ABSTRACT
Title of Dissertation: QUANTUM CONTROL AND MEASUREMENT
ON FLUXONIUMS
Haonan Xiong, Doctor of Philosophy, 2022
Dissertation Directed by: Professor Vladimir E. Manucharyan
Department of Physics
Superconducting circuit is a promising platform for quantum computing and quantum
simulation. A number of efforts have been made to explore the physics in transmon systems
and optimize the qubit performance. Compared to transmon, fluxonium is a relatively
new type of qubit and attracts more attention recently due to its high coherence time and
large anharmonicity. In this thesis, we summarize recent progress toward high fidelity two-
qubit gate and readout for fluxonium qubits. We report improved fluxonium coherence
either in cavity or cavityless environment. In the former case, we demonstrate single-shot
joint readout for two fluxonium qubits and explore various two-qubit gate schemes such
as controlled-Z(CZ) gate, controlled-phase(CP) gate, bSWAP gate and cross-resonance(CR)
gate. The CZ gate realized by near-resonantly driving the high transitions exhibits 99.2%
fidelity from randomized benchmarking. A continuous CP gate set can be implemented by
off-resonantly driving the high transitions and shows an average 99.2% fidelity from the
cross-entropy benchmarking technique. Other gates involving only computational states
are also explored to further improve the gate fidelity, which can take advantage of the high
coherence of the fluxonium lower levels. In the cavityless environment, we demonstrate
fluorescence shelving readout with 1.7 MHz radiative decay rate for the readout transition
while maintaining 52 ?s coherence time for the qubit transition. Our research explores the
basic elements for fluxonium-based quantum processors. The results suggest that fluxonium
can be an excellent candidate for not only universal quantum computation but also quantum
network and quantum optics studies.
QUANTUM CONTROL AND MEASUREMENT ON FLUXONIUM
by
Haonan Xiong
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
2022
Dissertation Committee:
Professor Vladimir E. Manucharyan, Chair
Professor Sankar Das Sarma
Professor Jay Deep Sau
Professor Benjamin S. Palmer
Professor Ichiro Takeuchi, Dean?s Representative
? Copyright by
Haonan Xiong
2022
ii
Acknowledgement
Pursuing a PhD for five years in a foreign land is never an easy journey. Fortunately, I was
surrounded by a group of people who care about me and supported me through the hard
times. Words are not enough to repay the kindness from them, which I know is priceless.
But right now at the final stage of my graduate study, I still thank all of you for making this
day come true.
I first thank my mom and dad, Dongmei and Qinglin, for their unconditional love and
support all the time. Even though they had a different opinion on my future career when
I chose to study physics in college nine years ago, they were always with me in no time
whenever I needed their help. They make me feel there is a place in this world called home
which is waiting for me wherever I go, with hot food and a comfy bed.
I would like to thank my advisor, Vladimir Manucharyan, without whom my accom-
plishments of all these years will be never possible. I still remember when I sent him my
derivation of a JPA circuit in my first year, he sent me back his derivation immediately on a
small piece of paper he managed to find nearby. I can also recall the time we spent together
soldering the skinny cables and testing the copper powders, not to mention the numerous
meetings on my projects. When a visitor came, he introduced me as ?the next generation?
and indeed, I learned a lot from him, not only the scientific knowledge and experimental
experiences, but also the ability to present the work. He tirelessly gave me feedbacks on
the slides of my talk and the figures in the paper. Thanks to the training of all these years,
I am much more professional on visualize the data and convey the ideas efficiently on a
iii
presentation. Vlad also gave me lots of suggestions when I was having trouble finding the
committee members. Besides, I express my gratitude to other members of the committee:
Professors Das Sarma, Sau, Palmer, and Takeuchi, for fitting my defense into their busy
schedule.
When I first came to the lab, Prof. Yen-Hsiang Lin, who was a postdoc at that time,
became my closest friend. Nothing can be more comforting than working with someone
speaking the native language during my first year in a foreign country. Yen-Hsiang is warm-
hearted, patient, and encouraging. I felt no pressure when discussing projects with him
and I could ask any stupid questions. He always explained to me nicely and leaded me
onto the right track. He was very experienced and taught me all most everything in the lab,
including the basics of fluxoniums, the measurement setup, designing cavities and waveg-
uides .... I am afraid I will run out of spaces if I list every details here. Yen-Hsiang was
also cared about my well-being and willing to offer any help and suggestions in my life.
Together, we improved the fluorescence experiment and got it published. This project was
launched by Dr. Nathanae?l Cottet. Nat left the lab before I came in. It is a pity that I do
not have the chance to work with him in person in the lab. But his experiment summary
was so well-organized that I could easily understand the basic principles. Nat and I had
several meetings when finalizing the work. Even though it is a short period of time, I was
still impressed by his patience and creativeness.
Dr. Long Nguyen was a senior graduate student when I joined the lab. He loves to com-
municate with others and soon we became friends. He had many good ideas and showed
me many good references for the decoherence and the gate. He was always ready to help
and gave us lots of suggestions in the fluorescence project. Later, he taught me how to
characterize the gate and handed over the two-qubit experiment to me. In addition, he was
also a humorous person and made the lab a fun place to stay.
I am grateful to Dr. Quentin Ficheux who worked closely with me in all the two-qubit
projects. Quentin is very knowledgeable both in theory and experiments. He had an agile
iv
mind. Whenever I was confused about a concept he brought, he said, ?OK, let me show
it to you?. Then he just started to derive on the while board without any hesitation and
preparation. He was an excellent example of a professional researcher and at the same time,
he was a good teacher with great pedagogical skill. He taught me many good experiment
habits. ?Whenever you see anything you don?t understand, you should try your best to
figure it out. Otherwise, it will eventually affect your experiment, in a way that you may
not expect or notice.? This is one of the most important lessons that I learned from him.
Dr. Ivan Pechenezhskiy was the one who guided me into the world of python. Before
I entered the lab, I rarely coded with python. With Ivan?s help, I quickly mastered it. We
spent many days writing and debugging the drivers for the instruments. We had many
fruitful discussions and I got familiar with professional coding and how the instruments
work and communicate. Thanks to Ivan, the whole lab has well-written python scripts to
use in daily experiments.
Unfortunately I am running out of spaces for my thanks to the group members. I also
would like to thank Dr. Roman Kuzmin for the help in fridge maintenance. I am thankful
to Dr. Aaron Somoroff for patiently teaching me all kinds of fab techniques, from silicon to
sapphire. I thank Raymond Mencia for always sharing with me the news in latest research.
I appreciate the help of Dr. Nitish Mehta and sharing the instruments with me. I express my
gratitude to Dr. Nicholas Grabon, Dr. Natalia Pankratova, and Dr. Hanho Lee for their help
in my fabrication and various little stuff in the lab. Wei-Ju Lin, Shukai Liu, and Dr. Arunav
Bordoloi are new members in the lab. I want to thank Wei-Ju in particular, for his timely
help on the experiments with no complaints while I was busy for graduation.
I appreciate the support from our collaborators. Dr. Konstantin Nesterov shared with
me his code and notes in two-qubit simulation. Prof. Maxim Vavilov gave me many good
suggestions in the gate projects. Prof. Chen Wang and his students Ebru Dogan and Dario
Rosenstock contributed in the gate project and shared with me their experimental skills
which are quite helpful. I thank you all for the discussions we had.
v
I want to thank other faculty members at UMD who offered help during my time here. I
thank Prof. Christopher Lobb for introducing me to Vlad and this wonderful team. I thank
Prof. Frederick Wellstood who gave me lots of advice on my scholarly paper. I thank Prof.
Benjamin Palmer for sharing their device with us. I also thank Prof. Johnpierre Paglione for
sharing their magnet data with us to debug our experiments.
I express my thank you to other labs who hosted my visits. I thank Prof. Michel Devoret
for his generous invitation and insightful discussions. I thank Prof. Andreas Wallraff for
showing me all the wonderful projects and acknowledging my ability. I thank Prof. Xiaobo
Zhu for introducing their experiments. I thank Dr. Haifeng Yu for inviting me to the new
labs. I also thank Dr. Fei Yan for hosting me at SIQSE.
When I first went to college, I had no idea about research. Prof. Chuangang Ning
showed me the way to entering the field of scientific research. After that, I joined Prof.
Luming Duan?s team. During the two years in the lab, Dr. Yipu Song, Dr. Hongyi Zhang,
and Prof. Luyan Sun introduced me to superconducting circuit community. During this
time, I had a short visit to Prof. Hong Tang?s group at Yale and spent a meaningful summer
with Wance, Mohan, Hanyu, Changling and Dr. Chang-Ling. I give my thanks to all of you
for your help and advice.
I would also like to thank all my old friends. I thank Wentao, Wei, Zhixin, Xueyue,
Xinghan, ... I was always looking forward to the gathering at the March meeting. I thank
Yu for being a close friend accompanying me starting from a small town in China all the
way to America. I thank Min for the suggestion and help in college and in my graduate
study. I thank Hsiang-Sheng and Junling for sharing their experience and opinion on the
industry.
It will be boring if I spent the five years without my friends at College Park. I thank
Zhiyu, Mingshu, Yihang, Yixu, Yalun, and Xiangyu for the fantastic parties and the supports
in my life. I also thank my roommates Shuyang, Kaixin, Haining, Fei,and Yizhou for facing
the problems together that we encountered in the house. I am grateful to Zhuo, Zeyu, and
vi
Peng for the entertainments in my holidays. I thank Mr. Jinlong Han, Riots, and Bilibili for
their wonderful products as my moral support.
Most importantly, I want to say thank you to my partner Tong. I appreciate your patience
over the five years and I am sincerely grateful to your support and understanding to help
me get through the difficult times. You also gave me lots of great advice and led to my
reflection on myself. You are the love of my life, forever and ever.
vii
Table of Contents
Page
ACKNOWLEDGEMENT ii
TABLE OF CONTENTS vii
LIST OF FIGURES x
LIST OF TABLES xiii
NOMENCLATURE xiv
1 INTRODUCTION 1
1.1 Quantum Computation and Simulation . . . . . . . . . . . . . . . . . . . 1
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 SUPERCONDUCTING CIRCUIT MODELS 5
2.1 Josephson Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Resonator, Transmon and Fluxonium . . . . . . . . . . . . . . . . . . . . . 8
2.3 Dispersive Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Coupled Fluxoniums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Energy Relaxation and Dephasing . . . . . . . . . . . . . . . . . . . . . . . 24
2.5.1 Energy relaxation mechanisms . . . . . . . . . . . . . . . . . . . . . 27
2.5.2 Dephasing mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 31
viii
2.6 Fluxonium in a Cavityless Environment . . . . . . . . . . . . . . . . . . . 33
3 EXPERIMENTS ON READOUT AND INITIALIZATION 37
3.1 Readout schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.1 Dispersive Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.2 Fluorescence readout . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 non-QNDness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2.1 non-QNDness for dispersive readout . . . . . . . . . . . . . . . . . . 48
3.2.2 non-QNDness for fluorescence readout . . . . . . . . . . . . . . . . . 52
3.3 Noise and Quantum Limited Amplifiers . . . . . . . . . . . . . . . . . . . 53
3.3.1 Added Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.2 JPA and JTWPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4 Initialization Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4.1 Preselection initialization . . . . . . . . . . . . . . . . . . . . . . . . 62
3.4.2 Cavity tone initialization . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.4.3 Initialization in the fluorescence experiment . . . . . . . . . . . . . . 65
3.4.4 Other fluxonium initialization schemes . . . . . . . . . . . . . . . . . 65
4 MICROWAVE GATE EXPERIMENTS WITH FLUXONIUM 67
4.1 Characterization techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.1.1 Quantum process tomography . . . . . . . . . . . . . . . . . . . . . . 68
4.1.2 Randomized benchmarking . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Single-qubit Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2.1 X and Y gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2.2 ac Stark shift and virtual Z gates . . . . . . . . . . . . . . . . . . . . 80
4.2.3 Cross-talk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3 Two-qubit Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.3.1 Off-resonance CPhase gate . . . . . . . . . . . . . . . . . . . . . . . . 86
ix
4.3.2 Near-resonance CZ gate . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3.3 bSWAP gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.3.4 Cross-resonance gate . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.4 Parameter Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5 SUMMARY 110
5.1 Fluorescence readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.2 Two-qubit gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
APPENDICES 114
A SIMULATION 115
A.1 Transmission simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
A.2 Coupling simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
A.3 Fluxonium simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
B EXPERIMENT TECHNIQUES 122
B.1 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
B.2 3D cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
B.3 Cryogenic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
B.4 Room temperature setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
C DERIVATIONS 129
C.1 Signal Gain for JPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
C.2 Schrieffer-Wolf Transformation . . . . . . . . . . . . . . . . . . . . . . . . 132
BIBLIOGRAPHY 135
x
List of Figures
Figure 2.1 Circuit diagrams and energy spectrum for a resonator, transmon and
fluxonium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Figure 2.2 Optical and SEM images for a single fluxonium . . . . . . . . . . . . 13
Figure 2.3 Spectrum and matrix elements for a fluxonium . . . . . . . . . . . . 15
Figure 2.4 A fluxonium capacitively coupled to a resonator . . . . . . . . . . . . 17
Figure 2.5 Two fluxoniums coupled with a capacitor . . . . . . . . . . . . . . . . 21
Figure 2.6 T1, Ramsey and spin echo measurements . . . . . . . . . . . . . . . . 26
