law interactions,? Phys. Rev. Lett. 113, 030602 (2014).
[170] M. Foss-Feig, Z.-X. Gong, C. W. Clark, and A. V. Gorshkov, ?Nearly Linear Light
Cones in Long-Range Interacting Quantum Systems,? Phys. Rev. Lett. 114, 157201
(2015).
[171] M. C. Tran, A. Y. Guo, Y. Su, J. R. Garrison, Z. Eldredge, M. Foss-Feig, A. M.
Childs, and A. V. Gorshkov, ?Locality and digital quantum simulation of power-law
interactions,? Phys. Rev. X 9, 031006 (2019).
[172] A. Y. Guo, M. C. Tran, A. M. Childs, A. V. Gorshkov, and Z.-X. Gong, ?Signaling
and Scrambling with Strongly Long-Range Interactions,? (2019), arXiv:1906.02662
[quant-ph].
[173] M. C. Tran, C.-F. Chen, A. Ehrenberg, A. Y. Guo, A. Deshpande, Y. Hong, Z.-X.
Gong, A. V. Gorshkov, and A. Lucas, ?Hierarchy of linear light cones with long-range
interactions,? Phys. Rev. X 10, 031009 (2020).
[174] B. Peropadre, A. Aspuru-Guzik, and J. J. Garc?a-Ripoll, ?Equivalence between spin
Hamiltonians and boson sampling,? Phys. Rev. A 95, 032327 (2017).
[175] M. S. Underwood and D. L. Feder, ?Bose-Hubbard model for universal quantum
walk-based computation,? Phys. Rev. A 85, 052314 (2012).
[176] T. J. Osborne, ?Efficient Approximation of the Dynamics of One-DimensionalQuan-
tum Spin Systems,? Phys. Rev. Lett. 97, 157202 (2006).
[177] R. Garc?a-Patr?n, J. J. Renema, and V. Shchesnovich, ?Simulating boson sampling
in lossy architectures,? Quantum 3, 169 (2019).
[178] N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, ?Computational Complexity
of Projected Entangled Pair States,? Phys. Rev. Lett. 98, 140506 (2007).
270
[179] J. Haferkamp, D. Hangleiter, J. Eisert, and M. Gluza, ?Contracting projected entan-
gled pair states is average-case hard,? Phys. Rev. Research 2, 013010 (2020).
[180] D. Gottesman, ?The Heisenberg Representation of Quantum Computers,? (1998),
arXiv:quant-ph/9807006.
[181] F. Verstraete and J. I. Cirac, ?Mapping local Hamiltonians of fermions to local Hamil-
tonians of spins,? J. Stat. Mech. 2005, P09012?P09012 (2005).
[182] S.-K. Chu, G. Zhu, J. R. Garrison, Z. Eldredge, A. V. Curiel, P. Bienias, I. B. Spiel-
man, and A. V. Gorshkov, ?Scale-Invariant Continuous Entanglement Renormaliza-
tion of a Chern Insulator,? Phys. Rev. Lett. 122, 120502 (2019).
[183] B. Gadway, ?An atom optics approach to studying lattice transport phenomena,? Phys.
Rev. A 92, 043606 (2015).
[184] G. M. Crosswhite, A. C. Doherty, and G. Vidal, ?Applying matrix product operators
to model systems with long-range interactions,? Phys. Rev. B 78, 035116 (2008).
[185] D. Barredo, S. de L?s?leuc, V. Lienhard, T. Lahaye, and A. Browaeys, ?An atom-by-
atom assembler of defect-free arbitrary two-dimensional atomic arrays,? Science 354,
1021?1023 (2016).
[186] S. Korenblit, D. Kafri, W. C. Campbell, R. Islam, E. E. Edwards, Z.-X. Gong, G.-D.
Lin, L.-M. Duan, J. Kim, K. Kim, and C. Monroe, ?Quantum simulation of spin
models on an arbitrary lattice with trapped ions,? New J. Phys. 14, 095024 (2012).
[187] R. Islam, C. Senko, W. C. Campbell, S. Korenblit, J. Smith, A. Lee, E. E. Ed-
wards, C.-C. J. Wang, J. K. Freericks, and C. Monroe, ?Emergence and Frustration
of Magnetism with Variable-Range Interactions in a Quantum Simulator,? Science
340, 583?587 (2013).
[188] M. Oszmaniec and Z. Zimbor?s, ?Universal extensions of restricted classes of quantum
operations,? Phys. Rev. Lett. 119, 220502 (2017).
[189] A. Bouland and M. Ozols, ?Trading Inverses for an Irrep in the Solovay-Kitaev The-
orem,? in Proceedings of the 13th Conference on the Theory of Quantum Computation,
Communication and Cryptography (TQC 2018), Leibniz International Proceedings in
Informatics (LIPIcs), Vol. 111 (Schloss Dagstuhl?Leibniz-Zentrum fuer Informatik,
Dagstuhl, Germany, 2018) pp. 6:1?6:15.
[190] Z. Eldredge, Z.-X. Gong, J. T. Young, A. H. Moosavian, M. Foss-Feig, and A. V.
Gorshkov, ?Fast Quantum State Transfer and Entanglement Renormalization Using
Long-Range Interactions,? Phys. Rev. Lett. 119, 170503 (2017).
[191] D. J. Brod and A. M. Childs, ?The computational power of matchgates and the XY
interaction on arbitrary graphs,? Quantum Inf. Comput. 14, 0901?0916 (2013).
271
[192] A. Lakshminarayan, S. Tomsovic, O. Bohigas, and S. N. Majumdar, ?Extreme statis-
tics of complex random and quantum chaotic states,? Phys. Rev. Lett. 100, 044103
(2008).
