ABSTRACT
Title of dissertation: Building and Programming
a Universal Ion Trap Quantum Computer
Caroline Margaret Figgatt
Doctor of Philosophy, 2018
Dissertation directed by: Professor Christopher Monroe
Department of Physics
Quantum computing represents an exciting frontier in the realm of informa-
tion processing; it is a promising technology that may provide future advances in
a wide range of fields, from quantum chemistry to optimization problems. This
thesis discusses experimental results for several quantum algorithms performed on
a programmable quantum computer consisting of a linear chain of five or seven
trapped 171Yb+ atomic clock ions with long coherence times and high gate fideli-
ties. We execute modular one- and two-qubit computation gates through Raman
transitions driven by a beat note between counter-propagating beams from a pulsed
laser. The system’s individual addressing capability provides arbitrary single-qubit
rotations as well as all possible two-qubit entangling gates, which are implemented
using a pulse-segmentation scheme. The quantum computer can be programmed
from a high-level interface to execute arbitrary quantum circuits, and comes with a
toolbox of many important composite gates and quantum subroutines.
We present experimental results for a complete three-qubit Grover quantum
search algorithm, a hallmark application of a quantum computer with a well-known
speedup over classical searches of an unsorted database, and report better-than-
classical performance. The algorithm is performed for all 8 possible single-result
oracles and all 28 possible two-result oracles. All quantum solutions are shown to
outperform their classical counterparts.
Performing parallel operations will be a powerful capability as deeper circuits
on larger, more complex quantum computers present new challenges. Here, we
perform a pair of 2-qubit gates simultaneously in a single chain of trapped ions.
We employ a pre-calculated pulse shaping scheme that modulates the phase and
amplitude of the Raman transitions to drive programmable high-fidelity 2-qubit
entangling gates in parallel by coupling to the collective modes of motion of the
ion chain. Ensuring the operation yields only spin-spin interactions between the
desired pairs, with neither residual spin-motion entanglement nor crosstalk spin-
spin entanglement, is a nonlinear constraint problem, and pulse solutions are found
using optimization techniques. As an application, we demonstrate the quantum full
adder using a depth-4 circuit requiring the use of parallel 2-qubit operations.
BUILDING AND PROGRAMMING
A UNIVERSAL ION TRAP QUANTUM COMPUTER
by
Caroline Margaret Figgatt
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
2018
Advisory Committee:
Professor Chris Monroe, Chair/Advisor
Professor Gretchen Campbell
Professor Andrew Childs
Professor Trey Porto
Professor James Williams
©c Copyright by
Caroline Figgatt
2018
Dedication
to
Alan Purring, Admiral Grace Murray Hoppurr, Mr. Grayson,
and most of all, Fermi
ii
Acknowledgments
I am incredibly grateful to have received so much support from so many won-
derful folks over my years in grad school. I can hardly do them all justice, and hope
my humble thanks is enough.
First and foremost, my advisor, Chris Monroe, provided me a wonderful op-
portunity to work in his lab at UMD. I came to grad school primarily interested
in making quantum computers a reality, and my work with Chris has been doing
precisely that. (I still sometimes come in to the lab, realize that this is i fact my
life, and can hardly believe I’m lucky enough to be doing exactly what I want to
be doing.) Chris has provided me with useful guidance throughout grad school and
pushed me to do my best work, and I am grateful for his support.
I would also like to thank my thesis committee for taking the time to evaluate
my candidacy: Gretchen Campbell, Jimmy Williams, Andrew Childs, and Trey
Porto. Gretchen and Jimmy in particular have provided mentorship and support
during my time at Maryland.
I have been fortunate to work some incredibly kind, patient, and brilliant col-
leagues throughout my time at Maryland. Norbert Linke, my postdoc for the last
several years, has provided me with invaluable mentorship. He has been an incred-
ibly patient teacher through endless bouts of questions, encouraged me to take on
new challenges, stayed cheerful and pushed me to persevere when morale was low,
and always made time to help me. I couldn’t ask for better and I am deeply grate-
ful. Kevin Landsman has been a fellow grad student in the lab with me for the
iii
past few years, and he has been an exemplary colleague. He has made outstanding
contributions to the lab, and has been a conscientious team player. I am grateful
for his competence and enthusiasm during the good times, and his level-headedness
and forebearance during the bad. Daiwei Zhu joined our project as a grad student
a few months ago, but is already making excellent progress and has been a pleasure
to work with. My immediate predecessor, Shantanu Debnath, taught me every-
thing I know about optics, and always had a few moments to cheerfully answer a
younger grad student’s questions. Ken Wright has been a friend as well as a great
colleague. By bringing the Igor control program with him from Georgia Tech, he
saved me from having to work with Labview throughout my graduate career, an
act of kindness I deeply appreciate. Andrew Manning and Taeyoung Choi helped
introduce me to the Gates project when I joined as a first year grad student, and I
appreciate the time they took to help me learn my way around. I’ve also appreciated
advice and insight from a number of older students and postdocs during my time in
the Monroe lab, and am grateful for my time working with all the members of the
group I’ve intersected over the years, including Kristi Beck, Jiehang Zhang, Marty
Lichtman, Marko Cetina, Guido Pagano, Michael Goldman, David Wong-Campos,
Clay Crocker, Harvey Kaplan, Ksenia Sosnova, Antonis Kyprianidis, Allison Carter,
Patrick Becker, Wen Lin Tan, Kate Collins, Laird Egan, Sophia Scarano, Eric Bir-
ckelbaw, Micah Hernandez, Nate Dudley, Steven Moses, Kai Hudek, Paul Hess,
Hanna Ruth, Jason Amini, Jonathan Mizrahi, Volkan Inlek, Kale Johnson, Jake
Smith, Aaron Lee, Geoffrey Ji, David Hucul, Brian Neyenhuis, Phil Richerme, Gra-
hame Vittorini, Crystal Senko, Chenglin Cao, Susan Clark, and Dave Hayes. I am
iv
also grateful for the mentorship and advice I’ve received from Shelby Kimmel, Kristi
Beck, Emily Edwards, Qudsia Quraishi, Steve Olmschenk, Kathy-Anne Soderberg,
Dave Moehring, John Berberian, and so many other folks.
Working on a world-class experiment with flexible capabilities has afforded
me incredible opportunities to collaborate with a large number of wonderful and
brililant theorists. Aaron Ostrander has been a fantastic collaborator on the par-
allel gates experiment as gate optimizer extraordinaire, and it has been a pleasure
to work with him. Dmitri Maslov has been an important collaborator on several
experiments, most particularly the Grover experiment, and I have greatly appreci-
ated his inexhaustible knowledge of quantum computing applications and his help
in getting our experiment more exposure in computer science circles. Shengtao
Wang was a great collaborator on the phonon hopping project. He was also kind
enough to share with me a paper draft containing a detailed derivation of the XX
entangling gate fidelity that proved incredibly helpful while I was deriving parallel
XX gate outcomes, and for that I will forever be grateful. I enjoyed working with
James Leung for the frequency modulated gates project, and appreciated his taking
time to answer some questions about theory problems while I was working on the
parallel gates project. I thank Zhexuan Gong for taking time on several occasions
to answer questions about XX gates and pulse shaping, and enjoyed working him
him for the 2014 pulse shaping paper. The quantum scrambling project involved
some fascinating physics, and I thank Norman Yao for his energy and incisiveness,
Beni Yoshida for his thoughtful insights and endless patience with explanations, and
Tommy Schuster for his hard work on simulations; working with them has been a
v
wonderful learning experience. Curtis Volin provided very helpful insights when I
was designing the individual control optical system. I have appreciated kind advice
and support from Jungsang Kim and Ken Brown, and appreciated Ken’s keen in-
sights during collaborations. I have also been grateful to work with a number of
other talented theorists, including Sonika Johri, Alireza Seif, Neal Solmeyer, Rad-
hakrishnan Balu, Luming Duan, Martin Roetteler, Mauricio Gutierrez, Yunseong
Nam, Marcello Benedetti, and Alejandro Perdomo-Ortiz.
My undergraduate career at MIT provided an important foundation for my
success in grad school, and to that end I’d like to that those who made that possible.
Martin Zwierlein, my undergrad research advisor, has a boundless and infectious
enthusiasm for all things physics that makes him a wonderful teacher and a patient
mentor. I’m incredibly grateful to him for his mentorship in the lab and his support
of me through some difficult times, as well as to his group members for teaching
me how to be a useful person in a lab. Nergis Mavalvala, Deepto Chakrabarty, and
Sean Robinson also provided incredible support and understanding through various
hardships, and I am grateful to them for the role all of them played in my eventual
success in grad school.
I am also incredibly grateful for the support of my friends and family through-
out this process. My parents, Mr. Grayson, my brother Thomas and sister-in-law
Liz, and the rest of my family have been very supportive. Gina Quan, Megan
Marshall, and Lora Price have been incredibly supportive friends and gym buddies
during my years here, and I couldn’t ask for a kinder, more sarcastic group with
which to share silly lifting videos and commiserate about the day to day frustra-
vi
tions of being a physics grad student. Zach Skigen, Alan Purring, and Admiral
Grace Murray Hoppurr have provided wonderful and fluffy companionship through
long days of writing. Alan in particular has been most helpful with data analysis,
and Grace has a knack for oversight. My roommates over the past few years have
been incredibly supportive and understanding: Richard Friend, Escher, Kearci Car-
son, Ella, Tiffany Calvert, and in loving memory of Nola. Dawit Girma has been
incredibly supportive as my powerlifting coach, and has been endlessly patient and
understanding with providing coaching around an often inconsisten schedule. To all
my other friends, of whom there are far to many to list here: thank you for your
hugs and understanding while I’ve gone through this process. And of course, my
own little Fermi has been a constant companion through this entire process, and
has been there for me during good times and bad.
I have been privileged to serve as first treasurer and then president of Women
in Physics at UMD during my time here, and I’m incredibly grateful for the support
my fellow women in physics have given me and each other over these years. I
particularly appreciate my fellow officers’ support in stepping in while I focused on
my thesis this semester. I’m also grateful to the department for being supportive of
WIP. I’m proud of the work the group has done during my leadership tenure and I
look forward to seeing where the upcoming leadership takes it.
vii
Table of Contents
Dedication ii
Acknowledgments iii
List of Tables xi
List of Figures xii
List of Abbreviations xiv
1 Introduction 1
2 Quantum Computer Hardware 7
2.1 The 171Yb+ Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Coherence Times . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.2 Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.3 Qubit Initialization . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.4 Qubit Readout . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 The Ion Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Coherent Individual Addressing System . . . . . . . . . . . . . . . . . 21
2.3.1 Raman Beam Geometry . . . . . . . . . . . . . . . . . . . . . 22
2.3.2 32-Channel AOM Characteristics . . . . . . . . . . . . . . . . 25
2.3.3 32-Channel AOM Control . . . . . . . . . . . . . . . . . . . . 26
2.3.4 Optics for Individual Addressing . . . . . . . . . . . . . . . . 30
2.4 Individual Addressing System Upgrades . . . . . . . . . . . . . . . . 38
2.5 Expansion to 7-Ion System . . . . . . . . . . . . . . . . . . . . . . . . 40
3 Quantum Control 43
3.1 Single-Qubit Rotations R . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.1 Rabi Flopping on 5 or 7 ions . . . . . . . . . . . . . . . . . . . 45
3.1.2 Z Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Two-Qubit Entangling XX Gates . . . . . . . . . . . . . . . . . . . . 48
viii
3.2.1 Pulse-Shaping Scheme . . . . . . . . . . . . . . . . . . . . . . 49
3.3 Universality of the Gate Set {R,XX} . . . . . . . . . . . . . . . . . 57
4 Control System 59
4.1 System Control Overview . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 The Igor Control Program . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3 Compiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5 Gate Library 79
5.1 Single-Qubit Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2 Multi-Qubit Circuit Construction . . . . . . . . . . . . . . . . . . . . 80
5.2.1 Optimization Strategy Adjustments with Experimental Up-
grades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3 Two-Qubit Composite Gates . . . . . . . . . . . . . . . . . . . . . . . 83
5.4 Toffoli Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.4.1 Toffoli-3 Characterization . . . . . . . . . . . . . . . . . . . . 88
5.4.2 Toffoli-4 Gate . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.5 The SWAP Gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.6 Controlled-SWAP Gate . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.7 Using Z-Rotations to Obviate χ Sign in XX Gates . . . . . . . . . . 96
5.8 The Square Root of CNOT . . . . . . . . . . . . . . . . . . . . . . . . 97
5.9 Quantum Scrambling Library . . . . . . . . . . . . . . . . . . . . . . 99
5.9.1 Deterministic Teleportation Protocol . . . . . . . . . . . . . . 101
5.9.2 Bell States and Bell Measurements . . . . . . . . . . . . . . . 102
5.9.3 Scrambling Unitary US . . . . . . . . . . . . . . . . . . . . . . 104
5.9.4 Scrambling Unitary UCZ . . . . . . . . . . . . . . . . . . . . . 106
5.9.5 Classical Scrambler . . . . . . . . . . . . . . . . . . . . . . . . 108
5.9.6 Circuit Optimization . . . . . . . . . . . . . . . . . . . . . . . 110
6 Complete 3-Qubit Grover Search 112
6.1 The Grover Search Algorithm . . . . . . . . . . . . . . . . . . . . . . 112
6.2 Oracles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.3 Circuit Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.5 Additional Iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.6 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7 Parallel 2-Qubit Operations 130
7.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.1.1 Fidelity of Parallel XX Operations . . . . . . . . . . . . . . . 137
7.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.3.1 Calculating Experimental Fidelities of 2-Qubit Entangling Gates162
ix
7.3.2 Fidelity of Parallel 2-Qubit Entangling Gates with Different
Degrees of Entanglement . . . . . . . . . . . . . . . . . . . . . 164
7.3.3 Independence of Parallel Gate Calibration . . . . . . . . . . . 166
7.4 Simultaneous CNOT Gates . . . . . . . . . . . . . . . . . . . . . . . 168
7.5 The Quantum Full Adder . . . . . . . . . . . . . . . . . . . . . . . . 169
7.6 Toward a Single-Operation GHZ State . . . . . . . . . . . . . . . . . 173
7.7 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
8 Outlook 181
8.1 Experimental Error Sources . . . . . . . . . . . . . . . . . . . . . . . 181
8.2 Scalability of Ion Traps . . . . . . . . . . . . . . . . . . . . . . . . . . 182
Bibliography 185
x
List of Tables
2.1 Final focal lengths for lenses chosen for the focusing lens system. . . . 35
6.1 Single-Solution Oracles . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.2 Two-Solution Oracles . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.1 Comparison of optical power for parallel and single XX gates. . . . . 160
xi
List of Figures
2.1 Energy levels of interest for 171Yb+. . . . . . . . . . . . . . . . . . . . 8
2.2 Qubit initialization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Qubit readout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Blade trap pictures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Vacuum chamber and optical access. . . . . . . . . . . . . . . . . . . 19
2.6 Raman transitions on the experiment. . . . . . . . . . . . . . . . . . . 23
2.7 Counterpropagating beam geometry. . . . . . . . . . . . . . . . . . . 24
2.8 Multi-channel AOM. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.9 Efficiency of the 32-channel AOM. . . . . . . . . . . . . . . . . . . . . 28
2.10 RF control of 32-channel AOM. . . . . . . . . . . . . . . . . . . . . . 29
2.11 DOE output. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.12 Individual addressing optics solution. . . . . . . . . . . . . . . . . . . 33
2.13 Individual addressing optical setup on table. . . . . . . . . . . . . . . 37
2.14 Focused beams for individual addressing. . . . . . . . . . . . . . . . . 37
3.1 Rotations on the Bloch sphere. . . . . . . . . . . . . . . . . . . . . . 44
3.2 Rabi flops on 5 ions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Rabi flops on 7 ions in a 9 ion chain. . . . . . . . . . . . . . . . . . . 46
3.4 Pulse-shaping scheme for XX gate. . . . . . . . . . . . . . . . . . . . 56
4.1 The computational stack. . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Experimental control. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3 Igor control system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4 Example experimental sequence. . . . . . . . . . . . . . . . . . . . . . 69
5.1 Single-qubit composite gates. . . . . . . . . . . . . . . . . . . . . . . 80
5.2 Two-qubit composite gates. . . . . . . . . . . . . . . . . . . . . . . . 83
5.3 Controlled-controlled unitary. . . . . . . . . . . . . . . . . . . . . . . 85
5.4 Three-qubit composite gates. . . . . . . . . . . . . . . . . . . . . . . . 86
5.5 Toffoli-3 gate data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.6 Toffoli-3 characterization. . . . . . . . . . . . . . . . . . . . . . . . . 89
5.7 Toffoli-4 gate circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
xii
5.8 Toffoli-4 gate data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.9 Controlled-SWAP gate implementation . . . . . . . . . . . . . . . . 94
5.10 CSWAP gate data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.11 XX gate χ-obviation circuit. . . . . . . . . . . . . . . . . . . . . . . . 97
5.12 Component circuits for CNOT and its square root gates . . . . . . . 98
5.13 General circuit to probe scrambling behavior. . . . . . . . . . . . . . 100
5.14 Grover protocol for quantum scrambling measurement. . . . . . . . . 102
5.15 Bell measurement circuit. . . . . . . . . . . . . . . . . . . . . . . . . 104
5.16 Circuits for the scrambling unitary US. . . . . . . . . . . . . . . . . . 106
5.17 Circuits for the scrambling unitary UCZ . . . . . . . . . . . . . . . . . 107
5.18 Circuits for the classically scrambling unitary UCS. . . . . . . . . . . 109
5.19 Unitary equivalence on EPR pair components. . . . . . . . . . . . . . 110
5.20 Optimization of XX gates in a scrambling experiment. . . . . . . . . 110
5.21 Optimization of XX gates in a scrambling experiment. . . . . . . . . 111
6.1 The Grover search algorithm. . . . . . . . . . . . . . . . . . . . . . . 113
6.2 General circuits for a Grover search algorithm. . . . . . . . . . . . . . 117
6.3 Grover search algorithm circuits. . . . . . . . . . . . . . . . . . . . . 119
6.4 Single-solution algorithm. . . . . . . . . . . . . . . . . . . . . . . . . 125
6.5 Two-solution algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.1 Pulse shape for (1,4), (2,5). . . . . . . . . . . . . . . . . . . . . . . . 151
7.2 Pulse shape for (1,2), (3,4). . . . . . . . . . . . . . . . . . . . . . . . 152
7.3 Pulse shape for (1,5), (2,4). . . . . . . . . . . . . . . . . . . . . . . . 153
7.4 Pulse shape for (1,4), (2,3). . . . . . . . . . . . . . . . . . . . . . . . 154
7.5 Pulse shape for (1,3), (2,5). . . . . . . . . . . . . . . . . . . . . . . . 155
7.6 Pulse shape for (1,2), (4,5). . . . . . . . . . . . . . . . . . . . . . . . 156
7.7 Parity curve for parallel 2-qubit gates on (1,4), (2,5). . . . . . . . . . 157
7.8 Parity curve for parallel 2-qubit gates on (1,2), (3,4). . . . . . . . . . 157
7.9 Parity curve for parallel 2-qubit gates on (1,5), (2,4). . . . . . . . . . 158
7.10 Parity curve for parallel 2-qubit gates on (1,4), (2,3). . . . . . . . . . 158
7.11 Parity curve for parallel 2-qubit gates on (1,3), (2,5). . . . . . . . . . 159
7.12 Parity curve for parallel 2-qubit gates on (1,2), (4,5(). ). . . . . . .(. ). 159
7.13 Parity curve for parallel 2-qubit gates on (1,5) XX π , (2,4) XX π .165
4 8
7.14 Independence of parallel gate calibration. . . . . . . . . . . . . . . . . 167
7.15 Independence of parallel gate calibration, continued. . . . . . . . . . . 168
7.16 Simultaneous CNOT gates. . . . . . . . . . . . . . . . . . . . . . . . 169
7.17 Full adder circuits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.18 Full adder data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
xiii
List of Abbreviations
ASP Algorithm Success Probability
AOM Acousto-Optic Modulator
AWG Arbitrary Waveform Generator
CNOT Controlled-NOT gate
DAC Digital to Analog Conveter
DDS Direct Digital Synthesizer
DOE Diffractive-Optic Element
ELU Elementary Logic Unit
EOM Electro-Optic Modulator
FPGA Field Programmable Gate Array
GUI Graphical User Interface
H Hadamard gate
NA Numerical Aperture
OTOC Out of Time Order Correlator
PMT Photo-Multiplier Tube
QCQP Quadratically Constrained Quadratic Program
R Rotation gate
RF Radio Frequency
SK1 1st order Solovay-Kitaev composite pulse
SPAM State Preparation And Measurement
SSO Squared Statistical Overlap
TTL Transistor-Transistor Logic
UHV Ultra-High Vacuum
VVA Voltage Variable Attenuator
XX Two-qubit σxσx entangling gate
Yb Ytterbium
xiv
Chapter 1: Introduction
The potential applications for quantum computers have been of great interest
since Manin and Feynman first proposed them in the early 1980s [1, 2]. By taking
advantage of properties unique to quantum systems like multi-state superposition
and multi-qubit entanglement, a large-scale quantum computer promises advances in
humanity’s ability to tackle complex chemistry problems that could revolutionize the
energy sector, solve optimization problems with uses across almost every industry,
accelerate breakthroughs in biology and medicine, and - depending on your point of
view - either wreak havoc on or revolutionize the security and data industries.
To simulate a two-level quantum mechanical system on a classical computer
(like the one you’re probably reading this thesis on), a system with N particles re-
quires keeping track of 2N amplitudes, an exponential growth in problem size that
prevents simulating large quantum systems; even the world’s biggest supercomput-
ers have not been able to simulate all amplitudes in a quantum system larger than
49 particles [3]. Future advances in computing will likely push past that, but most
predictions anticipate only getting to 60-70 particles. Simulating 300 quantum par-
ticles would require a classical computer the size of the known universe, which is
not practical. However, a quantum computer can simulate an N -particle quantum
1
system with much less than exponential growth, simply because it is a quantum sys-
tem as well; in principle, the problem size can be O(N) particles, with perhaps some
additional resources required for error correction. This power of “quantum paral-
lelism” [4] can be used to directly simulate quantum systems of interest, such as
for quantum chemistry applications, but can also be harnessed to solve other kinds
of problems, such as optimization problems. One particularly interesting potential
application in the field of quantum chemistry is to better understand biological ni-
trogen fixation [5], a process that is central to the creation of fertilizer. Our best
fertilizer creation methods require a substantial proportion of the human energy
budget, but bacteria can create the same chemicals far more efficiently. A quantum
computer could help us discover these efficient methods for producing fertilizer, mak-
ing it easier to feed humanity while reducing our energy usage, our carbon footprint,
and our effects on global climate change. The exact scope of quantum computing
technology’s effect on humanity remains tantalizingly unclear, but even pessimistic
outlooks envision significant benefits from large quantum computers. Consequently,
increasing numbers of government and industry players are investing in this tech-
nology - from U.S. government agencies like IARPA, the NSF, the Departments of
Defense and Energy, and the NSA, to companies like Google, Intel, Microsoft, and
startups like Rigetti and IonQ.
Of course, all of this being of any use at all hinges on an important question:
how do we build a quantum computer big enough to do something interesting? In
this thesis, I leave aside some important abstract issues that arise from this central
question (how do we define “big”? What is “interesting” in this context? What
2
constitutes a quantum computer, exactly?) to focus on the pragmatic and practical
aspects of the question: “building” and “doing”.
Ion trap quantum computers [6, 7] and superconducting quantum comput-
ers [8,9] represent the two proposed quantum computing platforms that so far have
made the most progress toward a practical and usable technology, and have opti-
mistic outlooks for scaling up. Recent advances in both of these platforms portend
optimism that a large-scale quantum computer will eventually be built, though chal-
lenges certainly remain. Additionally, it is not yet clear which platform will prove
itself to be the best candidate for a burgeoning technology, or whether the most
successful quantum computers will be some other hardware [10] that is currently
still being explored (such as neutral atoms [11, 12], quantum dots [13, 14], nitrogen
vacancy centers in diamond [15], or large photonic systems [16,17]), a hybrid of two
or more quantum systems [18], or even one that has yet to be proposed at all.
In this thesis, I will discuss an ion trap quantum computing machine we have
built [19, 20] and describe some results obtained with it. To demonstrate that our
machine is indeed a quantum computer, I use the DiVincenzo criteria [21] as a
practical measure for a highly programmable and usable device that is similar enough
to modern digital computers to be a familiar and user-friendly concept, but still takes
advantage of all available resources of quantum computing. The criteria, and our
adherence to them, are summarized below.
• Qubits: We use well-characterized 171Yb+ ions as qubits (see Section 2.1) in a
scalable trapped-ion quantum computing architecture [7].
3
• Qubit initialization: 171Yb+ qubits can be initialized to a known state with
high fidelity (see Section 2.1.3).
• Long coherence times: These qubits have typical coherence times longer than
1 s, which is many orders of magnitude greater than gate times of 20-200 µs
and hence allows for long coherent gate sequences (see Section 2.1.1).
• A universal gate set: The native gate set available, single-qubit R rotations and
two-qubit entangling XX gates, constitute a universal gate set (see Section
3.3).
• Qubit readout: State-dependent fluorescence of 171Yb+ qubits allows for high-
fidelity qubit measurement (see Section 2.1.4).
Two additional optional DiVincenzo criteria exist that address a system’s capabil-
ities for quantum communication; these are not addressed here, as the system is
not designed to communicate with other quantum systems and hence does not meet
these criteria at present. Other efforts exist to implement robust quantum communi-
cation and networking [22] and provide long-distance quantum communication [23]
using ion traps and other hardwares.
To be most useful, a quantum computer should be easily programmable and
flexible to be able to tackle many kinds of problems, as a classical computer can.
To this end, we have built an ion trap quantum computer with controls organized
into separate levels of abstraction, called a computing stack. Each level of the stack
represents a different level of control in the system, and is designed to be a black
box to the others; this architecture allows users to implement quantum algorithms
4
without needing to know much, if anything, about what qubits the system uses or
the physics and engineering details behind the hardware. This framework is very
similar to classical computers; I don’t need to know anything about NAND gates
or silicon microprocessors to read the news on the Internet or run data analysis
calculations for this thesis. Most previous quantum computing experiments have
been built to perform a single algorithm, with the hardware designed and tailored for
that purpose, and without flexibility to perform many other experiments; the stack
framework of this machine represents a step forward from a systems perspective, as
well as the previously-undemonstrated results we have generated on it.
The first half of this thesis, Chapters 2-5, will describe the functioning of
the ion trap quantum computer by going through the stack, from the low-level
hardware up to the high-level user interface. The second half, Chapters 6-7, will
describe several new results generated from the system over the course of this thesis
work, and can additionally be found in [24, 25]. Additional work performed on this
machine over the course of this thesis work but not discussed here in detail can be
found in [19,26–32].
In Chapter 2, I present the hardware level of the stack: trapped 171Yb+ ions
as qubits, with various lasers of initialization, readout, and coherent control.
In Chapter 3, I discuss the quantum control level of the stack, consisting of
pulse shaping schemes that effect high-quality quantum gates.
In Chapter 4, I explain the compiler level of the stack, which consists of a
computer control program that compiles quantum gate commands from the user into
the low-level operations necessary to execute the gate sequence on the experiment.
5
In Chapter 5, I enumerate the library of quantum gates we have assembled for
use in the compiler to execute various algorithms.
In Chapter 6, I report results for a complete 3-qubit Grover search algorithm.
In Chapter 7, I introduce a promising new tool for quantum control: parallel
2-qubit operations on pairs of ions in the same ion chain.
In Chapter 8, I conclude with a positive outlook on ion traps as quantum
computers, and discuss future challenges and suggest areas to improve.
6
Chapter 2: Quantum Computer Hardware
In this chapter, I will present an overview of the hardware of the ion trap
quantum computer used to perform the work in this thesis. The overview will be
brief, as much of this material has already been presented in [20], which concerns
the same system. I will also include some upgrades that have since been made to the
system, including to the individual addressing system allowing for better individual
ion controls crucial to work presented in Chapter 7, and expanding from 5 to 7 ions
for use in the work presented in [31] and future work.
2.1 The 171Yb+ Qubit
In this system, we have chosen as our qubit the first-order magnetic-field-
insensitive pair of clock states in the hyperfine-split 2S1/2 ground level manifold
of 171Yb+, with |0〉 ≡ |F = 0;mF = 0〉 and |1〉 ≡ |F = 1;mF = 0〉 (see Figure
2.1) [33]. The two qubit states have a 12.642821 GHz frequency difference, which
can be addressed with microwaves or Raman transitions via the 2P levels.
7
Figure 2.1: Energy levels of interest for the 171Yb+ qubit. The qubit is defined as
|0〉 ≡ |F = 0;mF = 0〉 and |1〉 ≡ |F = 1;mF = 0〉.
2.1.1 Coherence Times
The |F = 0;mF = 0〉 and |F = 1;mF = 0〉 states in 171Yb+ ions are so-called
“clock” qubits because they are insensitive to magnetic fields to first order. On our
experiment, our measured coherence time is 1.5(5) s, which is more than 3 orders
of magnitude greater than typical single-qubit gate times of 20 µs or two-qubit gate
times of 200 µs; for short algorithms with at most a few tens of two-qubit gates, this
coherence time is therefore sufficient. On this experiment, performance is primarily
limited not by qubit coherence, but by pointing noise on the Raman control beams
driving coherent transitions; this is further discussed in Chapter 8.
8
2.1.2 Cooling
The qubits are cooled to near their motional ground state in a two-step process.
First, the ion is cooled using Doppler cooling [34,35]. Light beams detuned from the
{2S 21/2, F = 1} to { P1/2, F = 0} and {2S1/2, F = 0} to {2P1/2, F = 1} transitions
is applied to the ions, with both π and σ polarizations added to address all of
the Zeeman states in the ground state manifold. The two transitions are applied by
supplying 369 nm light resonant with the {2S 21/2, F = 1} to { P1/2, F = 0} transition,
and adding a 14.7 GHz sideband to access the {2S1/2, F = 0} to {2P1/2, F = 1}
transition using the second-order sideband of an EOM at 7 GHz. The 369 nm light
is generated by doubling the output of a 739 nm laser that is locked to an iodine
vapor cell [36]. The output of the doubler is 300 MHz detuned from the transition, so
frequency shifts and TTL control are provided by one IntraAction AOM at 290 MHz
to give the optimal Doppler cooling frequency for a Γ/2π = 19.7 MHz linewidth.
A light beam red-detuned from resonance by 300 MHz (the “protection beam”)
acts on the ions during trapping only, and two more beams red-detuned by 10 MHz
with an AOM cool the ions further during setup before experimental procedures.
Red-detuned photons are absorbed by hot ions moving toward the beam source,
and then scattered in a random direction, reducing their net momentum (and con-
sequently, their kinetic energy) in the direction of the cooling beams. The three
beams are therefore at angles to each other to ensure cooling along all axes; the
main 10 MHz beam travels at a very slight angle from the ion chain axis in the
horizontal plane, the 300 MHz protection beam is applied in the horizontal plane at
9
an angle near perpendicular to the chain axis, and the low-power 10 MHz “oblique”
beam is applied with a vertical component at an oblique angle to the other two
(see Figure 2.5). On our experiment, we define the z axis as the horizontal direc-
tion along the trap axis, the x axis as the horizontal axis perpendicular to the trap
axis, and the y axis as the vertical direction, perpendicular to the trap axis and
the optical table. Consequently, the beams have components along the x, y, and z
directions, and while we don’t reach the Doppler limit, the ions are cooled enough
in all directions (n̄ = 10 to 15) that resolved sideband cooling can finish cooling the
ions to near their motional ground states.
Second, resolved sideband cooling [37] is necessary to further cool the ions
to the Lamb-Dicke regime, which is necessary to execute high-fidelity gates. The
Lamb-Dicke parameter η can be described as a measure of the coupling strength
between the ion’s internal or spin state (so called because we are only using the
two qubit states) and its motional state. In the Lamb-Dicke regime, this parameter
is small enough that transitions changing the motional quantum number by more
than 1 phonon are strongly suppressed. Specifically,
η2(2n+ 1) 1 (2.1)
where n is the motional quantum number of the ion. Resolved sideband cooling
is a process that couples to the motional modes of the ion to progressively remove
phonons while cycling between the |0〉 and |1〉 spin states [38]. This cools the ions
below the Doppler cooling limit, and on our experiment ends with a typical n̄ = 0.1
10
Figure 2.2: Qubits are initialized to the |0〉 state via optical pumping. The straight
blue arrows indicate photon absorption, and the squiggly purple arrows indicate
photon emission.
quanta.
2.1.3 Qubit Initialization
The qubits are initialized to their |0〉 state via optical pumping [33]. The ions
are illuminated with light resonant with the transition between the {2S1/2, F = 1}
and {2P1/2, F = 1}manifolds, as shown in Figure 2.2. An AOM provides a frequency
shift of 300 MHz to make the 369 nm beam on-resonant with the {2S1/2, F = 1} and
{2P1/2, F = 0} transition, and an EOM adds 2.1 GHz sidebands to then reach the
{2P1/2, F = 1} manifold to clear out the {2S1/2, F = 1,mF = 0} state. The beam
contains both π and σ polarization components to ensure successful pumping for ions
11
in any of the Zeeman states in the {2S1/2, F = 1} manifold. (π alone is sufficient,
but the beam path is shared with the detection beam discussed in Section 2.1.4,
which requires both components.) Once pumped to the {2P1/2, F = 1} manifold,
the ions then decay either back to the {2S1/2, F = 1} manifold, where they will
be pumped again, or to the {2S1/2, F = 0} target state. Once there, ions in the
{2S1/2, F = 0} state stay there, as the applied beam is now 14.7 GHz off-resonant
from a transition back to the {2P1/2, F = 1} manifold, so running the pump beam
for a short amount of time (5 µs) is sufficient to initialize all qubits to the |0〉 state
with high fidelity.
2.1.4 Qubit Readout
At the end of a quantum procedure, the qubits are measured and their states
read out using state-dependent fluorescence, as shown in Figure 2.3(a). Light res-
onant with the transition from {2S1/2, F = 1} to {2P1/2, F = 0} is applied to the
ions, using a 300 MHz AOM to shift a beam of 369 nm light to resonance. Qubits
in the |1〉 state are then excited to the {2P1/2, F = 0} state, which has a lifetime of
8.12 ns, and then decays back to the {2S1/2, F = 1} manifold, emitting a photon in
the process and hence termed “bright.” Polarization elements in π and σ are added
to ensure continued cycling from all of the Zeeman levels in the {2S1/2, F = 1}
manifold, allowing many photons to be scattered over the course of the detection
period. If the qubit is instead in the |0〉 state, the applied light is 2.1 GHz away
from the nearest dipole-allowed transition to the {2P1/2, F = 1} manifold, and so
12
(a) (b)
(c)
Figure 2.3: (a) Qubits are measured using state-dependent fluorescence. Ions in
the |1〉 state fluoresce in response to an applied detection beam; ions in the |0〉
state do not. The straight blue arrows indicate photon absorption, and the squiggly
purple arrows indicate photon emission. (b) A typical detection histogram showing
photon counts for bright (orange) and dark (purple) ions. The measurement fidelity
for a qubit in |0〉 is 99.74(3)% and for a qubit in |1〉 is 99.09(5)%. From [20]. (c)
Individual ions are detected on separate channels of a linear array of 32 PMTs. This
shows photon counts for 5 bright ions; the X-axis is the PMT channel number.
13
the qubit scatters no photons and remains dark. The detection process is run for
150µs to ensure sufficient photons are collected to discriminate between bright and
dark qubits. The entire experiment is repeated several hundred or thousand times
to build up statistics, and photons are collected with an imaging objective with a
numerical aperture (NA) of 0.37. Figure 2.3(b) shows a typical histogram showing
the number of photon counts for bright and dark states. The distributions are well-
distinguished, and we can discriminate between |0〉 and |1〉 by determining that any
qubit with 0 or 1 counts is dark, and any qubit with 2 or more counts is bright.
Measurement errors arise when a bright qubit is measured as dark or a dark
qubit is measured as bright during the measurement process. The former can occur
during the measurement due to off-resonant coupling between the F = 1 states of
2S and 21/2 P1/2, as the {2P1/2, F = 1} manifold is only 2.1 GHz away from the
{2P1/2, F = 0} state. Once pumped to the {2P1/2, F = 1} manifold, the ion can
decay to {2S1/2, F = 0}, where it ceases scattering photons. Off-resonant coupling
between |0〉 and the {2P1/2, F = 1} manifold is also possible, which would scatter
a photon upon decay, but with a detuning of 14.7 GHz, this is much less likely.
This results in the not-quite-Poissonian histograms in Figure 2.3(b); in particular,
the off-resonant coupling of |1〉 to {2P1/2, F = 1} is the largest effect, resulting in
visible histogram points at 0 and 1 counts for an ion prepared in the bright state.
Background scatter from the trap blades can also result in photon counts for dark
ions, but has a small effect. Consequently, the measurement fidelity for a single ion
in |0〉 is slightly higher than for |1〉: 99.74(3)% and 99.09(5)%, respectively.
To distinguish between the fluorescence of different ions, the ions are imaged
14
onto separate channels of a 32-channel linear PMT array1. Figure 2.3(c) shows typ-
ical data for N = 5 bright ions on the PMT array, where the X-axis is the channel
number. The ions are imaged onto every other channel to reduce electronic crosstalk
errors - neighboring channels have about 4% crosstalk error, and next-nearest neigh-
bors about 1%. The crosstalk is also asymmetric, as channels to the right see more
spillover from channels to their left than the other way around. Dark counts are 1-2
per channel per second and therefore negligible. When data is collected, each chan-
nel is analyzed separately, using the discriminator mentioned above to determine
which ions are in |0〉 or |1〉 after each experiment. After the experiment is repeated