Figure 2.7 A fluxonium in a waveguide . . . . . . . . . . . . . . . . . . . . . . . 34
Figure 2.8 Optical image of a fluxonium inside a waveguide . . . . . . . . . . . 35
Figure 3.1 Two-port cavity transmission . . . . . . . . . . . . . . . . . . . . . . . 39
Figure 3.2 Joint readout for two qubits in the same cavity . . . . . . . . . . . . . 42
Figure 3.3 Single-shot histograms . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Figure 3.4 Measured port coupling (solid black) vs. frequency and the calculated
transition dipoles (colored stars) . . . . . . . . . . . . . . . . . . . . . . . . 45
Figure 3.5 Conditional fluorescence readout . . . . . . . . . . . . . . . . . . . . 47
Figure 3.6 Non-QND process of the cavity readout pulse . . . . . . . . . . . . . 49
Figure 3.7 Rabi oscillations with and without readout errors correction . . . . . 51
Figure 3.8 Transient dynamics of fluorescence . . . . . . . . . . . . . . . . . . . 54
Figure 3.9 Circuit diagram of a JPA galvanically connected to a transmission line 58
Figure 3.10 Phase amplitude ?a at different pump frequency . . . . . . . . . . . . 59
xi
Figure 3.11 Signal gain Gs at different pump frequency and power . . . . . . . . 60
Figure 3.12 Signal gain Gs at different signal frequency and pump power . . . . 61
Figure 3.13 Initialization by preselection . . . . . . . . . . . . . . . . . . . . . . . 63
Figure 3.14 Initialization using the cavity pulse Non-QNDness . . . . . . . . . . . 64
Figure 3.15 Qubit population initialization by pumping |1?-|3? . . . . . . . . . . . 65
Figure 3.16 Other initialization diagrams . . . . . . . . . . . . . . . . . . . . . . . 66
Figure 4.1 ? matrix in quantum process tomography of a CZ gate . . . . . . . . 72
Figure 4.2 Different benchmarking techniques based on random Clifford gates . 75
Figure 4.3 Single-qubit gate RB for a two-qubit device . . . . . . . . . . . . . . . 84
Figure 4.4 Differential ac Stark shift in a two-fluxonium circuit . . . . . . . . . . 88
Figure 4.5 Tuning the ZZ-interaction . . . . . . . . . . . . . . . . . . . . . . . . 90
Figure 4.6 ZZ-interactions induced by driving around 6.5 GHz . . . . . . . . . . 91
Figure 4.7 Interleaved randomized benchmarking (IRB) for CZ gates . . . . . . 92
Figure 4.8 Cross-entropy benchmarking for arbitrary CPhase gates . . . . . . . . 93
Figure 4.9 Measured ZZ rates when applying the drive near the qubit transitions 94
Figure 4.10 Gate principle of the near-resonance CZ gate . . . . . . . . . . . . . . 96
Figure 4.11 IRB of different number of CZ gates . . . . . . . . . . . . . . . . . . . 98
Figure 4.12 Two-tone spectroscopy of the |00? ? |11? two-photon transition under
different drive strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Figure 4.13 Rabi oscillations of the |00? ? |11? two-photon transition . . . . . . . 101
Figure 4.14 Rotating frame transformation for bSWAP gate . . . . . . . . . . . . 103
Figure 4.15 bSWAP Ramsey measurements . . . . . . . . . . . . . . . . . . . . . 105
Figure 4.16 Relative fluctuation between two different RF sources . . . . . . . . . 106
Figure 4.17 Gate principle for a cross-resonance gate . . . . . . . . . . . . . . . . 107
Figure A.1 Waveguide model in HFSS . . . . . . . . . . . . . . . . . . . . . . . . 116
Figure A.2 S parameters simulation of a waveguide . . . . . . . . . . . . . . . . 116
xii
Figure A.3 Domino bandpass filter model in HFSS . . . . . . . . . . . . . . . . . 117
Figure A.4 Domino bandpass filter S parameters . . . . . . . . . . . . . . . . . . 117
Figure A.5 Lumped inductance in HFSS . . . . . . . . . . . . . . . . . . . . . . . 119
Figure A.6 Simulation of qubit capacitance and coupling strength to the cavity . 119
Figure A.7 Simulation of radiative decay rate of a qubit inside a waveguide . . . 120
Figure B.1 Low temperature wiring diagram of the two qubit experiment . . . . 125
Figure B.2 Thermal photon number simulation . . . . . . . . . . . . . . . . . . . 126
Figure B.3 Homemade Eccosorb filter . . . . . . . . . . . . . . . . . . . . . . . . 126
Figure B.4 Room temperature readout signal demodulation schemes . . . . . . . 128
xiii
List of Tables
Table 3.1 Thermal photon noise at different stages . . . . . . . . . . . . . . . . . 56
Table 4.1 Preparation and tomography pulses used for two-qubit quantum pro-
cess tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Table 4.2 Single-qubit Clifford gates . . . . . . . . . . . . . . . . . . . . . . . . . 109
Table 4.3 Energy relaxation time T1, Ramsey coherence time TR2 , and spin echo
coherence time TE2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
xiv
Nomenclatures
Abbreviations
AlOx Aluminum Oxide
AWG Arbitrary waveform generator
BOE Buffered Oxide Etch
CNC Computer numerical control
CNOT Controlled-NOT gate
CP, CPhase Controlled-Phase
CQED Cavity quantum electrodynamics
cQED Circuit quantum electrodynamics
CR Cross-resonance
CZ Controlled-Z
DRAG Derivative removal by adiabatic gate
EOM Equation of motion
IPA Isopropanol
xv
IQ In-phase, Quadrature
IRB Interleaved randomized benchmarking
JBA Josephson bifurcation amplifier
JC Jaynes-Cummings
JJ Josephson junction
JPA Josephson parametric amplifier
JPC Josephson parametric converter
JTWPA Josephson traveling wave parametric amplifier
LO Local oscillator
MLE Maximum likelihood estimation
NISQ Noisy intermediate-scale quantum
PB Purity benchmarking
QEC Quantum error correction
QND Quantum non-demolition
QPT Quantum process tomography
QST Quantum state tomography
RB Randomized benchmarking
RF Radio frequency
RWA Rotating wave approximation
xvi
SEM Scanning electron microscope
SMA SubMiniature version A
SNR Signal to noise ratio
SPAM State preparation and measurement
SQUID Superconducting quantum interference device
SSB Single sideband
TLS Two-level system
VNA Vector network analyzer
XEB Cross-entropy benchmarking
Greek Letters
? Anharmonicity
? Dispersive shift
?Al Superconducting energy gap of aluminum (Al), 1.8? 10?4 eV
? Quantum efficiency
? Relaxation rate
? Coupling/relaxation rate of cavity
? Rabi frequency
?p Josephson junction plasma frequency
??? Qubit transition frequency between states ? and ?
xvii
? Magnetic flux threading a loop
? Phase operator
? Density matrix
?I , ?0 Pauli identity matrix
?X , ?1 Pauli X matrix
?Y , ?2 Pauli Y matrix
?Z , ?3 Pauli Z matrix
Latin Letters
a?, a Creation and annihilation operators
C Capacitance
D Oscillator displacement operator
EC Electric charging energy
EJ Josephson energy
EL Magnetic inductive energy
G Gain
g Coupling constant
H Hamiltonian
I Pauli identity matrix
Ic Junction critical current
xviii
JC Capacitive coupling constant
L Inductance
Li Lindblad operators
LJ Josephson inductance
Me Error matrix of the readout
n Charge number operator
nth Thermal photon number
P Population
Q Quality factor
Qext External quality factor
Qin Internal quality factor
r Reflection coefficient
RN Normal resistance of a Josephson junction
S(?) Noise spectral density
Sij S parameter of a microwave network
T1 Relaxation time of qubit
T2 Coherence time of qubit
T echo2 Coherence time of qubit obtained from echo measurement
TRamsey2 Coherence time of qubit obtained from Ramsey measurement
xix
T? Pure dephasing time
Tc Critical temperature
X Pauli X matrix
xqp Quasiparticle density
Y Pauli Y matrix
Y (?) Admittance
Z Pauli Z matrix
Physical Constants
? Reduced Planck constant, 1.054? 10?34 J ? s
?0 Magnetic flux quantum, 2.067? 10?15T ?m2
?0 Reduced magnetic flux quantum, 0.3291? 10?15T ?m2
e Electronic charge, 1.602? 10?19C
h Planck constant, 6.626? 10?34 J ? s
kB Boltzmann constant, 1.38? 10?23 J/K
Citations to Published Work
Parts of Chapters 3, 4, and 5 have appeared in the following papers:
N. Cottet, H. Xiong, L. B. Nguyen, Y.-H. Lin, and V. E. Manucharyan, ?Electron shelving
of a superconducting artificial atom,? Nature Communications, vol. 12, p. 6383, 12 2021.
Q. Ficheux, L. B. Nguyen, A. Somoroff, H. Xiong, K. N. Nesterov, M. G. Vavilov, and V.
E. Manucharyan, ?Fast Logic with Slow Qubits: Microwave-Activated Controlled-Z Gate on
Low-Frequency Fluxoniums,? Physical Review X, vol. 11, p. 021026, 5 2021
H. Xiong, Q. Ficheux, A. Somoroff, L. B. Nguyen, E. Dogan, D. Rosenstock, C. Wang,
K. N. Nesterov, M. G. Vavilov, and V. E. Manucharyan, ?Arbitrary controlled-phase gate on
fluxonium qubits using differential ac Stark shifts,? Physical Review Research, vol. 4, p.
023040, 4 2022.

1
Chapter 1
Introduction
1.1 Quantum Computation and Simulation
Richard Feynman proposed the concept of a quantum computer at a conference on ?Sim-
ulating physics with computers? in 1981 [1]. He believes that a quantum system is too
hard to simulate for a classical computer, but it is affordable with quantum computer ele-
ments. Before this conference, Paul Benioff already studied a quantum model of the Turing
machine [2]. But Feynman described the quantum computer as
It?s not a Turing machine, but a machine of a different kind.
One of the main reason is that he thinks there are too many variables that define the status
of a large quantum system. This idea was shared by Yuri Manin
The quantum phase space is much bigger than classical.
So actually the original proposal of quantum computer is for quantum simulation tasks. It is
more like a quantum simulator, rather than a computer. The idea is to study (simulate) large
scale complex quantum systems with much fewer quantum elements (atoms, for example)
or at least in a controllable way such that the predominant interaction is maintained and
easy to debug.
Years after, David Deutsch, Umesh Vazirani, and Ethan Bernstein developed Feynman?s
idea of quantum computation. It was proposed that a quantum computer can solve prob-
lems that have nothing to do with quantum systems and it can have a speedup on some tasks
over a classical computer. Then Peter Shor came up with the famous algorithm for factoring
large numbers with a exponential speedup. Due to its close relation to cryptography, the
concept of quantum computation began its public fame.
2
At the same time, what a quantum computer should be like becomes clear and specific. It
should base on fundamentally quantum mechanical objects (qubits) and be able to perform
any algorithm written in the gate and circuit language. This machine is called a universal
quantum computer, which is the ultimate goal for this thesis. As Feynman already pointed
out, this is not an easy problem. The biggest obstacle in this field has been, and will still be,
decoherence in a quantum system. Two outstanding atomic physicists Serge Haroche and
Jean-Michel Raimond said in 1996
The large-scale quantum machine, though it may be the computer scientist?s dream,
is the experimenter?s nightmare.
Decoherence brings errors on the qubit. Its time scale determines the lifetime of quantum
information stored in the system. On the other hand, decoherence originates from the inter-
action between the qubit and the environment, which is an interesting topic itself. Figuring
out where the interactions come from and how to engineer them is crucial to understand
the microscopic world, which can also inspire new research directions in material science
or even cosmology.
One solution to fight against decoherence is quantum error correction. It requires a
large number of qubits, and more importantly, operation fidelity above a certain threshold.
The idea is to use multiple physical qubits to construct a logical qubit that has a much lower
error rate. Google published their 54-qubit quantum processor Sycamore in 2019 on which
they demonstrated quantum supremacy [3]. Unfortunately, it only showed advantage over
a specific task and its practical benefit is obscure. It can not efficiently run any advantageous
quantum algorithms due to the error rate on the gates. Following experiments [4] showed
the possibility to build logical qubits on the same device. Even though the total physical
qubit error rate is still above the threshold required by the surface code, there is still hope
that we can cross the line in the near future. This work pointed out two main limiting
factors on the current superconducting qubit device ? readout and two-qubit gate. These
two aspects will be the main topic of this thesis.
Nevertheless, Google?s experiment of quantum supremacy is a successful demonstration
that the current noisy quantum hardware is able to outperform classical computer in certain
situations, which sets a milestone of the Noisy Intermediate-Scale Quantum (NISQ) era. It
is not clear yet how a quantum processor without error correction can benefit us. At least,
it provides a new tool to study quantum systems. For example, NISQ devices can already
accomplish simulation tasks such as simulating energy spectrum of various systems [5]?[7].
These type of quantum simulators are closer to what Feynman imagines. The NISQ devices
do not look so powerful at this stage. However, as many competing efforts are devoted to
3
investigating the possibilities across different physical systems, any breakthroughs or new
ideas may be able to trigger an revolution that has a profound impact upon the society.
In principle, any quantum system with well-defined distinct two levels can be the poten-
tial building block of a quantum computer, i.e. qubit. The two levels can be the microscopic
degree of freedom, such as the spin or orbital states of the electron. They can also be the
collective modes, such as the electron wavefunction of a superconductor. If we use universe
to describe all the possible qubits in nature, there are some stars that are brighter than the
others after years of research. Superconducting qubits are defined by the states of the super-
conducting wavefunction and are the focus of this thesis. Ion trap systems confine ions in
vacuum through electromagnetic field. Progress has been reported in achieving high-fidelity
gates and scaling up the system [8]. The coherent photonic state of a microwave resonator
is another type of macroscopic qubit and has been successfully used to exhibit unique quan-
tum error correction techniques [9]?[11]. On the other hand, travelling photon in free
space can also be harnessed for quantum computing, but it belongs to another architecture
called the measurement-based quantum computing [12]. Recently, high-fidelity two-qubit
gates were reported on the silicon spin qubit system, pushing it further towards the logical
qubit era, which attracts more attention in this field.
1.2 Outline
The rest of this dissertation is organized as follows. In chapter 2, we introduce the circuit
models in circuit quantum electrodynamics, especially for the fluxonium. We derive the key
properties of fluxonium in different environments and briefly review the energy relaxation
and dephasing mechanisms in superconducting systems. We then present the first part
of the experimental results on the qubit readout and initialization in chapter 3. We will
discuss two different readout schemes. One is dispersive readout with a 3D cavity. The
other is fluorescence readout in a 3D waveguide, which we call a cavityless environment.
We will explain the concept of non-QNDness and show the corresponding data for both
cases. In chapter 4, we will present the experiments on microwave gates with fluxonium.
We begin with introducing the gate characterization techniques, including quantum process
tomography and randomized benchmarking based techniques (IRB, PB, XEB). Then we will
explain the principles of the single-qubit and two-qubit gates and show the experimental
data. In the end, we will conclude all the experiments in chapter 5. We will emphasize the
significance and provide insights for future research.