[193] Y. Li, X. Chen, and M. P. A. Fisher, ?Quantum Zeno effect and the many-body
entanglement transition,? Phys. Rev. B 98, 205136 (2018).
[194] B. Skinner, J. Ruhman, and A. Nahum, ?Measurement-Induced Phase Transitions in
the Dynamics of Entanglement,? Phys. Rev. X 9, 031009 (2019).
[195] A. Chan, R. M. Nandkishore, M. Pretko, and G. Smith, ?Unitary-projective entan-
glement dynamics,? Phys. Rev. B 99, 224307 (2019).
[196] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information:
10th Anniversary Edition (Cambridge University Press, 2010).
[197] S. Lloyd, ?Universal Quantum Simulators,? Science 273, 1073?1078 (1996).
[198] E. Knill, ?Fermionic linear optics and matchgates,? arXiv:quant-ph/0108033 (2001).
[199] M. Wimmer, ?Algorithm 923: Efficient numerical computation of the pfaffian for
dense and banded skew-symmetric matrices,? ACM T. Math. Software 38, 30 (2012).
[200] H.-P. Breuer and F. Petruccione, The theory of open quantum systems (Oxford University
Press, 2007).
[201] T. Prosen, ?Spectral theorem for the lindblad equation for quadratic open fermionic
systems,? J. Stat. Mech.-Theory. E. 2010, P07020 (2010).
[202] S. Bravyi and R. K?nig, ?Classical simulation of dissipative fermionic linear optics,?
Quantum Information & Computation 12, 925?943 (2012).
[203] C. Gardiner, P. Zoller, and P. Zoller, Quantum noise, Vol. 56 (Springer, 2004).
[204] A. Chenu, M. Beau, J. Cao, and A. del Campo, ?Quantum simulation of generic
many-body open system dynamics using classical noise,? Phys. Rev. Lett. 118, 140403
(2017).
[205] B. Misra and E. G. Sudarshan, ?The zeno?s paradox in quantum theory,? J. Math.
Phys. 18, 756?763 (1977).
[206] A. Degasperis, L. Fonda, and G. C. Ghirardi, ?Does the lifetime of an unstable system
depend on the measuring apparatus?? Il Nuovo Cimento A (1965-1970) 21, 471?484
(1974).
[207] J. D. Franson, B. C. Jacobs, and T. B. Pittman, ?Quantum computing using single
photons and the Zeno effect,? Phys. Rev. A 70, 062302 (2004).
[208] Y.-Z. Sun, Y.-P. Huang, and P. Kumar, ?Photonic nonlinearities via quantum zeno
blockade,? Phys. Rev. Lett. 110, 223901 (2013).
272
[209] A. J. Daley, M. M. Boyd, J. Ye, and P. Zoller, ?Quantum computing with alkaline-
earth-metal atoms,? Phys. Rev. Lett. 101, 170504 (2008).
[210] M. Anderlini, P. J. Lee, B. L. Brown, J. Sebby-Strabley, W. D. Phillips, and J. V.
Porto, ?Controlled exchange interaction between pairs of neutral atoms in an optical
lattice,? Nature 448, 452?456 (2007).
[211] P. J. Lee, M. Anderlini, B. L. Brown, J. Sebby-Strabley, W. D. Phillips, and J. V.
Porto, ?Sublattice addressing and spin-dependent motion of atoms in a double-well
lattice,? Phys. Rev. Lett. 99, 020402 (2007).
[212] C. Zhang, S. L. Rolston, and S. Das Sarma, ?Manipulation of single neutral atoms
in optical lattices,? Phys. Rev. A 74, 042316 (2006).
[213] A. D. Ludlow, N. D. Lemke, J. A. Sherman, C. W. Oates, G. Qu?m?ner, J. von
Stecher, and A. M. Rey, ?Cold-collision-shift cancellation and inelastic scattering in
a yb optical lattice clock,? Phys. Rev. A 84, 052724 (2011).
[214] R. Zhang, Y. Cheng, H. Zhai, and P. Zhang, ?Orbital feshbach resonance in alkali-
earth atoms,? Phys. Rev. Lett. 115, 135301 (2015).
[215] L. Riegger, N. Darkwah Oppong, M. H?fer, D. R. Fernandes, I. Bloch, and
S. F?lling, ?Localized magnetic moments with tunable spin exchange in a gas of ul-
tracold fermions,? Phys. Rev. Lett. 120, 143601 (2018).
[216] F. Scazza, C. Hofrichter, M. H?fer, P. C. De Groot, I. Bloch, and S. F?lling, ?Ob-
servation of two-orbital spin-exchange interactions with ultracold su(n)-symmetric
fermions,? Nat. Phys. 10, 779?784 (2014).
[217] R. Le Targat, X. Baillard, M. Fouch?, A. Brusch, O. Tcherbakoff, G. D. Rovera, and
P. Lemonde, ?Accurate optical lattice clock with 87Sr atoms,? Phys. Rev. Lett. 97,
130801 (2006).
[218] G. Cappellini, L. F. Livi, L. Franchi, D. Tusi, D. Benedicto Orenes, M. Inguscio,
J. Catani, and L. Fallani, ?Coherent manipulation of orbital feshbach molecules of
two-electron atoms,? Phys. Rev. X 9, 011028 (2019).
[219] N. Syassen, D.M. Bauer, M. Lettner, T. Volz, D. Dietze, J. J. Garc?a-Ripoll, J. I. Cirac,
G. Rempe, and S. D?rr, ?Strong dissipation inhibits losses and induces correlations
in cold molecular gases,? Science 320, 1329?1331 (2008).