many times, the data can then be assembled into a vector of output populations,
which gives the population in each of the N2 N -qubit states; for 5 ions, the state
vector gives the populations in each of {00000, 00001, 00010, 00011, . . .}.
Bright-to-dark pumping and crosstalk are the two main sources of error for
our multi-qubit readout process. We account for this and state preparation errors
by measuring our SPAM errors and then renormalizing our data appropriately. To
measure our SPAM errors, we trap a single ion, then use the DC voltage controls
to move that ion to each of the locations in the trap where an ion would be for
an N -ion chain. Two measurements are taken at each location: one where the ion
is initialized to the dark state, and another where the ion is initialized and then
coherently rotated to the bright state using a microwave pulse resonant with the
12.6 GHz qubit frequency. The measurements are repeated 80,000 times to build
robust statistics. The measured data is then assembled into a N2 × N2 matrix of
1Hamamatsu H7260-200
15
each of the possible states. Once data is taken for an algorithm of interest, this
SPAM matrix can then be inverted and multiplied with the corresponding N2×N2
state matrix for the data to remove SPAM errors [20, 39]. Additional methods of
measuring SPAM errors on our experiment are presented in [40].
2.2 The Ion Trap
The ions are held in an ion trap in a chamber held at room temperature
[41]. Our physical qubits are well-isolated from the environment in a stainless steel
chamber that has been evacuated to ultra-high vacuum (UHV) with pressure about
10−11 Torr, meaning a given ion will only experience a collision with a background
gas particle once every hour or so on average. This is therefore negligible on the
order of the coherence time, and primarily limits the lifetime of the ion in the trap.
Consequently, we can hold chains of 5 or 7 ions for about an hour at a time.
The trap used here is a blade trap, a variant of a linear Paul trap [42]. Four
gold-coated electrode substrates shaped like razor blades provide the confining po-
tential (see Figure 2.4 for several views of the blade trap). Two blades, diagonally
across from each other, provide an oscillating pseudopotential that confines the ions
along the radial directions of the trap. The RF signal supplying the pseudopo-
tential is generated by a DDS and amplified to ∼300V using a stabilized helical
resonator [43,44] at a driving frequency of 23.8167 MHz.
The other two blades are divided into 5 electrodes each that then have static
voltages applied. These DC electrodes provide an axial confinement of 296 kHz
16
(a) (b)
(c)
Figure 2.4: Several views of the ion trap blades. (a) A view along the blade, which
can be seen here to be divided into 5 electrodes for enhanced ion position control on
the DC blades. (b) A view of the 4 blades, looking directly down the trap axis. (c)
Another view of the 4 blades, shown at an angle from the trap axis. Image credit
S. Debnath.
17
for 5 ions and also allow for finer control of the spacing between the ions. Axial
confinement requires, at minimum, 3 electrodes on each blade, where the center
electrode on each blade is set to a negative voltage (lower potential for the positively-
charged ion), and the outer electrodes are set to a positive voltage (higher potential
for the ion). This creates a quadratic potential well confining the ion. Multiple
ions in such a trap will not be equally spaced; the inner ions will be pushed closer
together than the outer ions, and the spacing disparities increase with increasing ion
number [45]. Since our indvidual addressing beams are equally spaced, equal spacing
of ions would be ideal. Very nearly equal spacing can be achieved by adding at least
2 more electrodes to each blade, which allow for the addition of a quartic term to
the trap potential; appropriately selected trap voltages will create a potential with
a “flat” area in the center that will hold ions at approximately equal distances from
each other. In practice, however, simulations of our blade trap showed that such
a potential would require kilovolts on the outer electrodes.Equal spacing for large
numbers of ions will likely require a microfabricated surface electrode trap with
many DC electrodes to control the positioning of many ions. However, for 5 ions in
our blade trap, the spacing differential is small enough that the individual beams
are still fairly well aligned on the ions; the Rabi frequencies seen by the ions are
comparable (see Section 3.1.1.)
Below the trap, two small stainless steel tubes, or ovens, contain small amounts
of solid ytterbium metal that has been isotopically purified to be 95% 171Yb (natural
abundance 14%). Niobium wires wrapped around the ovens heat up when current
is applied, allowing a small amount of heated ytterbium to sublimate into a vapor
18
355nm Raman + 399nm 
   + 369nm cool
369nm cool, pump, detect 
   + 935nm
Microwave
Detection Collection
369 cool
Figure 2.5: A visualization of the vacuum chamber containing the ion trap, and the
optical access for the various necessary control components. The Raman control
beams (red) and one of the cooling beams are applied radially, orthogonal to the ion
chain, while another cooling beam, the detection and pump beams, and the loading
beam (blue) are applied along the chain axis. A third cooling beam is applied
through a small window at angles to both of the other cooling beams (turquoise.)
Microwaves are applied through a horn from below (purple.) Finally, detection
light is collected from above the trap (green.) Chamber CAD modified from S.
Debnath [19].
19
of neutral ytterbium atoms, some of which will spread into the trapping area. A
laser at 399nm, tuned to the 1S0 →1 P1 atomic transition in neutral 171Yb, is
directed through the trap area, so some atoms will absorb a photon and be excited
to the 1P1 state. An additional photon at ≤369 nm is required to achieve the first
photoionization energy and strip off one electron to create 171Yb+ [46]; the various
369nm laser beams already passing through the trap that will next be used for
cooling purposes serve to complete this ionization process. Once ionized, the atom
is now subject to the trap potential, and can be cooled to the motional ground
state by the cooling beams. The ions then form a linear Coulomb crystal along the
weakly-confined axial direction in the trap. On this experiment, once the ovens heat
up (a 3 minute process), 5-9 ions can typically be trapped within a minute or two.
As shown in Figure 2.1, states in the 2P1/2 manifold will decay to the metastable
2D3/2 level with a branching ratio of 200:1. To repump the ion, a 935 nm laser beam
is applied constantly to the ions. This repumper induces a transition to the 3[3/2]3/2
level, which then decays to the ground level. The repumper beam has 3.07 GHz
sidebands added by an EOM and has both π and σ polarization components to
address all states in the 2D3/2 manifold.
All of the optical controls discussed so far, as well as the Raman beams de-
scribed in the next section, present a challenge in delivering all of these controls to
the ions in the trap. Figure 2.5 shows a mockup of the vacuum chamber, which
has 8 windows around the trap providing ample optical access for cooling beams
(369nm) at 3 different angles, a pump/detection beam (369nm) transported through
a shared fiber at different times during the experimental sequence, the loading beam
20
(399nm), the repumper beam (935nm), counterpropagating 355nm Raman beams
applied from opposing sides (see Section 2.3), microwaves (12.6 GHz), and detection
light collection.
2.3 Coherent Individual Addressing System
Coherent transitions between the |0〉 and |1〉 states of the qubit can be driven
using direct microwave transitions at 12.642821 GHz, or via the P -levels using Ra-
man transitions driven by two beams with a beatnote of 12.642821 GHz. Microwaves
have two major drawbacks. First, they are not straightforward to focus in free space
to individually address an ion in a trap [47]. Second, microwaves provide very little
net momentum transfer to couple to the motional modes of the chain, which is what
we use to entangle multiple ions; doing so is possible, but again not straightfor-
ward, as it requires a surface trap with waveguides [48]. Individual addressing and
implementation of two-qubit gates on non-clock-state qubits can alternatively be
accomplished with microwaves by generating a very large magnetic field gradient,
which induces different Zeeman splittings in the qubits that can then be individu-
ally addressed by changing the microwave frequency and used to couple to motional
modes [49]. However, the magnetic field gradient required is not scalable to large
numbers of ions in the trap, as the gradient required for more than a handful of ions
quickly becomes infeasible. On our experiment, we do provide a microwave horn
that emits a 12.642821 GHz signal and globally addresses all ions, as a useful tool for
troubleshooting and performing diagnostics. Coherent operations during algorith-
21
mic processes are provided by individually-addressed Raman beams, as described in
this section.
2.3.1 Raman Beam Geometry
Coherent qubit operations are performed by a pair of counterpropagating Ra-
man beams on the ions in our trap (see Figure 2.6(a)). The Raman transition is
performed by a mode locked 355nm pulsed laser. The two Raman beams there-
fore each consist of a frequency comb [50], with components separated by the laser
repetition rate, about 118 MHz. To address the 12.6 GHz qubit splitting, the coun-
terpropagating Raman beams must be lined up in frequency space so that there is a
12.6 GHz beat note between each corresponding tooth in the two frequency combs.
To account for noise on the laser repetition rate, an AOM beat note lock (similar to
that described in [51]) is constructed, in which the repetition rate noise is inverted
and added to an AOM on one Raman beam. Consequently, one Raman beam can
coherently stimulate a transition from the |0〉 or |1〉 state to a virtual state partway
between the 2P 21/2 and P3/2 manifolds in
171Yb+, then the other beam induces a
transition to the other qubit state, as shown in Figure 2.6(b). The commercially-
available 355 nm laser wavelength is close to minimum spontaneous emission from
the 2P 21/2 and P3/2 manifolds, which is 1/3 of the way between the two manifolds
since 2P3/2 has more states in the manifold and therefore twice the coupling of the
2P1/2 manifold. Due to the counterpropagating geometry and the large amount of
energy carried by 355 nm photons (844 THz), stimulated Raman transitions impart
22
(a)
(b)
Figure 2.6: (a) An overview of the experimental setup, showing the counterpropagat-
ing Raman beams, the ion chain, and the multi-channel PMT detection apparatus.
The use of spin-motion couplings to enact 2-qubit entangling gates results in all
possible interaction pairs being available for entangling gates. Image from [19]. (b)
Raman transitions at 355 nm in 171Yb+.
23
(a) (b)
Figure 2.7: (a) Counterpropagating beams overlapping 5 example ions. The wide
beam is the global beam that interacts with all ions, while the narrow beam is one
individual beam aligned to interact with a single ion. (b) The two arms of the
Raman beam setup.
a large momentum kick to the ions (see Figure 2.7(a)); this will allow us to couple
strongly to the motional modes to implement 2-qubit gates (see Section 3.2.)
The beam path is split to provide the two arms of the counterpropagating
geometry, as shown in Figure 2.7(b). Each arm is then modulated by an AOM, one
of which is controlled by the beat note lock, and the other of which is is controlled by
the output of an AWG that modulates the frequency as needed, such as to address
the red and blue sidebands necessary to implement a Mølmer-Sørensen gate [52,53];
see Section 2.4 for discussion of AWG control of the AOMs, which was changed
partway through this thesis work.
The global beam is modulated with an IntraAction AOM at 130 MHz. After
the AOM, cylindrical lenses shape the beam to be wide along the trap axis and
narrow in the transverse direction, so that light from the beam falls on all of the
ions in the chain, while maintaining a narrower focus of ∼20 µm perpendicular to
24
the chain. The individual beam arm of the Raman setup is split horizontally into
10 beams; each individually couples to one of 10 neighboring channels of the 32-
channel AOM. The beams are deflected vertically, which is orthogonal to the ion
chain axis. Since the zeroth-order beams are not needed, a D-mirror is used after the
AOM to redirect them to a beam dump. The first-order beams continue to the ions,
where the AOM-provided modulation allows Raman transitions to occur. Since only
deflected beams reach the ions, shutting off the RF signal to a given channel will
shut off the beam that reaches the corresponding ion; hence, control over the RF
signal to each channel provides control over which ions perform operations.
2.3.2 32-Channel AOM Characteristics
AOMs are a tool used frequently in atomic physics experiments. Laser light of
frequency f and wavelength λ is passed through a transparent crystal, such as fused
silica. A piezoelectric transducer, controlled by an applied RF signal of frequency
F , transmits sound waves of wavelength Λ into the crystal, which varies the index of
refraction in the crystal. In the Bragg regime, the incoming light is then diffracted
into symmetric orders m (see Figure 2.8(a)), each of which is also modulated by its
order number times the applied RF frequency,
f → f +mF, (2.2)
25
and displaced by an angle mθ, where
mλ
sin θ = (2.3)
2Λ
Most of the light can be directed into the first diffraction order by adjusting the
incoming light’s angle of incidence in the plane of the acoustic waves.
The 32-channel AOM varies from the typical single-channel model in that it
consists of a single crystal with 32 evenly spaced transducers, as shown in Figure
2.8(b). The transducers are spaced about 450 µm apart, with a channel width of
100 µm. Each transducer is controlled by its own RF input channel; all of the
channels are designed for a drive frequency at 200 MHz frequency and 700 mW
in power. The channels are arranged horizontally, and activated channels deflect
beams vertically. The crystal and transducers are watercooled to accommodate
that much RF power while suppressing temperature-based signal fluctuations. We
characterized the efficiency of the AOM, and found that all channels had > 65%
efficiency into the first order, and a comfortable 20 MHz range over which the
efficiency was constant (for a typical channel, see Figure 2.9.) Crosstalk was also
found to be minimal, with < 1% optical and RF crosstalk.
2.3.3 32-Channel AOM Control
Providing RF power to the AOM channels has some important considerations
in order to work with the Raman beam setup, and so the circuit design and com-
ponent selection must be performed carefully. The circuit design is shown in Figure
26
(a)
(b)
Figure 2.8: (a) Diagram showing how a single channel AOM works, with two positive
and negative orders shown. (b) The 32-channel AOM differs from its more common
single-channel counterparts in that it has 32 piezoelectric transducers on a single
crystal, hence creating 32 channels that can independently modulate an input laser
beam.
27
Figure 2.9: Efficiency for channel A15 of the 32-channel AOM. The 20 MHz range
over which the efficiency is maximized is sufficient for experimental use, as the RF
control frequency signal does not vary more than 10%.
2.10. First, to control 10 beams, we need 10 independent channels. A two-way
power splitter (Minicircuit ZSCJ-2-1+) is used, and each output is then sent to a
five-channel control box with a five-way splitter (Minicircuits ZFSC-5-1-SB+) for a
total of 10 channels.
Second, we must provide a means of control over when a beam is deflected.
A TTL switch (Miniciruits ZASWA-2-50DR+) is used on the RF input for each
channel, which is in turn operated by the experimental control computer. Since the
switch time is at most 20ns, this allows a given ion’s Raman beam to be shut off
or turned on quickly relative to qubit rotations that take tens of microseconds, or
two-qubit gate times of hundreds of microseconds. We also provide a manual switch
option.
Third, the 32-channel AOM is rated by the manufacturer to take up to 700mW
28
Figure 2.10: Component layout for RF control of 32-channel AOM.
(+28.45dBm) of RF power per channel. Hence, the signal control on each channel
should be designed to produce no more than 700mW. To adhere to this limit, the
beat note power output was measured, insertion loss from the TTL and splitters were
calculated, and then amplifiers were chosen that would produce at least +29dBm
(Minicircuits ZX60-P103LN+, gain +18dB; Minicircuits ZHL-2-12, gain +26dB).
The channels were then individually calibrated by measuring the power output of
each channel with a spectrum analyzer, and then adding passive attenuators before
the pre-amplifier until the measured power output was less than +28.45dBm.
To ensure steady Rabi frequencies on each ion, there should be no power
fluctuations over time on any given channel. Such fluctuations can be caused by
fluctuations in the DC voltage provided to power the channel amplifier, so we used
low-ripple Acopian power supplies. For the large amplifier, we used a Gold Box
A24H850, which has a 0.25 mV ripple and provides +24VDC; for the pre-amplifier,
we used 5EB150, which has a 1 mV ripple and provides +5VDC. For additional
protection, bypass capacitors were installed on each power supply, the DC bulkhead
29
connection on the control box, and the DC power connection to the amplifier. These
bypass capacitors of 10 µF eliminate fluctuations by acting as a resistor to ground
for any AC components that may be in the signal, while the DC component sees
the capacitor as an open circuit and instead continues to the amplifier. It should
be noted that small power differences (< 1dB) from one channel to the next are
acceptable, as these differences can be measured and accounted for.
Another important consideration is that there cannot be phase variations be-
tween channels when individual phase control is not available. This requires that
the path length of the RF signal be the same for all channels. Hence, we must
be certain to use identical components for every channel, and same-length coaxial
cables to connect corresponding components. Additionally, RF components were
chosen that had very small phase imbalances.
2.3.4 Optics for Individual Addressing
In order to the use the 32-channel AOM for individual optical addressing,
an optical system must be designed and constructed to split 355nm light from the
pulse laser into 10 beams, focus those beams into the AOM with 450 µm beam
spacing between adjacent acoustic modes and < 50 µm waist on each beam, and
then focus those beams onto the ions in the trap, which requies 5 µm beam spacing
and < 2 µm beam waists. The problem is additionally complicated by the practical
constraints of a lab: the limited real estate on an optical table limits the allowable
beam path length, and due to commercial optics availability and quality, the total
30
Figure 2.11: An image of the 10 beams from the diffractive optic element, demon-
strating even beam spacing, even power distribution, and very low-power higher-
order beams.
beam diameter should not exceed 2 inches at any point on the beam path.
The 10-way beam splitting problem is solved with a diffractive optic element
(DOE) from Holo-Or, part number MS-244-U-Y-A. It splits one incoming collimated
Gaussian beam with waist at least 0.24 mm into 10 evenly spaced, collimated Gaus-
sian beams of equal power and with the same waist as the input beam (see Figure
2.11). It has an observed diffraction efficency of > 56% and a diffraction angle of
4.3 mrad.
Since the optical system to be designed has so many constraints, it was im-
portant to perform careful analysis of the optical path for each solution attempted.
Additionally, both the spacing between the beams and the waist of each beam are
important. Consequently, ray transfer matrix analysis was used to simultaneously
keep track of beam spacing and beam widths. In the ray picture, we assume that
each beam acts like a ray, and then keep track of one beam’s displacement x from
the optical axis to determine the spacing between beams. The optical system starts
with the 10-way diffractive optic beamsplitter (10BS) splitting the incoming beam
with a diffraction angle of θ = 4.3 mrad. For simplicity, we will examine the size
and displacement of a beam that is deflected θ = 4.3 mrad from the optical axis,
which is equivalent to examining the size and spacing between 2 beams that are
each deflected θ = 2.15 mrad on either side of the optical axis, which will be the
31
closest 2 beams for a diffractive optic element producing an even number of beams.
The initial ray vector is ( ) ( )
x0 0
r0 = = . (2.4)
θ0 0.0043
In the Gaussian picture, we instead assume each beam is a Gaussian beam, and
use the complex beam parameter q to keep track of one beam’s width and calculate
relevant waists. For a Gaussian beam of wavelength λ = 0.000355 mm, radius of
curvature R = ∞ for collimated beams, refractive index n ≈ 1 for air, and radius
w,
1 1 iλ 0.000113i
= + = . (2.5)
q0 R πnw2 w2
Using the ABCD matrix method, we can find the final complex beam parameter for
a given system using
Aq0 +B
q1 = (2.6)
Cq0 +D
and extract the beam radius from q1. For both the ray and Gaussian pictures, we
use ( ) ( )
A B 1 d
= (2.7)
C D 0 1
to represent a distance d through free space, and
( ) ( )
A B 1 0
= (2.8)
C D − 1 1f
to represent a thin lens with focal length f . A naive solution would involve simply
focusing the 10 beams into 10 channels of the AOM, and then using one lens to
32
(a)
(b)
Figure 2.12: The solution for the optics system. The system starts with the 10-way
diffractive optic element beamsplitter (10BS) and is then focused into the 32-channel
AOM (32AOM) with the correct spacing. After that, the system uses alternting pairs
of lenses to bring the separate beams closer together, while expanding each beam’s
radius to then focus very sharply with the final objective f5. The beam radius is
expanded between f3 and f
′
2 and between f
′
3 and f4, while the individual beams
are kept parallel. The individual beams are brought closer together between lenses
f and f , f ′2 3 2 and f
′
3, and f4 and f5, where each beam is also kept collimated. For
appropriate lens selection, this will achieve our desired waist of 1 µm and spacing
of 5 µm at the ions. (a) The Gaussian solution shows the width of a single beam
passing through the lens system. (b) The ray solution shows the spacing of 2 beams
(one red, one orange) passing through the lens system.
image the AOM onto the ions. However, this solution requires that the imagining
lens have a focal length of ∼ 8 meters, which is entirely impractical. Instead, using
the ray transfer analysis, we found a solution that uses two telescope arrangements
after the 32-channel AOM to achieve the desired waist and spacing on the ions,
while reducing the total length of the system. The solution is shown in Figure 2.12.
The first important step in the optics system is focusing the 10 beams into the
AOM onto 10 separate channels. Since the diffractive angle θ = 4.3 mrad for the
33
10-way beamsplitter is small, we can write
xAOM = f1θ (2.9)
for the spacing of the beams at the 32-channel AOM. So f1 must be chosen to achieve
the correct beam spacing at the AOM channels. After the AOM, the system’s
magnification can be calculated,
f4f
′
2f
m = 2′ = 28 (2.10)f1f3f3
where ray transfer matrix analysis shows that m = 28 is the ideal magnification to
yield a 5 µm spacing at the ions given a choice of a 32.7 mm objective (see next
paragraph.) Much like in Equation 2.9, the final spacing is
f5θ
xf = = 5µm. (2.11)
m
Taking into account contraints from commercially available lenses, the final
lenses chosen are listed in Table 2.1. f1 was chosen experimentally; the 90mm
Thorlabs lens allowed us to successfully line up 10 parallel beams into 10 channels.
For an input waist of 0.2 mm to the 10-way beamsplitter, this yields a waist of
55 µm; while this yields a spot slightly larger than the AOM channel, we do not
see any spillover since the channels are 450 µm apart, and the efficiency is still
high enough that the solution is workable. For the telescopes after the 32-channel
AOM, the magnification is m = 74.2, which is close to the ideal from equation 2.10.
34
Lens Focal length at 355 nm Lens used
f1 87.0 mm 90 mm Thorlabs plano-convex
f2 96.6 mm 100 mm Thorlabs plano-convex
f3 -29.0 mm -30 mm Thorlabs plano-concave
f ′2 120.8 mm 125 mm Thorlabs plano-convex
f ′3 -57.3 mm Doublet: -150 mm +meniscus and -100 mm
, Thorlabs plano-concave
f4 483.4 mm Doublet: 1000 mm and 1000 mm +meniscus,
Thorlabs plano-convex
f5 32.7 mm Ronar-Smith objective, 0.25 NA, working
distance 32.7 mm
Table 2.1: Final focal lengths for lenses chosen for the focusing lens system.
The final lens, f5, is a triplet objective made by Ronar-Smith with a 0.25 NA and
working distance 32.7 mm that was chosen because it will minimize abberations
on the ion. Since it will be focusing a large-diamater beam to a very small focal
point, it is important that f5 be chosen carefully. The total system length, from
10-way beamsplitter to ions, is 1.59 m. To save more space, it is possible to move
the objective f5 back toward f4, thus reducing the total length to slightly over 1
m. This is because f4 is a very slow lens; the individual beams have a very small
deflection angle as they converge after the lens, and so placing the objective closer
to f4 will still yield beams very close to parallel at the ions. The beam spacing will
still be correct at the working distance of the objective. This yields the additional
advantage that the individual beams can be focused on the ions without having to
move the entire optics system; instead, small adjustments in the objective will ensure
the individual beams are focused at the ions. While the system will no longer be
telecentric, meaning the beam spacing will vary with objective position, the amount
of variance is negligible.
The lens system was assembled on the table (see Figure 2.13) and a camera
35
placed at the objective focus to image the results. The assembly yielded 10 beams
separated by 5 µm and with waists of 1 µm, as shown in Figure 2.14(a-b). Once the
optics setup was achieved, a retroreflector was added to the global beam arm of the
Raman beam setup to adjust and match the optical path length of the global beam
arm to the individual beam arm. It is important to note that while the individual
beam spacing is fixed, the ion spacing can be adjusted using the five DC electrodes
on the blade trap; this will allow the beams to be aligned as well as possible to the
ions in the trap. Coarse beam alignment was achieved through a rough method of
observing direct scatter off the trap blades to align the Raman beams to the middle
of the trap. Daily fine-tuning consists of optimizing the alignment by maximizing
the observed Rabi frequency.
A higher-resolution 2D profile of 5 focused beams is shown in Figure 2.14(c),
which was imaged using the ion response. With one ion in the trap, the individual
beams were moved across the ion and the Rabi rate on the ion, which corresponds
to the square root of the optical intensity seen by the ion, was measured. The
beams were moved by moving the final imaging objective, which is mounted on a
translation stage controlled by picomotors along all 3 axes. The picomotors allow
the objective to be moved in steps of about 20 nm without opening the protective
enclousure around the Raman optics, which reduces air currents that induce beam
pointing instability and therefore noise on the experiment’s coherent operations.
These picomotors are also used to fine-tune the individual beam alignment on the
ions, which drifts on a daily timescale.
The final ion positions relative to the center ion were [−10.74,−5.05, 0, 5.16, 10.85]
36
Figure 2.13: Photograph of the 32-channel AOM on the optics table. Beam path is
shown in blue, along with the 10-way diffractive optic element (10BS), 32-channel
AOM, and the ion chamber. (This is a slightly earlier version of the setup, with
some different lenses.)
(a) (b)
(c)
Figure 2.14: Images of focused beams to be applied to ions. 10 beams are available
to shine on 10 ions; they can be turned off and on independently to achieve arbitrary
individual addressing. (a) Camera image of 10 focused beams with 1 µm waist and
4.5 µm spacing; this image was taken with an earlier lens setup, and was later
changed to a 5 µm beam spacing. (b) Camera image of beams for ions 1 and 2
turned on, and the rest off. (c) 2D profile across 5 beams focused to <1 µm waist
and separated by 5 µm, imaged by the ions themselves by moving the beams across
the ions and measuring the resulting Rabi frequencies experienced by the ions.
37
µm, yielding spacing differences of [5.69, 5.05, 5.16, 5.69] µm, measured with an error
of about 0.16 µm.
2.4 Individual Addressing System Upgrades
Some upgrades to the individual addressing system have been added since
the results reported in [19, 20], which will apply to some but not all of the results
reported in this thesis. The Grover search algorithm discussed in Chapter 6 was
performed before the following upgrades, but several more results, including those
in Chapter 7 were made possible by the following upgrades.
For the earlier work on the experiment, a single AWG output provided RF
control to the global beam AOM, supplying sidebands and pulse-shaping to con-
trol two-qubit entangling gates. The multi-channel AOM controlling the individual
beams was controlled by a single RF source from the beatnote lock output; only
on/off TTL control was available for individual beams.
A significant upgrade was enabled by the addition of a 4-channel AWG2, bring-
ing to 5 the total number of AWG channels available. This allowed for total phase,
frequency, and amplitude control for the signal on each ion, not just on/off TTL
control. To use these new capabilities, the AWG controls were transferred to the RF
signals to the individual beam channels on the multi-channel AOM, and the beat-
note lock signal was applied to the AOM on the global beam. The controls for the
AWG were then integrated into the Igor control program using the NI-VISA com-
2Model WX1284C-1 1.25 GS/s Four Channel Arbitrary Waveform Generator, PN: 126182,
Tabor Electronics Ltd.
38
munication protocol; a suite of utility functions were added in a separate procedure
file that managed communication with the device and ensured uploaded waveforms
adhered to the device’s parameters.
The additional, individually addressed controls provided by the AWG enabled
several procedures previously impossible to implement, including simultaneous ro-
tations, the simultaneous π pulses on different sidebands required to implement
phonon hopping and blockades in [29], and the parallel two-qubit operations demon-
strated in Chapter 7. Improved phase control for individual ions also enhanced ex-
perimental performance for all subsequent work by better compensating for differ-
ences in accumulated phases as operations are performed on different ions. Finally,
the arbitrary phase control on each ion allowed for the classical implementation of
Z rotations, further discussed in Section 3.1.2.
For further scaling in ion traps, however, continuing to rely on AWG control
for individual addressing is not ideal; at a cost of about $10,000 per channel, this is
not cost effective. Efforts such as Sinara [54] have made important progress in this
area, designing alternative control signal generators using an FPGA and a fast DAC
to create an effective dynamic AWG; this design is estimated to lower costs by an
order of magnitude to ∼$1000 per channel. Using only commercially available non-
AWG signal generators to generate control signals is unfortunately not currently
possible, due to the need to change frequencies while maintaining phase coherence.
Direct control over phase coherence also allows for compensation of effects like Stark
shifts, which can reduce fidelities if left unaddressed.
39
2.5 Expansion to 7-Ion System
One experiment performed over the course of this thesis work required a 7-ion
system (specifically [31]), and we anticipate performing more 7-qubit experiments
in future. This presented several challenges. First and foremost was arranging for
7 ions positioned in the trap to align well with the individual addressing beams,
which have fixed equal spacing due to the optical setup (see Section 2.3.4). As
discussed in Section 2.2, perfectly equal spacing of more than 3 ions in this trap is
not feasible. While 5 ions are still pretty close to equal spacing, more ions in the
trap will have bigger and bigger spacing differentials between the outer and inner
ions. Consequently, 7 ions will be less well-positioned relative to the individual ad-
dressing beams than for 5 ions. However, this can be mitigated by instead trapping
9 ions, and ignoring the outer 2 ions. The inner 7 ions are then pushed together
by the sacrificial outer ions, resulting in a subchain of 7 ions with smaller spacing
differentials than would occur with just 7 ions in the trap.
With this strategy, several candidate DC voltage sets based on simulations
were tested on 9 ions in the trap to find the best experimental values. Once a
voltage set was chosen, we next needed to align the individual beams on the not-
quite-equally-spaced center 7 ions. Some experimentation demonstrated that the
best alignment strategy was to try to maximize the alignment of all ions, but to
particularly prioritize ions 2-6. This yielded reasonably comparable Rabi flops
on all ions, further discussed in Section 3.1.1. The selected ion spacing yielded
motional sidebands at [2.9424, 2.9604, 2.9774, 3.0044, 3.0149, 3.0264, 3.0384, 3.0534]
40
MHz detuned from the carrier. The ion positions relative to the center ion were
[−23.24,−16.49,−10.69,−5.21, 0, 5.27, 10.59, 16.49, 23.40] µm, corresponding to dif-
ferential spacings of [6.76, 5.80, 5.48, 5.21, 5.27, 5.32, 5.90, 6.91] µm, measured with
an error of about 0.16 µm.
As discussed in Section 2.4, we now have 5 AWG channels for full phase,
frequency, and amplitude control on each individual addressing beam, but expanding
to 7 qubits means we no longer have 1 AWG channel per qubit. We instead had
some qubit controllers share AWG channels. For our initial setup, we had ions
1 and 7 share an AWG channel, and ions 2 and 6 share an AWG channel. This
consisted of installing a TTL-controlled switch on the two shared AWG outputs
that directed the AWG output to one qubit or the other, and adding software to
the control program to seamlessly handle the shared controllers by queueing control
sequences to the AWG and managing TTL timing accordingly. The tradeoff was
that ions 1 and 7, and ions 2 and 6, could not be controlled at the same time,
ruling out any 2-qubit gates between those pairs. For purposes of [31], we did not
need those gates, so this was sufficient. Future experiments may require these gates,
in which case we can set up a slightly more complicated TTL-controlled switching
system to supply two different AWG channels to a given pair of ions to perform a
2-qubit entangling gate. Finally, we also wanted to maintain our existing 5-qubit
infrastructure for ongoing 5-qubit experiments, so the control program was modified
to allow easy switching between controls for 5 and 7 qubits; a few button pushes
change the relevant parameters in the control program, 7 wires for the individual
addressing RF signals need to be shuffled, and the alignment of the readout objective
41
is adjusted slightly to better align the ions to the PMT array.
42
Chapter 3: Quantum Control
In this chapter I discuss our two mechanisms for quantum control: single qubit
rotations, or R gates, and two-qubit entangling interactions, or XX gates. These
two operations form the gate set native to our hardware, and together they form a
universal gate set that can implement any desired operation.
3.1 Single-Qubit Rotations R
In the matrix representation of quantum computation, we define the two qubit
states as
( ) ( )
1 0
|0〉 = |1〉 = . (3.1)
0 1
Superposition states can be represented as points on the surface of the Bloch sphere
(Figure 3.1), where |0〉 and |1〉 define the poles of the Z axis of the sphere. The
polar angle θ represents the angular distance from the Z axis, and the azimuthal
angle φ gives the projection in the XY plane running orthogonal to the Z axis.
We define the azimuthal angle as φ = 0 at the X axis; φ = π at the Y axis.
2
Consequently, coherent transformations of a qubit can be described as rotations on
43
z
y
x
Figure 3.1: Quantum states can be represented as points on the surface of the Bloch
sphere with the radial variables (θ, φ). Single-qubit operations can be represented
as rotations performed about a selected axis on the Bloch sphere.
the Bloch sphere.
Rotations of an angle θ about an axis described by the azimuthal angle φ in
the XY plane are given by
( )
cos θ −ie−iφ sin θ
R(θ, φ) = 2 2 , (3.2)
−ieiφ sin θ cos θ
2 2
with rotations about the X and Y axes given by
( )
cos θ −i sin θ
R (θ) = R(θ, φ = 0) = 2 2x (3.3)
−i sin θ cos θ
2 2
and ( π) ( )cos θ − sin θ
Ry(θ) = R θ, φ = =
2 2 . (3.4)
2 sin θ cos θ
2 2
Such rotations can be implemented on the experiment by applying a Raman
pulse and adjusting the duration and phase of the pulse accordingly. The duration of
44
1
Ion 1
0.9 Ion 2
Ion 3
0.8 Ion 4Ion 5
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1 3 5 7 9 11 13 15
Duration ( s)
Figure 3.2: Sideband-cooled Rabi flops on 5 ions.
the pulse governs the magnitude of the rotation θ, to which it is directly proportional
by way of the Rabi frequency. The duration required to transfer all population from
the |0〉 to the |1〉 state, which corresponds to a R(θ = π) rotation, is known as the π
time. Rotation calibration consists of measuring the π time for each ion, and using
it to calculate the pulse duration necessary to implement each rotation: τ θθ = · τπ π.
The rotation angle φ is set by the phase of the control signal, with φ = 0 defined to
be the X axis.
3.1.1 Rabi Flopping on 5 or 7 ions
With individual Raman addressing beams in place, we can coherently cycle
between the |0〉 and |1〉 states of each ion using Rabi flopping. Sideband-cooled
Rabi flopping on 5 ions is shown in Figure 3.1.1. Applying Raman pulses for a
45
Probability |1
1
Ion 1
0.9 Ion 2
Ion 3
0.8 Ion 4Ion 5
Ion 6
0.7 Ion 7
0.6
0.5
0.4
0.3
0.2
0.1
0
1 6 11 16 21 26 31 36 41 46 51 56 61 66
Duration ( s)
Figure 3.3: Sideband-cooled Rabi flops on 7 ions in a 9 ion chain at low power. Ion
1 is directly behind ion 5 in the plot.
controlled duration with a controlled phase allows us to implement coherent rotation
gates. Typical rotation fidelities are 98-99%. Rotation fidelities are limited largely
by intensity fluctuations on the Raman beams and imperfect cooling. Intensity
fluctuations also inhibit our ability to accurately measure and calibrate the π times
for gate implementation.
Rabi flopping on 7 ions is shown in Figure 3.1.1. after sideband cooling, the
third Rabi flop peak was still above 90% bright for all ions, which is comparable
to sideband-cooled Rabi flopping with 5 ions. The Rabi flops decay faster for ions
that are less well aligned to the individual beams, which is primarily due to beam
pointing fluctuations on the individual addressing beams; while this is still manage-
able for 7 ions, this problem gets worse as more ions are added to the chain and
46
Probability |1
the non-equally-spaced ions become less and less well matched to the equally-spaced
individual beams.
3.1.2 Z Rotations
Qubit rotations about the Z axis on the Bloch sphere, given by
( θ )
e−i 2 0
Rz(θ) = , (3.5)θ
0 ei 2
can be implemented on the ions in one of 2 ways. The first way is through a
composite series of rotations about axes in the XY plane, as discussed in Section
5.1. The second way is by adjusting the phase of individual controllers relative to the
ion. With this method, instead of performing a rotation on the qubit and moving
it on the Bloch sphere, the controller is rotating the Bloch sphere around the qubit.
This effectively rotates the XY plane around the Z axis and redefines the locations
of the X and Y axes, and is equivalent to rotating the ion around the Z axis of the
Bloch sphere. Subsequent operations are then implemented relative to the rotated
axes.
The upgrades discussed in Section 2.4 permitted individual phase offsets to be
applied differentially to individual ions. Because they can be implemented purely
classically by simply shifting the phase of the frequency source, the error introduced
by Z rotations is the same as the error of the frequency source itself. Consequently,
Z rotation contributions to the error in fidelity during an algorithm are negligible
compared to the error introduced by an R(θ, φ) of the same θ and can be disregarded.
47
This has implications for composite sequence optimization further discussed in Sec-
tion 5.2.1.
3.2 Two-Qubit Entangling XX Gates
We implement two-qubit entangling gates using a Mølmer-Sørensen scheme
[41, 52, 55, 56] that creates an XX spin-spin interaction between two ions using the
normal motional modes of the chain as an information bus. In this experiment, we
couple to the transverse phonon modes [57,58] to drive the interaction.
The two-qubit XX entangling gate is
  cos(χ) 0 0 −i sin(χ)  0 cos(χ) −i sin(χ) 0 XX(χ) =   , (3.6)0 −i sin(χ) cos(χ) 0 
−i sin(χ) 0 0 cos(χ)
where the parameter χ can be varied continuously, 0 < χ ≤ π , by adjusting the
4
overall power applied to the gate. The gate is maximally entangling for χ = ±π ,
4
where
( π) 
   