In this thesis, we will use the following symbols for Pauli matrices {?0, ?1, ?2, ?3}
4
= {?I , ?x, ?y, ?z}(= {I,X), Y, Z} a(nd ) ( ) ( )
1 0 1 0 0 1 0 ?i
?0 = , ?1 = , ?2 = , ?3 = . (1.1)
0 1 0 ?1 1 0 i 0
5
Chapter 2
Superconducting Circuit Models
In this chapter, we will introduce the basic concepts and models in circuit quantum elec-
trodynamics (cQED)[13]?[15]. Starting from the observation of coherent quantum super-
position and oscillation in superconducting circuits [16], [17], it has been realized that
the coherent electron wavefunction of the superconductor can be used to create a macro-
scopic quantum object, thanks to the Bose-Einstein condensation of the Cooper pairs. One
key ingredient to make superconducting circuits nontrivial compared to the conventional
circuits is the nonlinearity given by the Josephson junction, which will be introduced in
Sec. 2.1. Without nonlinearity there is no way to create a quantum superposition and the
system can always be described classically [9]?[11]. Josephson junction is the core for all
the superconducting qubits such that they can be an artificial atom. The invention of circuit
quantization technique [18]?[20] provides a useful tool to study cQED systems. It becomes
straightforward to design new qubit structures and understand how it behaves. There used
to be three types of superconducting qubits [20], the charge qubit [17], the flux qubit [21]
and the phase qubit [22]. After years of improvement and exploration, the superconducting
qubits are experiencing an evolution and modern types of qubits are realized experimen-
tally during the process including transmon [23], fluxonium [24], C-shunt flux qubit [25],
0?? qubit [26], Blochnium [27], etc. These new qubits come with the abilities to suppress
the noise from the environment and the environment itself is engineered in a more sophis-
ticated way than ever. We will introduce fluxonium as the main topic in the cQED context
and discuss the interaction with the environment and in the presence of another fluxonium.
6
2.1 Josephson Junction
Josephson junction (JJ), first introduced in [28], is the essential building block of any type
of superconducting qubits. It is a trilayer structure created by two superconducting layer
sandwiching an insulating layer. For most superconducting qubits, the junction is made of
Al/AlOx/Al. This is the simplest way to make Josephson junctions and it turns out to be
quite robust. Aluminum is not necessarily the best material for making Josephson junc-
tions. One criteria is the critical temperature Tc. Even though superconductor has zero dc
resistance at T ? Tc, it exhibits non-zero ac losses and surface resistance at finite tem-
perature [15], [29]. This is because electrons can be thermally excited from the Cooper
pair state with a binding energy 2? = 3.52kBTc and become ?quasiparticles?. These quasi-
particles add a loss channel in the superconducting system, which will also be discussed
in Sec. 2.5.1. Therefore, a higher Tc means lower resistive loss and will be beneficial for
the qubit lifetime. For example, tantalum (Tc = 4.48 K) and niobium (Tc = 9.5 K) have
higher Tc than aluminum (Tc = 1.14 K). However, aluminum still has the advantage in
terms of the fabrication feasibility such as the stability of the oxide layer and the uniformity
of the edges. On the other hand, the property of the interface between the metal and the
substrate (the air or the oxide layer) is also important, as the defects in these interfaces
can be detrimental to the qubit lifetime. So one need to consider multiple aspects and it is
not straightforward to judge whether a material is better than the other without any tests.
There are ongoing efforts to introduce new materials in the superconducting circuit system,
which has always been a hot topic [30]?[38].
The most useful property of a Josephson junction as a circuit element is relations be-
tween the phase difference across the junction ?, the junction current I and the junction
voltage V . They can be described by the Josephson equations
I = Ic sin?, (2.1)
d?
V = ?0 , (2.2)
dt
where ?0 = ?0/2? = ?/2e is the reduce flux quantum. (?0 is the flux quantum.) Combining
the two equations, we can have
?0 dI
V = . (2.3)
Ic cos? dt
This is similar to the property of a classical inductor V = LJ(dI/dt), where
?0
LJ = . (2.4)
Ic cos?
7
Therefore, the Josephson junction can be treated as a non-linear inductor classically. This
feature will be utilized when we derive the gain of a Josephson parametric amplifier (JPA)
in Sec. 3.3.
For a classical linear inductor, the voltage equals the changing rate of the flux ? thread-
ing the loop
d?
V = . (2.5)
dt
So the phase difference in the context of Josephson junction is equivalent to the reduced
flux in the context of the loop inductance ? = 2??/?0. This is not surprising because the
flux and phase difference are linked with the Aharonov-Bohm effect [39]. This is why ? is
still called the reduced flux in some cases where the Josephson junction is not connected to
a loop and there is actually no well-defined physical ?flux? in the system.
The inductance of a classical linear inductor comes from the fact that the energy in the
circuit can be converted to the magnetic field, which is determined by the wire geometry.
This type of inductance is called the geometric inductance. However, the inductance of a
Josephson junction LJ can be much larger considering its small geometry. This is because
the source of LJ is the inertia of the electrons at the junction and it is called kinetic induc-
tance which has a different origin from the inductance of a coil. The existence of kinetic
inductance makes it possible that a large inductance can be created without fabricating a
huge coil on the chip.
The energy of a linear inductor is HL = E 2L? /2, where E 2L = ?0/L is called the induc-
tive energy. The energy of a Josephson junction is HJ = ?EJ cos? (droping a constant),
where EJ = ?20/LJ is the Josephson energy. EJ is also related to the normal resistance RN
of the junction at room temperature EJ = ? 2AlRQ/(2RN ), where RQ = h/(2e) is the resis-
tance quantum. RN and LJ is inversely proportional to the junction area A. Therefore we
have EJ ? A. There also exists a small capacitance CJ between the two superconducting
layer of the junction and the energy associated with it is HC = 4ECn2, where n is the charge
number and E 2C = e /(2CJ) is the charging energy of the junction. For a Al/AlOx/Al junc-
tion, the empirical value for CJ per area is about 45 fF/?m2. Another important property
of a Josephson junction is the plasma frequency
1? 1
?p = 8EJEC = ? . (2.6)? LJCJ
From the definition, we can find that ?p does not depend on the junction and only depends
on the transparency of the insulating layer. For Al/AlOx/Al junctions, the oxidation process
is the most important factor. Typically its plasma frequency can be 10?40 GHz [40]. In our
8
experiment, ?p is usually around 20 GHz.
2.2 Resonator, Transmon and Fluxonium
There are three major elements in cQED: capacitor, inductor, and Josephson junction, which
can be characterized with the charging energy EC , the inductive energy and the Josephson
energy. Different combinations of these three elements can create the common circuit mod-
els: a resonator, a transmon, or a fluxonium.
A resonator is a capacitor shunted by an inductor as show in Fig. 2.1. In the experiment,
it can be the resonant mode of a metal structure. For example, the lowest mode of a on-chip
metal strip (2D coplanar waveguide cavity), the lowest mode of a copper box (3D cavity),
or a piece of metal pad (capacitor) connected with a Josephson junction chain (inductor)
[27], [41] can all be treated as a resonator. Its Hamiltonian can be written as
1
Hresonator = 4E n
2 + E ?2C L . (2.7)
2
Now n and ? are canonically conjugate quantum operators that satisfy the canonical com-
mutation relation [?, n] = i. n and ? are just like the momentum p and the position x
operator of a mass on the spring. 4E n2C is the kinetic energy and 12E
2
L? is the potential
energy. A resonator is essentially a harmonic oscillator. We can introduce the annihilation
and creation operators a and a? in the second quantization formalism.
( ) 1
4
? = ?1 8EC (a?( ) + a), (2.8)2 EL 1
i EL 4
n = ? (a? ? a). (2.9)
2 8EC
Eq. 2.8 and Eq. 2.9 are important when numerically simulating a arbitrary circuit Hamilto-
nian in the harmonic oscillator basis (using the Qutip python package for example). With
these equations, the Hamiltonian can be written as
1
Hresonator = ??(a?a+ ), (2.10)
2
? ?
where ? = 8ELEC/? = 1/ LC is the resonant frequency. The energy of the eigenstates
is En = ??(n+1/2), n=0, 1, 2, .... The vacuum state energy E0 = ??/2 will conventionally
be offset to 0 for simplicity in the following text. As shown in Fig. 2.1d, because of the
quadratic potential given by the inductor, the resonator spectrum is equally spaced. We can
9
define the anharmonicity of the system using the first three levels ? = (E2?E1)?(E1?E0),
which is obviously zero for a resonator. For this reason, a resonator itself can not be used as
a qubit. There is no way one can prevent the population leaking to higher two levels without
introducing a non-linear element in the circuit. If a drive with frequency ? is applied to the
resonator, it can be imagined that the population will keep climbing the equally-spaced
level ladder. Strictly speaking, for any linear classical drive, the evolution of the resonator
is described by the displacement operator D(?) = exp(?|?|2) exp(?a?) exp(??a). ? is a
complex number determined by the drive parameters. Applying the displacement operator
to the vacuum state, we can get a coherent state [42]
|?? = D(?) |0? ? (2.11)?n
= exp(?|?|2) ? |n? . (2.12)
n n!
The coherent state is the eigenstate of the annihilation operator a |?? = ? |?? and the mean
photon number is n? = ??| a?a |?? = |?|2. The parameters of the drive can only determines
the mean photon number and the phase of the coherent state, but can not change the fact
that the final state is a classical-like superposition of a series of Fock states and can not be
a qubit. This problem can only be solved after introducing a non-linearity by coupling the
resonator to a qubit. A cat state, a superposition of coherent states, can be created and used
as a qubit. Several error correction schemes have been explore in this type of system.
Now if we connect a Josephson junction in parallel with a large capacitor (which is
usually a large piece of metal pad), we can create a transmon. It is worth noticing that in
this case the extra capacitance is much larger than the capacitance of the junction itself,
that is, C ? CJ . The charging energy is dominated by the extra capacitance EC = e2/(2C).
In order to be in the transmon regime, EJ should be much larger than EC . This is necessary
to suppress the charge noise. In general, the transmon Hamiltonian can be written as
H 2transmon = 4EC(n? ng) ? EJ cos?, (2.13)
where ng is the offset charge on the Cooper pair island (which is the metal pad connecting
to the junction). ng can fluctuate due to the quasiparticle tunneling. In the case of large
EC (small capacitance), the energy spectrum has a strong dispersion on the offset charge.
Tunneling events can significantly change the energy levels and cause decoherence. With a
small EC , ng barely affect the energy and can be set at 0 in the spectrum analysis. But the
suppression of charge noise comes with a price. The system anharmonicity also becomes
smaller. This can be more clear in the second quantization language. First, we need to
10
expand cos? in Taylor series and substitute Eq. 2.8 and Eq. 2.9 into the Hamiltonian. Af-
ter dropping the fast rotating terms and the constants, the transmon Hamiltonian can be
written as
? ??
H = ?(? + )a?a+ (a?a)2transmon . (2.14)
2 2
?
where ? = 8EJEC/?, ? = ?EC is the anharmonicity. For a typical transmon, ?/2? is on
the order of 100 or 200 MHz [43] (the 0? 1 frequency is on the order of 5 GHz).
The existence of the anharmonicity ? makes it possible to use the transmon 0? 1 tran-
sition as a qubit and ? sets a limit on how fast we can perform a gate on this qubit. For
example, the Fourier spectrum of a Gaussian ? pulse can not have a line width exceeding
|?|. Otherwise the population can be excited to the higher levels. Therefore in the time
domain, the width of the ? pulse should be longer than 1/|?|. 100 MHz anharmonicity
corresponds to a time scale of 10 ns and this is the typical single-qubit gate time for trans-
mon qubits on state-of-the-art superconducting quantum processors. One way to suppress
the leakage at large drive strength is the Derivative Removal by Adiabatic Gate (DRAG)
technique. By adding an additional derivative component on the orthogonal quadrature of
the control pulse, the Fourier spectrum of the pulse can be engineered to avoid the leakage
transition. The pulse can be more adiabatic and there will be less leakage at the end of the
pulse.
Now let?s combine all the three circuit elements and connect them in parallel. In this
way, a fluxonium can be created and it is the main topic of this thesis. Fig. 2.2 shows
the design of a fluxonium. The fluxonium has a pair of antenna to couple to the external
environment, which also provides the charging energy. The inductance is provided by the
long Josephson junction array. Typically the number of junctions in the chain N is on the
order of 100. The junction size is much larger than the small junction on top of the loop.
The area ratio is on the order of 50, meaning that each junction has a very large Josephson
energy EchainJ . Because the phase across each junction in the chain is ?/N ? 1, the energy
contribution from the chain is
?N ? (N ( ) )2
? ? 1 ?Echain cos ? ?EchainJ J 1? (2.15)N 2 N
i i
chain
= ?NEchain 1E+ J ?2J . (2.16)2 N
Eq. 2.16 demonstrates that in the limit of small phase difference across each junction,
the Josephson junction chain behaves like a linear inductor with inductive energy EL =
EchainJ /N . One reason for large chain junctions is to suppress the phase slip which can
11
a b c
d e f
Figure 2.1: Circuit diagrams and energy spectrum for a resonator, transmon and flux-
onium. (a) A resonator. It can be modeled as an inductor and a capacitor connected in
parallel. (b) A transmon. The Josephson junction behaves like a non-linear inductor and
makes the system anharmonic. (c) A fluxonium. The Josephson junction is shunted by a
large linear inductor. This forms a loop where an external flux can be applied to tune the
circuit Hamiltonian. (d, e, f) The potential energy and energy spectra for a typical res-
onator, transmon, and a fluxonium at half flux quanta. The resonator has equally spaced
spectrum. The transmon spectrum has smaller spacing between higher levels due to the
weak anharmonicity. On the contrary, fluxonium system can have much larger anharmonic-
ity that comes from the non-trivial potential.
12
?
cause decoherence. This requires exp(? 8Echain chainJ /EC ) ? 1 [40].
A long chain with hundreds of junctions may look difficult to fabricate. However, it
turns out the yield is quite high and the recipe is as easy as making a single junction. The
junction chain is deposited together with the small Josephson junction in the same double-
angle deposition process and no extra steps are needed. Yet, one need to avoid a chain that
is too long, because the chain modes can be lower down to the GHz range and have a strong
interaction with the typical fluxonium spectrum [44], [45]. Using the Josephson junction
chain as the linear inductor is not the only choice. In fact, there have been some studies
exploring the fluxonium fabricated by replacing the junction chain with granular aluminum
[46] or geometric inductor [47].
The junction chain forms a closed loop with the small Josephson junction. The external
magnetic flux ?ext threading the loop corresponds to an extra phase difference ?ext =
2??ext/?0. The fluxonium Hamiltonian can hence be expressed as
1
Hfluxonium = 4E n
2
C ? EJ cos(?? ? 2ext) + EL? . (2.17)
2
Compared to the transmon Hamiltonian, the potential in a fluxonium Hamiltonian is more
complicated and can be tuned with the external flux. The potential has a quadratic en-
velope coming from the inductor and multiple small wells coming from the cos term of
the small Josephson junction that sit inside the parabola. At integer flux quanta (?ext =
...,?2?, 0, 2?, ...), the wavefunctions of |0? and |1? are localized in the same well. The tran-
sition of these type of states is called the plasmon transition. On the other hand, the inter-
well transition is called the fluxon transition. At half flux quanta (?ext = ...,??, ?, 3?...),
the potential has a double-well shape as shown in Fig. 2.1f. The local wavefunctions of the
two wells get hybridized and form the lowest two levels which are close in frequency. The
|0? state wavefunction has even parity and the |1? wavefunction has odd parity. The rules
are similar for higher levels: the parity of state number indicates the parity of the wave-
function. The transition between the states with the same parity is called an even transition
and the transition between the states with opposite parity is an odd transition. The parity
rule is quite useful when judging whether a matrix element is zero or not. For example, ? is
an odd function. Hence the integrand in the calculation of ?0|? |2? is also an odd function,
which gives ?0|? |2? = 0. The matrix element of n is proportional to the matrix element of
?