[220] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, ?Feshbach resonances in ultracold
gases,? Rev. Mod. Phys. 82, 1225?1286 (2010).
[221] S. Giorgini, L. P. Pitaevskii, and S. Stringari, ?Theory of ultracold atomic fermi
gases,? Rev. Mod. Phys. 80, 1215?1274 (2008).
[222] G. Barontini, R. Labouvie, F. Stubenrauch, A. Vogler, V. Guarrera, and H. Ott,
?Controlling the dynamics of an open many-body quantum system with localized
dissipation,? Phys. Rev. Lett. 110, 035302 (2013).
273
[223] R. Labouvie, B. Santra, S. Heun, S. Wimberger, and H. Ott, ?Negative differential
conductivity in an interacting quantum gas,? Phys. Rev. Lett. 115, 050601 (2015).
[224] R. Labouvie, B. Santra, S. Heun, and H. Ott, ?Bistability in a driven-dissipative
superfluid,? Phys. Rev. Lett. 116, 235302 (2016).
[225] H. P. L?schen, P. Bordia, S. S. Hodgman, M. Schreiber, S. Sarkar, A. J. Daley,
M. H. Fischer, E. Altman, I. Bloch, and U. Schneider, ?Signatures of many-body
localization in a controlled open quantum system,? Phys. Rev. X 7, 011034 (2017).
[226] Y. S. Patil, S. Chakram, and M. Vengalattore, ?Measurement-induced localization of
an ultracold lattice gas,? Phys. Rev. Lett. 115, 140402 (2015).
[227] J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and L. Luo, ?Observation of
parity-time symmetry breaking transitions in a dissipative floquet system of ultracold
atoms,? Nat. Commun. 10, 855 (2019).
[228] R. Bonifacio, F. Casagrande, and M. Milani, ?Superradiance and superfluorescence in
josephson junction arrays,? Phys. Lett. A 101, 427?431 (1984).
[229] M. C. Cassidy, A. Bruno, S. Rubbert, M. Irfan, J. Kammhuber, R. N. Schouten, A. R.
Akhmerov, and L. P. Kouwenhoven, ?Demonstration of an ac josephson junction
laser,? Science 355, 939?942 (2017).
[230] P. E. Dolgirev, J. Marino, D. Sels, and E. Demler, ?Non-gaussian correlations im-
printed by local dephasing in fermionic wires,? Phys. Rev. B 102, 100301 (2020).
[231] T. E. Lee and C.-K. Chan, ?Heralded magnetism in non-hermitian atomic systems,?
Phys. Rev. X 4, 041001 (2014).
[232] T. E. Lee, S. Gopalakrishnan, and M. D. Lukin, ?Unconventional magnetism via
optical pumping of interacting spin systems,? Phys. Rev. Lett. 110, 257204 (2013).
[233] C. Joshi, F. Nissen, and J. Keeling, ?Quantum correlations in the one-dimensional
driven dissipative xy model,? Phys. Rev. A 88, 063835 (2013).
[234] S. Diehl, W. Yi, A. J. Daley, and P. Zoller, ?Dissipation-induced d-wave pairing of
fermionic atoms in an optical lattice,? Phys. Rev. Lett. 105, 227001 (2010).
[235] K. Yamamoto, M. Nakagawa, K. Adachi, K. Takasan, M. Ueda, and N. Kawakami,
?Theory of non-hermitian fermionic superfluidity with a complex-valued interaction,?
Phys. Rev. Lett. 123, 123601 (2019).
[236] J. A. S. Louren?o, R. L. Eneias, and R. G. Pereira, ?Kondo effect in a PT -symmetric
non-hermitian hamiltonian,? Phys. Rev. B 98, 085126 (2018).
[237] M. Nakagawa, N. Kawakami, and M. Ueda, ?Non-hermitian kondo effect in ultracold
alkaline-earth atoms,? Phys. Rev. Lett. 121, 203001 (2018).
274
[238] M. S. Rudner and L. S. Levitov, ?Topological transition in a non-hermitian quantum
walk,? Phys. Rev. Lett. 102, 065703 (2009).
[239] T. E. Lee, ?Anomalous edge state in a non-hermitian lattice,? Phys. Rev. Lett. 116,
133903 (2016).
[240] D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, ?Edge modes,
degeneracies, and topological numbers in non-hermitian systems,? Phys. Rev. Lett.
118, 040401 (2017).
[241] S. Yao and Z.Wang, ?Edge states and topological invariants of non-hermitian systems,?
Phys. Rev. Lett. 121, 086803 (2018).
[242] Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, ?Topo-
logical phases of non-hermitian systems,? Phys. Rev. X 8, 031079 (2018).
[243] H. Shen, B. Zhen, and L. Fu, ?Topological band theory for non-hermitian hamilto-
nians,? Phys. Rev. Lett. 120, 146402 (2018).
[244] K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, ?Symmetry and topology in non-
hermitian physics,? Phys. Rev. X 9, 041015 (2019).
[245] T. Yoshida, K. Kudo, and Y. Hatsugai, ?Non-hermitian fractional quantum hall
states,? Scientific Reports 9, 16895 (2019).
[246] K. B. Efetov, ?Directed quantum chaos,? Phys. Rev. Lett. 79, 491?494 (1997).
[247] I. Y. Goldsheid and B. A. Khoruzhenko, ?Distribution of eigenvalues in non-hermitian
anderson models,? Phys. Rev. Lett. 80, 2897?2900 (1998).
[248] S. Longhi, D. Gatti, and G. Della Valle, ?Non-hermitian transparency and one-way
transport in low-dimensional lattices by an imaginary gauge field,? Phys. Rev. B 92,
094204 (2015).