 1 0 0 −i 1 0 0 i1 0 1 −i 0  ( π) 1 0 1 i 0
XX χ = = √   , XX χ = − = √   .
4 2  0 −i 1 0  4 2 0 i 1 0
−i 0 0 1 i 0 0 1
(3.7)
48
When applied to two ions initilized to |00〉, this yields a maximally entangled state:
( π) 1
XX (χ =
√
) |00〉 = (|00〉 − i|11〉)4 2π
XX χ = |00〉 √1= (|00〉+ i|11〉) . (3.8)
4 2
The sign of χ for a given gate depends on the interactions between the two ions
in question and the modes of motion primarily excited by the gate to produce the
spin-spin entanglement. The sign of the gate parameter χ is measured empirically
for each XX gate pairing. Accommodation for the difference in χ signs is handled
by the compiler at the gate library level (see Chapter 5). For ease of discussion, I
will only reference the χ = π case for the remainder of this chapter.
4
3.2.1 Pulse-Shaping Scheme
We implement entangling XX gates using a pulse-shaping scheme designed to
ensure that the motional modes are fully detangled from the qubits at the end of
the gate, leaving only spin-spin entanglement [53, 57, 59]. Two ions in a chain of N
ions are illuminated with red and blue sidebands that are detuned near the normal
transverse modes of motion and couple the modes to the qubit spins. The pulse-
shaping scheme is also robust against detuning errors, and the number of pulses
needed to implement gate solutions grow linearly with the number of ions in the
chain. The detuning µ and gate time τ are independent parameters that can be
chosen before calculating a gate pulse shape, and a unit fidelity gate is possible at
any detuning [53].
49
Because we couple to the normal motional modes of the chain to induce XX
gates, all ions participate in the motional modes, and so couplings are available
between all possible ion pairs. This fully connected architecture has significant ad-
vantages over less-connected architectures, which require additional gates to transfer
quantum information between unconnected qubits [27].
On this experiment, we have also implemented entangling XX gates using a
pulse-shaping scheme that modulates the frequency of the gate driver, rather than
the amplitude [28].
In order to entangle ions i and j in a chain of N ions with N motional modes
ωk using red and blue sidebands with detuning µ, we have the following 2-qubit
unitary:
( ∑ ∑ )
U xXX(τ) = exp i φi(τ)σi + i χij(τ)σ
xσxi j
( [ i i<j ])
= exp i φi(τ)σ
x
i + φ
x x x
j(τ)σj + χij(τ)σi σj (3.9)
where τ is the gate time, the spin-motion interaction φi(τ) is
φ † ∗i(τ) = αi,k(τ)âk − αi,k(τ)âk, (3.10)
â†k and âk are the raising and lowering operators for the motional phonons, the
spin-motion parameter αi,k(τ) is
∫ τ
αi,k(τ) = ηi,kΩ (t) sin(µt)e
iωkt
i dt, (3.11)
0
50
ηi,k is the spin-motion coupling or Lamb-Dicke parameter, Ωi(t) is the Rabi frequency
of the optical field applied to ion i, and the spin-spin interaction term χij(τ) is
∫ τ ∫ t′ ∑
χij(τ) = 2 dt
′ dt η ′ ′i,kηj,kΩi(t)Ωj(t) sin(µt) sin(µt ) sin(ωk(t − t)). (3.12)
0 0 k
At the end of the gate, the spin-motion terms must go to zero, ensuring that all
mode trajectories in phase space return to the origin. So, we require all 2N spin-
motion parameters (2 ions, N modes) α{i,j},k(τ) = 0. Entangling the ion pair (i, j)
requires that the spin-spin coupling term χij = χ
ideal
ij , where 0 < χ
ideal ≤ π . χidealij ij is4
typically π for a maximally entangling XX gate but can be set to smaller values to
4
implement partially-entangling gates. This then yields a set of 2N + 1 parameters
to control for when calculating pulse sequences to implement an XX gate:
α{i,j},k(τ) = 0
χ ideal ideal
π
ij(τ) = χij , 0 < χij ≤ . (3.13)4
To provide full control during the gate and fulfill the constraints in Equation
3.13, we divide up the gate amplitude Ωi(t) into S segments of equal duration τ/S,
and vary the amplitude in each segment Ωs as an independent variable. The gate
51
amplitude on a the ions then becomes a piecewise-constant function,
Ω1 0 ≤ t < τ/SΩ2 τ/S ≤ t < 2τ/S.. .. ..
Ωi(t) =  (3.14)