??|n |?? i????= ??|? |?? , (2.18)
8EC
where ??? = ?? ? ?? is the transition frequency between state |?? and |??. This relation
13
a
b c
Figure 2.2: Optical and SEM images for a single fluxonium (a) The optical image of
a single fluxonium. The metal strip connecting to the loop is the antenna that provides
the charging energy EC and also couples the fluxonium to the external environment. A
small Josephson junction (JJ) sits On the upper side of the loop and provides the Josephson
energy EJ . The lower part of the loop is a chain of Josephson junctions. Because of the
large area of the junction here, the chain behaves like a linear inductor and provides the
inductive energy EL in the circuit model. (b) The SEM image of the small Josephson
junction. The Al-AlOx-Al junction is fabricated with the standard Dolan bridge technique.
(c) The SEM image of the junction chain. The chain usually contains hundreds of large
Josephson junctions that are fabricated together with the small junction in the same batch.
14
holds for any circuits that can be described with the Hamiltonian 4E n2C + V (?). To prove
this, one needs to use the commutation relation
[H,?] = [4E n2 + V (?), ?] = [4E n2C C , ?] = ?8iECn. (2.19)
We can calculate the matrix element for both sides. The left hand side gives
??| [H,?] |?? = ??| (H?? ?H) |?? = (E? ? E?) ??|? |?? , (2.20)
and the right hand side gives ?8iEC ??|n |??. It is easy to arrive at Eq. 2.18 by equalling
the two sides. The matrix elements for a typical fluxonium at different flux are shown in
Fig. 2.3. For the 0? 1 transition at half flux quanta, the frequency can be 10 times smaller
than that at integer flux. The charge matrix element reaches minimum and the phase matrix
element reaches maximum at half flux quanta. From the state parity, we can know that the
even transitions have zero matrix elements for n and ? at half and integer flux quanta. This
is called the selection rule and will be frequently used when working at these flux points.
Unlike transmon, fluxonium is naturally insensitive to the offset charge noise because
the two sides of the small Josephson junction are galvanically connected by the inductor.
This ability is achieved without the price of sacrificing the anharmonicity. On the contrary,
the anharmonicity is much large than that of a transmon. For typical fluxonium parameter
regime, EJ/EC ? 1? 10 and EJ > EL [48]. The anharmonicity ? = (E2?E1)? (E1?E0)
can easily achieve few GHz at half flux quanta, which is one order of magnitude larger
than the transmon. The non-trivial potential can push 0 ? 1 transition down to sub GHz
level while keep 1? 2 transition still at several GHz. Therefore the 0? 1 transition is well-
isolated with other higher transitions and can be treated like a real spin. This feature is
quite convenient for quantum simulation tasks on spin-based models. On the other hand,
for using fluxonium as a qubit, the high anharmonicity also relieves the constraint on the
single qubit gate speed. Basically there is no need to worry about the population going to
the higher levels while driving the 0? 1 transition even with few nanosecond long ? pulse.
In this case, other factors need to be taken into account when ramping up the single-qubit
gate rate. For example, a strong drive on a low-frequency qubit can break the rotating wave
approximation. One need to consider this effect when calibrating the pulses. On the other
hand, low-frequency fluxonium qubits have small charge matrix element (they are also
called heavy fluxoniums ) and the charge matrix element determines how strong it couples
to the external microwave drive. This means it is harder to drive these low-frequency qubits.
In order to do fast gates, one need to crank up the RF power, which increases the heat load
15
Figure 2.3: Spectrum and matrix elements for a fluxonium. The left panel shows the
1 ? 0 (blue), 2 ? 0 (red), and 3 ? 0 (green) transitions of a typical fluxonium with EC =
1 GHz, EJ = 4 GHz, EL = 0.5 GHz. The right panels show the charge matrix element
n?? and the phase matrix element ??? accordingly. At 0 flux, fluxonium?s 1 ? 0 transition
frequency and matrix elements can be similar to that of a transmon. At half flux quanta
?ext = ?, the 1 ? 0 transition can be much lower and the charge matrix element also
becomes much smaller. The 2 ? 0 transition has vanishing matrix elements at half flux
quanta because of the even parity.
16
of the dilution refrigerator. Another solution is to use flux pulse to activate single-qubit
rotations [49].
The most important advantage of fluxonium is its high coherence. It has been reported
that the coherence time can achieve hundreds of microseconds and one device achieved
1 ms [50], [51]. This is mainly because the small charge matrix element suppresses the
dielectric loss, which will be introduced in Sec. 2.5.1.
2.3 Dispersive Coupling
In order to readout a qubit state while protecting the qubit from radiative decay, one com-
mon strategy is to let the qubit interact with the environment through a resonator. This
well-studied in cavity quantum electrodynamics (CQED). In our experiment, the qubit is
located at the center of a 3D copper cavity as shown in Fig 2.4b. In the GHz regime, the
copper box can filter out the noise and prevent the qubit from directly emitting photons
into the transmission line. It only lets in the photons that are on resonant with the modes in
the cavity through the SMA connectors. These photons can be used to readout qubit states.
For clarification, the Jaynes-Cummings Hamiltonian can be a simple example: a two-level
system with frequency ?q coupled to a resonator with frequency ?r. The coupling strength
is g.
? ??qHJC = ??ra a+ ? ?z + ?g(a?+ + a ??) (2.21)
2
In the dispersive regime, the qubit-resonator detuning ?qr = ?q ? ?r is much larger than
the coupling strength |?qr| ? g. Eq. 2.21 can be simpl[ified with]perturbation theory [15],[52]
?? 2q g 1
HJC = ?? ?ra a+ ?z + ? a?a+ ?z. (2.22)
2 ?qr 2
On one hand, the Hamiltonian(can be rewri)tten as ( )
g2 ? ? g2HJC = ? ?r + ?z a a+ ?q + ?z, (2.23)
?qr 2 ?qr
indicating the resonator frequency is dependent on the qubit state. The frequency shift is
called the dispersive shift
? = 2g2/?qr. (2.24)
17
a b
c
Figure 2.4: A fluxonium capacitively coupled to a resonator (a) A spin inside a cavity.
The qubit interacts with the harmonic oscillator through a dipole coupling, which is com-
monly described by the Jaynes-Cummings Hamiltonian. (b) A fluxonium device inside a
copper 3D cavity. The two halves of the cavity are sealed with indium [not shown]. The
size of the fluxonium in this picture is exaggerated for better visibility. (c) The circuit di-
agram for a fluxonium capacitively coupled to a resonator. The blue part is the resonator
and the green part is the fluxonium. The two parts are coupled with a capacitance Cc.
18
This mechanism can be utilized to readout a qubit. The experimental implementation is
further discussed in Sec. 3.1.1. On the other hand, we can also rewrite the Hamiltonian in
the form of
? g2? 2g2H ?JC = ??ra a+ (?q + + a a)?z. (2.25)
2 ?qr ?qr
The qubit frequency now depends on the photon number through the Stark shift term
2n?g2/?qr. There have been experiments carried out to use a Cooper pair box to detect
the photon number in a cavity [53].
To simulate a full spectrum of a fluxonium coupled to a resonator, one can consider a
circuit model shown in Fig. 2.4 with a capacitive coupling. For small coupling capacitance
Cc ? Cr, Cf , the Hamiltonian can be approximated as [52], [54]
Hrf = Hr +Hf + Vrf , (2.26)
H = 4E n2
1
r Cr r + EL ?
2
2 r r
, (2.27)
1
Hf = 4EC n
2 2
f ? EJ cos(?? ?ext) + EL ?f , (2.28)f 2 f
Vrf = JCnrnf (2.29)
where J 2C = 4e Cc/(CrCf ). For a 3D cavity, it is usually unnecessary to use three inde-
pendent parameters Lr, Cr, Cc to model the resonator and the coupling. We can introduce
the annihilation and creation operators of the resonator from Eq. 2.8 and Eq. 2.9 first to
simplify the notations.
Hr = ??ra?a, (2.30)
V ?rf = ign((a ? a)n)f , (2.31)1
JC EL 4
gn = ? r . (2.32)
2 8ECr
Now we can just use two parameters ?r and gn to describe the resonator and the cou-
pling. Together with the fluxonium parameters EC , E , and Ef Lf J , one can simulate the full
spectrum with 5 independent parameters.
Without an accurate numerical simulation, we can still use perturbation theory to get
an idea about what determine the dispersive shift of the resonator photon on the fluxonium
19
[40], [55]. ?
?? = g
2
n |n??|2
2???
?2 ? . (2.33)?2
?=? ? ?? r
?? is the dispersive shift felt by the fluxonium state ? when there is an extra photon in
the resonator. Because gn > 0, the frequency shift can be thought of as the repulsion
between all the related fluxonium transitions and the resonator transition. ??? > ?r means
the resonator pushes the upper level ? up and the lower level ? down, and vice versa
for ??? < ?r. In experiment, we are more concerned about differential dispersive shift
??? = ?? ? ??. For exampl?e, the dispersive shift of the 0? 1 transition?is
2 ?? ?| 2?1? 2?0??10 = gn n |21? 2 ? ? |n0?|2 ?2 . (2.34)? ?2 2
? ?=1 1? r
? ? ?
? ?=0 0? r
Therefore for a fluxonium qubit, even though the qubit frequency can be very far away
from the resonator frequency with a detuning on the order of GHz, the dispersive shift on
the qubit can be large thanks to the contributions from the higher levels. One can compare
Eq. 2.34 with Eq. 2.24 to treat the 0 ? 1 transition as a spin and write down the effective
coupling gJC in th?e Jaynes-Cummings model??? ?? ? ?
gJC = gn?(?10 ? ?r)? | | ?2 1?n ? | ?n 2 0?1? 2 ? 2 0?| ??1? ?r ?20? ? . (2.35)?2? ?=1 ? ?=0 r
For inductive coupling, the coupling term Vrf takes the form of Vrf = g?(a? + a)?. This
corresponds to the case where the resonator and the fluxonium have shared junctions. One
can arrive at a similar result but replacing the charge matrix element n?? with the phase
matrix element ???.
Apart from the frequency shift, the hybridization between the resonator mode and the
fluxonium mode makes it possible to drive the side-band transitions. For clarification, we
can denote the fluxonium-resonator state as |?, n?. ? is the state index of the fluxonium
and n is the photon number in the resonator. There are two types of side-band transitions.
Assuming ? > ?, the red side-band transition |?, n? ? |?, n? 1? locates at the frequency
|?r????|. The blue side-band transition |?, n??|?, n? 1? locates at the frequency |?r+???|.
In a fluxonium-resonator spectrum, the fluxonium transitions are offset by the resonator
frequency to form the side-band transitions. That makes the resonator transition look like a
20
mirror, as the blue side-band and the red side-band transitions are distributed symmetrically
with respect to the resonator. Another way to identify the side-band transition is that their
spectral lines disappear at half flux quanta due to vanishing matrix elements. The side-band
transitions can be used to initialize the qubit, which will be discussed in Sec. 3.4.
2.4 Coupled Fluxoniums
Now let?s move on to study a fluxonium that is coupled to another fluxonium, which is a
requirement to realize a two-qubit gate. We can start from the simplest case of two coupled
spins (set ? = 1 for simplicity)
??A ?BH = ?A Bz ? ?z + J?A Bx ?x . (2.36)2 2
Without loss of generality, we can assume the bare frequencies ?B > ?A > 0. The Hamilto-
nian can be diagnosed
? ??A ??BH? = ??Az ? ??Bz . (2.37)2 2
The coupled system can still be described with two independent spins in the new dressed
state basis (the Pauli matrices are labeled as ??i) and the energies are shifted to ???, ? = A,B.
Let?s define ?? = ?B ? ?A, ??? = ??B ? ??A. The relation between the bare energies and the
dressed energies is ( )
?? 2 (? )? ? 2
= + J2. (2.38)
2 2
The coupling makes the total energy of the two spins higher and also pushes the two spin
frequencies away from each other. In the bare basis, the driving term can be expressed as
V = F cos?dt and
F = ?A?
A
x +?
B
B?x (2.39)
?A(B) is the driving strength on spin A(B). The tensor product with the identity matrix of
the other qubit is omitted in the each term above. Without the coupling, the external drive
can only manipulate each spin individually. Now with the coupling, the actual qubits are
the dressed eigenstates and the external drive can create an entanglement in the new basis.
21
a
b c
Figure 2.5: Two fluxoniums coupled with a capacitor. (a) The optical image of two ca-
pacitively coupled fluxoniums. The capacitance mainly comes from the antenna of the two
fluxoniums. (b) The circuit diagram of the system. CM is the capacitance between the two
fluxoniums. (c) The energy diagram considering the coupling term. The two fluxoniums
are coupled by the charge operators. This coupling term pushes state |21? against |12? and
lifts the degeneracy between transition |20?? |10? and |21?? |11?. The energy splitting is ?.
In the dressed basis, V ? V? = F? cos?dt and
F? = ?A cos(?? ? ?+)??Ax +?A sin(?? ? ?+)??Az ??Bx
? +?B cos(?? + ? )??
B
+ x ? ?B?sin(?? + ?+)??
A B
x ??z , (2.40)
where cos ?? = (1 + ??/???)/2 and sin ?? = (1? ??/???)/2. The extra ZX(XZ)
term indicates that the drive applied at spin A(B) can also flip qubit B(A) with a rate that is
dependent on the state of qubit A(B). This mechanism can be exploited to construct multiple
types of two-qubit gates that will be introduced in Sec. 4.3.
Now let?s consider a more complicated case where two fluxoniums are coupled together.