[249] B. Horstmann, J. I. Cirac, and G. Giedke, ?Noise-driven dynamics and phase transi-
tions in fermionic systems,? Phys. Rev. A 87, 012108 (2013).
[250] M. ?nidari?, ?Exact solution for a diffusive nonequilibrium steady state of an open
quantum chain,? J. Stat. Mech.?Theory E. 2010, L05002 (2010).
[251] V. Eisler, ?Crossover between ballistic and diffusive transport: the quantum exclusion
process,? J. Stat. Mech.?Theory E. 2011, P06007 (2011).
[252] S. Bravyi, B. M. Terhal, and B. Leemhuis, ?Majorana fermion codes,? New J. Phys.
12, 083039 (2010).
[253] O. Shtanko, A. Deshpande, P. S. Julienne, and A. V. Gorshkov, ?Limits on Classical
Simulation of Free Fermions with Dissipation,? (2020), arXiv:2005.10840 [cond-mat,
physics:physics, physics:quant-ph].
275
[254] G. Lindblad, ?On the generators of quantum dynamical semigroups,? Comm. Math.
Phys. 48, 119?130 (1976).
[255] E.-M. Laine, J. Piilo, and H.-P. Breuer, ?Measure for the non-markovianity of quan-
tum processes,? Phys. Rev. A 81, 062115 (2010).
[256] J. Colpa, ?Diagonalisation of the quadratic fermion hamiltonian with a linear part,? J.
Phys. A-Math. Gen. 12, 469 (1979).
[257] P. Aliferis and B. M. Terhal, ?Fault-tolerant quantum computation for local leakage
faults,? Quantum Information & Computation 7, 139?156 (2007).
[258] D. Aharonov and M. Ben-Or, ?Fault-tolerant quantum computation with constant
error,? in Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Com-
puting - STOC ?97 (ACM Press, El Paso, Texas, United States, 1997) pp. 176?188.
[259] E. Knill, R. Laflamme, and W. H. Zurek, ?Resilient quantum computation,? Science
279, 342?345 (1998).
[260] B. M. Terhal, ?Quantum error correction for quantum memories,? Rev. Mod. Phys.
87, 307?346 (2015).
[261] M. Hebenstreit, R. Jozsa, B. Kraus, S. Strelchuk, and M. Yoganathan, ?All pure
fermionic non-Gaussian states are magic states for matchgate computations,? Phys.
Rev. Lett. 123, 080503 (2019).
[262] T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller,
R. Blatt, and C. F. Roos, ?Probing R?nyi entanglement entropy via randomized mea-
surements,? Science 364, 260?263 (2019).
[263] V. Alba and F. Carollo, ?Spreading of correlations in Markovian open quantum sys-
tems,? (2020), arXiv:2002.09527 [cond-mat, physics:hep-th, physics:quant-ph].
[264] D. Poulin, ?Lieb-Robinson Bound and Locality for General MarkovianQuantum Dy-
namics,? Phys. Rev. Lett. 104, 190401 (2010).
[265] T. Barthel and M. Kliesch, ?Quasilocality and Efficient Simulation of Markovian
Quantum Dynamics,? Phys. Rev. Lett. 108, 230504 (2012).
[266] E. Pednault, J. A. Gunnels, G. Nannicini, L. Horesh, and R. Wisnieff, ?Lever-
aging secondary storage to simulate deep 54-qubit sycamore circuits,? (2019),
arXiv:1910.09534 [quant-ph].
[267] G. Kalai and G. Kindler, ?Gaussian noise sensitivity and BosonSampling,? (2014),
arXiv:1409.3093.
[268] J. J. Renema, ?Marginal probabilities in boson samplers with arbitrary input states,?
(2020), arXiv:2012.14917 [quant-ph].
[269] S. Aaronson, ?Quantum supremacy, now with bosonsampling,? (2020).
276
[270] S. Aaronson, ?Chinese bosonsampling experiment: the gloves are off,? (2020).
[271] S. J. Freedman and J. F. Clauser, ?Experimental test of local hidden-variable theories,?
Phys. Rev. Lett. 28, 938?941 (1972).
[272] A. Aspect, P. Grangier, and G. Roger, ?Experimental Realization of Einstein-
Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell?s Inequali-
ties,? Phys. Rev. Lett. 49, 91?94 (1982).
[273] B. Hensen, H. Bernien, A. E. Dr?au, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg,
R. F. Vermeulen, R. N. Schouten, C. Abell?n, et al., ?Loophole-free bell inequal-
ity violation using electron spins separated by 1.3 kilometres,? Nature 526, 682?686
(2015).
[274] M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer,
K. Phelan, F. Steinlechner, J. Kofler, J.-?. Larsson, C. Abell?n, et al., ?Significant-
loophole-free test of bell?s theorem with entangled photons,? Phys. Rev. Lett. 115,
250401 (2015).
[275] L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M. J.
Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, et al., ?Strong loophole-
free test of local realism,? Phys. Rev. Lett. 115, 250402 (2015).
[276] W. Rosenfeld, D. Burchardt, R. Garthoff, K. Redeker, N. Ortegel, M. Rau, and
H. Weinfurter, ?Event-ready Bell test using entangled atoms simultaneously closing
detection and locality loopholes,? Phys. Rev. Lett. 119, 010402 (2017).
[277] J. Haferkamp, D. Hangleiter, A. Bouland, B. Fefferman, J. Eisert, and J. Bermejo-
Vega, ?Closing gaps of a quantum advantage with short-time Hamiltonian dynamics,?
Phys. Rev. Lett. 125, 250501 (2020).