Ωs (s− 1)τ/S ≤ t < sτ/S

.. .. ..
ΩS (S − 1)τ/S ≤ t < τ.
Consequently, Equations 3.11 and 3.12 can be re-written as
∑ [ ]S ∫ sτ/S
αi,k(τ) = Ω η sin(µt)e
iωkt
s i,k dt
∑s=1 (s−1)τ/SS
= Ω isCk,s (3.15)
s=1
and
∑S ∑S ∫ sτ/S ∫ s′τ/S ∑
χij(τ) = Ω Ω dt
′
s s′ dt ηi,kηj,k sin(µt) sin(µt
′) sin(ωk(t
′ − t))
∑ ′s=1 s∑′=1 (s−1)τ/S (s −1)τ/S kS S
= ΩsΩs′Ds,s′ , (3.16)
s=1 s′=1
where the terms
∫ sτ/S
Ci iωktk,s = ηi,k sin(µt)e dt (3.17)
(s−1)τ/S
52
and
∫ sτ/S ∫ s′τ/S ∑
Dijs,s′ = dt
′ dt ηi,kηj,k sin(µt) sin(µt
′) sin(ω ′k(t − t)) (3.18)
(s−1)τ/S (s′−1)τ/S k
are pre-calculated constants that are functions only of the motional mode frequencies
ωk, the detuning µ, and the segment number s, and so can be arranged into S ×N
and S × S matrices for each ion, respectively. Note that the time ordering of the
double integral in Equation 3.18 requires that t < t′, so the time-segmented scheme
requires s ≤ s′. In the case s = s′, we must force the t < t′ constraint, yielding
∫ sτ/S ∫ t′ ∑
Dij ′s=s′ = dt dt ηi,kηj,k sin(µt) sin(µt
′) sin(ω (t′k − t)). (3.19)
(s−1)τ/S (s′−1)τ/S k
If we now arrange the segment amplitudes Ωs into the vector Ωij, we can now
write our constraint equations from Equation 3.13 as
( )
Ci
Ωij = 0
Cj
ΩTDijΩ = χidealij ij ij (3.20)
{ }
where Ci,Cj are the S×N spin-motion interaction matricies for each segment on
each ion and Dij is the S × S spin-spin interaction matrix for each segment. With
the constraints in Equation 3.20, we can calculate a pulse shape on a given pair of
ions that will create an XX entangling interaction between them.
An XX gate implemented with such a scheme has a theoretical maximum unit
53
fidelity. From [59], the gate fidelity as a function of the spin-motion closure terms
α{i,j},k(τ) for a fully entangling gate with χ
ideal = πij is4
( π) 1
FXX χ = = [2 + 2(Γi + Γj) + Γ+ + Γ−] , (3.21)
4 8
where
( )
−1
∑
Γ± = exp( βk |αi,k ± α |
2
j,k
2 ∑k1 ∣∣ ∣∣ )− 2Γi(j) = exp βk αi(j),k (3.22)
2
k
and the inverse mode temperature βk is
( ) [ ( )]
h̄ωk 1 1
βk = coth = coth ln 1 + , (3.23)
kBT 2 n̄k
where n̄k is the average phonon number in the k
th mode. Errors on the gate arise if
the motional modes are not fully closed in phase space, in which case the number and
temperature of any phonons in these modes determine the extent of experimental
errors.
Since we are also interested in partially-entangling gates, we re-calculate this
fidelity for a general value of χij, and obtain
1
FXX(χij) = [2 + 2(Γi + Γj) cos(2∆χ) + Γ+ + Γ−] (3.24)
8
where ∆χ = χidealij −χij. Plugging in the ideal-case parameters, where α{i,j},k(τ) = 0
54
and χ = χidealij ij , we indeed get FXX = 1.
An XX gate bewteen a given pair of ions is implemented by finding a pulse
shape solution that fulfills the constraints in Equation 3.13. While in principle
2N + 1 segments are needed for a given gate to control for the 2N + 1 constraints,
in practice gates can be constructed with fewer segments. The modes closest to the
detuning µ have the biggest participation in the spin-motion coupling, but those
far from the detuning participate very little and don’t move far from the origin in
phase space. Consequently, fewer segments can be used, and the solution optimized
to close phase space for the high-participation modes where phase space closure
matters most.
Several solutions at different detunings are tested on the experiment to ensure
a high-quality gate is found. While all solutions in principle have high fidelity,
anharmonicities in the experiment mean that not all solutions will perform well on
our experiment. The process must be repeated for each ion pairing one wishes to
use. While this process of pre-calculating and testing solutions incurs some overhead,
once a good solution has been found, no further tweaks to the solution are required.
The only day-to-day calibration needed is to measure the overall scaling factor to the
gate necessary to correctly imiplement a fully-entangling gate, which is a function
of the Rabi frequency and hence subject to drifts in the beam intensity seen by the
ions.
An example pulse shape with 9 segments is shown in Figure 3.4, including
plots of the trajectory of each mode in phase space over the course of the gate, and
a parity scan from which fidelity can be calculated. The pulse-shaping scheme was
55
Figure 3.4: We use a pulse-shaping scheme to implement robust, high-fidelity XX
gates. Here, we show a pulse shape used to implement a gate on ions 1 and 2 in
a previous hardware setup; the spin-motion trajectories in phase space successfully
close at the end of the gate. Parity scan data shows the gate fidelity here is 95(2)%
[53]. Improvements in the hardware and analysis allowing for SPAM correction have
since increased typical XX gate fidelities to 98-99%.
56
implemented on two ions in a five-ion chain in a previous experimental setup with a
different trap [53]. The hardware improvements gained using our blade trap setup,
improved individual addressing, and SPAM correction analysis have since raised typ-
ical 2-qubit gate fidelities to 98-99%. Fidelities for XX gates can be measured by
scanning the phase of a global π rotation applied after the XX gate and calculating
2
the parity, as described in Section 7.3.1. Solutions were calculated with gate time
τ = 230µs. The gate time can be adjusted on the experiment a little to maximize
gate fidelities, yielding experimental values of 210-260 µs. We measure motional
sidebands at ωk/(2π) = [2.946, 2.978, 3.005, 3.027, 3.045] MHz detuned from the car-
rier.
I will revisit this pulse shaping method to implement a scheme for parallel
XX gates in Chapter 7.
3.3 Universality of the Gate Set {R,XX}
With quantum control consisting of single-qubit rotation R gates and two-
qubit entangling XX gates, we now want to be able to construct more interesting
unitaries. In fact, to be a fully programmable quantum computer, we require that
we have access to a universal gate set that can implement any possible unitary, just
as classical computers have access to a universal gate set that can implement any
possible classical computation. Using the quantum circuit model of computation
[60], 2-level unitaries have been shown to be universal, meaning that any unitary
can be built from 1- and 2-qubit gates [61]; however, deterministic ways of building
57
such unitaries do not necessarily yield the most efficient gate sequence for doing so.
Several gate sets have been shown to be universal, including the set of all 1-qubit
rotations and a CNOT gate [62, 63], as well as the more commonly cited universal
gate set of {CNOT,H, T} [64, 65].
Any single-qubit rotation can be constructed using a sequence requiring two
orthogonal Pauli rotations [65]. Since we have access to all rotations R(θ, φ) in
the XY plane of the Bloch sphere, we can construct any single-qubit rotation R.
Additionally, as will be shown in Figure 5.2(a), a CNOT gate can be constructed
from an XX gate and several single-qubit rotations R. Consequently, our available
gate set of {R,XX} is universal, and we can use it to implement any quantum
algorithm on this programmable quantum computer.
A toolbox of useful quantum gates have been developed in the literature, and
their unitaries decomposed into already-known quantum gates [65, 66]. In the next
chapter, I will discuss the gate library of composite gates that has been developed
and implemented on our machine so far.
58
Chapter 4: Control System
Chapters 2-5 presented the hardware, quantum control, and gate library layers
of a full-stack ion trap quantum computer. The highest level of the stack is the
implementation of quantum algorithms, and the user interface where this level of
control happens. While our control system software is hardly a production-ready
product, it nevertheless uses principles of abstraction and compartmentalization
to control the quantum computer, and make it easy for a knowledgeable user to
implement a quantum algorithm without having to worry about lower levels of
control. Here, I present in some detail the workings of the control software, built in
Igor, and show how all the parts of the system are stitched together by the control
software using a systems-level perspective.
In this chapter, I will discuss the compiler layer of the stack, as well as the
experimental control program it is embedded in. This will provide a systems-level
perspective on the experimental setup, as the control program manages every step
in the experimental process. The control program is written with Wavemetrics
IgorPro, a data processing tool with some instrumentation interfacing and GUI
construction capabilities. Our control program uses Igor in ways it wasn’t really
designed for, and as such we’ve had to add some creative workarounds to push the
59
boundaries of the program’s capabilities. Other projects, such as ARTIQ [67] and
the pyIonControl project through Sandia National Labs [68], have produced open-
source ion trap control programs in more versatile and widely-used languages like
Python that may prove more suitable with sufficient development.
In broad strokes, the Igor program provides the GUI the user interacts with,
the compiler that interprets algorithm sequences, and ongoing control of certain
low-level experimental processes necessary to trap and hold a small chain of ions,
When the user initiates an experiment, the Igor program compiles the user’s inputs,
assembles the complete experimental procedure, and hands it off to the FPGA for
real-time execution. The experiment requires a number of steps to be performed
by disparate experimental elements quickly and with precise timing, so an FPGA,
which can initiate and terminate TTL signals to other instrumentation on a 20ns
clock cycle, is ideal for such a role. When the user writes an algorithm for the
experiment to execute, they are writing a melody. The compiler then provides the
harmonies for the melody, the control program arranges the score for a full orchestra,
and when the user starts the experiment running, Igor hands the score off to the
FPGA, which then conducts the symphony orchestra that is our ion trap quantum
computer, ensuring each instrument - lasers, RF sources, electronic controllers, and
so on - comes in at the right time to create a harmonious piece of music.
60
User
interface Quantum algorithms: Grover search, etc.
Quantum Universal gates: Hadamard, CNOT, etc.
compiler Native gates: XX gates, R gates
Quantum 
control Pulse shaping: optimization of XX and R gates
Optical addressing: qubit manipulation / detection
Hardware
Qubit register: ion trap, Yb ion chain, etc.
Figure 4.1: A diagram of the computational stack, from the low-level hardware up
to algorithm implementation through a user interface. Image modified from [19].
4.1 System Control Overview
Figure 4.2 provides a systems-level view of the experimental control structure.
The Igor control program, run on a commercial PC, provides a user interface that
accepts user instructions and translates them into an experimental sequence. This
sequence and related control information is then passed to peripherals that execute
the instructions on the physical system. During an experimental run, an FPGA
is used to initiate operations in real time. The 20 ns clock cycle allows for fast
and consistently precise timing of experimental operations, which happen on mi-
crosecond timescales; a typical consumer computer is too slow and inconsistent for
our purposes. Consequently, the Igor program pre-sets any instrumentation that
uses static values before an experimental sequence begins. During experiments, the
61
Algorithm decomposition
(software)
Figure 4.2: An overview of the control setup for all major pieces of the experiment.
A note that all beams after AOMs are deflected first orders; the dumped 0th order
and the angle of deflection are omitted here for simplicity of viewing.
62
FPGA sends TTL pulses at pre-specified times to trigger the instrumentation in the
correct order for the correct length of time.
The Igor program sets the trap voltages using a DAC (digital to analog con-
verter.) When the user changes the DC electrode voltages, the new voltage values
are sent to the DAC, which then outputs those voltages to the corresponding trap
electrodes. The DAC also sets the amplitude of the RF drive, by outputting a volt-
age that controls a VVA (voltage variable attenuator) on the RF drive signal. The
trap drive signal is provided by a DDS frequency source, which is set through the
Igor program. The drive signal is amplified by a helical resonator [44], colloqui-
ally referred to as an RF can, and its output is actively stabilized with a lock [43].
Real-time shuttling can be implemented by triggering the DAC with the FPGA.
The Doppler cooling, optical pumping, and detection operations are all im-
plemented similarly. The Igor control program statically sets the frequency and
amplitude of a PTS1 frequency synthesizer for each operation. Each PTS channel
then outputs an RF signal at the specified frequency and amplitude that, when ap-
plied to the corresponding AOM, deflects the incoming 369 nm beam. The deflected
beams are aligned along the rest of the optical pathway to the ions and the zero
orders are not, so applying the driving RF to the AOM turns the beam “on” from
the points of view of the ions. A TTL-controlled switch on the RF signal serves
as an on-off switch for each beam. During an experiment, the FPGA triggers each
operation by sending a TTL pulse to the switch, thereby applying the beam to the
ions.
1Programmed Test Sources, Inc., model PTS 3200
63
The AWG provides the RF control signal used to implement coherent opera-
tions. Before an experiment starts, the Igor program calculates the AWG waveform
to be used in the experiment, which requires specifying the voltage the AWG should
output at every point in time. The program calculates the voltage output with
14-bit resolution for every 1 ns timestep. The waveform is then uploaded to the
AWG, and repeated for all channels. During run time, the FPGA sends a single
TTL trigger to the AWG, at which point it plays back the entire waveform stored
in each channel. The FPGA also sends TTL pulses to“turn on” individual beams,
timed to correspond with the AWG waveform needing to be applied to that channel.
As with the 369 nm AOM controls, this works by having the TTL from the FPGA
control a switch on the RF drive applied to each AOM channel in the 32-channel
AOM. These switches and TTLs proved particularly useful for controlling 7 ions
with only 5 AWG channels.
Data is collected by a multi-channel PMT, which counts photons collected
through an objective lens with an NA of 0.37. The detection light from each ion
is imaged onto a different PMT channel. After each experimental shot, the photon
counts for each channel are sent to the FPGA, which collects it for the specified
number of experimental repetitions per data ponit, then sends the compiled photon
counts back to the Igor control program for analysis. A flip mirror controls whether
the detection light is directed onto the PMT channels, or instead to an ICCD camera,
which is used for 2D imaging of ions while loading. The camera is controlled entirely
by the Igor program; a GUI provides a user interface for setting the gain, exposure,
and other parameters, and the camera data is sent back to the control program for
64
display.
4.2 The Igor Control Program
The Igor control program provides a user interface consisting of GUIs that
control nearly all aspects of the experiment. The front end allows the user to
perform several activities through GUIs. The user can create new experimental
sequences, consisting of a series of experimental operations. They can load such
a sequence and adjust the parameters of experimental operations in the sequence.
They can change static controls on certain peripherals, such as the ion trap voltage
controls and the frequency and amplitude settings for the RF signals controlling the
cool, pump, and detect AOMs. Finally, the user can initiate an experimental run
and select from various options for data display. Two modes of experimental run
are available: scanning and alignment sweeper. In a scan, the user designates one
or more experimental operation parameters to scan over, and provides a start value,
stop value, and scan increment. The program collects and displays a data point
at each scan value and then stops. In alignment sweeper mode, the experiment
is performed with the same values for each data point, and repeats until the user
stops it. This mode is particularly useful for continually running and updating
experimental outcomes while aligning or adjusting an apparatus on the experiment,
hence the name.
The back end of the Igor control program consists of several Igor procedure
(.ipf) files that handle each modular aspect of the control system. An overview
65
Figure 4.3: The Igor control system.
66
is shown in Figure 4.3. The experiment constructor file handles experimental pa-
rameters and execution. The data handler handles data analysis and display. The
compiler consists of the algorithm constructor (interpreter) and the AWG controller
(see Section 4.3). The pulse GUI provides the user interface for creating and exe-
cuting experimental procedures. The initialization module creates GUIs for the user
to control various pheripheral settings, including the PTS settings controlling the
cool, pump, and detect AOM drive signals. The trap voltage control provides a GUI
interface for modifying the trap potential with the DC and RF electrodes, as well
as some utility functions for loading saved voltage sets and recrystallizing the ion
chain. Additional modules control the camera peripheral and the DDS frequency
source.
The control program manages experimental procedures through the use of an
experiment object. The experiment object stores an array of experimental operation
objects in the order in which they should be executed. An example such sequence is
shown in Figure 4.4. The experiment also has two metadata variables related to the
experiment: the number of experimental operations to be executed, and the number
of experimental shots to be performed for each data point. Each experimental
operation object consists of an operation definition and a list of parameters relevant
to that particular definition. For example, the Cool experimental operation has a
parameter list that consists of the duration of the cooling to be applied. A Rotation
experimental operation has a list of 3 parameters: θ, φ, and the ion channel(s)
specifying the ions to be rotated. The experimental operation also stores variables
used for scanning the experimental operation parameters in a scanning experiment.
67
A full list of experimental operations is given in Figure 4.3.
The experiment constructor module creates an experiment object when the
user initiates an experiment with the operation sequence and parameters specified
by the user. The program parses through each experimental operation in the ex-
periment to construct a list of TTL names (pointing to one or more specific TTLs
that correspond to specific operations) and a corresponding list of TTL durations
(corresponding to how long that operation should be run.) The list of TTL’s and
durations is handed to the FPGA for execution during the experiment. If there is
an algorithm chapter, it is handed to the compiler. The experiment constructor
module also manages the experimental run, tracks and increments scan variables,
and listens for a user command to stop the experiment execution.
One limitation to this experiment object architecture is that Igor limits ar-
rays of objects to 100 items. Consequently, we cannot execute experiments with
more than 100 experimental operations. For most experiments performed over the
course of this thesis work, gate sequences were short enough that this was not a
problem. However, the measurement of Renyi entropy performed in [30] required
multiple repetitions of a circuit block of 20 operations, and for longer sequences,
the 100 operation limit was exceeded. To overcome this obstacle, the control was
moved down the stack. Instead of sending the compiler a set of 20 R and XX gates
per repetition, a new experimental operation was created at the control level. The
Renyi Block experimental operation executes an entire circuit, consisting of several
R and XX gates, as a single experimental operation. To do so, the AWG Controller
calculates a very long waveform, comprised of multiple sections of what would nor-
68
Figure 4.4: An example sequence to be performed on the experiment. This will
implement a Bell measurement on qubits 3 and 4.
69
mally be standalone R and XX gate waveforms. Thus, many repetitions of the gate
sequence became possible, allowing data collection to proceed.
The data handler receives photon counts from the PMT via the FPGA and
analyzes the contents for display in the data frame. It uses a discriminator on
each PMT channel to determine which ions in the chain are bright and dark for
each experimental shot. This data is then aggregated to calculate birght and dark
populations on each ion. Before running the experiment, the user selects what
analyses will be displayed in the 4 panels of the data frame from several options.
Per-ion analysis options include the average photon counts seen for each ion and
probability each ion is bright. A histogram of the photon counts seen for each
ion is also displayed as a monitor; deviations from the usual histogram distribution
indicate something is wrong in the ion with the ion control. For multi-qubit analysis,
the data is aggregated into a state vector for each data point that contains the
population in each of the possible multi-qubit states. For example, the user can
specify that they want to see populations in the 4-qubit state vector for ions 1, 3,
4, and 5. The data frame will then display a plot in the data frame showing the
populations in (0000, 0001, 0010, . . .) for each data point. Another option analyzes
the populations in 2-qubit state vectors for each sub-pair in a set of 3 or more
ions. For example, if the user selects ions 1, 2, 4, and 5 for this analysis, then the
populations in (00, 01, 10, 11) will be displayed for each of the 6 ion pairs (1,2), (1,4),
(1,5), (2,4), (2,5), and (4,5). This option is particularly useful for looking at the
result of parallel 2-qubit operations. Finally, parity can be calculated and displayed
for a pair of ions, or for multiple pairs of ions within a set of 3 or more. After an
70
experiment is ended, the data handler can also save data from the experiment into
a series of related text files for the user, including the raw data, analyzed data files,
and the experiment parameters used.
The AWG Freeform experimental operation can be used in lieu of an algorithm
chapter. It consists of an editable procedure file that calculates and assembles AWG
waves directly from user input. The file is provided with pre-written instructions
and templates to ease programming new sequences; the user simply fills in the
blanks with desired parameters. This setup allows for more flexibility and can
be used to implement schemes outside the standard R/XX gate set; for example,
it was used to implement phonon hopping and blockades [29]. This module may
prove useful for implementing other kinds of quantum simulation experiments that
would particularly benefit from our individual addressing capabilities not available
on many other experiments. The module could benefit from improvements such as
a user-friendly GUI control interface.
The Igor control program for ion experiments was originally developed at Geor-
gia Tech. The program has since been significantly expanded, with many new func-
tionalities added to create a program capable of controlling a programmable ion
trap quantum computer.
4.3 Compiler
When a user provides an algorithm for the quantum computer to execute, the
compiler breaks it down from a high-level gate specification into a quantum machine
71
language - a series of pulses that are then executed on the qubit. The high-level gate
specification is provided by the user as a string of abbreviated control statements
separated by colons; for example, “CN24” indicates that a CNOT gate should be
performed with control ion 2 and target ion 4. Generic native R and XX gates
can also be included. Each control statement requires the inclusion of certain gate
parameters, depending on the specific operation or composite gate being called.
The interpreter of the compiler reads each control statement in turn, and then
creates one or more new experimental operations that are added to the experiment
object corresponding to the control statement. In our “CN24” example, the inter-
preter identifies that a CNOT gate is needed on the ions in question. It then adds
to the experiment object the series of R and XX experimental operations that will
execute this composite gate (see Section 5.3), specifying the required parameters for
each R/XX gate. For rotation R gates, the interpreter specifies the rotation angle
θ, the rotation axis φ, and the ion(s) to be rotated. For classical Z rotations, the
interpreter specifies the rotation angle θ and the ion(s) to be rotated. For two-qubit
entangling XX gates, the interpreter provides the ion pair to be entangled, and the
absolute value of the entanglement parameter χ. For parallel 2-qubit XX gates,
the interperter specifies two pairs of ions to be entangled, and two χ values. For
composite gates containing XX gates, the interpreter also accesses the sign of each
XX gate’s χ parameter in the XX gate parameter lookup table, which may in-
form the parameters of R gates in the composite operation to ensure fully modular
operations. How this is handled for specific gates is further discussed in Chapter 5.
With the algorithm section of the experiment object consisting entirely of R
72
and XX native gates, the experiment is handed off to the AWG controller, which
compiles the waveforms needed to implement the gates. For each gate, the AWG
controller calculates a waveform for every AWG control channel; channels not used
are simply set to 0 V for the duration. For channels in use, the AWG controller
accesses calibration data from lookup tables to correctly calculate the needed wave-
form using the parameters specified in the experimental operation. For R gates, the
program looks up the π time(s) on the ion(s) in question to calculate the necessary
pulse duration to execute a rotation angle θ, and then calculates a sine wave of the
specified duration with phase set by the rotation angle φ and the phase tracked by
the phase counter. For XX gates, the program calls information about the pulse
shape specific to the ion pair in question from the lookup table, including the gate
time, sideband detuning, and the normalized pulse shape. It also accesses the cal-
ibrated gate power scaling factor, which scales the overall power of the gate pulse
shape to ensure the correct value of χ is imparted to the ions. The AWG con-
troller then calculates a bichromatic signal using these parameters, changing the
amplitude over the course of the gate in accordance with the pulse shape. Two sine
waves are calculated, red and blue of the carrier by the specified gate detuning, and
then summed, with the phase set by the phase counter. Parallel two-qubit XX
gate parameters are similar to those of standard XX gates, but with two ion pairs,
pulse shapes, calibrated gate powers, and χ values. The AWG controller calculates
2 separate bichromatic waveforms, one applied to each ion pair.
A phase counting scheme serves to track the individual phase of each ion as it
evolves over the course of the experiment. During an operation, an ion experiences
73
a small Stark shift due to the applied laser field, accumulating phase at a slightly
different rate than in the dark. This phase accumulation must be carefully tracked
for phase-coherent operations. For each operation performed, the program calculates
how much phase each ion accumulates and adds it to a table of running phase
counters, which are then used to set the phase of the signal for the next operation.
Classical Z rotations are performed by adding a phase of θ to this counter for the
ion(s) undergoing the rotation.
The calculated waveforms consist of a sequence of values corresponding to
plotting a point of the waveform function every 1 ns. The sine waves are initially
calculated for a normalized maximum amplitude of 1. The waveforms for each
channel are then formatted to voltage values by the AWG utility functions; the Chase
and Tabor AWGs have different formattings, so the initial waves are calculated in
a normalized manner first and then formatted separately for ease of modularity.
A set of utility functions written to control the 4-channel Tabor AWG using
the NI-VISA communication protocol initialize the instrument’s settings. It also
provides functionality to check that a calculated waveform is within uploadable
parameters, format it for the AWG, and manage the upload to each channel over
a USB connection. The Chase AWG is directly connected to the control computer
through a PCI connection, and requires a much more limited set of utility functions
to serve a similar purpose. Once uploaded to the AWG channels, a TTL signal
from the FPGA triggers all channels to start outputting their waveforms at the
same time. The waveforms act as RF controls for the individual channels of the
32-channel AOM, which sends deflected beams onto their corresponding ions for
74
coherent control.
4.4 Outlook
The control program described here has proved very capable at executing small
quantum algorithms with short control sequences on a few qubits. However, scaling
up will present challenges. The most immediate need is to automate calibration
functions. Currently, experimental operators spend a signficant amount of time cal-
ibrating the π times for R gates and the scaling power needed for XX gates by
hand as the Rabi frequency of the individual controllers drifts on a scale of hours.
Automating these procedures will speed the process significantly, as well as reduc-
ing the burden on experimental operators. Ongoing efforts include implementing
automated calibration, as well as automated loading of 5 or 9 ions. This will be
necessary for more ambitious projects, such as applying machine learning techniques
to build quantum circuits using a method that will require many repetitions of a
feedback loop that provides experimental data to a machine learning module [69].
Implementing longer gate sequences will require several structural changes to
the control systems. As discussed in Section 4.2, the experiment object in Igor
is limited in the number of experimental operations it can perform as it is cur-
rently designed. While workarounds can be implemented on an ad hoc basis2, a
structural redesign allowing for unlimited experimental operations will be needed.
A better solution, however, will be to switch to a broader, more robust control
software scheme altogether, such as those provided with ARTIQ [67] or Sandia’s
2This is sufficient for an experiment in an academic lab, right up until it isn’t.
75
pyIonControl software [68], which use a Python, a more well-known and versatile
programming language. They also have the benefit of being an open-source project
with contributions from other users who can help build new functionality and fix
bugs. Other platforms may also prove to be good candidates to build such a control
system.3 On the hardware side, using the AWG for control waveforms is not scalable
not only in terms of cost, but also in terms of time. Pre-calculating and uploading
waveforms to the AWG is the slowest part of the experimental sequence; a sequence
with 10-20 XX gates may take tens of seconds to calculate and upload. Hardware
upgrades such as those provided in the Sinara project [54], which calculates and
outputs waveforms on the fly, will obviate this problem.
Another challenge will be managing the growth in the number of two-qubit
gate connections. For a fully-connected system with N qubits, the number of two-
qubit gates is ( )
N N ! N2 −N
= = ∼ O(N2), (4.1)
2 2!(N − 2)! 2
yielding a quadratic growth rate in the number of connections. While this is still
polynomial (and not exponential), and larger ion traps will likely be missing some
connections, it still presents a challenge. Our current strategy of using lookup tables
to supply gate parameters may prove slow in the long term; more nimble database
architectures may be needed. Additionally, this presents significant growth in the
overhead required to set up all available gate connections by pre-calculating pulse
shapes for each ion pairing and then selecting one that performs well in practice.
3Just do yourself a favor and don’t use Labview.
76
While this is indeed overhead, and does not slow down operational execution once
gates are set up, the growing overhead will present costs for those needing to iterate
through many machines for development purposes. Finding methods to automatize
gate solution selection may be helpful here. Additionally, several of our two-qubit
gates use the same pulse shape and/or detuning; characterizing why and how this
happens may facilitate finding experimentally optimal solutions.
Although we already have quite a few composite operations in our gate li-
brary (see Chapter 5), further expanding the library will be important for providing
pre-optimized subroutines for larger and larger algorithms. Also useful will be a
user-friendly interface for defining new additions to the gate library; this currently
must be done by writing new code in an Igor procedure file. Improvements to the
overall user-friendliness and robustness of a quantum computing control program
will be important for allowing more users to operate the machine without having
to know either how to program changes to the control software or how to avoid the
little quirks and bugs always present in ad hoc software with no quality control pro-
cess. Integrating a universal language for gate entry that functions across multiple
platforms - just as programming languages like Python and Java can be used on
many hardwares and operating systems in the classical world - will be crucial for
collaboration as quantum computers scale up. Finally, integrating software that can
automatically optimize new user-entered gate sequences before implementing it on
the hardware will be incredibly useful. Efforts such as Microsoft’s LIQUi| > [70]
and Rigetti’s Forest [71] have started to address these needs.
While the control program described here certainly has limitations, and more
77
streamlined approaches will be needed as the size of quantum computers grows, this
can certainly serve as a demonstration that, moving forward, quantum computer
control software will require modularity, flexibile construction, and well-thought-
out program design.
78
Chapter 5: Gate Library
Now that we have a programmable quantum computer with high-quality qubits
(Chapter 2) and coherent single- and two-qubit operations yielding a universal gate
set (Chapter 3), the next step is to build a gate library. These gates are used
in Chapters 6-7 and [30, 31] to construct complex, multi-qubit algorithms, and the
modular nature of these gates will be crucial to flexible ion-qubit mappings and allow
great ease of implementation. In this chapter, I present detailed circuit diagrams
for all of the operations used in the course of the research for this thesis, shown in
terms of the R(θ, φ) and XX(χ) gates directly implemented by the experiment.
5.1 Single-Qubit Gates
From Chapter 3, the single-qubit rotation is defined as
( )
cos θ −ie−iφ sin θ
R(θ, φ) = 2 2 . (5.1)
−ieiφ sin θ cos θ
2 2
Rotations about the X-axis (Rx(θ)) are achieved by setting φ = 0, and rotations
about the Y -axis (Ry(θ)) are achieved by setting φ =
π . Rotations about the Z
2
axis (Rz(θ)) can be constructed from three rotations about axes in the XY plane,
79
(a)
|q〉 Rz(θ) = Ry(π ) Rx(θ) Ry(−π )2 2
(b)
|q〉 H = Ry(π )2 R
π
x(π) = Rz(π) Ry( )2
Figure 5.1: (a) Rz(θ) gate implementation using Rx(θ) and Ry(θ) gates. (b)
Hadamard (H) gate implementations using Rx(θ), Ry(θ), and Rz(θ) gates.
as demonstrated in Figure 5.1(a).
A Hadamard (H) gate,
( )
√1 1 1H = , (5.2)
2 1 −1
can be performed with 2 rotations, as shown in Figure 5.1(b), with a few implemen-
tation options. The fidelities of these composite rotations gates are comparable to
the fidelities of individual rotations, typically 98-99%.
5.2 Multi-Qubit Circuit Construction
From Chapter 3, the two-qubit XX entangling gate is
 
 cos(χ) 0 0 −i sin(χ)  0 cos(χ) −i sin(χ) 0 XX(χ) =   . (5.3)0 −i sin(χ) cos(χ) 0
−i sin(χ) 0 0 cos(χ)
The parameter χ can be varied continuously, 0 < χ ≤ π , by adjusting the overall
4
power applied to the gate, but the gates used here require only χ = ±π or χ =
4
80
±π . The ability to vary χ continuously has proved useful in implementing circuits
8
for other applications not discussed in depth here, including the quantum Fourier
transform [19] and a Bayesian quantum game [32]. The gate is maximally entangling
for χ = ±π .
4
Two-qubit XX gates are combined with rotation R gates to construct com-
posite gates. The parameter χ can be positive or negative, depending on what ion
pair is chosen and the particulars of the pulse segmentation solution chosen for the
ion pair in question; the sign of χ (sgn(χ)) is determined experimentally for each ion
pair. Consequently, some composite gate circuits include rotations with parameters
that depend on sgn(χ); all composite gates are constructed to account for both signs
of χ for each XX gate used, so that composite gates are always fully modular.
Composite gates are constructed by starting with known circuits, converting
constituent parts into R and XX gates using lower-level constructions, and then
optimizing the circuit. First, the number of XX gates is minimized (as in the Toffoli-
3 gate, described in Section 5.4). Second, the single-qubit gates are optimized by
minimizing the sum of all rotation angles θ, as this minimizes the total time for the
experiment.
Several composite circuits used in algorithms performed in this thesis are pre-
sented in this chapter. Other quantum algorithms may be implemented on this
system in a similar fashion. First, decompose the algorithms subroutines into high-
level circuits. Second, optimize those circuits to minimize the number of two-qubit
interactions required. Third, decompose the high-level circuits into physical-level R
and XX gates. Finally, perform further optimizations to first minimize the num-
81
ber of two-qubit XX gates required, and then to minimize the total rotation time
(the sum of all rotation angles θ) across all R gates. However, the optimization of
quantum circuits is in the QMA-Hard complexity class, which is the quantum ana-
log of the classical NP-Hard complexity class. We therefore anticipate that future
improvements in algorithm design, circuit synthesis, and circuit optimization tech-
niques may result in more efficient circuit implementations, facilitating increased
experimental performance.
5.2.1 Optimization Strategy Adjustments with Experimental Up-
grades
This chapter features composite gates developed for use before and after the
improvements described in Section 2.4 were implemented, and consequently are opti-
mized slightly differently. In particular, the introduction of individual ion phase con-
trol allowing for the purely classical implementation of Z rotations (see Section 3.1.2)
leads to different optimization strategies for minimizing rotations. Pre-upgrade, any
Z rotations had to be implemented via a combination of several rotations in the XY
plane. Consequently, composite gates were optimized by minimizing the total rota-
tion angle θ for all rotations R(θ, φ) in the XY plane; Z rotations were to be avoided
where possible, and compiled into X and Y rotations for further optimization when
not. Post-upgrade, however, Z rotations became effectively error-free to perform
relative to R(θ, φ) rotations. Therefore, when optimizing a given composite gate,
the total rotation angle θ over all R(θ, φ) rotations can be further minimized by
82
(a)
|q 〉 • R (απ ) R (−απc y y ) Rz(−π )2
= XX(απ
2 2
)
| 〉 4qt Rx(−π )2
(b)
|qc〉 • R π π πy(− ) R2 x(− ) Ry( )2 2
= XX(απ )
|q 〉 R (απ ) R (απ 4t Z y x ) Ry(−απ )2 2 2
Figure 5.2: χct is the parameter for the XX gate between the two qubits. Let
α = sgn(χct). (a) CNOT gate implementation using XX(χ), Rx(θ), Ry(θ), and
Rz(θ) gates. (b) Controlled-Z gate implementation using XX(χ), Rx(θ), and Ry(θ)
gates.
employing Z rotations instead wherever possible. In particular, the following Pauli
rotation identities were very useful for converting between X, Y , and Z rotations
to minimize rotations other than Z:
(π) ( ) ( ) ( )
Rz(θ) = Ry( ) ·
π π π
Rx(θ) ·Ry(− ) = Rx(− ) ·Ry(θ) ·Rx (5.4)2 2 2 ( 2π π π π)
Rx(θ) = Rz ( ) ·Ry(θ) ·Rz (− )= Ry2 2π π (
−
2
π)
·Rz(θ) ·Ry ( 2π)
(5.5)
Ry(θ) = Rx ·Rz(θ) ·Rx − = Rz − ·Rx(θ) ·Rz (5.6)
2 2 2 2
It is also useful to note that X rotations commute with XX gates, and can therefore
be moved around more to minimize rotations. In this chapter, I will identify which
composite gates were optimized under which set of circumstances.
5.3 Two-Qubit Composite Gates
The controlled-NOT , or CNOT , gate, is a two-qubit interaction used as the
building block for many quantum algorithms. A control qubit |qc〉 acts on a target
83
qubit |qt〉 by flipping the target qubit’s state if the control qubit is in the |1〉 state;
otherwise, it does nothing. This has the unitary evolution operator
 1 0 0 00 1 0 0UCNOT =  . (5.7)0 0 0 1
0 0 1 0
Another useful two-qubit interaction is the controlled-Z, or C(Z), gate, which flips
the phase of the two-qubit state if the two qubits are in the |11〉 state:
 1 0 0 00 1 0 0 UC(Z) =  . (5.8)0 0 1 0
0 0 0 −1
Circuits for the CNOT and (C()Z) gates are shown in Figures 5.2(a-b) respectively.
They each require one XX π gate and several rotations. Both gates were first
4
demonstrated on this system in [19], with typical fidelities of 96-98% [19, 20]. The
upgrades described in Section 2.4 have resulted in typical CNOT gate fidelities
increasing to 98-99%. In this work, the CNOT gate will be used in Chapters 6 and
7, while the controlled-Z gate will only be needed for Chapter 6.
5.4 Toffoli Gates
Next, we scale up to a three-qubit composite gate by implementing a controlled-
controlled-NOT (C2(NOT )), or Toffoli-3, gate. The Toffoli gate [72] requires two
control qubits (q1 and q2) and one target qubit (qt). Like the CNOT gate, it flips
84
• • • •
• = • •
U V V † V
Figure 5.3: Any doubly-controlled unitary U can be constructed out of 5 two-qubit
interactions using CNOT gates and controlled-V operations, where V 2 = U [63].
the target qubit state if and only if both control qubits are in the |1〉 state:
 
1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0UT3 = 

0 0 0 0 1 0 0 0 . (5.9)0 0 0 0 0 1 0 00 0 0 0 0 0 0 1
0 0 0 0 0 0 1 0
( )
Our(T)offoli-3 gate is constructed from 5 two-qubit gates (three XX
π and
8
two XX π gates) in a manner similar to the Toffoli gate demonstrated in [73].
4
As shown in Figure 5.3, any doubly-controlled unitary C2(U) operation can be
performed with 5 two-qubit interactions (two CNOT s, two C(V )s, and one C(V †))
if a controlled-V operation is available such that V 2 = U [63]. Since
[ (π)]2 (π)
XX = XX , (5.10)
8 4
we can add single-qubit rotations to construct a Toffoli-3 gate with minimal use
of two-qubit gates, as shown in Figure 5.4(a). This compares favorably to t(he)6
two-qubit gates that would be necessary if only CNOT (or equivalently, XX π )
4
gates were available. These constructions also provide for the implementation of
85
(a)
|q π1〉 • Ry(−β ) Rx(β 3π ) XX(β π )2 4 8
|q2〉 • = Ry(−γ π ) R πx(γ )2 2
XX(γ π )
8
|qt〉 R (πx ) XX(β π )4 8
|q1〉 · · ·
XX(απ )
|q2〉
4
· · · R(−2π , (γ+1)π − P )
3 2 R(−αβγ
2π , (αβ+1)π − αβγP )
3 2
|qt〉 · · ·
|q1〉 · · · R πy(β )
π 2XX(α )
|q2〉 · · ·
4
R(π,−αβγ π )
XX(γ π
4
)
|qt〉 · · ·
8
(b)
|q1〉 • Ry(−β π ) R 3πx(β ) XX(β π )2 4 8
|q2〉 • = Ry(−γ π ) R (γ πx )2 2
XX(γ π )
8
|qt〉 Z Ry(π ) Rx(π ) XX(β π )2 4 8
|q1〉 · · ·
XX(απ )
|q 〉 4· · · R(−2π2 , (γ+1)π − P )
3 2 R(−αβγ
2π , (αβ+1)π − αβγP )
3 2
|qt〉 · · ·
|q1〉 · · · Ry(β π )
XX(απ
2
)
|q2〉 · · ·
4
R(π,−αβγ π )
4
XX(γ π )
|qt〉 · · ·
8
R πy(− )2
Figure 5.4: Three-qubit composite gates using XX(χ), Rx(θ), Ry(θ), a(n√d R)(θ, φ)
gates. Let α = sgn(χ12), β = sgn(χ1t), γ = sgn(χ2t), and P = arcsin
2 . (a)
3
Toffoli-3 gate implementation. (b) Controlled-controlled-Z (CCZ) gate implemen-
tation.
86
1
1
0.5
0 0.5
000 111
111 000
detected input 0
Figure 5.5: Measured truth table for a Toffoli-3 gate. The average process fidelity is
89.6(2)%, corrected for a 1.5% average state preparation and measurement (SPAM)
error.
C2(Z) gates, which can be constructed by adding a few single-qubit rotations to a
Toffoli-3 gate (see Figure 5.4(b)) and has the unitary evolution operator
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0 
0 0 0 1 0 0 0 0
UC2(Z) =  

0 0 0 0 1 0 0 0 

. (5.11)


0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0 
0 0 0 0 0 0 0 −1
For all circuits, the single-qubit rotations are further optimized to minimize total
rotation time [74], optimized under pre-upgrade constraints.
Here, we show results for a Toffoli-3 gate, with a process fidelity of 89.6(2)%
(see Figure 5.5). Toffoli-3 gates have been previously performed in NMR systems [75]
and ion traps [76], including this system [24,27]. We employed a limited tomography
procedure to verify that the Toffoli-3 gate performed had no spurious phases on the
87
probability
outputs (see Section 5.4.1). This gate will be used for the Grover search algorithm
in Chapter 6.
5.4.1 Toffoli-3 Characterization
We employed a limited tomography procedure to characterize the outputs of
the Toffoli-3 gate performed. A global rotation into the X basis was applied to
all 3 ions before and after the Toffoli-3 gate for each input (see Figure 5.6(a)):
R (πy ) for the even inputs (000, 010, 100, 110) and R
π
y(− ) for the odd inputs (001,2 2
011, 101, 111). An ideal Toffoli-3 gate will result in an anti-diagonal input-output
matrix in the Z basis when this procedure is applied. The experimental results
of this verification procedure are shown in Figure 5.6(b) with an average success
probability of 82.1(2)%, indicating the Toffoli-3 is faithful for arbitrary input states.
88
(a) ( ) ( )
|q 〉 R ±π1 y • R ±π( 2 ) y ( 2 )
|q2〉 Ry ±π • R ±π( 2 ) y ( 2 )
|q 〉 R ±πt y Ry ±π2 2
(b)
1
1 0.8
0.8
0.6 0.6
0.4
0.2 0.4
0
000 0.2
011 111
011
111 0
detected 000 input
Figure 5.6: (a) Circuit for implementing the Toffoli-3 limited tomography procedure.
The global rotations are positive for even input states and negative for odd input
states. (b) Limited tomography check performed on the Toffoli-3 gate to verify
phases. The average success probability is 82.1(2)%, corrected for a 2.4% average
SPAM error.
89
probability
|q1〉 • Ry(α1α π2 ) XX(α π1 )2 4
|q2〉 • Ry(π ) XX(α π2 )2 4
|q3〉 • =
|qt〉
|qa〉 Ry(α πt ) XX(α π2 ) R(−αtπ,Qπ) XX(α π1 ) Ry(α π4 4 4 t )4
|q1〉 · · ·
|q π2〉 · · · XX(α2 )4
|q 〉 · · · R (−β π ) R (β π3 y 2 x )2
XX(β π )
· · · π 8|qt〉 Rx( )4
XX(α πt )
|qa〉 · · · Rx(−
8
α π2 ) XX(α
π
2 ) R
π 3π
2 4 y
(−αt )4 Rx(αt )4
|q1〉 · · ·
|q2〉 · · ·
|q3〉 · · · R(−2π , (β+1)π − P ) XX(α π ) R(−α 2π α3αt+13 2 3 4 3αtβ , ( )π − α3αtβP )3 2
|qt〉 · · ·
|qa〉 · · · XX(α π3 )4
|q1〉 · · · Rx(−α1π)
|q2〉 · · · XX(α π2 )4
|q3〉 · · · R(π,−α π π3αtβ ) XX(α )4 3 4
XX(β π )
|qt〉 · · ·
8
|qa〉 · · · XX(α π ) R (α π ) XX(α π3 4 y t 4 2 )4
|q 〉 · · · XX(α π ) R π1 1 4 y(−α1α2 )2
|q2〉 · · · XX(α π2 ) Ry(−π )4 2
|q3〉 · · ·
|qt〉 · · ·
|qa〉 · · · R(−αtπ,Qπ) XX(α π1 ) R (α π π π π4 y t ) Rx(−α4 2 ) XX(α ) R2 2 4 y(αt )4
Figure 5.7: Toffoli-4 gate implementation using XX(χ) and R(θ, φ) gates.
90
1
1 0.8
0.8
0.6 0.6
0.4
0.2 0.4
0
0000
1111 0.2
1000 1000
1111 0
detected 0000 input
Figure 5.8: Measured truth table for a Toffoli-4 gate performed with 3 controls, 1
target, and 1 ancilla qubit. The average process fidelity is 70.5(3)%, corrected for a
1.9% average SPAM error.
91
probability
5.4.2 Toffoli-4 Gate
We use a related strategy to construct a Toffoli-4 gate as was done for the
Toffoli-3 gate, where the Toffoli-4 gate is a triply-controlled NOT gate,
 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
 