This can be done either through the capacitance between the antenna or sharing junctions
22
in the junction. The former case corresponds to capacitive coupling shown in Fig. 2.5, which
will be discussed here. The details of the latter case corresponding to inductive coupling
can be found in Ref. [54], [56]. The system Hamiltonian is
HAB = HA +HB + VAB. (2.41)
The individual fluxonium Hamiltonian is
1
H? = 4E
2
C,?n? ? EJ,? cos(?? ? ?ext,?) + EL,??2?, (2.42)2
where ? = A,B. The coupling term is
VAB = JCnAnB, (2.43)
2 CMJC = 4e . (2.44)
CACB
Due to fluxonium?s multi-level nature, the interaction from the higher levels changes the
energies of the computational states. If we only consider the lowest two levels of both
fluxoniums, the Hamiltonian is
H = ??A?A ? ?B B ?zz? + ?A Bz z z ?z . (2.45)2 2 4
The last term is the static ZZ interaction coming from the level pushing of the higher tran-
sitions, which is absent in the previous spin models. The effect of the ZZ interaction can be
thought of as conditional Z rotations. This means that the qubits can get entangled auto-
matically even if no external drive is applied. One solution is to inducing a ZZ interaction
with opposite sign to cancel it out as discussed in Sec. 4.3.1. Another way is to design
the qubit with lower frequencies. It has been observed that the low-frequency qubits have
smaller static ZZ interaction in general with the same coupling strength JC .
We can also extend the concept of static ZZ interaction to non-computational states such
as the 1 ? 2 manifolds. Due to the level repulsion between |21? and |12?, the degeneracy
between |10? ? |20? and |11? ? |21? is lifted and the energy splitting is denoted by ? as
demonstrated in Fig. 2.5c. This type of ZZ interaction is order of magnitude larger than that
of the computational states. Therefore it is possible to use these high energy transitions for
fast two-qubit gates shown in Sec. 4.3.
Similar to the spin model, the external drive can entangle the two qubits. Again, the
23
driving term is V = F cos?dt and
F = ?AnA +?BnB. (2.46)
In order to see its behavior in the computational space, we can calculate the matrix elements
of nA and nB in the coupled fluxonium basis. This can be done perturbatively and by
keeping the terms with major contributions [57]. We can define V ? ?kl,k?l? = ?kl|VAB |k l ? =
JC ?kA|nA |l ? ?A? ?kB|nB |lB? and express the dressed states in the bare fluxonium basis,
| ? ? | ? | ? ? V11,00 | ? | ? ? V31,00 V13,0000 0A 0B 1A 1B |3A? |1B? ? |1A? |3B?+ ? ? ? ,
?A B A10 + ?10 ?30 + ?
B ?A B10 10 + ?30
(2.47)
| ? ? | ? | ? V00,11 | ? | ? ? V20,11 | V02,1111 1A 1B + 0 0 2A B A B A ? B A? |0B?+ ? |0A? |2B?+ ? ? ? ,? + ? ? A B10 10 21 ?10 ?10 ?21
(2.48)
| ? ? | ? | ? ? V10,01 | ? | ? ? V12,0101 0A 1B ? 1A 0B |1A? | ? ?
V30,01
2B |3A? |0B?+ ? ? ? ,
?A B A B A B10 ?10 ?10 + ?21 ?30 ? ?10
(2.49)
| ? ? | ? | ? V01,10 | ? | V21,10 V03,1010 1A 0B + ? 0A 1A B B? ? |2A? |1B?+? A B A10 ?10 ?21 + ?10 ?10 ?
|0A? |3B?+ ? ? ? .
?B30
(2.50)
We keep the first order terms with non-zero matrix elements and truncate the Hilbert space
up to the |3? states. From the expansion, we can find the parity is conserved in the dressed
states, which indicates that matrix elements between the same parity states are zeros. For
example, ?00|nA |00? = ?10|nA |01? = ?00|nA |11? = 0.
Because ?k|n? |l? is pure imaginary in our convention, we can define n?kl = ?i ?k?|n? |l??
to simplify the expression and calculate other matrix elements of nA and nB. For example,
24
the cross matrix elements for transition[s of(the )other qubit are ]
A 2
( )
A A 2 A
?00|nA |01? ? ? B ( n)01 (?10 ) ( n2iJ n 03 ?30C 0[1 2 2 + )2 ( )2 , (2.51)(?A10 ) ? ?B A B10 ?30 ? ?10( n) ( ) ]A 2(?A A 2 A?10|n |11? ? 2iJ nB 01 10 ) ? ( n)12 (?21A C 01 [ ) , (2.52)?A( 210 ?) ?B 2 A 210 ?2(1 ?( ) 2( ) ( ) )
B 2?10 ]
B B B 2 B
? n ? n ?00|nB |10? ? 2iJ nA 01 10 03 30C 01 [ + , (2.53)?A10(2 ( )?) 2 2?B A10 ?10( ?)?B 230 ]2 2
? | | ? ? ? A ( nB ?B B B01 n 11 2iJ n )01 ( 10 ) ? ( n)12 ?21B C 01 2 2 2 ( ) . (2.54)
?A10 ? ?B A B
2
10 ?10 ? ?21
One can use the matrix elements to express n? in terms of Pauli operators when only con-
sidering computational states. For example,
comp ?XI A B ?XZ A B ?IX A B ?ZXnA = ?x ?0 + ?x ?z + ?0 ?x + ?
A?B. (2.55)
2 2 2 2 z x
The absence of other terms such as IZ, ZZ,XX etc. is because of the zero matrix element.
?XI = ?00|nA |10?+ ?01|nA |11? , (2.56)
?XZ = ?00|nA |10? ? ?01|nA |11? , (2.57)
?IX = ?00|nA |01?+ ?10|nA |11? , (2.58)
?ZX = ?00|nA |01? ? ?10|nA |11? . (2.59)
?XI is the zeroth order term. This is trivial term corresponding to direct drive on qubit A.
?IX and ?ZX are linear in JC , meaning that the electric field on qubit A can excite qubit B.
?XZ is quadratic in JC . This term indicates that the drive directly applied on qubit A can
also slightly depend on the state of qubit B. We can compare the result to Eq. 2.40 and find
that the higher level of fluxonium give rise to the extra IX and XZ terms that are absent
in the simple spin model. In Sec. 4.3, we are going to explain how we can use these terms
for two-qubit gates with only computational states.
2.5 Energy Relaxation and Dephasing
Before introducing the last circuit model of a fluxonium in a waveguide, it is necessary
to first discuss the concepts in noise and decoherence. The noise from the environment
25
is always inevitable and can cause energy relaxation and dephasing in superconducting
circuits. These effects can be treated with the master equation in Lindblad form [52], [58],
[59] in order to simulate the evolution of a quantum system.
d?t ? i
?
= [H, ?
? t
] + D[Li](?t) (2.60)
dt
i
D[L](?t) is called dissipator and is defined as
D[L ](? ) = L ? L? ? 1 ? 1 ?i t i t i LiLi?t ? ?tLiLi. (2.61)? 2 2 ?
Li are the Lindblad operators. L1?= ???? is for the downward transition. L2 = ???+ is
for the upward transition. L3 = ??/2?z is for pure dephasing. We can consider a simplest
example by assuming H = 0 and only keeping the dissipators. The differential equation can
be written in Pauli coordinates ( )
?1
dxt = ?( + ??)xtdt (2.62)2
? ?1dyt = + ?? ytdt (2.63)
2
dzt = ??1 (zt ? zth) dt. (2.64)
Even though the upward and downward transitions are in opposite directions, the two
transition rates contribute to the energy relaxation rate with the same sign ?1 = ?? + ??.
The transition direction makes a difference when it comes to the expectation value of ?z at
thermal equilibrium zth = (?? ? ??)/?1. ?1 determines how fast the Z component of the
qubit Bloch vector decays to the equilibrium value and the decoherence rate ?2 = ?1/2+??
determines how fast the transverse components decay to zero.
The energy relaxation time T1 = 1/?1 and the coherence time T2 = 1/?2 can be mea-
sured experimentally with the pulse sequence shown in Fig. 2.6. To measure T1, one needs
to flip the qubit, wait for a variable amount of time and measure the qubit population. As-
suming the qubit is always at the ground state |0? at the beginning of the sequence and the
qubit pulse length can be neglected, the excited state population measured at the end of
the sequence is P ?t/T| ? = e 11 . It is worth mentioning that sometimes the T1 curve can take
double-exponential form, which can come from T1 fluctuations caused by quasiparticles
[60].
To measure T2, there are two common pulse protocols and they can give different T2
values. One is the Ramsey sequence, where the qubit is first flipped to the equator on the
26
Figure 2.6: T1, Ramsey and spin echo measurements. X? and X? are ? and ?/2 pulses
2
around X axis. Assuming the qubit starts from the ground state and the qubit pulses are
much faster than the decay time, the T1 and Ramsey curves starts from P|1? = 1 and the
spin echo curve starts from P|1? = 0. The Ramsey curve oscillation rate equals the detuning
of the drive.
27
Bloch sphere with a ?/2 pulse, evolve for time t and applied a second ?/2 pulse. The final
?t/TRamseypopulation is P|1? = e 2 cos (??t) + 1/2, where T
Ramsey
2 is the measured T2 with this
Ramsey sequence and ?? is the detuning of the qubit drive. The other pulse protocol is the
spin echo sequence. It is similar to the Ramsey sequence, but there is an extra ? pulse in the
middle of the sequence to refocus the qubit phase on the XY plane. The ?z rotations coming
from the drive detuning before and after the ? pulse are cancelled out. There will be no
echo
oscillations on the spin echo trace and the population is P|1? = ?e?t/T2 + 1/2. Typically
T echo Ramsey2 > T2 because the spin echo sequence can filter out low frequency noise on the
qubit phase. Imagine the case where the qubit frequency is drifting randomly over a long
time scale. In experiment one need to average the Ramsey traces from multiple repetitive
measurements to get a good contrast. Because these traces oscillates at different frequency,
the average curve decays faster than each of the single trace. But averaging echo traces will
not introduce this problem because each single trace does not oscillate and decays with the
same rate. T2 can typically be longer using the dynamical decoupling technique with more
? pulses in the sequence. One can have more rigorous comparison on the filtering effect by
calculating the filter function for different pulse sequence [61]. It is worth noticing that T2
curves can also have non-exponential behavior depending on the noise type [62].
2.5.1 Energy relaxation mechanisms
Generally speaking, the energy relaxation rate can be calculated with Fermi?s golden rule
[63], [64]. The relaxation rate from state ? to state ? is
1
??? = | ??| O? |?? |2S(???) (2.65)?2
O? is the operator that couples the quantum system to the noise source. S(???) is the noise
spectral density at frequency ???. In most cases, the spectral density satisfy the relation
S(??)/S(?) = e???/kBT in thermal equilibrium, where S(?) corresponds to the downward
transition and S(??) corresponds to the upward transition. This leads to the detailed
balance relation in thermal equilibrium
??/? = e
???/k
? B
T . (2.66)
The upward transition is more significant for low frequency qubits such as fluxoniums at
half flux quanta. Assuming the temperature of a dilution refrigerator at base plate is 20
mK. This corresponds to a frequency at 417 MHz, which is the typical frequency of the
fluxonium 0? 1 transition.
28
Superconducting qubit T1 are susceptible to several noise source, such as dielectric loss,
Purcell effect, quasiparticle loss and 1/f flux noise. These are the typical relaxation mecha-
nism considered for superconducting circuits.
Dielectric loss
Dielectric loss comes from the defects in the environment. These defects behave like lossy
two-level systems (TLSs) exchanging energy with the superconducting circuit and they exist
on different interfaces such as the metal-substrate interface, substrate-air interface, etc [65].
This can be modeled by assuming the capacitor in the circuit is filled with lossy dielectric
materials such that Ceff = C(1 + tan ?C). tan ?C is called the loss tangent. In this way, the
dielectric loss rate can be written as
?diel
1
?? = | ??|? |??? |2Sd(iel(???), )? (2.67)(2e)2 ? ???? ?
Sdiel(?) = 2??Re[Ydiel(?)] ??coth + 1 ?? , (2.68)2kBT
Re[Ydiel(?)] = ?C tan ?C . (2.69)
The temperature dependence comes from the stimulated emission by thermal photons in
the lossy environment [18], [50]. Combine the equ?ations(above,)the r?elaxation rate is??2
diel ??
?
? = | ??| | ? |2 ?? ??? ? tan ? coth + 1??? C ?? (2.70)8EC 2kBT
This formula demonstrates that 1/(T diel1 | ??|? |?? |2) can have a quadratic dependence on
qubit frequency at zero temperature assuming the loss tangent is constant. It also has been
observed experimentally that the loss tangent can have a weak dependence on frequency
[50]. Using Eq. 2.18, the relaxation rate can also b?e expressed in term of the charge matrixelement ( ) ?
diel 8EC ? ???? ???? = | ??|n |?? |2 tan ? ?coth + 1? , (2.71)? C ? 2k T ?B
where the ?2?? dependence is absorbed in the charge matrix element. Therefore fluxoniums
can have suppressed dielectric loss with low qubit frequency and small matrix elements.
At the same time, there are on-going efforts to experiment with different materials [30]?
[38]. Because the total loss tangent comes from different parts of the material, it is also
possible to make it smaller by optimizing the qubit design and engineering the participation
ratios [65]. The dieletric loss introduced above is considered to be an averaged effect from
29
many defects in the material. Recent work also shows that there are some two-level systems
that can be coupled strongly with the qubit. Qubit T1 can drop significantly When the qubit
frequency is near these two-level systems [66]. These defects are distributed randomly over
the spectrum, can drift over time and can be quite different between different cool-downs.
Purcell effect
Another energy relaxation channel is Purcell effect [67] when the qubit is coupled to a lossy
resonator. The qubit can lose energy to the environment when exchanging excitation with
the resonator. Assuming the resonator line width is ?, the coupling strength is g and the
detuning is ?qr. The Purcell decay rate can be derived with the Jaynes-Cummings model
[52]
? 2
?
Purcell = |g| . (2.72)
?2 2qr + (?/2)
On the other hand, one can also derive the relaxation rate from a circuit model for a qubit
coupled to a coplanar waveguide resonator [40], [68]. Because fluxonium qubit frequency
is much lower than the resonator frequency, the Purcell effect can be neglected in this case.
However, the high energy transitions such as the 0?3 transition can be close to the resonator
and limited by the Purcell effect. In the more extreme case where the cavity has a strong
coupling to the external environment and becomes a waveguide, the high transition can
have a large spontaneous emission rate and emit fluorescence signal that depends on the
ground state population (See Sec. 2.6 and Sec. 3.1.2).
1/f flux noise
1/f flux noise is one of the main noise source for dissipation in flux qubits. There has
been research demonstrating that the absorbed molecular O2 at the surface is the main
contributor of the magnetic noise [69]. The flux noise is coupled to the qubit through the
term H??ext = IL??ext. IL = ?/L is the current flow around the loop [25], [70]
??
1
?? = | ??|? |?? |2S?(???), (2.73)2e2L2
A2
S?(?) =
? . (2.74)
?
Flux qubits have faster relaxation rate at low frequency due to the 1/? dependence. How-
ever, this effect is not observed in fluxonium system, which can be explained by the large
inductance provided by the junction chain. For typical fluxonium parameters, the corre-
sponding relaxation rate can be smaller by a factor of 102 ? 104 compared to flux qubits.