[278] M. Oszmaniec, N. Dangniam, M. E. S. Morales, and Z. Zimbor?s, ?Fermion Sam-
pling: A robust quantum computational advantage scheme using fermionic linear
optics and magic input states,? (2020), arXiv:2012.15825 [math-ph, physics:quant-
ph].
[279] J. E. Bourassa, R. N. Alexander, M. Vasmer, A. Patil, I. Tzitrin, T. Matsuura, D. Su,
B. Q. Baragiola, S. Guha, G. Dauphinais, K. K. Sabapathy, N. C. Menicucci, and
I. Dhand, ?Blueprint for a scalable photonic fault-tolerant quantum computer,?Quan-
tum 5, 392 (2021).
[280] S. Bartolucci, P. Birchall, H. Bombin, H. Cable, C. Dawson, M. Gimeno-Segovia,
E. Johnston, K. Kieling, N. Nickerson, M. Pant, F. Pastawski, T. Rudolph, and
C. Sparrow, ?Fusion-based quantum computation,? (2021), arXiv:2101.09310 [quant-
ph].
[281] T. R. Bromley, J. M. Arrazola, S. Jahangiri, J. Izaac, N. Quesada, A. D. Gran,
M. Schuld, J. Swinarton, Z. Zabaneh, and N. Killoran, ?Applications of near-term
277
photonic quantum computers: software and algorithms,? Quantum Sci. Technol. 5,
034010 (2020).
[282] H. Qi, D. J. Brod, N. Quesada, and R. Garc?a-Patr?n, ?Regimes of classical simula-
bility for noisy Gaussian boson sampling,? Phys. Rev. Lett. 124, 100502 (2020).
[283] A. Bj?rklund, B. Gupt, and N.Quesada, ?A faster hafnian formula for complex matri-
ces and its benchmarking on a supercomputer,? ACM J. Exp. Algorithmics 24, 1?17
(2019).
[284] A. Deshpande, A. Mehta, T. Vincent, N.Quesada, M. Hinsche, M. Ioannou, L. Mad-
sen, J. Lavoie, H. Qi, J. Eisert, D. Hangleiter, B. Fefferman, and I. Dhand, ?Quan-
tum Computational Supremacy via High-Dimensional Gaussian Boson Sampling,?
(2021), arXiv:2102.12474 [quant-ph].
[285] R. J. Lipton, ?New directions in testing,? in Distributed Computing and Cryptography,
Proceedings of a DIMACS Workshop, Princeton, New Jersey, USA, October 4-6, 1989,
DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 2,
edited by J. Feigenbaum and M. Merritt (DIMACS/AMS, 1989) pp. 191?202.
[286] Y. Kondo, R. Mori, and R. Movassagh, ?Fine-grained analysis and improved robust-
ness of quantum supremacy for random circuit sampling,? (2021), arXiv:2102.01960
[quant-ph].
[287] K. Fujii, ?Noise Threshold of Quantum Supremacy,? (2016), arXiv:1610.03632
[quant-ph].
[288] M. Oszmaniec and D. J. Brod, ?Classical simulation of photonic linear optics with lost
particles,? New J. Phys. 20, 092002 (2018).
[289] S. Rahimi-Keshari, A. Scherer, A. Mann, A. T. Rezakhani, A. I. Lvovsky, and B. C.
Sanders, ?Quantum process tomography with coherent states,? New Journal of Physics
13, 013006 (2011).
[290] S. Boixo, V. N. Smelyanskiy, and H. Neven, ?Fourier analysis of sampling from noisy
chaotic quantum circuits,? (2017), arXiv:1708.01875 [quant-ph].
[291] L. Valiant, ?The complexity of computing the permanent,? Theor. Comput. Sci. 8,
189?201 (1979).
[292] D. Hangleiter, M. Kliesch, J. Eisert, and C. Gogolin, ?Sample Complexity of Device-
Independently Certified ?Quantum Supremacy?,? Phys. Rev. Lett. 122, 210502
(2019).
[293] C. Oh, Y. Lim, B. Fefferman, and L. Jiang, ?Classical simulation of bosonic linear-
optical random circuits beyond linear light cone,? (2021), arXiv:2102.10083 [quant-
ph].
278
[294] S. Gharibian, Y. Huang, Z. Landau, and S. W. Shin, ?Quantum Hamiltonian Com-
plexity,? Found. Trends? Theor. Comput. Sci. 10, 159?282 (2015).
[295] S. A. Cook, ?The complexity of theorem-proving procedures,? in Proceedings of the
Third Annual ACM Symposium on Theory of Computing - STOC ?71 (ACM Press,
Shaker Heights, Ohio, United States, 1971) pp. 151?158.
[296] B. Trakhtenbrot, ?A Survey of Russian Approaches to Perebor (Brute-Force Searches)
Algorithms,? IEEE Annals Hist. Comput. 6, 384?400 (1984).
[297] D. Aharonov and T. Naveh, ?Quantum NP - A Survey,? (2002), arXiv:quant-
ph/0210077.
[298] D. Aharonov, I. Arad, Z. Landau, and U. Vazirani, ?The detectability lemma and
quantum gap amplification,? in Proceedings of the 41st Annual ACM Symposium on
Symposium on Theory of Computing - STOC ?09 (ACM Press, Bethesda, MD, USA,
2009) p. 417.
[299] D. Aharonov, I. Arad, and T. Vidick, ?Guest column: The quantum PCP conjecture,?
SIGACT News 44, 47?79 (2013).
[300] S. Aaronson and G. Kuperberg, ?Quantum versus classical proofs and advice,? Theory
Comput. 3, 129?157 (2007).