0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
U = 

0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
T4  . (5.12)0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
Using the methods described in [77], we construct a circuit between 3 control qubits
(q1, q2, and q3), one target qubit (qt), and an additional ancilla qubit (qa), shown
in Figure 5.7. In the circuit diagram, α1 =(√sgn)(χ1a), α2 = sgn(χ2a), α3 = sgn(χ3a),
αt = sgn(χta), β(=)sgn(χ3t), P( =) arcsin
2 , and Q = 1(4 − 3α2αt). By again3 8
using both XX π and XX π gates, we are able to save one two-qubit gate
4 8
relative to a construction limited to CNOT gates [77]. When executing the gate on
the system, we report an average process fidelity of 70.5(3)% (see Figure 5.8) [24].
92
5.5 The SWAP Gate
A SWAP gate exchanges the quantum states of two qubits, like so:
 1 0 0 00 0 1 0USWAP =  . (5.13)0 1 0 0
0 0 0 1
If one assumes that qubits are fixed wires, one could naively implement a SWAP
gate as follows [65]:
• • × .
= (5.14)
• ×
However, if one has a fully-connected system - for example, an ion-trap quantum
computer where the normal modes of motion yield multi-qubit entangling gates with
all possible pairwise interactions available - then one can simply swap the qubit
assignments, a purely classical operation that saves us 3 two-qubit interactions.
This will come in handy in Section 5.9. A comparison to another architecture,
the IBM Quantum Experience, shows that our system benefits from the absence of
SWAP -gate overhead due to its complete connectivity graph [27].
5.6 Controlled-SWAP Gate
93
(a)
|qc〉 • •
|q2〉 × = •
|q3〉 × • •
(b)
|qc〉 • Rx(γ π ) R (−π )2 z 2
|q2〉 × = Rz(−β π ) R (πx + (1− β)π )2 4 4
XX(β π ) XX(β π )
4 8
|q3〉 × Ry(β π ) R π π π2 z(− ) Rx(−β + )2 2 4
|q 〉 · · · XX(γ π ) R(−2π , (γ+1c 8 )π − P )3 2
XX(απ )
|q2〉
4
· · ·
|q3〉 · · · XX(γ π )8
|qc〉 · · · R(−αβγ 2π , (αβ+1)π − αβγP ) XX(γ π ) R(π,−αβγ π )3 2 8 4
|q2〉 · · ·
|q3〉 · · · XX(γ π )8
|qc〉 · · ·
XX(απ )
| 〉 · · · 4q π2 Rz(−β )2
XX(β π )
| 〉 · · · 4q3 R (β πy ) R πy(−β ) Rz(−π )2 2 2
Figure 5.9: Controlled-SWAP (or Fredkin) gate implementation. (a) Controlled-
SWAP gate in terms of CNOT and Toffoli gates, used for optimized circuit con-
struction. (b) Controlled-SWAP gate circuit optimized for use on the experi-
ment using XX(χ), Rx(θ), Ry(θ), Rz(θ), an√d R(θ, φ) gates, where α = sgn(χ12),
β = sgn(χ23), γ = sgn(χ13), and P = arcsin
2 .
3
94
1
1
0.9 0.8
0.8
0.7
0.6
0.5 0.6
0.4
0.3
0.2 0.4
0.1
0
000 0.2
010 110
100 100
110 010
detected state 000 0input state
Figure 5.10: Measured truth table for a Controlled-SWAP gate. The average pro-
cess fidelity is 86.8(3)%, corrected for a 1% average state preparation and measure-
ment (SPAM) error.
The controlled-SWAP (CSWAP , or Fredkin) gate [78] is a three-qubit in-
teraction that operates by swapping the last two qubits |q2〉 and |q3〉 if the control
qubit |qc〉 is in the state |1〉, and has the unitary evolution operator
 
1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0
 0 0 1 0 0 0 0 00 0 0 1 0 0 0 0
UCSWAP =   . (5.15)0 0 0 0 1 0 0 00 0 0 0 0 0 1 0

0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
It has been previously demonstrated on NMR [79] and photonic systems [80, 81];
this represents its first demonstration on a trapped ion system.
95
probability
The CSWAP can be implemented using two CNOT gates and a Toffoli gate,
as shown in Figure 5.10(a). The final composite circuit, shown in Figure 5.10(b), was
constructed by assembling the pre-optimized CNOT and Toffoli circuits in Figures
5.2(a) and 5.4(a) respectively, and then further optimizing the combined rotations.
The Toffoli and CNOT gates used here were optimized for use on the machine pre-
upgrade, but the CSWAP was optimized for use post-upgrade; consequently, it is
possible further rotation savings could be achieved were a more optimal Toffoli gate
determined first. Future uses of the CSWAP may benefit from a more thorough
optimization procedure taking advantage of the upgraded experimental capabilities.
The CSWAP gate was used as a readout module to measure the Renyi-2
entropy in a two-site Fermi-Hubbard model system [30]. Data for all bitwise inputs
and outputs is shown in Figure 5.10, with an average process fidelity of 86.8(3)%.
For use in the Reyni-2 entropy measurement scheme, only the outcome of the control
qubit is needed; its process fidelity is 94.0(2)%.
5.7 Using Z-Rotations to Obviate χ Sign in XX Gates
With Z rotations implemented classically (see Section 5.2.1), an opportunity
now arises to abstract away the considerations made( fo)r the eff(ects )of the sign of χ
on the constuction of a composite gate. Since XX π 6= XX −π , a mechanism
4 4
is needed to ensure correct gate sequences. Previously, a parameter for the sign of
the χ for each XX gate had to be introduced, and affected the properties of other
rotations in the circuit in order to ensure the gate was fully modular and could be
96
|q1〉 Rz(π) Rz(−π)
XX(+χ) = XX(−χ)
|q2〉
|q1〉
XX(+χ) = XX(−χ)
|q2〉 Rz(π) Rz(−π)
Figure 5.11: Adding an appropriate Z rotation on one qubit on either side of an
XX(χ) gate flips the sign of its χ parameter, obviating the need to consider all
possible combinations of χ signs when optimizing a modular, programmable gate.
When implementing a gate on a given set of ions, simply add Z rotations on either
side of any XX gates on ion pairs with experimentally-verified negative χ signs.
used on any qubit selection, regardless of the χ-signs of the XX gates in use. This
made circuit optimization considerably more complicated, and led to some circuits
(such as the Toffoli-3 in Figure 5.4(a)) that were not necessarily optimal for all
possible χ sign combinations.
Conveniently, it turns out that adding a classical Z rotation on the same qubit
before and after an XX(χ) gate flips the sign of its χ parameter, as shown in Figure
5.11. By adding these classically-implemented rotations whenver an XX gate with
a negative χ parameter is used, all XX gates are rendered identical without an error
penalty, making composite gate optimzation much simpler (I optimized all of the
following gate sequences by hand) and ensuring that all composite gates are always
optimal.
5.8 The Square Root of CNOT
In Section 7.5, to construct the full adder circuit from [82] we will need to
construct a Controlled-V (C(V )) gate, where C(V ) is the square root of a CNOT
gate, or C(V )2 = CNOT , and consists of a controlled-X(θ) operation. As discussed
97
(a)
|qc〉 • R π π πy( ) Rx(− ) R (− )2 2 y 2
= XX(π )
|qt〉
4
R (−πx )2
(b)
|qc〉 • R (−π πy ) R ( )2 y 2 Rz(π)
= XX(π )
| 4q 〉 R (πt x )2
(c)
|qc〉 • Ry(π ) R πx(− ) R (−π )2 y
= XX(π
4 2
)
|qt〉
8
V Rx(−π )4
(d)
|qc〉 • Ry(π ) Rx(π ) R πy(− )2 4 2
= XX(−π )
|qt〉
8
† R πV x( )4
(e)
|qc〉 • • • • •
= =
|q 〉 V V V †t V †
Figure 5.12: Component circuits for CNOT and its square root gates using XX(χ),
Rx(θ), Ry(θ), and Rz(θ) gates.. (a) and (b) Alternate CNOT gate implementa-
tions. (c) Controlled-V (C(V )) gate implementation. (d) Controlled-V † (C(V †))
gate implementation. (e) C(V ) and C(V †) gates are equivalent to the square root
of a CNOT gate.
98
in Section 5.4, we can adjust the χ parameter of the XX(χ) gate to achieve such a
relationship, shown in Equation 5.10. Figure 5.12(a) shows a variation on the CNOT
√
gate from Figure 5.2(a) that allows for the creation of a CNOT by adjusting
the χ of the XX gate, and concurrently adjusting the length of the following X
rotations on each ion; the resulting circuit for a C(V ) gate is shown in Figure
5.12(c). Additionally, C(V †) is needed, shown in Figure 5.12(d); from Section 5.7,
we can easily implement the XX(−π ) gate by adding Z rotations on either side of
8
the XX gate. By squaring the C(V ) or C(V †) gates, a CNOT gate is performed
(C(V †) (Figure 5.12(e)).
√
The unitary for the C(V ) = CNOT gate is
 1 0 0 00 1 0 0 UV =  . (5.16)0 0 1(1− i) 1(1 + i)
2 2
0 0 1(1 + i) 1(1− i)
2 2
5.9 Quantum Scrambling Library
Quantum scrambling is a phenomenon of interest that describes how infor-
mation spreads through a system. It has direct implications for out of time order
correlators (OTOCs) and has implications for the study of quantum information in
black holes [83]. We have implemented scrambling unitaries on our experiment and
probed their properties in several ways [31], as proposed in [84]. Here, I discuss the
library of quantum circuits used for implementing and probing scrambling unitaries.
The general circuit diagram to test whether a unitary U is a quantum scram-
99
bling unitary is shown in Figure 5.13. In the case of a scrambling unitary, the
input state |ψ〉 will teleport to the output state |ϕ〉 with fidelity F = |〈ψ|ϕ〉|2 = 1,
conditioned on the outcome of the Bell measurement. The Bell measurement it-
self will yield the correct Bell state |00〉 + |11〉 with post-selection probability P .
Three possible pairs of qubits can be used for Bell measurements: (3,4), (2,5), and
(1,6). If successful teleportation with the correct post-selection probability P for
all possible input states and all possible choices of Bell pair, quantum scrambling is
verified. Using the teleportation fidelity F and the post-selection probability P , we
can probe the relationship between scrambling, OTOCs, and decoherence [84].
|ψ〉
|0〉 EPR U
|0〉
EPR Bell Bell Bell
|0〉
|0〉 EPR U∗P
|0〉
EPR
|0〉 |ϕ〉
Figure 5.13: General figure for testing the scrambling properties of a unitary U .
The unitary U∗P is a permutation of the unitary U∗ that permutes the first
100
and third input, such that U∗P = PU∗P , where
 
1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0
P =  

. (5.17)
0 1 0 0 0 0 0 00 0 0 0 0 1 0 00 0 0 1 0 0 0 0
0 0 0 0 0 0 0 1
This can also be modeled by simply swapping the inputs of qubits 4 and 6 into the
unitary U∗, and swapping them back afterwards.
5.9.1 Deterministic Teleportation Protocol
An alternative protocol that teleports |ψ〉 deterministically, eliminating the
need to perform post-selection, can be performed using a variant of the Grover search
protocol [85] and implemented as shown in Figure 5.14(a). The Grover operations,
GD and GA, have the form



0 0 0 −1
G = I − 2|EPR〉〈EPR| =  0 1 0 0  , (5.18)0 0 1 0 
−1 0 0 0
which is a SWAP operation and a few rotations. The circuit is shown in Figure
5.14(b). We can implement the SWAP by simplly re-assigning the qubits to the
other ion; no quantum operation needed.
101
(a)
|ψ〉
|0〉 EPR U
|0〉
EPR GD
|0〉
|0〉 EPR U∗ UT
|0〉
EPR GA
|0〉 |ϕ〉
(b)
|q1〉 Rz (π) Rx (π) × Rz (π)
|q2〉 Rx (π) ×
Figure 5.14: (a) General figure for testing the scrambling properties of a unitary U ,
using a Grover protocol requiring G. (b) Circuit for Grover protocol G.
5.9.2 Bell States and Bell Measurements
For the quantum scrambling experiments presented in [31], we will need to
create and measure Bell pairs. Below are the schemes used for dealing with Bell
pairs, optimized under the post-upgrades capabilities.
The four Bell states can be created using the following recipe (or related vari-
ations) as a method of constructing the EPR pairs required by the scramble-tester
102
circuit in Figure 5.13:
| +〉 √1
( ) (
| 〉 | 〉 π π
)
Φ = ( 00 + 11 ) = RZ,1 (α X) X α( |)00〉 (5.19)2 2 4
| 1 π πΦ−〉 = √ (|00〉 − |11〉) = RZ,1 (−α) XX α |(00〉 ) (5.20)2 2 4
|Ψ+〉 1= √ (| 〉 | 〉 π π01 + 10 ) = RZ,1 (α R)X,2 (π)XX α( |)00〉 (5.21)2 2 4
| −〉 √1 π πΨ = (|01〉 − |10〉) = RZ,1 −α RX,2 (π)XX α |00〉 (5.22)
2 2 4
Our native XX(χ) gates create the entangling state √1 (|00〉 − i|11〉). As such,
2
XX(π ) gates can be used by themselves to create EPR pairs. Since the resulting
4
output in Figure 5.13 simply adds a global phase to the teleported state |ϕ〉, the
bare XX(π ) gates can be used without modification in the circuit. Note that proper
4
Bell measurement as described next is still necessary - we can’t use bare XX gates
there.
Bell measurements can be performed using the circuit shown in Figure 5.15(a)
[65]. This takes as input a state |q1q2〉 in the Bell basis, so
|q q 〉 = a|Φ+〉+ b|Ψ+1 2 〉+ c|Φ−〉+ d|Ψ−〉, (5.23)
where a2 + b2 + c2 + d2 = 1, and outputs the result
|q1q2〉 = a|00〉+ b|01〉+ c|10〉+ d|11〉. (5.24)
This circuit can be optimized using our experimentally available gates R(θ, φ) and
103
(a)
|q1〉 • H
|q2〉
(b) ( ) ( )
|q π1〉 (Rz ) R
π
2 y 2 ( )
XX απ
4 ( ) ( )
|q2〉 Rz (1− α) π Rz (α− 1) π R −π2 2 x 2
Figure 5.15: (a) Simple Bell measurement circuit implemented with a CNOT gate
and a Hadamard gate. (b) Optimized Bell measurement gate sequence using R and
XX gates.
XX(χ) to the Bell measurement circuit shown in Figure 5.15(b).
5.9.3 Scrambling Unitary US
The scrambling unitary US used for much of the data discussed in [31],
 −1 0 0 −1 0 −1 −1 0 

0 1 −1 0 −1 0 0 1 
0 −1 1 0 −1 0 0 1 
1  1 0 0 1 0 −1 −1 0 US =   , (5.25)2  0 −1 −1 0 1 0 0 1 1 0 0 −1 0 1 −1 0 1 0 0 − − 1 0 1 1 0 
0 −1 −1 0 −1 0 0 −1
104
is verified to be a scrambling unitary by showing
U †(X ⊗ I ⊗ I)U = X ⊗ Z ⊗ Z
U †(I ⊗X ⊗ I)U = Z ⊗X ⊗ Z
U †(I ⊗ I ⊗X)U = Z ⊗ Z ⊗X
U †(Y ⊗ I ⊗ I)U = Y ⊗X ⊗X
U †(I ⊗ Y ⊗ I)U = X ⊗ Y ⊗X
U †(I ⊗ I ⊗ Y )U = X ⊗X ⊗ Y
U †(Z ⊗ I ⊗ I)U = Z ⊗ Y ⊗ Y
U †(I ⊗ Z ⊗ I)U = Y ⊗ Z ⊗ Y
U †(I ⊗ I ⊗ Z)U = Y ⊗ Y ⊗ Z (5.26)
where {X, Y, Z, I} are the Pauli operators. The circuit to implement this unitary
consists of 3 XX gates followed by 3 Y Y gates; the latter can be implemented by
adding rotations to an XX gate to move the interaction to the Y Y basis. Because
this unitary is fully real, US = U
∗
S. The gate sequence to produce this unitary is
shown in Figure 5.16(a). A circuit with a continuously-adjustable parameter θ that
scans how much scrambling the unitary performs is shown in Figure 5.16(b); for
θ = 0, the unitary is the identity I, and for θ = π , the unitary is the maximally-
2
scrambling unitary US. We note that the 3 XX gates in each group of consecutive
XX gates can be done in any order.
105
(a) ( ) ( )
|q 〉 XX R (π1 z ) XX Rz −π2 2XX XX ( )
|q2〉 R πz Rz −π2 2
XX ( ) XX ( )
|q 〉 XX R π π3 z XX Rz −2 2
(b)
|q1〉 XX Rz (θ) XX Rz (−θ)
XX XX
|q2〉 Rz (θ) Rz (−θ)
XX XX
|q3〉 XX Rz (θ) XX Rz (−θ)
Figure 5.16: (a) Circuit to produce the unitary US. (b) Circuit for U(θ), with an
adjustable parameter θ that goes from the id(ent)ity U(θ = 0) = I to the maximally-scrambling U(θ = π ) = US, allowing for a scan of the amount of scrambling per-2
formed by the unitary. XX indicates a XX π gate.
4
5.9.4 Scrambling Unitary UCZ
An additional scrambling unitary, UCZ , is also probed in [31] for use with the
Grover protocol, where
 