30
Quasiparticle loss
Quasiparticle loss is also an important relaxation channel for superconducting circuits.
Quasiparticles are unpaired electrons and can affect qubit lifetime when they tunnel through
the junctions. In fluxonium system, the tunneling can happen at the small junction and the
junction chain. The corresponding r?elaxation rates are [71]
qp,SJJ 8EJ 2?g | ? ?? ?ext??? = ? ?| sin( ) |?? |
2xqp, (2.75)
?? ???? 2
?qp,chain
8EL 2?g
?? = (? |)??|
? |?? |2xqp, (2.76)
?? ???? 2
xqp = 2?T/? e
??g/T
g . (2.77)
?g is the superconductor energy gap. xqp is the normalized quasiparticle density. These
equations are based on two assumptions. First, the quasiparticle is in thermal equilibrium.
In reality, the observed quasiparticle density is higher than that corresponding to the en-
vironment temperature, which means the non-equilibrium quasiparticles is quite common.
Still, the above relations hold with some effective xqp and Teff [72], [73]. The second
assumption is that the qubit is in the high-frequency regime ??? ? ?E, where ?E is the
characteristic energy of a quasiparticle. In thermal equilibrium, ?E ? T . For a low fre-
quency qubit, if we still use Eq. 2.75 and 2.76, the accurate expression of xqp can have
a dependence on frequency [71]. But the good new is that, when ??? ? T which is the
typical case for fluxonium qubit, the deviation of Eq. 2.77 is still acceptable. In order to
measure the quasiparticle density xqp, one can fit the T1 curve with double exponential for-
mula [74], measure T1 variations [75], [76], monitor the quantum jumps [77] or measure
the frequency jump of the charge-sensitive Cooper pair box [76], [78].
At half flux quanta ?ext = ?, the parity of the operator in Eq. 2.75 leads to ?
qp,SJJ
10 =
0. The quasiparticle loss is dominated by the tunneling event happened in the junction
chain. It has been reported that cosmic rays [76], high energy photons [79]?[81] and
strong microwave pulses [75], [77] can generate quasiparticles on the sample. The general
strategy of suppressing quasiparticle loss is to have good shielding and filtering. Infrared
filters such as Eccosorb filters are widely used to filter out the high energy photons in the
measurement line. It is believed that these photons (over 100 GHz) are a common source
in superconducting experiments and these filters need to be few centimeters long. One has
to make sure that the sample holder is well sealed and there are no apertures or slits on
the last few stage of shielding where the infrared filters are located. Otherwise the high
31
energy photons can leak into the measurement line after the infrared filter through the slits
at the connectors. There are also some other techniques to mitigate quasiparticle loss such
as trapping quasiparticles with vortices[75], quasiparticle pumping[74], engineering the
superconducting gap and using normal metal to absorb quasiparticles [82]?[84].
2.5.2 Dephasing mechanism
The coherence time T2 is limited by T1 and the dephasing time T? = ??. By definition,
1 1 1
= + . (2.78)
T2 2T1 T?
This indicates T2 should be always shorter than 2T1. Because T2 determines how long the
quantum information can survive in qubits, it is worth the efforts to suppress the dephasing
rate in order to push T2 to the 2T1 limit. Dephasing process is often modeled as the noise on
the qubit phase (or qubit frequency) [54], [62]. In terms of the noise spectral density, the
noise can be classified as white noise, 1/f noise, long-time correlated noise and telegraph
noise. The mathematical form of the noise spectral density determines the shape of the
decay curves measured in Ramsey or spin echo experiments. Most of the time, the decaying
envelope takes the form of e?t/T? , which corresponds to a white noise. For other types of
noise, the decaying envelope can be more complicated and can be usually approximated as
?(t/T )2e ? . It worth noticing that the Gaussian type of decay does not necessarily indicate
non-white noise. Non-accurate ? pulse in the spin echo sequence shown in Fig. 2.6 can also
lead to Gaussian decay envelope.
One common dephasing mechanism for flux-tunable qubits is flux noise. It is especially
important when the qubit is not at flux sweet spot. A fluctuation on the external flux
can be translated to qubit frequency fluctuation through the derivative with respect to the
external flux ??/??ext of the spectrum. It is also possible that the flux noise can affect
the qubit through the second order effect where the frequency fluctuation depends on the
?2?/??2ext. But it is not the dominate dephasing mechanism for recent qubit experiments
[50].
The white flux noise induced dephas(ing tim)e has a quadratic dependence on ??/??
?? 2 Swhite
?? = . (2.79)
??ext 2
Swhite is a constant. For white noise, Ramsey and spin echo measurements will give the
same dephasing rate. It is reported that the flux noise spectral density has 1/f behavior
32
[25], [70]. In this case, the spin echo decay envelope is Gaussian e?(??t)
2
and the dephasing
rate is ?
?echo
??
? = S1/f ln 2. (2.80)??ext
S1/f is the 1/f noise spectral density strength. The decay envelope for Ramsey measurement
is more complicated in theory but Gaussian is still a good approximation [62]
Ramsey ? ???? S1/f . (2.81)??ext
Because Ramsey measurement is sensitive to the noise at low frequency, or equivalently
speaking, the long-time scale frequency fluctuations. ?Ramsey under 1/f noise can be de-
pendent on the total measurement duration and it is faster than ?echo.
At flux sweet spot, ??/??ext = 0, which means fluxonium can be protected against
flux noise at this point. Therefore fluxoniums are usually biased at the sweet spot for gate
operation and readout. But in order to do flux-activated gates, the external flux will be
tuned away and the flux noise need to be considered during this flux pulse [49], [85].
Similar to flux noise, offset charge noise has been the dominate dephasing channel
for charge qubits, because the qubit frequency is sensitive to offset charge ng [23]. The
invention of transmon solved this problem though a large capacitance. On the other hand,
fluxonium avoid the charge noise by shunting the circuit with a junction chain, such that
the offset charge can flow through the chain and will not accumulate on the metal pads
[24].
At flux sweet spot, thermal photon shot noise becomes an important dephasing mech-
anism. When the qubit is dispersively coupled to a resonator, the qubit frequency can be
shifted according to the resonator photon number as shown in Sec. 2.3. In thermal equilib-
rium, the photon number can fluctuate randomly and lead to qubit frequency fluctuation.
The dephasing rate of the 0? 1 transition due to these thermal photons is
n 2th??
? = 01? . (2.82)
?2 + ?201
nth is the thermal photon number. ? is the resonator line width. ?01 is the dispersive shift
of the 0? 1 transition. If we fix ?, the larger ?01 is, the larger ?? will be. If we fix ?01, the
dephasing rate reaches maximum at ? = ?01. Because the parameters ? and ?01 are also
related to the readout fidelity. One need to be careful when optimizing these parameters in
order to keep a balance between a good readout fidelity and the thermal photon dephasing
rate.
33
Another way to address this dephasing channel is to make the resonator colder. This
does not just mean that the resonator itself should be thermalized well. For example, the
3D copper cavity is directly mounted on the cold fingers. The copper body should be as
cold as the base plate. However, the measured thermal photon temperature is typically
higher than the base temperature. This is because the thermal photons mainly come from
the measurement line as explained in Appendix B.4. The thermal photon number can be
related with the effective temperature using the Bose-Einstein distribution
1
nth = ? . (2.83)e ?r/kBTeff ? 1
Similar dephasing mechanism can appear in a cavityless environment described in Sec. 2.6.
There is no cavity mode and dispersive coupling in this case. Instead, the thermal photons
can directly excite the high energy transitions that are strongly coupled to the transmis-
sion line and have large downward transition rates. The qubit absorbs a photon at a rate
? = ? e???/k? ? B
Teff and decays back to the 0 ? 1 manifold immediately due to the large
fluorescence rate. The emitted photon carries the qubit phase information and flies away.
Therefore the qubit loses its coherence during this process. Because the qubit can be ex-
cited by the thermal photon at 0 state or at 1 state depending on the environment, the
corresponding dephasing rate?can be written as [86]
? ? (? e???i0/kBTeff + ? e???i1/kBTeff?,10 i0 i1 ). (2.84)
i>1
2.6 Fluxonium in a Cavityless Environment
One fundamental challenge in quantum technology is to interface qubits with travelling
photons. This is essential in constructing quantum network where qubits can get entangled
remotely by exchanging photons. In this architecture, waveguide is the key component to
propagate microwave photons from one qubit to another with low loss [87]. One simple
case to study the interaction between the qubit and the photons in the waveguide is the
reflection setup where the qubit sits inside a waveguide with a closed end. In experiment,
we can replace the small pin of the cavity to a gigantic launcher to ensure free access of the
microwave from the transmission line to the space inside the box as shown in Fig. 2.7. The
port can be viewed as a broadband coaxial-to-waveguide adapter [41]. Therefore the box
does not behave like a single mode cavity and a continuous spectrum of modes can exist
in the box. The photons go into the box and bounce back into transmission line. Standing
waves are formed inside the box and the qubit can interact with the microwaves which do
34
a b
c
Figure 2.7: A fluxonium in a waveguide. (a) A spin inside a waveguide with a closed end.
Within certain frequency range, the photons can travel inside a waveguide and interacts
with the qubit. (b) A fluxonium device inside a copper waveguide. The two halves of the
waveguide are sealed with indium [not shown]. The size of the fluxonium in this picture
is exaggerated for better visibility. (c) The circuit diagram for a fluxonium capacitively
coupled to a transmission line. The blue part is the transmission line and the green part is
the fluxonium. The two parts are coupled with a capacitance Cc.
not have a node at the qubit position.
The circuit diagram is depicted in Fig. 2.7. The fluxonium circuit is directly coupled
to the transmission line where the incoming photon mode is ain and the outgoing photon
mode is aout. For clarification, we are going to consider a semi-classical model and treat the
fluxonium transition as a spin, instead of solving the full fluxonium spectrum [88].
The spin evolution in the presence of the input field can be described by the Hamiltonian
?? ?q
H = ?z + i? ? ?r(ain + ain)(?+ ? ??). (2.85)2
?r is the radiative Purcell rate that quantifies how strong the spin is coupled to the trans-
mission line. Here we treat the photon field classically ain = ? e?i?dtin , where ?in is a real
number without loss of generality. In the rotating frame at ?d and with the Rotating Wave
35
Figure 2.8: Optical image of a fluxonium inside a waveguide. In order to increase the
coupling between the fluxonium and the itinerant photons, a pair of couplers is designed
next to the fluxonium. The couplers are 3 mm long Al strips that are deposited with the
qubit within the same batch.
Approximation (RWA), we arrive at the textbook spin Hamiltonian in the presence of drive
? ?
HR = ? ?z ? ? ?y, (2.86)
2 2
?
where ? = ?q ? ?d is the drive detuning and ? = 2 ?r?in is the drive strength. It can
be noticed that the same Hamiltonian is used for driving a qubit inside a cavity. In both
cases, the driving term comes from the direct coupling between the qubit and the input
port. What is different in the waveguide setup is that the bulky launcher together with the
coupler design shown in Fig. 2.8 significantly increases the Purcell rate ?r.
At the same time, the spin is coupled to the outgoing field through the same Purcell rate.
This can be treated as an incoherent channel in the spin evolution. In general, the qubit
energy relaxation rate ?1 = ?r + ?nr, where ?nr includes all the non-radiative channels
such as the dielectric loss, quasiparticle loss, etc. We can use master equation to model this
system. ?
d?t ? i
? ??
= [H, ?t] +D[ ?1??](?t) +D[ ?z](?t). (2.87)
dt ? 2
36
In the steady state where t ? ?, the components of the Bloch vector are
x? = ?1?2?/(?1(?
2 2
2 + ? ) + ? ?
2
2 ), (2.88)
y? = ?1??/(?1(?
2
2 + ?
2) + ? 22? ), (2.89)
z? = ?1 + ? 22? /(? 2 2 21(?2 + ? ) + ?2? ). (2.90)
This can be calculated using x? = ??x? = tr(?t?x), y? = ??y? = tr(?t?y), and z? = ??z? =
tr(?t?z).
The outgoing field aout can be calculated in terms of the i?ncoming field and the qubit
operator from the input-output formalism [20] aout = ain ? ?r??. In circuit language,
this is equivalent to the boundary condition of the incoming wave and the outgoing wave
at the end of a transmission line with a lumped circuit Vout + Vin = V . The outgoing field
can also be considered as the result of the interference between the directly reflected field
that does not interact with the qubit and the field radiated by the qubit. In experiment,
one can measure the reflection coefficient r = ?aout?/?ain?. The expectation values for the
field operators are the oscillation amplitude that we introduced earlier ?ain(out)? = ?in(out).
Even though ?in is a real number in our convention, ?out can be complex. Substituting
Eq. 2.88?Eq. 2.90 into ???? = (??x? ? i??y?)/2 and assuming ?r ? ?nr,??, the reflection
coefficient is
? 1 + 2i?/?rr = 1 2 . (2.91)
1 + 4(?/? 2r) + 2(?/? )2r
This model can be applied to the high energy transitions (e.g. 0?3) of a fluxonium that are
strongly coupled to the continuum of photon states. In this case, the spin refers to the two
levels of that transition instead of the computational states in the usual context. The total
population p0 on these two levels adds a contrast factor in Eq. 2.91
r = 1? 1 + 2i?/?r2p0 . (2.92)
1 + 4(?/? 2 2r) + 2(?/?r)
This can serve as a novel readout scheme to measure fluxonium qubit population. The
experiment implementations are presented in Sec. 3.1.2.
37
Chapter 3
Experiments on Readout and
Initialization
The quantum error correction (QEC) codes on superconducting qubits require the qubit
readout need to be single-shot, high-fidelity, and fast [4], [89]?[91]. For whatever readout
scheme, the qubit signal is typically rather weak and needs multi-stage amplifications in
order to fight against the thermal noise picked up during the trip from the base plate to
the room temperature digitizer. Without a quantum amplifier [92]?[94] at the base tem-
perature, a single experiment needs to be repeated thousands of times and the measured
signal needs to be averaged over all the realizations to get a better signal-to-noise ratio
(SNR). Therefore this type of measurement gives the expectation value (e.g. tr(??z)) over
a ensemble and it is impossible to identify whether there is an error among one of the real-
izations. According to Copenhagen interpretation of quantum mechanics, the wavefunction
collapses instantaneously after a projective measurement and the measurement outcome
should be the eigenvalues of the measurement operator. In QEC codes like surface code,
the eigenvalues should be obtained in a single run instead of averaging over all the realiza-
tions. This is called single-shot measurement. The readout fidelity depends on the SNR the
single-shot histogram and whether extra transitions are induced during the readout pro-
cess. This puts a constraint on the readout pulse strength and duration. When one qubit is
being measured, other qubits are still suffering from decoherence. Hence the readout pulse
should be as short as possible. On the state-of-art quantum processors, the readout pulse is
much longer than the gate time. High-fidelity readout with short pulses have been reported
in a few demonstrations, indicating the current quantum processors can have better readout
performance potentially [59].