[301] B. Fefferman and S. Kimmel, ?Quantum vs. Classical proofs and subset verifica-
tion,? in 43rd International Symposium on Mathematical Foundations of Computer Sci-
ence, MFCS 2018, August 27-31, 2018, Liverpool, UK , LIPIcs, Vol. 117, edited by
I. Potapov, P. G. Spirakis, and J. Worrell (Schloss Dagstuhl - Leibniz-Zentrum f?r
Informatik, 2018) pp. 22:1?22:23.
[302] F. D. M. Haldane, ?Nonlinear Field Theory of Large-Spin Heisenberg Antiferro-
magnets: Semiclassically Quantized Solitons of the One-Dimensional Easy-Axis N?el
State,? Phys. Rev. Lett. 50, 1153?1156 (1983).
[303] E. Witten, ?Physical law and the quest for mathematical understanding,? Bull. Am.
Math. Soc. 40, 21?30 (2002).
[304] T. S. Cubitt, D. Perez-Garcia, and M. M. Wolf, ?Undecidability of the spectral gap,?
Nature 528, 207?211 (2015).
[305] M. B. Hastings, ?An area law for one-dimensional quantum systems,? J. Stat. Mech.
2007, P08024 (2007).
[306] Z. Landau, U. Vazirani, and T. Vidick, ?A polynomial time algorithm for the ground
state of one-dimensional gapped local Hamiltonians,? Nat. Phys. 11, 566?569 (2015).
[307] I. Arad, Z. Landau, U. Vazirani, and T. Vidick, ?Rigorous RG algorithms and area
laws for low energy eigenstates in 1D,? Commun. Math. Phys. 356, 65?105 (2017).
279
[308] D. Aharonov, M. Ben-Or, F. G. S. L. Brandao, and O. Sattath, ?The Pursuit For
Uniqueness: Extending Valiant-Vazirani Theorem to the Probabilistic and Quantum
Settings,? (2008), arXiv:0810.4840.
[309] R. Jain, I. Kerenidis, G. Kuperberg, M. Santha, O. Sattath, and S. Zhang, ?On the
Power of a Unique Quantum Witness,? Theory Comput. 8, 375?400 (2012).
[310] C. E. Gonz?lez-Guill?n and T. S. Cubitt, ?History-state Hamiltonians are critical,?
(2018), arXiv:1810.06528 [quant-ph].
[311] E. Crosson and J. Bowen, ?Quantum ground state isoperimetric inequalities for the
energy spectrum of local Hamiltonians,? (2018), arXiv:1703.10133 [quant-ph].
[312] A. M. Dalzell and F. G. S. L. Brandao, ?Locally accurate MPS approximations
for ground states of one-dimensional gapped local Hamiltonians,? Quantum 3, 187
(2019).
[313] R. Babbush, D.W. Berry, I. D. Kivlichan, A. Y.Wei, P. J. Love, and A. Aspuru-Guzik,
?Exponentially more precise quantum simulation of fermions in second quantization,?
New J. Phys. 18, 033032 (2016).
[314] A. M. Childs, R. Kothari, and R. D. Somma, ?Quantum Algorithm for Systems of
Linear Equations with Exponentially Improved Dependence on Precision,? SIAM J.
Comput. 46, 1920?1950 (2017).
[315] M. Aguilera Sammaritano, M. Gonz?lez Vera, P. Marcelo Cometto, T. Nicola Tejero,
G. F. Bauerfeldt, and A. Mellouki, ?Temperature dependence of rate coefficients for
the gas phase reaction of OH with 3-chloropropene. A theoretical and experimental
study,? Chem. Phys. Lett. , 137757 (2020).
[316] C. Hubig, J. Haegeman, and U. Schollw?ck, ?Error estimates for extrapolations with
matrix-product states,? Phys. Rev. B 97, 045125 (2018).
[317] S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cambridge University Press, Cam-
bridge, 2011).
[318] Y. Atia and D. Aharonov, ?Fast-forwarding of Hamiltonians and Exponentially Precise
Measurements,? Nat. Commun. 8, 1572 (2017).
[319] B. Fefferman and C. Y.-Y. Lin, ?A Complete Characterization of Unitary Quan-
tum Space,? in 9th Innovations in Theoretical Computer Science Conference (ITCS
2018), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 94, edited by
A. R. Karlin (Schloss Dagstuhl?Leibniz-Zentrum fuer Informatik, Dagstuhl, Ger-
many, 2018) pp. 4:1?4:21.
[320] A. Kitaev and J. Watrous, ?Parallelization, amplification, and exponential time simula-
tion of quantum interactive proof systems,? in Proceedings of the Thirty-Second Annual
ACM Symposium on Theory of Computing - STOC ?00 (ACM Press, Portland, Oregon,
United States, 2000) pp. 608?617.
280
[321] M. N. Vyalyi, ?QMA = PP implies that PP contains PH,? in ECCCTR: Electronic
Colloquium on Computational Complexity, Technical Reports (2003).
[322] C. Marriott and J. Watrous, ?Quantum Arthur?Merlin games,? Comput. Complex.
14, 122?152 (2005).
[323] E. W. Allender and K. W. Wagner, ?Counting Hierarchies: Polynomial Time And
Constant Depth Circuits,? in Current Trends in Theoretical Computer Science (World
Scientific, 1993) pp. 469?483.
[324] S. Gharibian, M. Santha, J. Sikora, A. Sundaram, and J. Yirka, ?Quantum Gen-
eralizations of the Polynomial Hierarchy with Applications to QMA(2),? in 43rd
International Symposium on Mathematical Foundations of Computer Science (MFCS
2018), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 117 (Schloss
Dagstuhl?Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2018) pp. 58:1?
58:16.
[325] T. Morimae and H. Nishimura, ?Merlinization of complexity classes above BQP,?
Quantum Inf. Comput. 17, 959?972 (2017).