 1 1 1 −1 1 −1 −1 −1  1 −1 1 1 1 1 −1 1  1 1 −1 1 1 −1 1 1 √1 −1 1 1 1 −1 −1 −1 1 UCZ = 2 2 

 . (5.27) 1 1 1 −1 −1 1 1 1 −1 1 −1 −1 1 1 −1 1−1 −1 1 −1 1 −1 1 1 
−1 1 1 1 1 1 1 −1
The gate sequence yielding this unitary is shown in Figure 5.17(a), and is
based on an arrangement of 6 Control-Z gates. An optimized gate sequence is
shown in Figure 5.17(b). Additionally, a gate sequence with the same number of
106
operations as the unitary but instead performing the identity is shown in Figure
5.17(c), allowing for direct comparisons and error characterization. As with US, we
note that the 3 XX gates in each group of consecutive XX gates can be done in
any order. Additionally, the rotations before the first XX gate and after the last
XX gate can all have their signs flipped without changing the unitary performed.
This comes in handy when optimizing the overall circuit with the Grover protocol,
and allows for some XX gate cancellations (see Section 5.9.6.) Finally, we note that
U = U∗ = UTCZ CZ CZ .
(a) UCZ
|q1〉 • • H • •
|q2〉 • Z H Z •
|q3〉 Z Z H Z Z
(b) UCZ, opti(mi)zed ( ) ( )
|q1〉 Rz (π) Ry (π) XX R (πy ) XX R π2 2 y 2XX XX ( )
|q2〉 Rz (π) Ry (π2 ) R πy ( 2 ) Ry (π2XX XX )
|q3〉 Rz (π) R πy XX R π R π2 y 2 XX y 2
(c) Identity I( ) ( )
|q π1〉 Rz (π) Ry ( ) XX Rz (π) XX R π2 y 2XX XX ( )
|q2〉 R πz (π) Ry ( ) Rz (π) R π2 y 2XX XX ( )
|q3〉 R (π) R πz y 2 XX Rz (π) XX R
π
y 2
Figure 5.17: (a) Basic circuit to produce the scrambling unitary UCZ . (b) Opti-
mized circuit to produce the scrambling unitary UCZ . (c) Identity circuit requiring
the s(am)e number of gates as UCZ , allowing for error characterization and direct com-parisons between maximally- and minimally-scrambling unitaries. XX indicates a
XX π gate.
4
107
5.9.5 Classical Scrambler
Some unitaries will scramble quantum information non-maximally. This means
they can transform 1-body information into 3-body information in some bases, but
not others. Maximally-scrambling unitaries transform 1-body information into 3-
body information in all bases. One such unitary is the “classical” scrambling unitary
UCS, which scrambles information in the X and Y bases, but not information in Z.
Here,  
1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0 
0 0 0 −1 0 0 0 0 
UCS = 0 0 0 0 1 0 0 0 
 (5.28)
0 0 0 0 0 −1 0 0 0 0 0 0 0 0 −1 0 
0 0 0 0 0 0 0 −1
and we have
U †CS(X ⊗ I ⊗ I)UCS = X ⊗ Z ⊗ Z
U †CS(I ⊗X ⊗ I)UCS = Z ⊗X ⊗ Z
U †CS(I ⊗ I ⊗X)UCS = Z ⊗ Z ⊗X
U †CS(Y ⊗ I ⊗ I)UCS = Y ⊗ Z ⊗ Z
U †CS(I ⊗ Y ⊗ I)UCS = Z ⊗ Y ⊗ Z
U †CS(I ⊗ I ⊗ Y )UCS = Z ⊗ Z ⊗ Y. (5.29)
108
However, U †CS(Z ⊗ I ⊗ I)UCS, U
†
CS(I ⊗ Z ⊗ I)UCS, and U
†
CS(I ⊗ I ⊗ Z)UCS are
not equal to any combination of three Pauli matrices, and therefore no scrambling
occurs in that basis. The gate sequence yielding this unitary (controlled-Z gates
connecting each pair of qunits) is shown in Figure 5.18(a), with its optimized version
in Figure 5.18(b).
Because this unitary only scrambles classical information, the circuit in Figure
5.13 will only teleport classical information; in other words, information about pop-
ulations will be teleported from the input to the output state, but not information
about the phase on each state. For an input state |ψ〉 = a|0〉 + b|1〉 on qubit 1, a
measurement on the output qubit ϕ will yield a2 population in |0〉 and b2 population
in |1〉 conditioned on the Bell pair |00〉 + |11〉. Additionally, the probability P of
getting the Bell pair |00〉+ |11〉 will be 50% rather than 25%.
(a)
|q1〉 • •
|q2〉 Z •
|q3〉 Z Z
(b) ( ) ( ) ( )
|q 〉 R (π) R −π ( ) XX π R π1 z y ( 2 ) 4 y 2XX π4 ( )
|q2〉 Rz (π) R −π ( ) R πy ( 2 ) ( ) y ( 2XX π4 )
|q3〉 Rz (π) R πy − XX π R π2 4 y 2
Figure 5.18: (a) Basic circuit to produce the unitary UCS, which scrambles classical
but not phase information. (b) Optimized circuit to produce the unitary UCS.
109
5.9.6 Circuit Optimization
We can eliminate several XX gates by cancelling XX gates repeated between
repeated scrambling unitaries on the same qubits. Additionally, we can use prop-
erties of EPR pairs that allow us to “move” gates from one half of an EPR pair
to the other half, subject to a transformation. Specifically, any unitary U applied
on one half of an EPR pair is the same as that unitary’s transpose UT applied to
the other half (see Figure 5.19). As an example of how this improves circuit perfor-
mance, Figures 5.20 and 5.21 showsthe gates eliminated with these techniques for
UCZ implemented in the circuit from Figure 5.13.
|q1〉 U
EPR = EPR
|q2〉 UT
Figure 5.19: A unitary U performed on one half of an EPR pair is the same as its
transpose UT performed on the other half of the EPR pair.
( ) ( ) ( )|q1〉 R πy Rz (−π) R π2 y − 2 ( )
XX π XX π
4 ( ) ( )
|q 〉 R π π
4
2 y R2 z (−π) Ry − 2
( ) Rx (π) ( )
= XX π XX π4 4 =
Rx (π)
Figure 5.20: Equivalence used to eliminate XX gates in neighboring, repeated UCZ
unitaries.
110
( )
|q1〉 Rz (π) Ry (π · · ·2 )
|q2〉 EPR Rz (π) Ry (π2 ) ( ) · · ·XX π4
|q 〉 R (π) R (π3 z y ) · · ·2EPR
|q4〉 Rz (π) Ry (π2 ) ( ) · · ·XX π4
|q 〉 EPR R (π) R (π5 z y ) · · ·2
|q6〉 R (π) R πz y · · ·2
EPR
|q7〉 · · ·( )
R (π) R πz y · · ·2
EPR · · ·
· · ·
EPR
= ( ) ( ) ( )RTy
XX π
T
4 (
π
2 ) RT πz (π) Rz (π) Ry ( 2 ) ( ) · · ·XX π4
EPR RT πy ( 2) RTz (π) Rz (π) R πy · · ·2
Rz (π) R
π
y · · ·2
EPR
· · ·
( )
R (π) R πz y · · ·2
EPR · · ·
· · ·
EPR
= Rx (π) · · ·
EPR Rx (π) ( ) · · ·
Rz (π) R
π
y · · ·2
EPR
· · ·
Figure 5.21: Equivalence used to eliminate XX gates from stacked UCZ unitaries
using the EPR property in Figure 5.19.
111
Chapter 6: Complete 3-Qubit Grover Search
The Grover quantum search algorithm is a hallmark application of a quan-
tum computer with a well-known speedup over classical searches of an unsorted
database. Here, we report results for a complete three-qubit Grover search algo-
rithm using the scalable quantum computing technology of trapped atomic ions,
with better-than-classical performance. Two methods of state marking are used for
the oracles: a phase-flip method employed by other experimental demonstrations,
and a previously-undemonstrated Boolean method requiring an ancilla qubit that
is directly equivalent to the state-marking scheme required to perform a classical
search. The data presented here is also presented in [24].
6.1 The Grover Search Algorithm
Searching large databases is an important problem with broad applications.
The Grover search algorithm [86,87] provides a powerful method for quantum com-
puters to perform searches with a quadratic speedup in the number of required
database queries over classical computers. It is an optimal search algorithm for
a quantum computer [88], and has further applications as a subroutine for other
112
Initialize Oracle Amplification
Figure 6.1: Evolution of relative amplitudes for each state during a Grover search
algorithm. The initialization stage creates an equal superposition of all possible
input states, so the amplitude αx = 1 for all basis states |x〉. The oracle stage marks
the desired state, so the amplitude αm of the marked state |m〉 becomes negative
while the amplitudes αb∑of the unmarked states |b〉, b 6= m remain∑unchanged. Theamplification stage performs a reflection about the mean vector N−1x=0 |x〉, which
has amplitude A = 1 N−1 1x=0 αx = (−αm + (N − 1)αN N b), to amplify the marked
state. An appropriate number of repetitions of the oracle and amplification stages
will maximize the amplitude of the correct answer. All qubit states are normalized
by the factor √1 . The algorithm can also be generalized to mark and amplify the
N
amplitude of t desired states.
quantum algorithms [89, 90]. Searches with two qubits have been demonstrated on
a variety of platforms [91–97] and proposed for others [98], but larger search spaces
have only been demonstrated on a non-scalable NMR system [73].
The Grover search algorithm has 4 stages: initialization, oracle, amplifica-
tion, and measurement, as shown in Fig∑ure 6.1. The initialization stage creates
an equal superposition of all states, √1 N−1x=0 |x〉 for all basis states |x〉. The or-N
acle stage ma(rks th∑e solution(∑s) |m〉 by fl)ipping the sign of that state’s amplitude,
yielding √1 −αm m |m〉+ b=6 m αb|b〉 for marked state amplitude(s) αm andN
non-marked state amplitudes αb. Inverting an amplitude α about an average am-
plitude A can be written as −α + 2A or A + (A −∑α). Here, the amplification
stage performs a reflection about the mean vector N−1x=0 |x〉, thus increasing the
113
amplitude of the marked state:
[ ∑ ∑ ]
|Ψamp〉 √
1
= (2A+ αm) |m〉+ (2A− αb) |b〉 , (6.1)
N m b 6=m
where A is the amplitude of the mean vector. Finally, the algorithm output is
measured. For a search database of size N with t possible solutions, the single-
shot probability of measuring the correct answ√er is maximized to near-unity by
repeating the oracle and amplification stages O( N/t) times before measurement.
The probability of failing to measure a correct answer is guaranteed to be less than t
N
if the optimal number of iterations, j = b π c, is used, where sin2(θ) = t , 0 < θ ≤ π ;
4θ N 2
for t N , the probability of failure is negligible [86,87]. By comparison, a classical
search algorithm will get the correct answer after an average of N/2 queries of the
oracle, and in the worst case may require up to N queries to find the correct answer.
For large databases, this quadratic speedup represents a significant advantage for
quantum computers.
Here, we implement the Grover search algorithm on n = 3 qubits, which
corresponds to a search database of size N = 2n = 8. All searches are performed
with a single iteration (j = 1). After a single iteration, the initial∑amplitudes
αx = 1 on all basis states |x〉, so the mean vector has amplitude A = 1 N−1x=0 αx =N
1 (−αm · t+ (N − t)α N−2tb) = . Plugging in to Equation 6.1 above and rewritingN N
114
the inversion about the mean vector as A+ (A− α), we have
[ ∑ ∑ ]
| 1Ψamp(j = 1)〉 = √ [(A+ (A+ αm)) |m〉+ (A+ (A− αb)) |b〉N ( ( m )) b 6=m
√1 N − 2t N − 2t
∑
= + + 1 |m]〉N ( N ( N )) m
N − 2t N − 2t ∑
+ [ + − 1 |b〉( N N ) b6=m ( ) ]
√1 N − 2t 2(N − t
∑ ∑
= + | N − 2t 2tm〉+ − ( |b〉 .
N N N N Nm b6=m
(6.2)
Since there are t possible correct solutions |m〉, the probability of measuring one
correct solution after one iteration is
([ ] )2
· N − 2t 2(N − t) √1Pm = t + . (6.3)
N N N
In contrast, the optimal classical search strategy consists of a single query (equiva-
lent to a single Grover iteration) followed by a random guess in the event the query
failed. Therefore, the total probability of finding a correct solution is P (success) =
P (query correct) + P (query incorrect) ∗ P (guess correct|query incorrect), where
P (query correct) = t , P (query incorrect) = N−t , and P (guess correct|query incorrect)
N N
= t− . In general, for t solutions in the classical case,N 1
t N − t t
P (success) = + · . (6.4)
N N N − 1
115
Consequently, for a single-solution algorithm (t = 1),(t[he algorithmic]prob)abil-2
(ity of)measuring the correct state after one iteration is t · N−2t + 2(N−t) √1 =N N N2
√5 = 78.125% [87], compared to t + N−t · t 1 7 1− = + · = 25% for the optimal4 2 N N N 1 8 8 7
classical search strategy.
In the two-solution case (t = 2), where two states are marked as correct answers
during the oracle stage and both states’ amplitudes are amplified in the algorithm’s
am(p[lification stage], the)probabi(lity o)f measur(ing)one of the two correct answers is2 2 2
t · N−2t + 2(N−t) √1 = 2 · 1√6 = 2 · √1 = 100% for the quantum case,
N N N 8 8 2
as compared to 2 + 6 · 2 = 13 ≈ 46.4% for the classical case.
8 8 7 28
The algorithm is performed with both a phase oracle, which has been previ-
ously demonstrated on other experimental systems, and a Boolean oracle, first re-
ported here, which requires more resources but is directly comparable to a classical
search. All quantum solutions are shown to outperform their classical counterparts.
6.2 Oracles
We examine two alternative methods of encoding the marked state within the
oracle. While both methods are mathematically equivalent [65], only one is directly
comparable to a classical search. The previously-undemonstrated Boolean method
requires the use of an ancilla qubit initialized to |1〉, as shown in Figure 6.2(a). The
oracle is determined by constructing a circuit out of NOT and Ck(NOT ) (k ≤ n)
gates such that, were the oracle circuit to be implemented classically, the ancilla bit
would flip if and only if the input to the circuit is one of the marked states. By using
116
(a)
Init Amplification
|q1〉 : |0〉 H H X • X H
|q2〉 : |0〉 H H X • X H
| 〉 | 〉 Oracleq3 : 0 H H X Z X H
|qa〉 : |1〉 H H
(b)
Init Oracle Amplification
|0〉 H X • X H X • X H  ∑
|0〉 • •  √5 1H H X X H |011〉+ √4 2 4 2 x 6=011 |x〉
|0〉 H • H X Z X H
|1〉 H H |1〉
(c)
Init Amplification
|q1〉 : |0〉 H H X • X H
|q2〉 : |0〉 H Oracle H X • X H
|q3〉 : |0〉 H H X Z X H
(d)
Init Oracle Amplification
|0〉 H • H X • X H 
|0〉 H • H X • 1X H √ (|011〉+ |101〉)2
|0〉 H Z Z H X Z X H
Figure 6.2: (a) General circuit diagram for a Grover search algorithm using a
Boolean oracle, depicted using standard quantum circuit diagram notation [65].
The last qubit qa is the ancilla qubit. (b) Example single-solution Boolean oracle
marking the |011〉 state. (c) General circuit diagram for a Grover search algorithm
using a phase oracle. (d) Example two-solution phase oracle marking the |011〉 and
|101〉 states.
classically available gates, this oracle formulation is directly equivalent to the classi-
cal search algorithm, and therefore can most convincingly demonstrate the quantum
algorithm’s superiority. On a quantum computer, because the initialization sets up
an equal superposition of all possible input states, the Cn(NOT ) gate targeted on
the ancilla provides a phase kickback that flips the phase of the marked state(s) in
117
the data qubits. An example oracle is shown in Figure 6.2(b) to illustrate this. The
phase method of oracle implementation does not require the ancilla qubit. Instead,
the oracle is implemented with a circuit consisting of Z and Ck(Z) (k ≤ n−1) gates
that directly flip the phase(s) of the state(s) to be marked (see Figures 6.2(c-d)).
6.3 Circuit Implementation
The Grover search algorithm is implemented using circuits that are equivalent
to those shown in Figures 6.1(b,d), but with the initialization and amplification
stages optimized to minimize gate times, as shown in Figures 6.3(a-b). The circuits
shown are for use with Boolean oracles; in the phase oracle case, the ancilla qubit
qa is simply omitted. To preserve the modularity of the algorithm, the initialization
stage and amplification stage were each optimized without regard to the contents of
the oracle, so each possible oracle can simply be inserted into the algorithm without
making any changes to the other stages.
Oracles for the Grover search algorithm were constructed using a combination
of reversible and classical logic synthesis techniques. For Boolean oracles, reversible
logic synthesis was employed to find a set of X,CN(NOT ) gates that marked the
desired state(s) for each oracle. For phase oracles, EXOR polynomial synthesis was
used to find a set of Z,CN(Z) gates that marked the desired state(s) for each oracle.
For example, for Boolean oracles, the selection was limited to the classically available
X (or NOT ) and CN(NOT ) gates, and a reversible circuit was constructed such
that the output bit (corresponding to the ancilla qubit in the quantum oracle) would
118
(a)
|q1〉 : |0〉 Rx(π) Ry(−π )2
|q2〉 : |0〉 Rx(π) Ry(−π )2
|q3〉 : |0〉 Rx(π) R πy(− )2
|qa〉 : |0〉 Ry(−π )2
(b) ( )
|q1〉 Ry ((1− β)π) Rx(β 3π ) XX(β π )2 4 8|q2〉 Ry (1− γ)π R (γ π ) 2π γ+12 x 2 R(− , ( )π − P )3 2
XX(γ π )
|q 83〉 Ry(π) Rx(π ) XX(β π )4 8
|qa〉 R (πy )2 Rx(π)
· · ·
XX(απ )
4
· · · R(−αβγ 2π , (αβ+1)π − αβγP ) R(π,−αβγ π )
3 2 4
XX(γ π )
8
· · ·
· · · ( )
· · · Ry (β − 1)π2
XX(απ )
4
· · · Ry(−π )2
· · · Ry(π)
· · ·
Figure 6.3: Grover search algorithm implementation by substage using XX(χ),
Rx(θ), Ry(θ), and R(θ, φ) gates. The circuits shown are for use with Boolean ora-
cles; removing the ancilla qubit |qa〉 produces the necessary circuits for use(w√ith)a
phase oracle. Let α = sgn(χ12), β = sgn(χ1t), γ = sgn(χ2t), and P = arcsin
2 .
3
(a) Grover initialization stage implementation. (b) Grover amplification stage im-
plementation.
119
be flipped if and only if a marked state was used as the input to the circuit. While
there are many possible circuit constructions for each oracle, the oracle chosen for
implementation was one that first minimized the number of two-qubit interactions
required, and then minimized the number of single-qubit interactions needed. The
synthesis techniques used are scalable and can be applied to oracles of any size. The
oracles used here were implemented as per the circuit diagrams shown in Table 6.1
for single-solution oracles and Table 6.2 for two-solution oracles. The algorithm is
executed for all 8 possible single-result oracles and all 28 possible two-result oracles.
Mark Boolean Oracle Phase Oracle Mark Boolean Oracle Phase Oracle
000 |q 1001〉 X • X |q1〉 X • X |q1〉 • |q1〉 •
|q2〉 X • X |q2〉 X • X |q2〉 X • X |q2〉 X • X
|q3〉 X • X |q3〉 X Z X |q3〉 X • X |q3〉 X Z X
|qa〉 |qa〉
001 |q1〉 X • X |q1〉 X • X 101 |q1〉 • |q1〉 •
|q2〉 X • X |q2〉 X • X |q2〉 X • X |q2〉 X • X
|q3〉 • |q3〉 Z |q3〉 • |q3〉 Z
|qa〉 |qa〉
010 |q1〉 X • X |q1〉 X • X 110 |q1〉 • |q1〉 •
|q2〉 • |q2〉 • |q2〉 • |q2〉 •
|q3〉 X • X |q3〉 X Z X |q3〉 X • X |q3〉 X Z X
|qa〉 |qa〉
011 |q1〉 X • X |q1〉 X • X 111 |q1〉 • |q1〉 •
|q2〉 • |q2〉 • |q2〉 • |q2〉 •
|q3〉 • |q3〉 Z |q3〉 • |q3〉 Z
|qa〉 |qa〉
Table 6.1: Table of all oracles used for the single-solution Grover
search algorithm.
120
Marked Boolean Oracle Phase Oracle
000, 001 |q1〉 X • X |q1〉 Z •
|q2〉 X • X |q2〉 Z Z
|q3〉 |q3〉
|qa〉
000, 010 |q1〉 X • X |q1〉 Z •
|q2〉 |q2〉
|q3〉 X • X |q3〉 Z Z
|qa〉
000, 011 |q1〉 X • X |q1〉 Z • •
|q2〉 X • • X |q2〉 Z Z
|q3〉 • |q3〉 Z Z
|qa〉
000, 100 |q1〉 |q1〉
|q2〉 X • X |q2〉 Z •
|q3〉 X • X |q3〉 Z Z
|qa〉
000, 101 |q1〉 X • • X |q1〉 Z •
|q2〉 X • X |q2〉 Z Z •
|q3〉 • |q3〉 Z Z
|qa〉
000, 110 |q1〉 X • • X |q1〉 Z •
|q2〉 • |q2〉 Z •
|q3〉 X • X |q3〉 Z Z Z
|qa〉
000, 111 |q1〉 X • • • • X |q1〉 Z • •
|q2〉 • |q2〉 Z Z •
|q3〉 • |q3〉 Z Z Z
|qa〉
001, 010 |q1〉 X • X |q1〉 • •
|q2〉 • • |q2〉 Z Z
|q3〉 • |q3〉 Z Z
|qa〉
Table 6.2 – Continued on next page
121
Table 6.2 – Continued from previous page
Marked Boolean Oracle Phase Oracle
001, 011 |q1〉 X • X |q1〉 •
|q2〉 |q2〉
|q3〉 • |q3〉 Z Z
|qa〉
001, 100 |q1〉 • • |q1〉 Z •
|q2〉 X • X |q2〉 Z •
|q3〉 • |q3〉 Z Z
|qa〉
001, 101 |q1〉 |q1〉
|q2〉 X • X |q2〉 •
|q3〉 • |q3〉 Z Z
|qa〉
001, 110 |q1〉 • • • • |q1〉 • •
|q2〉 X • X |q2〉 Z •
|q3〉 • |q3〉 Z Z Z
|qa〉
001, 111 |q1〉 X • • X |q1〉 •
|q2〉 • |q2〉 •
|q3〉 • |q3〉 Z Z Z
|qa〉
010, 011 |q1〉 X • X |q1〉 •
|q2〉 • |q2〉 Z Z
|q3〉 |q3〉
|qa〉
010, 100 |q1〉 • • |q1〉 Z •
|q2〉 • |q2〉 Z •
|q3〉 X • X |q3〉 Z Z
|qa〉
010, 101 |q1〉 • • • • |q1〉 • •
|q2〉 • |q2〉 Z Z •
|q3〉 X • X |q3〉 Z Z
|qa〉
Table 6.2 – Continued on next page
122
Table 6.2 – Continued from previous page
Marked Boolean Oracle Phase Oracle
010, 110 |q1〉 |q1〉
|q2〉 • |q2〉 Z •
|q3〉 X • X |q3〉 Z
|qa〉
010, 111 |q1〉 X • • X |q1〉 •
|q2〉 • |q2〉 Z Z •
|q3〉 • |q3〉 Z
|qa〉
011, 100 |q1〉 • • • • |q1〉 Z • •
|q2〉 • |q2〉 Z •
|q3〉 • |q3〉 Z Z
|qa〉
011, 101 |q1〉 • • |q1〉 •
|q2〉 • |q2〉 •
|q3〉 • |q3〉 Z Z
|qa〉
011, 110 |q1〉 • • |q1〉 •
|q2〉 • |q2〉 Z •
|q3〉 • |q3〉 Z
|qa〉
011, 111 |q1〉 |q1〉
|q2〉 • |q2〉 •
|q3〉 • |q3〉 Z
|qa〉
100, 101 |q1〉 • |q1〉 Z •
|q2〉 X • X |q2〉 Z
|q3〉 |q3〉
|qa〉
100, 110 |q1〉 • |q1〉 Z •
|q2〉 |q2〉
|q3〉 X • X |q3〉 Z
|qa〉
Table 6.2 – Continued on next page
123
Table 6.2 – Continued from previous page
Marked Boolean Oracle Phase Oracle
100, 111 |q1〉 • |q1〉 Z • •
|q2〉 X • • X |q2〉 Z
|q3〉 • |q3〉 Z
|qa〉
101, 110 |q1〉 • |q1〉 • •
|q2〉 • • |q2〉 Z
|q3〉 • |q3〉 Z
|qa〉
101, 111 |q1〉 • |q1〉 •
|q2〉 |q2〉
|q3〉 • |q3〉 Z
|qa〉
110, 111 |q1〉 • |q1〉 •
|q2〉 • |q2〉 Z
|q3〉 |q3〉
|qa〉
Table 6.2: Table of all oracles used for the two-solution Grover
search algorithm.
Detailed information and individual fidelities for constituent composite gates
can be found in Chapter 5.
6.4 Data
Figures 6.4 and 6.5 show the results, respectively, of single- and two-solution
Grover search algorithms, each using both the Boolean and phase marking methods.
All possible oracles are tested to demonstrate a complete Grover search (see Tables
6.1 and 6.2). Two figures of merit are provided with the data for each oracle. The
124
Boolean Oracles ASP(%) SSO(%) Phase Oracles ASP(%) SSO(%)
50
000 34(1) 80(2) 34.1(7) 79(1)
001 38(1) 83(2) 50.2(7) 89(1) 40
010 40(1) 85(2) 33.3(7) 78(1)
011 41(1) 86(2) 46.4(7) 87(1) 30
100 31(1) 76(2) 42.7(7) 86(1) 20
101 35(1) 81(2) 55.1(7) 91(1)
110 41(1) 85(2) 41.4(7) 85(1) 10
111 50(1) 90(2) 46.5(7) 84(1)
0
000 010 100 110 000 010 100 110
detected state detected state
Figure 6.4: Results from a single iteration of a single-solution Grover search algo-
rithm performed on a 3-qubit database. Data for the Boolean oracle formulation is
shown on the left, and data for the phase oracle formulation is shown on the right.
The plots show the probability of detecting each output state. All values shown are
percents, with a theoretical ASP of 78.1% and theoretical SSO of 100%. Data is
corrected for average SPAM errors of 1%.
algorithm success probability (ASP) is the probability of measuring the marked
state as the experimental outcome. For the two-solution algorithm, the ASP is
calculated by summing the probabilities of measuring each of the two marked states.
The squared statistical overlap (SSO) measures the statis(t∑ical overlap b)etween the√ 2
measured and expected populations for all states: SSO = Nj=0 ejmj , where ej
is the expected population and mj is the measured population for each state j [99].
Additionally, all of the data shown in this paper is corrected to account for state
preparation and measurement (SPAM) errors (see figure captions for values), similar
to the method proposed in [39] while also accounting for multi-ion crosstalk [19]. All
uncertainties given are statistical uncertainties based on the number of experiments
performed.
The single-iteration, single-solution Grover search algorithm shown in Figure
6.4 has a theoretical ASP of 78.1%, as discussed above. The SSO takes into account
125
marked state
Boolean Oracles ASP(%) SSO(%) Phase Oracles ASP(%) SSO(%)
50
000,001 67(1) 67(1) 77(1) 77(1)
000,010 66(1) 66(1) 79(1) 79(1)
000,011 66(1) 66(1) 77(1) 75(1)
000,100 68(1) 68(1) 74.7(9) 72(1)
000,101 65(1) 65(1) 79(1) 79(1)
000,110 62(1) 62(1) 69(1) 69(1) 40
000,111 66(1) 66(1) 74(1) 73(1)
001,010 64(1) 64(1) 79(1) 79(1)
001,011 71(1) 71(1) 72(1) 71(1)
001,100 68(1) 67(1) 79(1) 78(1)
001,101 69(1) 69(1) 80(1) 79(1)
001,110 60(1) 60(1) 82(1) 81(1) 30
001,111 65(1) 65(1) 73(1) 72(1)
010,011 75(1) 75(1) 73(1) 73(1)
010,100 66(1) 65(1) 70(1) 70(1)
010,101 65(1) 64(1) 73.8(9) 69.7(9)
010,110 72(1) 72(1) 74(1) 73(1)
010,111 73(1) 72(1) 75(1) 74(1) 20
011,100 65(1) 65(1) 74(1) 74(1)
011,101 66(1) 66(1) 74(1) 74(1)
011,110 73(1) 73(1) 75(1) 75(1)
011,111 74(1) 74(1) 80(1) 78(1)
100,101 74(1) 74(1) 72(1) 71(1) 10
100,110 67(1) 66(1) 66.7(9) 67(1)
100,111 64(1) 64(1) 75.5(9) 71(1)
101,110 65(1) 65(1) 72(1) 71(1)
101,111 70(1) 69(1) 77(1) 77(1)
110,111 74(1) 74(1) 80(1) 80(1)
0
000 010 100 110 000 010 100 110
detected state detected state
Figure 6.5: Results from the execution of a two-solution Grover search algorithm
performed on a 3-qubit database. Data for the Boolean oracle formulation is shown
on the left, and data for the phase oracle formulation is shown on the right. The
plots show the probability of detecting each output state. All values shown are
percents. The ASP is the sum of the probabilities of detecting each of the two
marked states. Data is corrected for average SPAM errors of 1%.
126
marked state
that the 7 unmarked states then have equal expected probabilities totaling 21.9%
of being measured. For all Boolean oracles, the average ASP is 38.9(4)% and the
average SSO is 83.2(7)%, while phase oracles have an average ASP of 43.7(2)% and
an average SSO of 84.9(4)%; the reduced use of resources in the phase oracles (10
XX(χ) gates and 3 qubits for phase oracles compared to 16 XX(χ) gates and 5
qubits for Boolean oracles) results in better performance, as expected. These results
compare favorably with the classical ASP of 25%.
The two-solution Grover search algorithm shown in Figure 6.5 has a theoretical
ASP of 100%, as discussed above. For all Boolean oracles, the average ASP is
67.9(2)% and the average SSO is 67.6(2)%, while phase oracles have an average
ASP of 75.3(2)% and an average SSO of 74.4(2)%; the reduced use of resources in
the phase oracles (6-8 XX(χ) gates and 3 qubits for phase oracles compared to 10-
14 XX(χ) gates and 4 qubits for Boolean oracles) results in better performance, as
expected. For all oracles in both cases, the two states with the highest measurement
probability are also the two marked states. These results compare favorably with
the classical ASP of 46.4%.
6.5 Additional Iterations
Performing an additional iteration on the single-solution Grover search algo-
rithms was inhibited by circuit-depth limitations in the experimental control pro-
gram, which will be fixed for future work. Here, we estimate the impact of an
additional iteration on algorithm performance. While a single iteration of the single-
127
solution Grover search algorithm has a maximum ASP of 78.125%, performing two
iterations raises the maximum ASP to 94.5312%. Applying the error estimation
models used in [27], the likely performances of two Grover search algorithm cases
were examined: the Boolean 000 oracle and the phase 111 oracle, which correspond
to the worst- and best-case oracles by gate count. The random error estimation
√
model assumes random error propagation for each operation of the form (1−  ) Ng ,
and the systematic error estimation model assumes coherent over- or under-rotations
for each operation and has the form (1− g)N , where N is the number of operations
and g is the error per operation. Based on the analysis in [27] on this same system,
we expect the actual results to fall somewhere between these two models. For a
single iteration of the phase 111 oracle, we estimate an SSO of 86% and an ASP of
41% using the random error model, or an SSO of 61% and an ASP of 16% using the
systematic error model; as in the analysis in [27], we compare this to the measured
SSO of 84(1)% and ASP of 46.5(7)% and see that the experiment performs slightly
worse than the random error model, and better than the systematic error model.
Extending the analysis to two iterations of the phase 111 oracle, we estimate an
SSO of 81% and ASP of 60% using the random error model, or an SSO of 40% and
an ASP of 19% using the systematic error model.
Similarly, for a single iteration of the Boolean 000 oracle, we estimate an
SSO of 83% and an ASP of 37% using the random error model, or an SSO of 45%
and an ASP of 6% using the systematic error model; as before, we compare this
to the measured SSO of 80(2)% and ASP of 34(1)% and see that the experiment
performs slightly worse than the random error model. Extending the analysis to two
128
iterations of the phase 111 oracle, we estimate an SSO of 77% and ASP of 55% using
the random error model, or an SSO of 22% and an ASP of 6% using the systematic
error model. We expect the experiment would perform somewhere between these
two models, although we do not know how well these error models hold for very
deep circuits; it is not clear whether the experiment would have outperformed the
best classical strategy with two iterations, which has a success probability of 37.5%.
6.6 Outlook
We note that this implementation of the Grover search algorithm scales linearly
in the two-qubit gate count and ancilla count for increasing search database size as
a function of the number of qubits n, and for a constant number of solutions t. For
a database of size N = 2n stored on n qubits, the amplification stage requires one
Toffoli-n gate, and the t-solution oracle stage requires at worst t Toffoli-n (for a phase
oracle) or Toffoli-(n+ 1) (for a Boolean oracle) gates; optimal oracles for particular
sets of marked states may require even fewer two-qubit gates. The method used in
Section 5.4.2 to construct the Toffoli-4 circuit scales to Toffoli-n gates as 6n− 13 in
the two-qubit gate count and as dn−3e in the ancilla count [77]. This paves the way
2
for more extensive use of the Grover search algorithm in solving larger problems on
quantum computers, including using the circuit as a subroutine for other quantum
algorithms.
129
Chapter 7: Parallel 2-Qubit Operations
In any computer, quantum or classical, parallel operations are highly desir-
able. Parallel operations save considerable time over performing operations in series.
Current trends in the (classical) computer industry include a focus on developing
multi-core processors and developing software with multiple parallel threads. Quan-
tum computers have a long way to go before such a scale will be possible, but the
ability to perform parallel operations is nevertheless crucial to our ultimate ability
to scale up.
In ion-trap quantum computers, two-qubit interactions are mediated by the
normal modes of motion in the ion chain. However, as the chain grows in size, so do
the number of modes of motion, and spectral crowding makes sideband resolution
more difficult. Two-qubit interactions can be implemented in less time by using
more optical power in the Raman beams, but this has the consequence of reducing
sideband resolution, which degrades gate fidelity. Consequently, gate speed is lim-
ited by sideband resolution, a limitation that gets worse as the processor size grows.
Parallel two-qubit operations are a tool to speed up computation that avoids this
problem. Parallel gate operations also reduce the overall gate depth of a given pro-
cess, permitting more operations before decoherence and error accumulation obviate
130
any meaningful outputs. Several quantum computing subroutines and composite
gates have been shown to benefit directly from parallel entangling gates, including
the quantum Fourier transform [100,101], multiply controlled Toffoli gates, and sta-
bilizer circuits [101]. Quantum algorithms that will similarly benefit from parallel
entangling gates include Shor’s integer factoring [102], solving the discrete logarithm
problem over the elliptic curve group [103], simulating Hamiltonian dynamics using
the Suzuki-Trotter formula [104], and quantum chemistry [105].
Here, we present experimental results for a pair of two-qubit gates performed
simultaneously in a single chain of trapped ions. We employ a pulse shaping scheme
that modulates the phase and amplitude of the Raman laser to drive programmable
high-fidelity 2-qubit XX gates in parallel by coupling to the collective modes of
motion of the ion chain. Ensuring the interaction produced yields only spin-spin in-
teractions between the desired pairs with neither residual spin-motion entanglement
nor crosstalk spin-spin entanglement between non-desired ion pairs is a nonlinear
constraint problem, and optimal pulse shapes are found using optimization tech-
niques. As an application of these parallel operations, we demonstrate the quantum
full adder using a depth-4 circuit requiring the use of parallel 2-qubit operations as
well as single- and two-qubit operations previously demonstrated on this system.
7.1 Theory
We perform parallel gates by modulating the Rabi frequency of the individ-
ual Raman beams. This is accomplished in a similar manner to the 2-qubit gates
131
already implemented on this experiment [19,53] (see Section 3.2), which implement
entangling XX gates using a Mølmer-Sørensen scheme [41,52,55,56], where red and
blue sidebands are applied to entangle the ion spins of illuminated ions with the
radial normal motional modes of the ions in the trap. Using the motional modes
as an information bus, the spin states of separate ions become entangled, and the
pulse shape is engineered so that at the end of the gate, the motional modes are
entirely disentangled from the spins, leaving only spin-spin entanglement [53,57,59].
The pulse shape is controlled by dividing the gate time into several equal-length
segments, and adjusting the amplitude of the gate modulation in each segment.
Alternative possible schemes include varying the detuning µ [106] or the beatnote
phase [107] to engineer high-fidelity gates; the former scheme has been demonstrated
to generate 2-qubit entangling XX gates on this system [28].
In order to perform parallel entangling operations involving 2 pairs of qubits
(i, j) and (m,n) in a chain of N ions with N motional modes ωk using red and blue
sidebands with detuning µ, we have a 4-qubit unitary, as follows:
( ∑ ∑ )
U x x x||(τ) = exp i φi(τ)σi + i χij(τ)σi σj
( [ i i<j
= exp i φi(τ)σ
x
i + φ
x x
j(τ)σj + φm(τ)σm + φn(τ)σ
x
n
+ χij(τ)σ
x
i σ
x x
j + χmn(τ)σmσ
x + χ (τ)σx xn im i σm])
+ χ x xin(τ)σi σn + χjm(τ)σ
xσx + χ (τ)σxσxj m jn j n (7.1)
132
where τ is the gate time, the spin-motion interaction φi(τ) is
φi(τ) = α
†
i,k(τ)âk − α
∗
i,k(τ)âk, (7.2)
â†k and âk are the raising and lowering operators for the motional phonons, the
spin-motion parameter αi,k(τ) is
∫ τ
α (τ) = η Ω (t) sin(µt)eiωkti,k i,k i dt, (7.3)
0
ηi,k is the spin-motion coupling or Lamb-Dicke parameter, Ωi(t) is the Rabi frequency
of the optical field applied to ion i, and the spin-spin interaction term χij(τ) is
∫ τ ∫ t′ ∑
χij(τ) = 2 dt
′ dt η ′i,kηj,kΩi(t)Ωj(t) sin(µt) sin(µt ) sin(ω (t
′
k − t)). (7.4)
0 0 k
At the end of the gate, the spin-motion terms must go to zero, ensuring that all mode
trajectories in phase space return to the origin. So, we require all 4N spin-motion
parameters (4 ions, N modes) α{i,j,m,n},k(τ) = 0. Since we wish to entangle only the
ions pairs (i, j) and (m,n), we do not wish to entangle the “crosstalk” pairs (i,m)
(i, n), and so on. Consequently, for the entangling pairs, we require χ = χidealij ij and
χ = χidealmn mn , where 0 < χ
ideal ≤ π . χideal is typically π for a maximally entangling
4 4
XX gate but can be set to smaller values to implement partially-entangling gates.
For the crosstalk pairs, we require χim = χin = χjm = χjn = 0 to ensure no
unwanted entanglement at the end of the gate. This then yields a set of 4N + 6
parameters to control for when optimizing pulse sequences to implement parallel
133
XX gates:
α{i,j,m,n},k(τ) = 0
χ (τ) = χidealij ij
χ (τ) = χidealmn mn
χim(τ) = χin(τ) = χjm(τ) = χjn(τ) = 0. (7.5)
To provide optimal control during the gate and fulfill the constraints in Equa-
tion 7.5, we divide up the gate amplitude Ωi(t) into S segments of equal duration
τ/S, and vary the amplitude in each segment Ωs as an independent variable. In or-
der to implement independent XX gates, we implement independent signals on the
two ion pairs we want to entangle; ions (i, j) see one pulse shape, while ions (m,n)
see another. Separate signals on the two ion pairs are necessary to provide suffi-
cient control to simultaneously entangle the desired ion pairs but not the crosstalk
pairings. Without two different signals, there is no way to provide cancellation to
crosstalk entanglement. On the experiment, this ion-specific signal shaping is pro-
vided by the upgrade to a multi-channel AWG, as discussed in Section 2.4. The
134
gate amplitude on a given ion then becomes a piecewise-constant function,
Ω1 0 ≤ t < τ/SΩ2 τ/S ≤ t < 2τ/S... ...
Ωi(t) =  (7.6)Ωs (s− 1)τ/S ≤ t < sτ/S
.. .. . .ΩS (S − 1)τ/S ≤ t < τ.
Consequently, Equations 7.3 and 7.4 can be re-written as
∑ [S ∫ ]sτ/S
αi,k(τ) = Ωs ηi,k sin(µt)e
iωktdt
∑s=1 (s−1)τ/SS
= Ω isCk,s (7.7)
s=1
and
∑S ∑S ∫ sτ/S ∫ s′τ/S ∑
χij(τ) = ΩsΩs′ dt
′ dt η ′ ′i,kηj,k sin(µt) sin(µt ) sin(ωk(t − t))
∑ ′s=1 s∑′=1 (s−1)τ/S (s −1)τ/S kS S
= ΩsΩs′Ds,s′ , (7.8)
s=1 s′=1
where the terms
∫ sτ/S
Ci iωktk,s = ηi,k sin(µt)e dt (7.9)
(s−1)τ/S
135
and
∫ sτ/S ∫ s′τ/S ∑
Dij ′s,s′ = dt dt ηi,kη sin(µt) sin(µt
′ ′
j,k ) sin(ωk(t − t)) (7.10)
(s−1)τ/S (s′−1)τ/S k
are pre-calculated constants that are functions only of the motional mode frequencies
ωk, the detuning µ, and the segment number s, and so can be arranged into S ×N
and S × S matrices for each ion and ion-ion pair, respectively. Note that the time
ordering of the double integral in Equation 7.10 requires that t < t′, so the time-
segmented scheme requires s ≤ s′. In the case s = s′, we must force the t < t′
constraint, yielding
∫ sτ/S ∫ t′ ∑
Dij ′s=s′ = dt dt ηi,kηj,k sin(µt) sin(µt
′) sin(ωk(t
′ − t)). (7.11)
(s−1)τ/S (s′−1)τ/S k
If we now arrange the segment amplitudes Ωs into two vectors Ωij and Ωmn,
one for each entangling pair (i, j) and (m,n), we can now write our constraint
equations from Equation 7.5 as
 CiCj
( )
 Ωij = 0Cm Ωmn
( )( Cn )( )Dij 0 Ω
T T ijΩij Ωmn( )( = χ
ideal
ij
( ) 0 0 Ωmn)0 0 Ω
T T ijΩ Ω = χidealij mn mn
( )( 0 Dmn)(Ωmn)0 Dcross Ω
ΩT T
ij
ij Ωmn = 0, (7.12)0 0 Ωmn
136
{ }
where Ci,Cj,Cm,Cn{ are th}e S × N spin-motion interaction matricies for each
segment on each ion, Dij,Dmn are the two S×S spin{-spin interaction mat}ricies for
each segment on the entangling ion pairs, and Dcross = Dim,Din,Djm,Djn are the
four S×S spin-spin interaction matricies for each segment on the crosstalk ion pairs.
While the C-constraints are linear, the 6 D constraints on the spin-spin interaction
terms are not. With multiple quadratic constraints and no evident guarantees the
constraint matricies are positive or negative semidefinite, this is now a non-convex
quadratically constrained quadratic program (QCQP). In the general case, this kind
of problem is NP-hard. As a result, analytical approaches are intractable, and so
we use optimization techniques to find solutions that fit the constraints in Equation
7.12 as well as possible.
7.1.1 Fidelity of Parallel XX Operations
Next, we calculate the fidelity of simultaneous XX gate operations as a func-
tion of the control parameters in Equation 7.5. The fidelity is given by
F|| = 〈ψ †init|UidealρrUideal |ψinit〉 , (7.13)
where ρr is the density matrix for the experimental operation traced over the motion,
[ ]
ρ †r = Trm Uexpt |ψinit〉 〈ψinit|Uexpt . (7.14)
For convience of calculations, this derivation will be performed in the X basis.
137
Consequently, we use the initial state
( ) ( ) ( ) ( )
| 1ψinit〉 =  |0〉i + |1〉i ⊗ |0〉j + |1〉j ⊗ |0〉m + |1〉m ⊗ |0〉n + |1〉4 n