In this chapter, the main topic will be on qubit readout and introduce relevant concepts.
38
The last section is focused on initialization schemes for fluxonium qubits. On one hand, QEC
requires qubit reset during the algorithm and fluxonium qubits have long T1. On the other
hand, fluxonium qubits have low frequency, resulting in significant thermal populations.
The characteristic frequency of 20 mK is 417 MHz, corresponding to a thermal population
of P|1? = 0.27. Therefore high-fidelity fast initialization techniques are quite beneficial in
fluxonium experiments and several initialization schemes are discussed in this section.
3.1 Readout schemes
Fluxonium qubits can be measured with or without cavity. When the qubit coupled to the
cavity, we can perform dispersive readout by measuring the cavity transmission. The cavity
mode frequency depends on the qubit state and the transmitted signal can change in phase
and amplitude accordingly. When the fluxonium is strongly coupled to the transmission line
using a waveguide, we can perform fluorescence readout taking advantage of the higher
transitions. We used 0 ? 3 transition as the fluorescence transition and 0 ? 1 transition as
the qubit. The change of qubit population is manifested in the reflection coefficient.
In both cases, the outgoing signal is down-converted and recorded by the digitizer.
We performed heterodyne detection in which both quadratures, named I and Q, of the
oscillating signal are simultaneously measured. The two quadratures correspond to the real
and the imaginary part of the complex amplitude. It is a common technology in microwave
measurements. The heterodyne setups are depicted in Sec. B.4.
3.1.1 Dispersive Readout
As introduced in Sec. 2.3, dispersive readout is the most common way to readout a qubit.
The qubit can be detuned away from the cavity frequency for protection against the Purcell
effect while dispersively coupled to the cavity mode for readout. It has been demonstrated
that the cavity frequency is dependent on the qubit state in the JC model. How does give us
the state-dependent IQ signal in the experiment? We need to first understand how a single
cavity respond to external drives.
In general, the cavity?s response is measured in the form of transmission or reflection.
Depending on how the cavity is connected to the external circuits, the response can be dif-
ferent. For a 3D cavity, the setup can either be one-port (reflection) or two-port (transmis-
sion). In 2D architecture, the widely adopted hanger geometry is another type of reflection
setup. Here in this thesis, we focus on the 3D architecture.
39
a b
c d
Figure 3.1: Two-port cavity transmission. (a) Transmission coefficient amplitude |S21| of
a cavity at 7 GHz calculated from Eq. 3.2. ?1 = 1 MHz, ?2 = 5 MHz, ?L = 0.5 MHz. The
parameters are close to the case for a copper cavity in the experiment. (b) Transmission
coefficient on the complex plane (IQ plane) at different drive frequency. The trace forms a
circle. The points with large detuning are close to the origin. (c) Cavity transmission when
dispersively coupled to a qubit. ? = 6 MHz. The blue peak corresponds to the qubit at |0?
state. The red peak corresponds to the qubit at |1? state. (d) The readout signal on the IQ
plane when reading at the frequency indicated by the light green dashed line in (c). The
blue and the red blobs correspond to the qubit at |0? and |1? state. The angle spanned by
the two blobs ? ? arctan?/?tot when the readout frequency is at the middle of the two
peaks and ? is small.
40
The S parameters are defined as the transmission and reflection coefficients for a multi-
port system. For a 3D cavity, the reflection S11 and the transmission S21 are
?1 ? ?2 ? ?L + 2i(? ? ?r)
S11(?) = (3.1)
?1 + ?2 + ??L ? 2i(? ? ?r)
2 ?1?2
S21(?) = (3.2)
?1 + ?2 + ?L ? 2i(? ? ?r)
?r is the cavity frequency. ?1 (?2) is the coupling rate to port 1 (2), corresponding to the
external quality factor Qexti = ?r/?i. ?L is the cavity internal loss rate, corresponding to
the internal quality factor Qin = ?r/?L. One may notice that the phase definitions can be
different in the literature. For example, the definition can be Sij ? ?S?ij in some other
references. ?tot = ?1 + ?2 + ?L is the total damping rate. Q = ?r/?tot is the cavity quality
factor. We use Eq. 3.1 to measure the external quality factors for the multi-port cavity and
use Eq. 3.2 to fit the total damping rate.
As shown in Fig. 3.1, the transmission amplitude reaches maximum at the resonance
frequency. The amplitude square has a Lorentzian shape. On the IQ plane, the transmission
data at different drive frequency fall on a circle. When it is far detuned, the transmission
is at the origin. When it is on resonance, the transmission is at the other end of the circle
on the imaginary axis. If the cavity is coupled to a qubit, there will be a frequency shift
depending on the state of the qubit. For example in Fig. 3.1c, if the qubit is at ground state,
the transmission is the blue curve. If the qubit is at the excited state, the transmission is
the red curve. If the population is equally distributed on the two states, the transmission
would be the summation of two complex transmission numbers weighted on the population
on each state. This could be a way to measure qubit population in principle. In practice,
the measurement line itself can have non-trivial frequency-dependent transmission due to
attenuation, reflection or amplification. The height of the Lorentzian can not be directly
converted to population. A better way to measure population would be the single-shot
readout (Sec. 3.1.1).
If the readout frequency is set the position indicated by the green dashed line Fig. 3.1c,
the S21 in the two cases correspond to the position of the two blobs on the IQ plane in
Fig. 3.1d. When ? ? ?tot, the angle spanned by the two blobs ? ? arctan?/?tot. In the
experiment, we usually directly work with the transmitted signal instead of S21 which is
the ratio between the input and the output. We can assume that the positions of the two
blobs on the IQ plane of the signal are bk = xk + iyk and k = 0, 1 and the populations are
p0, p1. The averaged signal would be m = p0b0 + p1b1 and p0 + p1 = 1. When the qubit
population changes, the point m moves on the line connected by the two blobs. Therefore
41
its quadratures are linearly dependent on the qubit population. We can rewrite the the
population with the expectation value of the Pauli matrix p0 = (??I? + ??z?)/2 and p1 =
(??I?? ??z?)/2. Then we have m = ?I??I?+ ?Z??z?, where ?I = 1/2 and ?Z = (p0 ? p1)/2.
From this formula, we can see that the averaged signal m is actually the expectation value
of the measurement operator
M = ?I?I + ?Z?z, (3.3)
and m = ?M? = tr(?M) [59].
Joint Readout
When a cavity is coupled to two qubits simultaneously, we perform joint readout to the two-
qubit system. Assuming the dispersive shifts are ?A and ?B for the two qubits, the shifted
cavity frequencies are
00 ?A ?B?r = ?r ? ? , (3.4)2 2
?01
?A ?B
r = ?r ? + , (3.5)2 2
?10
?A
r = ?r + ?
?B
, (3.6)
2 2
11 ?A ?B?r = ?r + + . (3.7)2 2
There are four blobs on the IQ plane, corresponding to the four computational states |00?,
|01?, |10?, and |11? as demonstrated in Fig. 3.2. One need to avoid the case where ?A??B ?
0, as this will result in overlap of two blobs and making the two states indistinguishable.
The optimal situation would be ?A ? ?2?B (assuming ? is large) and the four blobs get
maximum separation.
The measurement operator is
M = ?II?I ? ?I + ?IZ?I ? ?z + ?ZI?z ? ?I + ?ZZ?z ? ?z. (3.8)
Because the averaged signal now depends on the population of the four blobs, its trace can
be more complicated. It is not limited on a straight line but can be any where in the region
spanned by the four blobs. For example, when one measures the T1 of qubit A by looking
at one quadrature of the signal, the fitted decay rate can be inaccurate if qubit B is also
decaying at the same time. To solve this problem, one can apply a ?/2 pulse on qubit B to
equalize the population at the |0? state and |1? state. This only works when qubit B is not
42
a b
Figure 3.2: Joint readout for two qubits in the same cavity. (a) |S21| of a cavity in the
case for four computational states. The simulation parameters are the same as those in
Fig. 3.1. ?A = 4 MHz, ?B = 2 MHz. (b) The readout signal when reading at the frequency
indicated by the light green dashed line in (a). There are four different blobs corresponding
to the four computational states. The population of both qubits can affect the averaged
signal.
at a coherent superposition of |0? and |1?. That is, its density matrix can be described by
?B = (?I + (2p
B
0 ? 1)?z)/2. Otherwise, or if the ?/2 pulse is not accurate, the populations
may not equal.
Single-shot Readout
When the readout SNR is good enough, one can distinguish qubit state in a single acquisi-
tion without averaging over many traces. In this case, we are able to perform single-shot
readout which is essential for quantum error correction codes. Single-shot readout has been
demonstrated and well-studied for dispersive readout. As shown in Fig. 3.1, depending on
the qubit state, the cavity transmission can be at either of the two blobs on the IQ plane.
At low power, the probability of the data point falling on the positions around the blob
follows a Gaussian distribution. The width of the Gaussian is determined by the quantum-
limited noise from the uncertainty principle after amplification and the added noise from
the components on the readout line. Therefore the criteria to distinguish the two blobs is
the distance between the blobs d should be larger than the width of the blobs. The smaller
overlap the blobs have, the higher readout fidelity one can achieve.
43
a
b
Figure 3.3: Single-shot histograms. (a) Single-shot data taken from the experiment. The
four blobs correspond to the four computational states and have the same width. There
are some data points fallen between the blobs, which are caused by non-QNDness during
readout. (b) Fitted histograms. The slight difference between the fit and the data comes
from the readout non-QNDness.
44
Single-shot readout provides a simple way to measure the qubit population in CQED.
One can make a histogram of the single-shot data and fit it to multiple Gaussian functions
as shown in Fig. 3.3. The blobs have the same width and the heights are proportional to the
population at corresponding state. Another way is to separate the IQ into different regions
and determine the qubit state by the region where data point falls into. For example,
one can use a linear support vector machine which divides the plane with perpendicular
bisectors of the two blobs. Or one can draw a circle around each blob and discard the
points outside the circle as demonstrated in Fig. 3.13. After setting up the regions, one can
simply count the number of points in each region and the ratio give the population.
In practice, the non-QND effect can be observed in fluxonium single-shot experiments.
This can result in blob distortion as the population can change during the readout process.
Some of the data points can migrate to other blobs due to the cavity photons and some
other points get ?captured? midway and fall on the lines connecting two blobs. This effect
will be discussed in Sec. 3.2.
3.1.2 Fluorescence readout
Another way to readout the qubit is through the high energy transition fluorescence, which
is also called shelving readout. As introduced in Sec. 2.6, we mount the device chip inside
a rectangular waveguide with a specially designed single input/output port. Above the
waveguide cut-off frequency, given by its dimensions, the photons can enter or leave the
box freely and interact with the fluxonium strongly. As shown in Fig. 3.4d, the port coupling
is near unity in a few-GHz band. Below the cut-off, the port coupling drops rapidly with
frequency. It is designed that the 0?3 transition is within the passband to emit fluorescence
and other transitions are protected by the cut-off frequency. Therefore, the radiative lifetime
of state |1? can be orders of magnitude longer than that of state |3?, even though transitions
|0?-|1? and |0?-|3? have comparable dipoles ? matrix elements of the Cooper pair number
operator ?i??. Moreover, radiative decays |3? ? |1? and |3? ? |2? ? |1? are inhibited by the
parity selection rule and the weak port coupling, respectively. As a result, resonant driving
of |0?-|3? transition would continuously generate fluorescence if and only if the atom (and
hence the qubit) is initially in state |0?. Despite a periodic transit of population through the
non-computational state |3?, the atom rapidly relaxes back to state |0? on switching off the
drive. Likewise, the atom starting in state |1? remains unperturbed by the far off-resonant
readout drive (fluorescence is absent or ?shelved?).
We begin our experiment by probing the readout transition (Fig. 3.5a,b). The measured
reflection amplitude r as a function of drive frequency ? and amplitude ? fits well to the
45
0.6 * 0
0.4 * ? 20
0.2 * * ? 40
0.0
0 1 2 3 *4 5* 6 7 8
Frequency (GHz)
Figure 3.4: Measured port coupling (solid black) vs. frequency and the calculated
transition dipoles (colored stars). Note the absence of resonant modes in the adapter in
the entire 0? 8 GHz frequency range. The |0? ? |3? transition is within the passband of the
waveguide. Other transitions such as |0? ? |1? , |1? ? |2? , |2? ? |3? are protected against the
Purcell effect by the high waveguide cutoff frequency.
Eq. 2.92 model. Only four adjustable parameters are used in the global fit: ?03 = 2? ?
6.544 GHz, ? = 2??1.74 MHz, the thermal equilibrium ground state population pth0 = 0.78,
and the scaling factor converting the value of ? to the RF-source amplitude. We integrate
the reflection signal during a time ?m = 0.5 ?s after a delay ?w = 0.5 ?s. The wait time is
chosen so that it is much longer than the characteristic emission time 1/? = 91 ns but much
shorter than the time scale of non-radiative processes in the system. As a result the atom
is in the driven steady state while fluorescence is being integrated. At low power, ? ? ?,
the reflection amplitude r as a function of frequency becomes a circle in the parametric
IQ-plot, bounded by the values Re[r] = 1 for |???03| ? ? and Re[r] = 1? 2pth0 at ? = ?03.
As stronger driving saturates the |0?-|3? transition, the circle deforms into an ellipse and
progressively shrinks into a point at r = 1 for ? ? ?. This good agreement between
spectroscopy data and the Eq. 2.92 model indicates that transition |0?-|3? is coupled to
itinerant radiation with a nearly unit efficiency, similarly to previous works on artificial
two-level atoms.
The characterization presented in Fig. 3.5a,b provides absolute calibration of the mea-
surement records in terms of the ground state population p0. This is in regard with standard
superconducting qubit readout based on the dispersive interaction with a far-detuned cavity,
where the measurement records are only known up to a scaling factor to the qubit popula-
tion and have to be independently calibrated. In order to account for small thermal fluctu-
ations over the course of the experiment, we repeated the previous calibration over 5 days.
Transition dipole
| i |?? |j |
|0 -|1
|1 -|2
|2 -|3
|0 -|2
|1 -|3
|0 -|3
Port coupling (dB)
46
We find an average thermal equilibrium value of pth0 = 0.76 ? 0.026, which corresponds to
the atom being at thermal equilibrium with the effective temperature T = 45?5 mK, which
is indeed commonly encountered in superconducting qubits. In what follows, we operate
the readout at the optimal power/frequency settings (Fig. 3.5a ? black star) where the re-
flection amplitude becomes a real positive number between zero and unity, with p0 = 1? r.