[326] R. Jefferson and R. C. Myers, ?Circuit complexity in quantum field theory,? J. High
Energ. Phys. 2017, 107 (2017).
[327] C. D. White, C. Cao, and B. Swingle, ?Conformal field theories are magical,? (2020),
arXiv:2007.01303 [cond-mat, physics:hep-th, physics:quant-ph].
[328] D. Aharonov and A. Green, ?A quantum inspired proof of P#p ? IP,? (2017),
arXiv:1710.09078 [quant-ph].
[329] A. Green, G. Kindler, and Y. Liu, ?Towards a quantum-inspired proof for IP =
PSPACE,? (2019), arXiv:1912.11611 [quant-ph].
[330] A. Shamir, ?IP = PSPACE,? J. ACM 39, 869?877 (1992).
[331] J. Kempe and O. Regev, ?3-local Hamitonian is QMA-complete,?Quantum Inf. Com-
put. 3, 258?264 (2003).
[332] J. Kempe, A. Kitaev, and O. Regev, ?The Complexity of the Local Hamiltonian
Problem,? SIAM J. Comput. 35, 1070?1097 (2006).
[333] R. V. Mises and H. Pollaczek-Geiringer, ?Praktische Verfahren der Gleichungsaufl?-
sung,? Z. F?r Angew. Math. Mech. 9, 152?164 (1929).
[334] L. M. Adleman, J. DeMarrais, and M.-D. A. Huang, ?Quantum Computability,?
SIAM J. Comput. 26, 1524?1540 (1997).
[335] S. Aaronson, ?Quantum computing, postselection, and probabilistic polynomial-
time,? Proc. R. Soc. Math. Phys. Eng. Sci. 461, 3473?3482 (2005).
281
[336] J. R. Schrieffer and P. A. Wolff, ?Relation between the Anderson and Kondo Hamil-
tonians,? Phys. Rev. 149, 491?492 (1966).
[337] S. Bravyi, D. DiVincenzo, and D. Loss, ?Schrieffer-Wolff transformation for quantum
many-body systems,? Ann. Phys. 326, 2793?2826 (2011).
[338] B. Fefferman and C. Y.-Y. Lin, ?Quantum Merlin Arthur with Exponentially Small
Gap,? (2016), arXiv:1601.01975 [quant-ph].
[339] N. Schuch, I. Cirac, and F. Verstraete, ?Computational Difficulty of Finding Matrix
Product Ground States,? Phys. Rev. Lett. 100, 250501 (2008).
[340] T. Jones, S. Endo, S. McArdle, X. Yuan, and S. C. Benjamin, ?Variational quantum
algorithms for discovering Hamiltonian spectra,? Phys. Rev. A 99, 062304 (2019).
[341] D. Aharonov, I. Arad, and S. Irani, ?Efficient algorithm for approximating one-
dimensional ground states,? Phys. Rev. A 82, 012315 (2010).
[342] A. Molnar, N. Schuch, F. Verstraete, and J. I. Cirac, ?Approximating Gibbs states of
local Hamiltonians efficiently with projected entangled pair states,? Phys. Rev. B 91,
045138 (2015).
[343] S. Knysh, ?Zero-temperature quantum annealing bottlenecks in the spin-glass phase,?
Nat. Commun. 7, 12370 (2016).
[344] N. Usher, M. J. Hoban, and D. E. Browne, ?Non-Unitary Quantum Computation
in the Ground Space of Local Hamiltonians,? Phys. Rev. A 96, 032321 (2017).
[345] D. Nagaj, P. Wocjan, and Y. Zhang, ?Fast amplification of QMA,? Quantum Inf.
Comput. 9, 1053?1068 (2009).
[346] S. Aaronson, ?Quantum lower bound for the collision problem,? in Proceedings of the
Thiry-Fourth Annual ACM Symposium on Theory of Computing, STOC ?02 (Associa-
tion for Computing Machinery, New York, NY, USA, 2002) pp. 635?642.
[347] S. Aaronson, ?BQP and the polynomial hierarchy,? in Proceedings of the Forty-Second
ACM Symposium on Theory of Computing, STOC ?10 (Association for Computing
Machinery, New York, NY, USA, 2010) pp. 141?150.
[348] R. Raz and A. Tal, Oracle Separation of BQP and PH , Tech. Rep. 107 (2018).
[349] S. Bravyi, D. P. DiVincenzo, D. Loss, and B. M. Terhal, ?Simulation of Many-Body
Hamiltonians using Perturbation Theory with Bounded-Strength Interactions,? Phys.
Rev. Lett. 101, 070503 (2008).
[350] T. Cubitt and A. Montanaro, ?Complexity classification of local Hamiltonian prob-
lems,? SIAM J. Comput. 45, 268?316 (2013).
282
[351] D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd, and O. Regev, ?Adiabatic
Quantum Computation is Equivalent to Standard Quantum Computation,? SIAM J.
Comput. 37, 166?194 (2007).
[352] N. Schuch and F. Verstraete, ?Computational Complexity of interacting electrons and
fundamental limitations of Density Functional Theory,? Nat. Phys. 5, 732?735 (2007).
[353] T. S. Cubitt, A. Montanaro, and S. Piddock, ?Universal quantum Hamiltonians,?
Proc. Natl. Acad. Sci. USA 115, 9497?9502 (2018).
[354] J. Bausch, T. S. Cubitt, A. Lucia, D. Perez-Garcia, and M. M. Wolf, ?Size-driven
quantum phase transitions,? Proc. Natl. Acad. Sci. 115, 19?23 (2018).
[355] S. Piddock and A. Montanaro, ?Universal qudit Hamiltonians,? (2018),
arXiv:1802.07130 [quant-ph].