1
1 

1
1


1



1
1
1= 1

 
 (7.15)
1

1

1

1 11
1
and calculations of the unitary will be performed with σz instead of σx. The final
fidelity is independent of the basis in which it is calculated. In matrix form, the
138
general parallel 2-qubit gate unitary from Equation 7.1 is
eiΦ 0000 0 0 0 0 0 0 0 0 eiΦ0001 0 0 0 0 0 0 0 0 eiΦ0010 0 0 0 0 0 0 0 0 eiΦ0011 0 0 0 0 0 0 0 0 eiΦ0100 0 0 0 0 0 0 0 0 e
iΦ0101 0 0
 0 0 0 0 0 0 eiΦ0110 0 0 0 0 0 0 0 0 eiΦ0111Uexpt =  · · · 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 
· · ·  ,
eiΦ1000 0 0 0 0 0 0 0 
0 eiΦ1001 0 0 0 0 0 0 
0 0 eiΦ1010 0 0 0 0 0 
0 0 0 eiΦ1011 0 0 0 0 
0 0 0 0 eiΦ1100 0 0 0 
0 0 0 0 0 eiΦ1101 0 0 
0 0 0 0 0 0 eiΦ1110 0 
0 0 0 0 0 0 0 eiΦ1111
(7.16)
139
where
Φ0000 = φi + φj + φm + φn + χij + χim + χin + χjm + χjn + χmn
Φ0001 = φi + φj + φm − φn + χij + χim − χin + χjm − χjn − χmn
Φ0010 = φi + φj − φm + φn + χij − χim + χin − χjm + χjn − χmn
Φ0011 = φi + φj − φm − φn + χij − χim − χin − χjm − χjn + χmn
Φ0100 = φi − φj + φm + φn − χij + χim + χin − χjm − χjn + χmn
Φ0101 = φi − φj + φm − φn − χij + χim − χin − χjm + χjn − χmn
Φ0110 = φi − φj − φm + φn − χij − χim + χin + χjm − χjn − χmn
Φ0111 = φi − φj − φm − φn − χij − χim − χin + χjm + χjn + χmn
Φ1000 = −φi + φj + φm + φn − χij − χim − χin + χjm + χjn + χmn
Φ1001 = −φi + φj + φm − φn − χij − χim + χin + χjm − χjn − χmn
Φ1010 = −φi + φj − φm + φn − χij + χim − χin − χjm + χjn − χmn
Φ1011 = −φi + φj − φm − φn − χij + χim + χin − χjm − χjn + χmn
Φ1100 = −φi − φj + φm + φn + χij − χim − χin − χjm − χjn + χmn
Φ1101 = −φi − φj + φm − φn + χij − χim + χin − χjm + χjn − χmn
Φ1110 = −φi − φj − φm + φn + χij + χim − χin + χjm − χjn − χmn
Φ1111 = −φi − φj − φm − φn + χij + χim + χin + χjm + χjn + χmn. (7.17)
In an ideally-executed pair of parallel XX gates, with all parameters set as in
140
Equation 7.5, we get
( )
Uideal = U α{i,j,m,n},k = 0, χij = χidealij , χmn = χidealmn , χim = χin = χjm = χjn = 0eiχ ++ 0 0 0 0 0 0 0 0 eiχ+− 0 0 0 0 0 0 0 0 eiχ+− 0 0 0 0 0 0 0 0 eiχ++ 0 0 0 0 0 0 0 0 eiχ−+ 0 0 0 0 0 0 0 0 e
iχ−− 0 0
 0 0 0 0 0 0 eiχ−− 0 0 0 0 0 0 0 0 e
iχ−+
=  · · · 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 
· · ·  , (7.18)
eiχ−+ 0 0 0 0 0 0 0 
0 eiχ

−− 0 0 0 0 0 0 
0 0 eiχ−− 0 0 0 0 0
iχ− 0 0 0 e + 0 0 0 0 
0 0 0 0 eiχ++ 0 0 0 
0 0 0 0 0 eiχ+− 0 0 

0 0 0 0 0 0 eiχ+− 0
0 0 0 0 0 0 0 eiχ++
141
where
χ++ = χ
ideal
ij + χ
ideal
mn
χ = χideal − χideal+− ij mn
χ ideal−+ = −χij + χidealmn
χ ideal−− = −χij − χidealmn . (7.19)
Next, we calculate the density matrix ρr, given by Equation 7.14. As an
intermediate step, we calculate the matrix A to be traced over. The first 3 columns
of this 16× 16 matrix are shown here:
A = Uexpt |ψ †init〉 〈ψinit|Uexpt
 1 e2i(φn+χin+χjn+χmn) e2i(φm+χim+χjm+χmn) e−2i(φn+χin+χjn+χmn) 1 e2i(φm−φn+χim−χin+χjm−χjn) e−2i(φm+χim+χjm+χmn) e−2i(φm−φn+χim−χin+χjm−χjn) 1e−2i(φm+φn+χim+χin+χjm+χjn) e−2i(φm+χim+χjm−χmn) e−2i(φn+χin+χjn−χmn) e−2i(φj+χij+χjm+χjn) e−2i(φj−φn+χij−χin+χjm−χmn) e−2i(φj−φm+χij−χim+χjn−χmn) e−2i(φj+φn+χij+χ in
+χjm+χmn) e−2i(φj+χij+χjm−χjn) e−2i(φj−φm+φn+χij−χim+χin)
e−2i(φj+φm+χij+χim+χjn+χmn) e−2i(φj+φm−φn+χij+χim−χin) e−2i(φj+χij−χjm+χjn)1  e−2i(φj+φm+φn+χij+χim+χin) e−2i(φj+φm+χij+χim−χjn−χmn) e−2i(φj+φn+χij+χin−χjm−χmn)= 16  e−2i(φi+χij+χim+χin) e−2i(φi−φn+χij+χim−χjn−χmn) e−2i(φi−φm+χij+χin−χjm−χmn) e−2i(φi+φn+χij+χim+χjn+χmn) e−2i(φi+χij+χim−χin) e−2i(φi−φm+φn+χij−χjm+χjn)e−2i(φi+φm+χij+χin+χjm+χmn) e−2i(φi+φm−φn+χij+χjm−χjn) e−2i(φ i
+χij−χim+χin)
 e−2i(φi+φm+φn+χij+χjm+χjn) e−2i(φi+φm+χij−χin+χjm−χmn) e−2i(φi+φn+χij−χim+χjn−χmn) e−2i(φi+φj+χim+χin+χjm+χjn) e−2i(φi+φj−φn+χim+χjm−χmn) e−2i(φi+φj−φm+χin+χjn−χmn) e−2i(φi+φj+φn+χim+χjm+χmn) e−2i(φi+φj+χim−χin+χjm−χjn) e−2i(φi+φj−φm+φn)e−2i(φi+φj+φm+χin+χjn+χmn) e−2i(φi+φj+φm−φn) e−2i(φi+φj−χim+χin−χjm+χjn)
e−2i(φi+φj+φm+φn) e−2i(φi+φj+φm−χin−χjn−χmn) e−2i(φi+φj+φn−χim−χjm−χmn)
)
· · · etc. . (7.20)
Now we find the trace over the motion for this matrix A to find the density matrix.
142
Since the spin-spin chi terms have no motional components, they are unaffected by
the trace operation. The scalars on the diagonal are similarly unaffected. Hence,
we only have to worry about the spin-motion φ{i,j,m,n} terms. We further note that
the displacement operator D is
( ) ∗
D αk(τ) = e(α
k
i (τ)â
† k
k−αi (τ)âk)
i ( )
| k |2 − k∗αi (τ) /2= e e( αi (τ)â ) (αkk e i (τ)â
†
k) (7.21)
for a given parameter αk{i,j,m,n}. Consequently, as an example, tracing over the term
in row 4, column 1 of Equation 7.20 can be written as
[
Tr e−
] [ ]
2i(φm+φn+χim+χin+χjm+χjn) = e−2i(χim+χin+χjm+χjn)m T∏rm e
−2i(φm+φn)
= e−
[ ( )]
2i(χim+χin+χjm+χjn) Tr D 2αk + 2αkm m n .
k
(7.22)
In Fock space, a thermal state for the kth motional mode in a linear ion chain
can be written as
( )∑∞
ρ = 1− eh̄ωk/kBT e−nh̄ωk/kBTtherm |n〉 〈n| , (7.23)
n=0
where n is the phonon number. For a generic displacement operator D(α) using
Equation 7.21 and using the properties of the raising and lowering operators ↠and
143
â, we can write
Trm [D(α)] = T( r [D(α)ρther)m] ∑∞ [ ]
= 1− eh̄ω/kBT e|α|
2/2 e−nh̄ω/kBT
∗
Tr e−α âeαâ
†
( ) ∑n=0 [
|n〉 〈n|
∞ ∑∞ ∑ ]∞
= 1− eh̄ω/k
2 1 1
BT e|α| /2 e−nh̄ω/kBTTr ∗ p[ (−α â) (αâ
†)q |n〉 〈n|
∑ ∑ p! q!( ) n=0 p=0 q=0 ]∞ ∞
− h̄ω/k T |α|
2/2 −nh̄ω/k T (−1)p |α|
2p
= 1 e B e e B Tr âp(â†)q |n〉 〈n|
∑ (p!)
2
( ) n=0 p=0∞
= 1− eh̄ω/kBT e|α|
2/2 e−nh̄ω/kBT×
∑ n=0∞ (−1)p |α|2p
. . . (n+ 1)(n+ 2)× . . .× (n+m)
(p!)2
p=0
|α|2
= e− coth(h̄ω/2kBT )2
(7.24)
where we note that the trace is only nonzero for q = p. So continuing our example
144
from Equation 7.22, we can now write
[ ]
Tr e−2i(φm+φn+χim+χin+χjm+χjn)m ∏ [ ( )]
= e−2i(χim+χin+χjm+χjn) Tr k km D 2αm + 2αn
k
∏  | 22αkm+2αk− n| coth(h̄ωk/2k2 BT )
= e−2i(χim+χin+χjm+χjn) e
(k )
− 1
∑
= e−2i(χim+χin+χjm+χjn)e 2 k|
k k |22αm+2αn coth(h̄ωk/2kBT )
( ∑ )2
= e−
1 k k
2i(χim+χin+χjm+χjn) − β 2α +2αe 2 k k| m n|
= e−2i(χim+χin+χjm+χjn)Γ00++. (7.25)
Here, to better abbreviate the representation, we define a set of parameters,
( ∑ )1
ΓA A AmAn = exp − β
2
i j k |2 (Aiαi,k + Ajαj,k + Amαm,k + Anαn,k)| , (7.26)2
k
where the parameters {Ai, Aj, Am, An} can be {0,±1} and are indicated as {0 →
0,+1→ +,−1→ −}. The inverse mode temperature βk is
( ) [ ( )]
h̄ωk 1 1
βk = coth = coth ln 1 + , (7.27)
kBT 2 n̄k
where n̄k is the average phonon number in the k
th mode. By applying the trace
operation to all elements of Equation 7.20, we get the final density matrix.
Plugging in all values to the fidelity equation from Equation 7.13 and solving,
145
we derive the fidelity,
( )
F α , χ , χideal ideal|| {i,j,m,n},k ij ij , χmn, χmn , χim, χin, χjm, χjn =
1
(8 + Γ+−−− + Γ+−−+ + Γ+−+− + Γ+−++ + Γ++−− + Γ++−+ + Γ+++− + Γ++++
128
+ 2 (Γ0+−− + Γ+000) cos [2 (∆χij − χim − χin)]
+ 2 (Γ0++− + Γ+000) cos [2 (∆χij + χim − χin)]
+ 2 (Γ0+−+ + Γ+000) cos [2 (∆χij − χim + χin)]
+ 2 (Γ0+++ + Γ+000) cos [2 (∆χij + χim + χin)]
+ 2 (Γ0+00 + Γ+0−−) cos [2 (∆χij − χjm − χjn)]
+ 2 (Γ00++ + Γ+−00) cos [2 (χim + χin − χjm − χjn)]
+ 2 (Γ0+00 + Γ+0+−) cos [2 (∆χij + χjm − χjn)]
+ 2 (Γ00+− + Γ++00) cos [2 (χim − χin + χjm − χjn)]
+ 2 (Γ0+00 + Γ+0−+) cos [2 (∆χij − χjm + χjn)]
+ 2 (Γ00+− + Γ+−00) cos [2 (χim − χin − χjm + χjn)]
+ 2 (Γ0+00 + Γ+0++) cos [2 (∆χij + χjm + χjn)]
+ 2 (Γ00++ + Γ++00) cos [2 (χim + χin + χjm + χjn)]
+ 2 (Γ00+0 + Γ+−0+) cos [2 (χim − χjm + ∆χmn)]
+ 2 (Γ0+0− + Γ+0−0) cos [2 (∆χij − χin − χjm + ∆χmn)]
+ 2 (Γ00+0 + Γ++0+) cos [2 (χim + χjm + ∆χmn)]
+ 2 (Γ0+0+ + Γ+0+0) cos [2 (∆χij + χin + χjm + ∆χmn)]
+ 2 (Γ0+−0 + Γ+00−) cos [2 (∆χij − χim − χjn + ∆χmn)]
146
+ 2 (Γ000+ + Γ+−+0) cos [2 (χin − χjn + ∆χmn)]
+ 2 (Γ0++0 + Γ+00+) cos [2 (∆χij + χim + χjn + ∆χmn)]
+ 2 (Γ000+ + Γ+++0) cos [2 (χin + χjn + ∆χmn)]
+ 2 (Γ00+0 + Γ+−0−) cos [2 (χim − χjm −∆χmn)]
+ 2 (Γ0+0+ + Γ+0−0) cos [2 (∆χij + χin − χjm −∆χmn)]
+ 2 (Γ00+0 + Γ++0−) cos [2 (χim + χjm −∆χmn)]
+ 2 (Γ0+0− + Γ+0+0) cos [2 (∆χij − χin + χjm −∆χmn)]
+ 2 (Γ0++0 + Γ+00−) cos [2 (∆χij + χim − χjn −∆χmn)]
+ 2 (Γ000+ + Γ+−−0) cos [2 (χin − χjn −∆χmn)]
+ 2 (Γ0+−0 + Γ+00+) cos [2 (∆χij − χim + χjn −∆χmn)]
+ 2 (Γ000+ + Γ++−0) cos [2 (χin + χjn −∆χmn)]) . (7.28)
Here,
∆χ idealij = χij − χij
∆χ = χ − χidealmn mn mn . (7.29)
Plugging in the ideal-case parameters, where α{i,j,m,n},k(τ) = 0, χim = χin = χjm =
χjn = 0, χ = χ
ideal, and χ = χidealij ij mn mn , we indeed get F|| = 1.
The Mathematica notebook for this derivation is available upon request; it
includes the full matrix for Equation 7.20 and the final density matrix ρr.
See also Appendix D.3-D.5 of [108]), which provides a very nice treatment of
147
the fidelity calculation for a 2-ion XX gate.
7.2 Implementation
To find a sequence of pulses that would give a Rabi frequency modulation op-
timally satisfying the above constraints, an optimization scheme was implemented.
The built-in MATLAB unconstrained multivariable optimization function “fmin-
unc” was used, where the objective function included the above constraints on α
and χ parameters, as well as a term to minimize power. (The fidelity for a given
solution was not calculated until after the solution was optimized, as calculating the
fidelity with Equation 7.28 slowed the solution-finding process considerably; simply
minimizing the parameters in question was sufficient.) Sequences were calculated
for a gate time of τgate = 250 µs, which is comparable to the standard 2-qubit XX
gates already used on the experiment (see Section 3.2), and for a range of detunings
µ. This generated a selection of solutions, which were tested on the experimental
setup; the solution generating the highest-quality gate using the least amount of
power was chosen. The quick screening method used to determine comparative gate
quality consisted of running the gate and looking at the populations in the states
|00〉, |01〉, |10〉, and |11〉 for each of the 6 pairs available from the 4 ions in use. The
entangled pairs should show populations at 50% in each of |00〉 and |11〉 with no
population in |01〉 or |10〉, whereas the crosstalk pairs should have 25% population
in each of |00〉, |01〉, |10〉, and |11〉. The amount of odd parity (|01〉 and |10〉) pop-
ulation for the two entangled pairs provided a strong heuristic for gate quality; on
148
the experiment, high quality gates will show odd parity populations of 0-2% on the
entangled pairs (before correcting for SPAM errors.)
Experimental gates were found for 6 ion pair combinations: (1,4) and (2,5);
(1,2) and (3,4); (1,5) and (2,4); (1,4) and (2,3); (1,3) and (2,5); (1,2) and (4,5). For
each set of parallel 2-qubit gates, Figures 7.1-7.6 show the pulse sequence applied to
each entangled pair to construct the gate, as well as the trajectories in phase space
of each mode-pair interaction. Within each figure, the phase space plots are plotted
on axes of the same size, so relative engagement of each mode is shown. The phase
space trajectories start out at the blue open circle and follow the path to end at the
green star. Most phase space trajectories do end up back at the origin (indicating the
corresponding α = 0), but since these solutions were generated with a numerical op-
timization procedure, not all of them are perfect. The 5 transverse motional modes
in this 5 ion chain have sideband frequencies νx = {3.045, 3.027, 3.005, 2.978, 2.946}
MHz, and the plot captions include information on the detuning value used for that
solution; the phase space trajectories for each set show that the mode interactions
closest to the selected detuning exhibit the greatest activity, and contribute the
most to the final spin-spin entanglement by enclosing more of phase space.
Negative-amplitude pulses are implemented by inverting the phase of the con-
trol signal. This capability was crucial, as changing the control signal phase allows
the entangling pairs to continue accumulating entanglement while cancelling out
accumulated entanglement with cross-talk pairs. To that end, the initial guess used
for the gates in the optimization protocol was that one pair would have two pos-
itive “humps” (since regular XX gate pulse shapes frequently featured symmetric
149
increasing and then decreasing segment amplitudes), and the other pair would have
one positive “hump” and one negative “hump,” the idea being the two gates would
perform half of their entanglement process each in their first humps, and then per-
form the second half of the entanglement during the second humps while the relative
change in phase on pair 2 cancelled out the accumulated crosstalk entanglement.
The pulse shapes in Figure 7.1 provide a good example of this; most of the other
pulse solutions feature similar patterns with some kind of symmetry, increasing and
decreasing segment amplitudes, and phase flips on one pair to cancel out crosstalk
entanglement.
7.3 Experimental Results
Here, we present experimental results from implementing parallel 2-qubit en-
tangling gates on several ion pair selections. Fidelities are calculated by performing
the parallel gates followed by an analysis pulse, then using the calculated parity
to determine the fidelity (see Section 7.3.1.) The analysis pulses are rotations us-
ing the SK1 composite pulse for increased robustness against errors in the rotation
angle [109, 110]. For the four ions involved in each gates, the parity analysis was
performed for all 6 possible pairs within the set, allowing for analysis of the 2 en-
tangled ion pairs as well as the 4 crosstalk pairs. Parity curves are shown in Figures
7.7 - 7.12. Entangling gate fidelities were typically between 96-99%, with crosstalk
of a few percent. An exception is the (1,2), (4,5) gate, for which the (4,5) gate has
a fidelity of 91% (Figure 7.12); looking at its phase space closure diagram in Figure
150
Figure 7.1: Pulse shapes and phase space trajectories for parallel XX gates on ions
(1,4) and (2,5), with detuning µ = 2.962 MHz and theoretical fidelity 99.63%.
151
Figure 7.2: Pulse shapes and phase space trajectories for parallel XX gates on ions
(1,2) and (3,4), with detuning µ = 3.016 MHz and theoretical fidelity 99.97%.
152
Figure 7.3: Pulse shapes and phase space trajectories for parallel XX gates on ions
(1,5) and (2,4), with detuning µ = 2.992 MHz and theoretical fidelity 99.75%.
153
Figure 7.4: Pulse shapes and phase space trajectories for parallel XX gates on ions
(1,4) and (2,3), with detuning µ = 2.964 MHz and theoretical fidelity 99.85%.
154
Figure 7.5: Pulse shapes and phase space trajectories for parallel XX gates on ions
(1,3) and (2,5), with detuning µ = 3.036 MHz and theoretical fidelity 98.97%.
155
Figure 7.6: Pulse shapes and phase space trajectories for parallel XX gates on ions
(1,2) and (4,5), with detuning µ = 3.018 MHz and theoretical fidelity 95.91%.
156
Parity Scan (1,4), (2,5)
1
0.8
0.6
0.4
0.2
0
14 data
-0.2 14 fit
25 data
25 fit
-0.4 12 data
12 fit
15 data
-0.6 15 fit
24 data
24 fit
-0.8 45 data
45 fit
-1
0 /4 /2 3 /4 5 /4 3 /2 7 /4 2
Rotation Axis
Figure 7.7: Parity curve for parallel XX gates on ions (1,4) and (2,5), yielding
fidelities of 96.5(4)% and 97.8(3)% on the respective entangled pairs, with an average
crosstalk error of 3.6(3)% and corrected for 3% SPAM errors.
Parity Scan (1,2), (3,4)
1
0.8
0.6
0.4
0.2
0
12 data
-0.2 12 fit
34 data
34 fit
-0.4 13 data
13 fit
14 data
-0.6 14 fit
23 data
23 fit
-0.8 24 data
24 fit
-1
0 /4 /2 3 /4 5 /4 3 /2 7 /4 2
Rotation Axis
Figure 7.8: Parity curve for parallel XX gates on ions (1,2) and (3,4), yielding
fidelities of 98.4(3)% and 97.7(3)% on the respective entangled pairs, with an average
crosstalk error of 0.6(3)% and corrected for 3% SPAM errors.
157
Parity Parity
Parity Scan (1,5), (2,4)
1
0.8
0.6
0.4
0.2
0
15 data
-0.2 15 fit
24 data
24 fit
-0.4 12 data
12 fit
14 data
-0.6 14 fit
25 data
25 fit
-0.8 45 data
45 fit
-1
0 /4 /2 3 /4 5 /4 3 /2 7 /4 2
Rotation Axis
Figure 7.9: Parity curve for parallel XX gates on ions (1,5) and (2,4), yielding
fidelities of 96.8(3)% and 98.1(2)% on the respective entangled pairs, with an average
crosstalk error of 1.7(3)% and corrected for 2% SPAM errors.
Parity Scan (1,4), (2,3)
1
0.8
0.6
0.4
0.2
0
14 data
-0.2 14 fit
23 data
23 fit
-0.4 12 data
12 fit
13 data
-0.6 13 fit
24 data
24 fit
-0.8 34 data
34 fit
-1
0 /4 /2 3 /4 5 /4 3 /2 7 /4 2
Rotation Axis
Figure 7.10: Parity curve for parallel XX gates on ions (1,4) and (2,3), yielding
fidelities of 98.8(3)% and 99.0(3)% on the respective entangled pairs, with an average
crosstalk error of 1.4(3)% and corrected for <1% SPAM errors.
158
Parity Parity
Parity Scan (1,3), (2,5)
1
0.8
0.6
0.4
0.2
0
13 data
-0.2 13 fit
25 data
25 fit
-0.4 12 data
12 fit
15 data
-0.6 15 fit
23 data
23 fit
-0.8 35 data
35 fit
-1
0 /4 /2 3 /4 5 /4 3 /2 7 /4 2
Rotation Axis
Figure 7.11: Parity curve for parallel XX gates on ions (1,3) and (2,5), yielding
fidelities of 98.3(3)% and 97.5(2)% on the respective entangled pairs, with an average
crosstalk error of 0.8(4)% and corrected for 3% SPAM errors.
Parity Scan (1,2), (4,5)
1
0.8
0.6
0.4
0.2
0
12 data
-0.2 12 fit
45 data
45 fit
-0.4 14 data
14 fit
15 data
-0.6 15 fit
24 data
24 fit
-0.8 25 data
25 fit
-1
0 /4 /2 3 /4 5 /4 3 /2 7 /4 2
Rotation Axis
Figure 7.12: Parity curve for parallel XX gates on ions (1,2) and (4,5), yielding
fidelities of 97.2(3)% and 91.9(3)% on the respective entangled pairs, with an average
crosstalk error of 0.9(3)% and corrected for 2% SPAM errors.
159
Parity Parity
Parallel Gate Pairs R||, Pair 1 R||, Pair 2
(1,4) and (2,5) 4.3 1.8
(1,2) and (3,4) 7.9 5.0
(1,5) and (2,4) 2.1 1.6
(1,4) and (2,3) 4.3 3.8
(1,3) and (2,5) 0.9 1.5
(1,2) and (4,5) 2.2 2.2
Table 7.1: For each pair of parallel XX gates implemented, we compare the power
required to perform each component XX with its corresponding stand-alone 2-qubit
XX gate by calculating the power ratio R||.
7.6, however, it is clear that it did not close very well, the likely source of the low
fidelity.
Error bars and fidelity errors are statistical errors. Data has been corrected
for state preparation and measurement (SPAM) errors, as described in [19, 26] and
Section 2.1.4. Crosstalk errors were found by fitting the crosstalk pair parity scan
to a sine curve as if it were a normal parity flop, calculating its fidelity as an
entangling gate, and subtracting out the 25% base fidelity that represents a complete
statistical mixture; any fidelity above that represents a correlation or small amount
of entanglement that is considered an error here. All crosstalk fidelities for all pairs
was well below 50% (in fact, most were quite close to 25%), indicating that no
crosstalk pairs had verifiable entanglement.
While the gate time τgate = 250 µs for running 2 XX gates in parallel is
comparable to that of a single XX gate (and consequently, half the time it would
take to execute two XX gates in series), the parallel gates scheme requires somewhat
more optical power. The Rabi frequency Ω is proportional to the square root of
√
each beam intensity I, Ω ∝ I0I1, where I0 and I1 are the beam intensities for the
160
individual and global beams, respectively. We can therefore calculate the ratio R||
of the power for a gate executed in parallel to the(powe)r required for a single XX
P I Ω 2
gate on the same ions ions as R = || = || = |||| . Since intensity is powerPXX IXX ΩXX
per unit area, and the beam sizes do not not vary, this cancels out. Power ratios
for each gate are shown in Table 7.1. While some gates required rather more power
(for example, we had some trouble finding a solution for (1,2), (3,4) that was both
high-quality and low power), most gates performed in parallel require about two to
four times as much power as their singly-performed counterparts. However, a full
accounting of power requirements on this experiment must also take into account
power wasted by unused beams, and the total time required to perform equivalent
operations. Since the individual addressing system has all individual beams on at
all times and are dumped after the AOM when not in use (see Section 2.3), any
ion not illuminated corresponds to an individual beam wasting power. Running 2
XX gates in parallel takes τgate = 250 µs and uses beams each with power P to
illuminate 4 ions, but performing those same 2 gates in series using stand-alone
XX gates requires time 2τgate and uses 4 beams each with power P/4 to P/2 to
illuminate 2 ions, wasting 2 beams. This yields a choice of using twice the power
(or more) in half the time versus half the power in twice the time, coming out close
to equal cost depending on the gate in question; these parallel gates are then very
useful when faster calculation is a higher priority than minimizing laser power.
161
7.3.1 Calculating Experimental Fidelities of 2-Qubit Entangling Gates
The fidelity of a two-qubit XX(χ) entangling gate can be measured by scan-
ning the phase φ of a global π rotation applied after performing the XX gate and
2
calculating the parity at each point of the scan [58,97,111]. We start with a global
rotation on 2 qubits,
(π ) (π ) (⊗ π )RG , φ = R1 , φ R2 , φ2 2 2 1 −ie− iφ −ie
−iφ −e−2iφ
1 −ieiφ 1 −1 −ie−iφ=  , (7.30)2 −ieiφ −1 1 −ie−iφ
−e2iφ −ieiφ −ieiφ 1
and a general 2-qubit density matrix ρg that represents the density matrix produced
after experimentally performing an XX gate,
 
ρ00 ρ01 ρ02 ρ03ρ∗ 01 ρ11 ρ12 ρ13ρg = ρ∗ ρ∗  , (7.31)02 12 ρ22 ρ23
ρ∗03 ρ
∗ ∗
13 ρ23 ρ33
where ρ00 = |00〉 〈00|, ρ01 = |00〉 (〈01|, ). . ., ρ23 = |10〉 〈11|, ρ33 = |11〉 〈11|.( Aft)er
performing the analysis pulse R πG , φ , the new density matrix ρa = R
π
G , φ ·2 2
162
† ( )ρ ·R πg G , φ is used to calculate the parity:2
( ) ( )
Π (ρa, φ) = ρ
00 + ρ33a a − ρ11a + ρ22a
= 2A12 cosφ12 − 2A03 cos(2φ− φ03)
= 2A12 cosφ12 − AΠ cos(2φ− φ03), (7.32)
where parity is defined as the sum of the even parity populations minus the sum of
the odd parity populations and the coherences (off-diagonal density matrix elements)
from ρ are re-written in the form ρ = A e−iφxyg xy xy . Let us also define the parity
amplitude AΠ ≡ 2A03.
Now we calculate the fidelity of an XX(χ) gate. Using the XX(χ) gate unitary
(see Equation 5.3), we construct the ideal density matrix after an XX(χ) gate,
ρideal = XX(χ) · |00〉 〈00| ·XX(χ)
†
  cos
2(χ) 0 0 i cos(χ) sin(χ)
 0 0 0 0=  . (7.33)0 0 0 0
−i cos(χ) sin(χ) 0 0 sin2(χ)
The fidelity of the general fidelity matrix ρg with respect to the ideal fidelity matrix
ρideal is given by [ ]
F (χ) = Tr ρideal(χ) · ρg · ρ†ideal(χ) . (7.34)
163
Plugging in equations 7.31 and 7.33, using AΠ ≡ 2A03, and simplifying yields
F (χ) = ρ 200 cos (χ) + ρ33 sin
2(χ) + AΠ cos(χ) sin(χ) (7.35)
as the fidelity of an XX(χ) gate. Specifically for maximally entangling gates, we
plug in χ = π and get
4
( π) 1 1
F χ = = (ρ00 + ρ33) + AΠ. (7.36)
4 2 2
While ρ00 and ρ33 are simply the populations in |00〉 and |11〉 respectively after
an XX gate, we still need the AΠ term. We can extract this from a parity scan
using Equation 7.32. Given a perfect XX(χ) gate where A12 = 0, φ03 = −fracπ2,
and AΠ = 2A03 = 2 cos(χ) sin(χ) (from Equation 7.33), scanning the analysis phase
φ from 0 to 2π and measuring the parity at each point will yield a sine curve
of amplitude 2 cos(χ) sin(χ) with 2 periods in the range from 0 to 2π. (For a fully
entangling XX(χ = π ) gate, the sine curve should have amplitude 1.) Consequently,
4
by fitting a sine curve to thie measured parity curve, we can estimate the parity
amplitude AΠ and use it in Equation 7.35 to calculate the gate fidelity.
7.3.2 Fidelity of Parallel 2-Qubit Entangling Gates with Different
Degrees of Entanglement
Since the XX gates in this parallelization scheme have independent calibration
(see Section 7.3.3), the χ parameters of the two XX gates are independent. The
164
1
0.8
0.6
0.4
0.2
0
15 data
-0.2 15 fit
24 data
24 fit
-0.4 12 data
12 fit
14 data
-0.6 14 fit
25 data
25 fit
-0.8 45 data
45 fit
-1
0 /4 /2 3 /4 5 /4 3 /2 7 /4 2
Rotation Axis
Figur(e 7).13: Parity curve for parallel XX(χ) gates on(ion)s (1,5) and (2,4), where an
XX π gate is performed on ions (1,5), and an XX π gate is performed on ions
4 8
(2,4). This yields fidelities of 96.4(3)% and 99.4(3)% on the respective entangled
pairs, with an average crosstalk error of 2.2(3)% and corrected for 1% SPAM errors.
continuously-variable parameter χ is directly related to the amount of entanglement
generated between the two qubits, given by
XX (χ) |00〉 = √1 (cos (χ) |00〉 − i sin (χ) |11〉) , (7.37)
2
and can be adjusted on the experiment by scaling the power of the overall gate.
Consequently, we can simultaneously implement two XX gates with different de-
grees of entanglement, which may prove useful for some applications. For example,
the full a(dd)er implementation in Section 7.5 will requir(e s)imultaneously performing
an XX π gate on one pair of qubits, and an XX π gate on another pair of
4 8
qubits. To demonstrate this capability, Figure 7.13 shows parity scan data for a
165
Parity
( ) ( )
simultaneous XX π gate on ions (1,5) and an XX π gate on ions (2,4). The
4 8
data is analyzed as in Section 7.3, but while we use Equation 7.36 (setting χ = π )
4
to calculate the fidelity for the (1,5) gate, we use Equation 7.35 and set χ = π for
8
the (2,4) gate. The respective gate fidelities are therefore 96.4(3)% and 99.4(3)%,
with an average crosstalk error of 2.2(3)% and corrected for 1% SPAM errors.
7.3.3 Independence of Parallel Gate Calibration
Parallel gates can be calibrated independently from one another by adjusting
a scaling factor that controls the overall power on the gate without modifying the
pulse shape. Furthermore, adjusting a scaling factor that controls the power on
a single ion only affects the gate in which it participates by modifying the total
amount of entanglement, without any apparent ill effects on the gate quality. This
was confirmed experimentally using parallel operations on ions (1,2) and (3,4) by
scanning over the scaling factors associated with ions 1 and 2. Figure 7.14 shows
several such scans over the scaling factors for ions 1 and 2 while keeping the (3,4)
gate “on”, with the scaling factor for those two ions set near to a fully-entangling
gate; Figures 7.14(a,c) show scans with just the scaling factor for ion 1 while holding
the scale factor for ion 2 constant, and Figures 7.14(b,d) show scans over the scaling
factor for ions 1 and 2 together. Figure 7.15 shows scans over the scaling factors for
ions 1 and 2 while keeping the interaction on (3,4) “off”; the scaling factor for this
gate is set to 0, so the ions see no light and therefore perform no interaction during
the gate. Figure 7.15(a) scans the scale factor just on ion 2 while holding the scale
166
(a) (b)
Ions (3,4) on Ions (3,4) on
1 1
(1,2) 00 (1,2) 00
0.9 (3,4) 00 0.9 (3,4) 00
(1,2) 01 (1,2) 01
0.8 (3,4) 01 0.8 (3,4) 01
(1,2) 10 (1,2) 10
0.7 (3,4) 10 0.7 (3,4) 10
(1,2) 11 (1,2) 11
0.6 (3,4) 11 0.6 (3,4) 11
0.5 0.5
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Scale Factor for ion 1 Scale Factor for ions (1,2)
(c) (d)
Ions (3,4) on (zoom) Ions (3,4) on (zoom)
1 1
(1,2) 00 (1,2) 00
0.9 (3,4) 00 0.9 (3,4) 00
(1,2) 01 (1,2) 01
0.8 (3,4) 01 0.8 (3,4) 01
(1,2) 10 (1,2) 10
0.7 (3,4) 10 0.7 (3,4) 10
(1,2) 11 (1,2) 11
0.6 (3,4) 11 0.6 (3,4) 11
0.5 0.5
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0 0
0.7 0.75 0.8 0.85 0.7 0.75 0.8 0.85
Scale Factor for ion 1 Scale Factor for ions (1,2)
Figure 7.14: Parallel gates can be calibrated independently. Here, we perform an
entangling gate on ions (3,4) while simultaneously performing a gate on ions (1,2)
and scanning a scale factor, which determines the overall amplitude envelope on the
control signal applied to perform the gate on ions (1,2). This indicates we can inde-
pendently set the amount of entanglement created by each gate when performed in
parallel. (a) Scanning the scale factor on ion 1, with ions (3,4) performing an entan-
gling gate. (b) Scanning the scale factor on ions (1,2), with ions (3,4) performing an
entangling gate. (c) Scanning the scale factor on ion 1, with ions (3,4) performing
an entangling gate, with higher resolution zoomed in close to full entanglement for
ions (1,2). (d) Scanning the scale factor on ions (1,2), with ions (3,4) performing
an entangling gate, with higher resolution zoomed in close to full entanglement for
ions (1,2).
167
Population Population
Population Population
(a) (b)
Ions (3,4) off Ions (3,4) off
1 1
(1,2) 00 (1,2) 00
0.9 (3,4) 00 0.9 (3,4) 00
(1,2) 01 (1,2) 01
0.8 (3,4) 01 0.8 (3,4) 01
(1,2) 10 (1,2) 10
0.7 (3,4) 10 0.7 (3,4) 10
(1,2) 11 (1,2) 11
0.6 (3,4) 11 0.6 (3,4) 11
0.5 0.5
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0 0
0.7 0.75 0.8 0.85 0.7 0.72 0.74 0.76 0.78 0.8 0.82
Scale Factor for ion 2 Scale Factor for ions (1,2)
Figure 7.15: Parallel gates can be calibrated independently. (a) Scanning the scale
factor on ion 2, with no light on ions (3,4). (b) Scanning the scale factor on ions
(1,2), with no light on ions (3,4).
on ion 1 constant, and Figure 7.15(b) scans the overall scaling factor for ions 1 and 2
together. For all of these scans, as the scaling factors are increased, the population in
|11〉 for ions 1 and 2 increases (and the population in |00〉 decreases correspondingly),
while the |00〉 and |11〉 populations for the (3,4) gate remain unchanged.
7.4 Simultaneous CNOT Gates
As an example application of a parallel operation, we performed a pair of
CNOT gates in parallel on two pairs of ions. The CNOT gate sequence (compiled
version with R and XX gates shown in Figure 5.2(a)) was performed simultaneously
on the pair (1, 4), with ion 1 acting as the control and ion 4 acting as the target,
and on the pair (2, 3), with ion 2 acting as the control and ion 3 acting as the target.
Each constituent operation in the composite gate was performed in parallel, with
each rotation performed at the same time as the corresponding rotation on the other
pair, and the two XX gates performed using parallel XX gates on ion pairs (1, 4)
168
Population
Population
1
1 0.8
0.8
0.6 0.6
0.4
0.2 0.4
0
0000
0100
1101111
0.2
1000 0
1100 10000100 0
Detected 1111 0000 Input
Figure 7.16: Data for simultaneous CNOT gates on ions (1,4) and (2,3), with an
average process fidelity of 94.5(2)% and corrected for average SPAM errors of 5%.
and (2, 3) (see Figure 7.10 for fidelity data.)
The simultaneous CNOT gates were performed for each of the 16 possible
bitwise inputs, and population data for the 16 possible bitwise outputs is shown in
Figure 7.16 with an average process fidelity of 94.5(2)% and corrected for average
SPAM errors of 5%.
7.5 The Quantum Full Adder
The ability to add numbers is fundamental to classical computers, and indeed
was one of the first motivations for computing machines; arguably the very first
computing machine, the abacus, was a tool for adding and subtracting numbers. In
modern computing, a full adder is a basic circuit that can be cascaded to add many-
bit numbers, and can be found in processors as components of arithmetic logic units
169
Probability
(a) Feynman adder (b) Maslov adder
x • • x x • • x
y • • • y′ y • • y′
C • S Cin • • Sin
0 C 0 V V V V
† Cout
out
(c) Optimized adder circuit
x Rz(−3π ) R (πy )4 2
XX(π )
4
y Rz(−π ) Ry(π ) XX(π ) R π π4 2 8 y(− ) Rz(− )2 2
C π π πin Rz( ) Ry( ) R (− )4 2 z 2
XX(π )
8
0 Rx(−π ) XX(π )2 8
· · · XX(π ) Ry(−π )8 2 x
· · · R (πy )2 Rz(π) y′
XX(π )
4
· · · R π πy( ) Ry(− ) R (π )2 2 z 4 S
XX(−π )
8
· · · XX(π )8 Cout
Figure 7.17: (a) Feynman quantum full adder [112]. (b) Maslov adder with 2-
qubit gate depth 4 [82]. (c) Application-optimized full adder implementation using
XX(χ), Rx(θ), and Ry(θ) gates. The two parallel 2-qubit operations are outlined
in dashed boxes for both (b) and (c).
170
Figure 7.18: Data for full adder using sumiltaneous 2-qubit gates on ions (1,2,4,5),
with an average process fidelity of 83.3(3)% and corrected for average SPAM errors
of 3%.
(ALU’s) and performing low-level operations like computing register addresses.
A cascadable full adder is one that takes 3 inputs - two bits x and y you wish
to add, plus a carry bit Cin stemming from a previous addition - and has 2 outputs
that communicate the 2-bit value of the sum of the two inputs. To build a quantum
adder, one must design a circuit that accomplishes these goals while being reversible.
Feynman first designed such a circuit using CNOT and Toffoli gates [112], shown
in Figure 7.17(a).
However, with 5 2-qubit interactions per Toffoli meaning that this full adder
would require 12 XX gates to implement on an ion trap quantum computer, this
circuit is not efficient. A more efficient circuit requiring at most 6 2-qubit inter-
actions was shown in [82], one which has the further advantage of being reduced
to a gate depth of 4 if simultaneous two-qubit operations are available, as shown
by the dashed outlines in Figure 7.17(b). Like the Feynman circuit, the quantum
171
full adder requires 4 qubits, 3 for the inputs x, y, and the carry bit Cin, and the
fourth a qubit initialized to |0〉. The four outputs consist of the first input, x, simply
carrying through; y′, which carries x ⊕ y (an additional CNOT can be added to
extract y if desired); and the sum S and output carry Cout, which together comprise
the 2-bit result of summing x. y, and Cin, where Cout is the most significant bit and
hence becomes the carry bit to the next adder in the cascade, and S is the least
significant bit. We can also write sum as S = x⊕ y ⊕ Cin and the output carry as
Cout = (x · y)⊕ (Cin · (x⊕ y)).
The optimized full adder circuit to be implemented on the experiment, shown
in Figure 7.17(c), is constructed by combining the CNOT , C(V ), and C(V †) gates
from Figure 5.12 and further optimizing the rotations per the method described in
Section 5.2.1. The two parallel 2-qubit operations are outlined in dashed boxes for
Figures 7.17(b-c).
The full adder was implemented using 2 different parallel XX gate configu-
rations, as well as the rotations and additional XX gates shown in Figure 7.17(c).
The inputs x, y, Cin, and 0 were mapped to the qubits (1, 2, 4, 5) respectiv(ely).
Consequently, (the) first parallel gates were implem(en)ted on the p(air)s (1,2) XX
π
4
and (4,5) XX π (see Figure 7.12 for (1,2) XX π , (4,5) XX π fi(de)lity data),8 4 4
and the π( )second parallel gate was implem( e)nted on the p(ai)rs (1,5) XX and (2,4)4
XX π (see Figu(re )7.9 for (1,5) XX
π , (2,4) XX π( ) fidelity data and Figure8 4 4
7.13 for (1,5) XX π , (2,4) XX π fidelity data.) Figure 7.18 shows the resulting
4 8
data from implementing this algorithm, with all 8 possible bitwise inputs on the 3
input qubits, and displaying the populations in all 16 possible bitwise outputs on
172
the 4 qubits used. The data yielded an average process fidelity of 83.3(3)%, and was
corrected for average SPAM errors of 3%.
7.6 Toward a Single-Operation GHZ State
This control scheme for parallel 2-qubit entangling gates in ions also suggests
a method for performing multi-qubit entanglement in a single operation. Of partic-
ular interest is the creation of GHZ states [113], which are a class of non-biseparable
maximally-entangled multi-qubit states. To calculate parallel XX gates, as dis-
cussed in Section 7.1, we set the spin-spin interaction terms χ = π for the two
4
desired entangling interactions, and to 0 for the remaining 4 crosstalk interactions.
173
However, setting all 6 spin-spin interaction terms to χ = π yields the unitary
4
( )
U ideal
π
GHZ = U α{i,j,m,n},k = 0, χij = χim = χin = χjm = χjn = χmn = 4 
 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −i 0 1 0 0 0 0 0 0 0 0 0 0 0 0 −i 0  0 0 1 0 0 0 0 0 0 0 0 0 0 −i 0 0   0 0 0 1 0 0 0 0 0 0 0 0 −i 0 0 0 

0 0 0 0 1 0 0 0 0 0 0 −i 0 0 0 0 
  0 0 0 0 0 1 0 0 0 0 −i 0 0 0 0 0  0 0 0 0 0 0 1 0 0 −i 0 0 0 0 0 0  √1  0 0 0 0 0 0 0 1 −i 0 0 0 0 0 0 0 =  2  0 0 0 0 0 0 0 −i 1 0 0 0 0 0 0 0 

0 0 0 0 0 0 −i 0 0 1 0 0 0 0 0 0 
  0 0 0 0 0 −i 0 0 0 0 1 0 0 0 0 0 

 0 0 0 0 −i 0 0 0 0 0 0 1 0 0 0 0  0 0 0 −i 0 0 0 0 0 0 0 0 1 0 0 0   0 0 −i 0 0 0 0 0 0 0 0 0 0 1 0 0 0 −i 0 0 0 0 0 0 0 0 0 0 0 0 1 0 
−i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
(7.38)
and adding one Z rotation produces a 4-qubit GHZ state:
(
1 π
)
R · U idealz GHZ |
1
0000〉 = √ (|0000〉+ |1111〉) . (7.39)
2 2
Following a similar derivation as in Section 7.1, we therefore calculate the
objective fidelity function to be
( )
FGHZ α{i,j,m,n},k, χ
ideal, χideal, χideal ideal ideal idealij im in , χjm , χjn , χmn , χij, χim, χin, χjm, χjn, χmn =
1
(8 + Γ+−−− + Γ+−−+ + Γ+−+− + Γ+−++ + Γ++−− + Γ++−+ + Γ+++− + Γ++++
128
174
+ 2 (Γ0+++ + Γ+000) cos [2 (∆χij + ∆χim + ∆χin)]
+ 2 (Γ0+−+ + Γ+000) cos [2 (∆χij −∆χim + ∆χin)]
+ 2 (Γ0++− + Γ+000) cos [2 (∆χij + ∆χim −∆χin)]
+ 2 (Γ0+−− + Γ+000) cos [2 (∆χij −∆χim −∆χin)]
+ 2 (Γ0+00 + Γ+0++) cos [2 (∆χij + ∆χjm + ∆χjn)]
+ 2 (Γ00++ + Γ++00) cos [2 (∆χim + ∆χin + ∆χjm + ∆χjn)]
+ 2 (Γ0+00 + Γ+0−+) cos [2 (∆χij −∆χjm + ∆χjn)]
+ 2 (Γ00+− + Γ+−00) cos [2 (∆χim −∆χin −∆χjm + ∆χjn)]
+ 2 (Γ0+00 + Γ+0+−) cos [2 (∆χij + ∆χjm −∆χjn)]
+ 2 (Γ00+− + Γ++00) cos [2 (∆χim −∆χin + ∆χjm −∆χjn)]
+ 2 (Γ0+00 + Γ+0−−) cos [2 (∆χij −∆χjm −∆χjn)]
+ 2 (Γ00++ + Γ+−00) cos [2 (∆χim + ∆χin −∆χjm −∆χjn)]
+ 2 (Γ00+0 + Γ++0+) cos [2 (∆χim + ∆χjm + ∆χmn)]
+ 2 (Γ0+0+ + Γ+0+0) cos [2 (∆χij + ∆χin + ∆χjm + ∆χmn)]
+ 2 (Γ00+0 + Γ+−0+) cos [2 (∆χim −∆χjm + ∆χmn)]
+ 2 (Γ0+0− + Γ+0−0) cos [2 (∆χij −∆χin −∆χjm + ∆χmn)]
+ 2 (Γ0++0 + Γ+00+) cos [2 (∆χij + ∆χim + ∆χjn + ∆χmn)]
+ 2 (Γ000+ + Γ+++0) cos [2 (∆χin + ∆χjn + ∆χmn)]
+ 2 (Γ0+−0 + Γ+00−) cos [2 (∆χij −∆χim −∆χjn + ∆χmn)]
+ 2 (Γ000+ + Γ+−+0) cos [2 (∆χin −∆χjn + ∆χmn)]
175
+ 2 (Γ00+0 + Γ++0−) cos [2 (∆χim + ∆χjm −∆χmn)]
+ 2 (Γ0+0− + Γ+0+0) cos [2 (∆χij −∆χin + ∆χjm −∆χmn)]
+ 2 (Γ00+0 + Γ+−0−) cos [2 (∆χim −∆χjm −∆χmn)]
+ 2 (Γ0+0+ + Γ+0−0) cos [2 (∆χij + ∆χin −∆χjm −∆χmn)]
+ 2 (Γ0+−0 + Γ+00+) cos [2 (∆χij −∆χim + ∆χjn −∆χmn)]
+ 2 (Γ000+ + Γ++−0) cos [2 (∆χin + ∆χjn −∆χmn)]
+ 2 (Γ0++0 + Γ+00−) cos [2 (∆χij + ∆χim −∆χjn −∆χmn)]
+ 2 (Γ000+ + 2Γ+−−0) cos [2 (∆χin −∆χjn −∆χmn)]) , (7.40)
where
∆χ = χ − χidealij ij ij
∆χ = χ − χidealmn mn mn
∆χ = χ − χidealim im im
∆χin = χ − χidealin in
∆χjm = χjm − χidealjm
∆χjn = χ
ideal
jn − χjn . (7.41)
This indicates we may be able to use the same optimization approach to produce
pulse shapes that will create GHZ states when applied to the ions. Unlike with
parallel gates, however, it may be necessary to allow independent pulse shapes on
all 4 ions, rather than solving for pairwise solutions; this will provide more free pa-
176
rameters. Additional challenges will include finding effective calibration techniques
when implementing such gates on the experiment, since there will be 6 interactions
that will all need to be at the same strength, but only 4 control signals. Our current
approach of calibrating a 2-qubit gate by adjusting the overall power for the pulse
shape applied by the control signal may no longer work; new techniques with more
degrees of freedom may be needed, such as independently adjusting the power for
different segments of the pulse shape on each ion.
The benefits of implementing GHZ states with fewer gates would be significant,
as it would substantially reduce the circuit depth of several important algorithms.
With only 2-qubit gates available, building a GHZ state of size N requires O(N)
2-qubit gates. With parallel 2-qubit gates available, the gate depth required to
build a GHZ state is reduced to O(log(N)); this is accomplished with a binary tree
algorithm by dividing all qubits into pairs and entangling those pairs in parallel,
then entangling pairs of these pairs, and so on until all are entangled. A single-
operation GHZ state would drop this circuit depth to unity. Single-operation GHZ
state construction will greatly enhance the efficiency of several algorithms; for exam-
2
ple, arbitrary stabilizer circuits require O( N ) CNOT gates [114], but could be
log(N)
implemented in O(N) gates with single-operation GHZ state circuitry [115]. Single-
operation GHZ state creation will also benefit applications such as quantum secret
sharing [116], Toffoli-N gates, the quantum Fourier transform, and quantum Fourier
adder circuits [115].
177
7.7 Outlook
The scaling outlook on simultaneous gates is polynomial or better in the num-
ber of constraints to consider when calculating optimal solutions. As discussed in
Section 7.1, two parallel XX gates in a chain of N ions requires 4N+6 ∼ O(N) con-
straints, so the problem growth is linear in N . Entangling more pairs at once grows
quadratically: entangling M pairs involves the interactions of 2M ions, yielding the
number of spin-spin interactions we must control to be
( )
2M (2M)!
= = 2M2 −M ∼ O(M2) (7.42)
2 2!(2M − 2)!
and the number of spin-motion interactions to be the number of ions times the
number of modes, 2MN . Scaling both the number of entangled pairs M and the
number of ions N in the chain therefore gives a total problem growth rate of
2MN + 2M2 −M = M(2N − 1) + 2M2 ∼ O(M2 +MN). (7.43)
The work presented here represents a successful first attempt at implementing
parallel 2-qubit entangling operations in a chain of trapped ions, and further ex-
ploration will likely produce improved results. In particular, future research could
investigate better optimization techniques to produce solutions that have higher fi-
delities and require lower power on the experiment. Since the solutions are found
via optimization rather than analytically, the theoretical fidelities for these gates are
178
less than 1; most of the gates presented here have theoretical fidelities <99.9%, so
optimization approaches increasing the theoretical fidelities to ≥99.99% will help.
An additional problem is that testing the possible solutions generated by the op-
timization techniques is a time-consuming process; better systematics to pinpoint
the most promising candidates may help here. Techniques that further suppress
entanglement crosstalk between undesired pairs may also enhance experimental per-
formance. However, these problems are all ones of overhead. Once a high-quality
gate solution is implemented on the experiment, no further calculations are needed;
only a single calibration is required to determine the overall power on each XX gate
as the Rabi frequencies seen by the ions drift.
Several areas of exploration may help with increasing solution fidelity, both
theoretically and experimentally. One is measuring and using the experimental
Lamb-Dicke parameter ηi,k for each spin-mode coupling, rather than calculating
what they should be based on the ion spacings and measured motionial modes. The
trap in our experiment has some anharmonicities that cause asymmetries in the
motional mode eigenvectors, so the participation of each ion in each mode is a little
different from what one would theoretically expect. Calculating gate solutions with
experimentally measured parameters may result in solutions that perform better
on the experiment. Another parameter of interest is the gate time τ ; while we
settled on 250 µs as one comparable to our existing XX gate times, adjusting that
parameter - theoretically and on the experiment - may prove beneficial. Easing
constraints on the power needed may also allow for higher-fidelity solutions to be
calculated, although increasing power on the experiment can exacerbate errors due
179
to Stark shifts; an accounting of tradeoffs between power usage and gate fidelity will
optimize operational success given experimental conditions.
While the gate solution for N = 5 ions in principle has 4N + 6 = 26 degrees
of freedom that need to be solved (per Section 7.1), since solutions were found
via optimization rather than analytically, we provided the optimization program
additional degrees of freedom to increase the probability of finding good solutions;
most of the gate solutions shown here use 60 segments per pair, although the (1,4),
(2,3) optimization was performed with only 40 segments to no apparent ill effects.
Further explorations of this technique may benefit from determining the optimal
number of segments to use. A final line of future inquiry could include investigating
whether the constraint matricies in Equation 7.12 are in fact positive or negative
semidefinite, or can be modified to have such properties, as convex QCQP’s are
readily solved using semidefinite programming techniques and could allow for higher-
fidelity solutions.
180
Chapter 8: Outlook
Trapped ion systems are a promising candidate for constructing a large-scale
quantum computer. Here, we have reported results for several algorithms on a fully
programmable machine with modular gates, and exhibiting a stack architecture in
the control system that will be imperative for effective control of larger machines.
With high initialization and detection efficiencies, well-understood coherent controls,
and nature-provided, stable qubit frequencies providing long coherence times, scaling
up to larger systems has a positive outlook. While much work certainly remains to
be done, the biggest challenges can largely be tackled with engineering solutions.
8.1 Experimental Error Sources
The measured process fidelities for results discussed in this thesis are reduced
from the theoretical ideal fidelities primarily through technical imperfections in the
experimental system. The predominant source of error is beam pointing instability
on the individual Raman beams, causing laser intensity fluctuations at the ions. This
results in small random coherent errors, as the effective Rabi frequency seen by each
ion drifts randomly during the course of an experiment. Causes for this instability
181
include air currents through the free-space Raman optical setup and phase noise on
the laser itself. Future work on the experiment will include installing a piezo mirror
controlled by a quadrant photodiode to actively stabilize the power and position of
the individual addressing beams.
Another source of error is crosstalk between individual ion controls. While the
individual Raman beams are very tightly focused, abberations cause some spillover,
so a given ion will see small amounts of light from its neighbors’ control beams.
Additionally, there is also crosstalk between neighboring channels at the 32-channel
AOM. Overall, the ions typically experience ∼2% nearest-neighbor crosstalk from
their coherent optical controls, and ≤1% from the next nearest neighbors [20]. Other
error sources include inhomogeneous Stark shifts across the ion chain that could not
be perfectly compensated. These control problems can largely be solved through
improved engineering of key components.
8.2 Scalability of Ion Traps
Continuing challenges will include building apparatuses with more ions per
trap that can execute long sequences of high-fidelity quantum gates. More qubits will
be needed to implement algorithms larger than can be simulated classically. While
error correction schemes provide protection against experimental errors, baseline
gate fidelities will need to be higher than we have demonstrated here to achieve
fault tolerant computation.
Schemes to implement entangling gates in chains of a few hundred ions have
182
been proposed [117], although full connectivity will not be possible at such scaling;
co-desgning algorithms with available connection architecture will mitigate the costs
associated with reduced connectivity. Experimentally, improvements in multi-zone,
microfabricated ion trap technology provide control over tens of ions in a single
trap, and ongoing efforts in chip trap technology at Sandia National Labs, MIT
Lincoln Labs, and elsewhere will likely scale to hundreds of ions in future. Vacuum
chambers cooled to low temperatures using cryostats can operate at significantly
lower pressure than room-temperature ion traps, allowing for longer chains to be
held in a trap for longer periods of time before collisions with background gas disrupt
the crystal.
Fidelity gains can be achieved through improvements in the quantum control
hardware. Other experiments have shown up to four 9’s of fidelity for single-qubit
gates and up to three 9’s of fidelity for two-qubit gates in ions [118, 119]. However,
these were performed with only one or two ions in the trap; achieving these kinds
of fidelities for larger chains will present a significant engineering challenge. As
discussed in Section 8.1, a major source of experimental errors on our experiment
is noise on the coherent addressing beams. Optical intensity stabilization will be
imperative to achieving long, high-fidelity gate sequences, and can likely be achieved
with integrated photonics, minimal use of free-space optics, active stabilization of
the source power, and other engineering improvements. Coherence times for ions can
be extended using techniques like dynamic decoupling [120] and additional magnetic
shielding.
Further scalability may be achieved by connecting multiple ion trap modules
183
through ion shuttling [121] or photonic interconnects [6, 7]. In the latter scheme,
each qubit register would constitute an elementary logic unit (ELU), and consist of a
vacuum chamber containing an ion trap holding a single chain of ions, much like the
one discussed in this work. Many such ELU modules could be connected remotely
through coherent photon exchange over fibers. The qubits within a single register
would perform computations together, much like the bits of an ELU in a classical
computer. Quantum information that needs to be used in another register would
be imparted from an atomic qubit to a photon, and the photon sent to another
chamber to implement remote entanglement. The ion-photon information transfer
could occur via a different ion species that emits photons at frequencies more suitable
for fiber communication [122]. Optical switches would permit connections between
all qubit registers. The modularity of this design would enable scalability, allowing
for far bigger quantum computers than can be achieved with a single ion trap.
184
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