[356] T. Kohler and T. Cubitt, ?Toy models of holographic duality between local Hamilto-
nians,? J. High Energ. Phys. 2019, 17 (2019).
[357] T. Kohler, S. Piddock, J. Bausch, and T. Cubitt, ?Translationally-Invariant Universal
Quantum Hamiltonians in 1D,? (2020), arXiv:2003.13753 [quant-ph].
[358] J. Bausch and E. Crosson, ?Analysis and limitations of modified circuit-to-
Hamiltonian constructions,? Quantum 2, 94 (2018).
[359] L. Caha, Z. Landau, and D. Nagaj, ?Clocks in Feynman?s computer and Kitaev?s local
Hamiltonian: Bias, gaps, idling, and pulse tuning,? Phys. Rev. A 97, 062306 (2018).
[360] J. D. Watson, ?Detailed Analysis of Circuit-to-Hamiltonian Mappings,? (2019),
arXiv:1910.01481 [quant-ph].
[361] J. Watrous, ?Quantum Computational Complexity,? in Encyclopedia of Complexity and
Systems Science, edited by R. A. Meyers (Springer New York, New York, NY, 2009)
pp. 7174?7201.
[362] F. G. S. L. Brandao, ?Entanglement Theory and the Quantum Simulation of Many-
Body Physics,? (2008), arXiv:0810.0026 [quant-ph].
[363] H. Blier and A. Tapp, ?All Languages in NP Have Very Short Quantum Proofs,? in
Proceedings of the 2009 Third International Conference on Quantum, Nano and Micro
Technologies, ICQNM ?09 (IEEE Computer Society, USA, 2009) pp. 34?37.
[364] A. Pereszl?nyi, ?Multi-Prover Quantum Merlin-Arthur Proof Systems with Small
Gap,? (2012), arXiv:1205.2761 [quant-ph].
[365] T. Ito, H. Kobayashi, and J. Watrous, ?Quantum interactive proofs with weak error
bounds,? in Proceedings of the 3rd Innovations in Theoretical Computer Science Confer-
ence, ITCS ?12 (Association for Computing Machinery, Cambridge, Massachusetts,
2012) pp. 266?275.
283
[366] Z. Ji, ?Compression of quantum multi-prover interactive proofs,? in Proceedings of the
49th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2017 (ACM
Press, Montreal, Canada, 2017) pp. 289?302.
[367] J. Fitzsimons, Z. Ji, T. Vidick, and H. Yuen, ?Quantum proof systems for iterated
exponential time, and beyond,? in Proceedings of the 51st Annual ACM SIGACT Sym-
posium on Theory of Computing - STOC 2019 (ACM Press, Phoenix, AZ, USA, 2019)
pp. 473?480.
[368] B. Fefferman, H. Kobayashi, C. Y.-Y. Lin, T. Morimae, and H. Nishimura, ?Space-
Efficient Error Reduction for Unitary Quantum Computations,? in 43rd Interna-
tional Colloquium on Automata, Languages, and Programming (ICALP 2016), Vol. 55
(Schloss Dagstuhl?Leibniz-Zentrum fuer Informatik, 2016) pp. 14:1?14:14.
[369] L. Valiant and V. Vazirani, ?NP is as easy as detecting unique solutions,? Theor. Com-
put. Sci. 47, 85?93 (1986).
[370] A. Ambainis, ?On Physical Problems that are Slightly More Difficult than QMA,? in
Proceedings of the 2014 IEEE 29th Conference on Computational Complexity, CCC ?14
(IEEE Computer Society, Washington, DC, USA, 2014) pp. 32?43.
[371] S. Gharibian and J. Yirka, ?The complexity of simulating local measurements on quan-
tum systems,? Quantum 3, 189 (2019).
[372] S. Gharibian, S. Piddock, and J. Yirka, ?Oracle complexity classes and local measure-
ments on physical Hamiltonians,? (2019), arXiv:1909.05981 [quant-ph].
[373] L. Novo, J. Bermejo-Vega, and R. Garc?a-Patr?n, ?Quantum advantage from energy
measurements of many-body quantum systems,? (2019), arXiv:1912.06608 [quant-
ph].
[374] P. Wocjan, D. Janzing, and T. Beth, ?Two QCMA-Complete problems,? Quantum
Inf. Comput. 3, 635?643 (2003).
[375] C. H. Bennett, ?Time/Space Trade-Offs for Reversible Computation,? SIAM J. Com-
put. 18, 766?776 (1989).
[376] K.-J. Lange, P. McKenzie, and A. Tapp, ?Reversible Space Equals Deterministic
Space,? J. Comput. Syst. Sci. 60, 354?367 (2000).
[377] S. P. Jordan, H. Kobayashi, D. Nagaj, and H. Nishimura, ?Achieving perfect com-
pleteness in classical-witness quantum Merlin-Arthur proof systems,? Quantum Inf.
Comput. 12, 461?471 (2012).
[378] S. Dooley, G. Kells, H. Katsura, and T. C. Dorlas, ?Simulating quantum circuits by
adiabatic computation: Improved spectral gap bounds,? Phys. Rev. A 101, 042302
(2020).
284
[379] S. Arora and B. Barak, Computational Complexity: A Modern Approach (Cambridge
University Press, Cambridge ; New York, 2009).
[380] M. Sipser, Introduction to the Theory of Computation, 3rd ed. (Cengage Learning,
Boston, MA, 2012).
[381] P. W. Shor, ?Polynomial-Time Algorithms for Prime Factorization and Discrete Log-
arithms on a Quantum Computer,? SIAM J. Comput. 26, 1484?1509 (1997).
285