The protocol for time-domain qubit manipulations matches that of traditional circuit
quantum electrodynamics. For example, applying a drive at the qubit frequency ?01 =
2? ? 1.15 GHz prior to the readout pulse results in Rabi oscillations of p0 as a function of
drive duration (Fig. 3.5c - left panel). Operating the circuit slightly off the sweet-spot at
?ext/2? = 0.507, i.e. weakly breaking the parity selection rule, we could produce Rabi os-
cillations between states |0? and |2? by compensating the vanishing transition dipole with a
strong drive at frequency ?02 = 2??3.88 GHz. Importantly, the non-zero decay rates of even
transitions resulting from breaking the parity selection rule remain several orders of magni-
tude smaller than other decoherence processes. The reflection coefficient measured during
Rabi oscillations between |0? and |1? or |2? yields equilibrium ratios pth1 /pth0 = 0.329 and
pth2 /p
th
0 = 0.007 independently of the absolute calibration of p
th
0 . Assuming p
th+pth0 1 +p
th
2 = 1,
we obtain the equilibrium value of pth0 = 0.75, in close agreement with the spectroscopic
calibration.
The energy relaxation signal following the 0?1 swap is exponential with a characteristic
decay time T1 = 52 ?s, which is nearly three orders of magnitude longer than the radiative
decay time of the |3?-state (Fig. 3.5c - right panel). Yet, given the highly suppressed port
coupling at the qubit frequency, such value of T1 is still too short to be accounted for by
radiative decay. In fact, the qubit decay is likely due to the dielectric loss in the circuit?s
antenna capacitance with an effective quality factor Q 5diel = 5 ? 10 . The spin-echo signal
decay is also exponential with a characteristic coherence time T2 = 52 ?s. The additional
dephasing ?? = 1/T2 ? 1/2T1 is likely due to measurement induced dephasing caused by
thermal photons at readout frequency. In fact, the rate of thermal jumps |0? ? |3? can
be linked to the effective photon temperature T at the microwave port via the detailed
balance equation ?? = ?exp(???03/kBT ). Plugging in the same value T = 45 ? 5 mK
corresponding to the equilibrium ground state population leads to a close agreement with
the measured dephasing rate. Such an error process could be exponentially suppressed in
the future by a moderate improvement of circuit thermalization and increasing the readout
transition frequency.
47
(a) (b)
1 1
0.5 0.5
0 * 0
-0.5 -0.5 2
1
0
- 1-1 -0.5 0 0.5 1 - 1 6.540 6.545 6.550
(c)
1.0 0
0.8 0.2
0.6 0.4
0.4 0.6
0.2 0.8
0 0 50 100 150 200 1.00 0.2 0.4 0.6
Figure 3.5: Conditional fluorescence readout. (a) Complex plane representation and (b)
real part of the reflection coefficient r as a function of readout drive frequency ? at various
powers represented by the |0?-|3? transition Rabi frequency ? (inset). The experimental
data (colored dots) are fitted by the theory of Eq. 2.92 (lines) with ? = 2? ? 1.74 MHz
and p0 = 0.78 due to thermal occupation of excited states. In the limit ? ? 0, the data
in Fig. 2a becomes a circle with a radius p0 (gray circle, p0 = 1 and dashed circle, p0 =
0.78). The optimal readout settings are highlighted by the black star and arrow (see text).
(c) Rabi oscillations (left panel), energy relaxation (right panel, yellow circles), and spin-
echo coherence decay (green triangles) measured using conditional fluorescence readout at
optimal settings. Fitting to exponential decay yields (coincidentally) T1 = T2 = 52 ?s.
48
3.2 non-QNDness
Quantum non-demolition (QND) measurement means that the quantum state is read out
without being ?destroyed? in contrast with the destructive measurement. A classical example
for destructive measurement is when detecting a flying photon with a photon-counter. The
photon is absorbed and destroyed at the detector. On the contrary, a strong QND measure-
ment projects the qubit state to the eigenstates of the measurement operator and another
immediate QND measurement after that will generate the same measurement result. Ide-
ally, the readout should be single-shot and QND to reach high fidelity. The readout fidelity
is
F = 1? P (e|g)? P (g|e), (3.9)
where P (e|g) is the probability of measuring the qubit to be at the excited state when
preparing the qubit at the ground state assuming perfect preparation and vice versa for
P (g|e). Ideally, it is related to the overlap of the blobs in the single-shot histogram. In-
tuitively, a larger readout power can lead to a greater separation of the blobs and thus
smaller overlap. However, a large photon number in the resonator can induce extra transi-
tion rates between qubit levels through measurement-induced qubit state mixing [95], [96]
and non-RWA coupling [97]. This unwanted dynamics adds an additional error rate to the
probabilities of P (x|y). There is an trade-off between the overlap error and the non-QND
error when choosing the readout power. On the other hand, another important property of a
readout pulse is the integration time. The trade-off should also considered when determine
the integration time. On a multi-qubit quantum processor, a longer integration time means
the other qubits that are not been measured are more susceptible to the decoherence error
while the measured qubit is suffering from the non-QND error. A shorter integration time,
however, will cause wider spread of the blob (the integration process can be considered as
another type of averaging but in a single-shot manner). This can be compensate with larger
power. Therefore one need to optimize the power and integration time simultaneously to
minimize the summation of the effects from overlap error, non-QND error, and decoherence
error during the readout, because they can all contribute to the logical error in a QEC code.
3.2.1 non-QNDness for dispersive readout
The non-QND process is observed by measuring the qubit population after a readout pulse
as demonstrated in Fig. 3.6 for a two-qubit device. The populations of four computational
states are changed by the first readout pulse. The measurement-induced transitions move
49
Figure 3.6: Non-QND process of the cavity readout pulse. Non-QNDness is usually ob-
served in two-fluxonium joint readout experiment at large power. A pulse at the readout
frequency and with the same amplitude and varying duration is applied to the qubits fol-
lowed by a joint single-shot readout. The qubit populations tends to be moved to the excited
states after applying a long cavity pulse. The readout pulse duration is 6 ?s in this experi-
ment.
the population to the |eg? state. The system is prepared with ?/2 pulses on both qubits.
Because the second readout pulse integration time is 6 ?s, it can be seen that already a
significant non-QND effect at the beginning of the traces (t = 0).
In our two-qubit experiment, the qubit population is measured by fitting the single-shot
histogram acquired with many repetitions. In this case, one can compensate the non-QND
error on the population measurement empirically. We assume the measured populations p?ij
are linked to populations pij at the?begin?ning of ?the re?adout by a linear transformation Me,??p? p???
gg gg
p? ?ge??? ? ?? ????pge= M ??? e . (3.10)peg p ?eg?
p?ee pee
50
The error matrix Me has 6 parameters that account for the incorrect mapping |0? ? |1?
and |1? ? |0? for both qubits and entangling transitions between the two qubits during the
readout. The?error matrix reads ?
??1? a1 ? b1 b2 a2 0???
?
b1 1? a1 ? b2 ? c1 c2 a2 ?
M = ?e . (3.11)
a1 c1 1? a2 ? b1 ? c ?2 b2 ?
0 a1 b1 1? a2 ? b2
Among the 6 parameters, a1 and a2 represent incorrect mappings |0?A ? |1?A and |1?A ?
|0?A. b1 and b2 represent incorrect mappings |0?A ? |1?B and |1?B ? |0?A. c1 and c2
describe the readout cross-talk where the two qubits swap excitations. In addition to these
six parameters, the initial populations for the two qubits p0gA, p
0
gB are also two unknown
variables that need to be determined. To calibrate the eight parameters, we perform Rabi
measurements on each qubits while the other qubit is left in the |+? state, as shown in
Fig. 3.7. Assuming that the s?ystem?star?ts with the populations
??pgg?? ??
?
p0gAp
0
? ? ? gB? ? ? ?p p0 0? ge? ?= ? gA(1? pgB) ? , (3.12)peg (1? p0gA)p0 ?gB ?
pee (1? p0gA)(1? p0gB)
one can predict the Rabi oscillation amplitudes and offsets in the two Rabi measurements
of Fig. 3.7 as a function of the 6 parameters of the error matrix plus 2 parameters for the
qubit initial population. In each Rabi measurement, there are 4 equations for the amplitude
and 4 equations for the oscillation offset. Even though there are 16 equations in total, 8 of
them are redundant in this model. Therefore the number of equations matches the number
of unknown variables. Me can be deduced by solving the 8 equations. The corrected
populations are obtained by applying M?1e to the measured populations (p
?
gg, p
? ? ? T
ge, peg, pee) .
With this calibration technique, the populations for both qubits oscillates around 0.5 and
only the rotated qubit displays oscillations during the Rabi experiment.
Alternatively, one can include the information in the readout non-QNDness measure-
ment of Fig. 3.6 and add more parameters in the error matrix to match the number of
51
a b
c d
Figure 3.7: Rabi oscillations with and without readout errors correction. Oscillations
of qubit A (a, b) and qubit B (c, d). Before correction (left), the Rabi oscillations are not
centered around 0.5 because of qubit transitions occuring during the readout. We also
correct for readout cross-talk: qubit B readout signal oscillates when qubit A is rotated
before correction, and vice-versa.
52
constrain?ts. For example, we can come up with a 10-parameter matrix ?
??1? a ? b ? d b a d
M = ???
1 1 1 2 2 2 ?
b1 + e 1? a1 ? b2 ? c1 c2 a2 ??
e . (3.13)
a1 ? e c1 + f 1? a ? b ? c ?2 1 2 b2 ?
d1 a1 ? f b1 ? f 1? a2 ? b2 ? d2
In this matrix, d1 and d2 describe the bSWAP-type of readout cross-talk connecting state
|00? and |11?, in contrast with the SWAP-type parameters c1 and c2. The parameter e (f)
describes the population that falls into the blob of |01? (|10?) when immigrating from blob
|00? (|01?) to blob |10? (|11?). The meas?urem?ent of Fig. 3.6 gives the fixed point P?
? of the
non-QND effect
?p?gg? ?
P? ? ???= ?p ?ge??? = M P? ?? e . (3.14)peg
p?ee
For this model, 9 constraints in the Rabi measurements are independent. Plus the 3 inde-
pendent equations in the fixed point definition Eq. 3.14, the 12 constraints can be solved for
the 12 unknow variables. Because of the extra information from the non-QNDness measure-
ment in this model, the correction on the population is more robust than the 6-parameter
matrix. It needs to be emphasized that these models are purely empirical because the
exact dynamics for the non-QNDness in fluxonium-cavity system are not fully understood
yet. More theoretical efforts are needed to investigate the accurate physical picture in the
presence of cavity photons.
3.2.2 non-QNDness for fluorescence readout
The non-QND effect also exists in the conditional fluorescence measurement, which is un-
derstood better than the fluxonium dispersive readout. To test the readout QNDness, we
drive the |0?-|3? transition for a duration ? ? 1/?, pause for a brief period for state |3? to re-
lax, swap the state of interest with state |0?, and perform the readout (Fig. 3.8a - top panel).
As a result, we acquire the population transients p0(t), p1(t), and p2(t) of states |0?, |1?, and
|3?, respectively, during the readout. The fidelity of this population measurement is limited
by the swap gate fidelities, which is above 99% for |0??|1? and above 90% for |0??|2?. The
p0(t)-data indicates a gradual leakage of population outside the cycling manifold {|0?, |3?}
on a characteristic time scale ?cyc = 9.6 ?s. Importantly, the leakage involves only states |1?
and |2? as the data satisfies p0+p1+p2 = 1 within the measurement uncertainty (Fig. 3.8a -
53
bottom panel). The average number of fluorescence cycles Ncyc = ?? ?cyc ? 105 quantifies
the deviation of our readout from ideal QND behavior.
The finite fluorescence lifetime is expected in presence of non-radiative transitions out-
side the cycling manifold. Additional information about energy relaxation in our circuit was
obtained by measuring populations of states |0?, |1?, and |2? as a function of time following
the |0?-|2? swap (Fig. 3.8b). Combining the two data sets represented on Fig. 3.8a and b,
we constructed a relaxation model based on two additional decay mechanisms, dielectric
loss and quasiparticle tunneling across the weak junction. Indeed, such model adequately
fits the six experimental curves with only three adjustable parameters: the dielectric loss
quality factor Q 5diel = 5 ? 10 , previously introduced to explain the |1? ? |0? decay, the
dimensionless quasiparticle density xqp = 10?6, and a fudge-factor of 2.5 in front of the
|3? ? |2? decay rate. The presence of additional decay for this precise transition could be
due to a nearly resonant two-level system [66]. From this model we conclude that fluores-
cence lifetime originates from two error processes. A smaller contribution comes from the
direct decay |3? ? |1? with a characteristic time 36 ?s due to quasiparticle tunneling. Such
process may break the parity selection rule for ?ext ?= 0, a prediction that has not been tested
so far. The dominant error though comes from the decay |3? ? |2? with a characteristic time
of 3.6 ?s due to dielectric loss in the fluxonium capacitance. Therefore, improving our fabri-
cation procedures will likely increase the fluorescence lifetime and hence the QND-fidelity.
Alternatively, with a modified choice of circuit parameters, one can increase the frequency
of |1?-|2? transition and use it for cycling, in which case the fluorescence lifetime would be
limited by the qubit T1.
3.3 Noise and Quantum Limited Amplifiers
3.3.1 Added Noise
The non-linearity of a Josephson junction makes it inherently an core component to con-
struct amplifiers that are essential for detecting weak microwave signals of qubit systems.
Over decades, many different amplifiers are invented with Josephson junctions, includ-
ing the Josephson Parametric Amplifier (JPA), the Josephson Parametric Converter (JPC),
Josephson bifurcation amplifier (JBA), the Josephson Traveling Wave Parametric Amplifier
(JTWPA), etc. In this section, the basic principle of JPA will be explained as a simple exam-
ple among all these amplifiers. After that, we are going to compare it with the JTWPA that
we used in our experiments. JPA and JTWPA can be used phase insensitive amplifiers (i.e.
phase preserving amplifiers) with equal amplification on both quadratures of the signal in
54
(a) (b)
I I
X01? X
01
?
@!03
X02 02 02? X X? ?
1. 1.
0.8 0.8 p0
0.6 p1 0.6 p0
p1
p
0.4 2p 0.40 p0 + p1 + p2p1
0.2 0.2
p2 p2
0. 0.
0 10 20 30 40 50 0 50 100 150 200 250
Figure 3.8: Transient dynamics of fluorescence. (a) Time-domain evolution of popula-
tions p0 (down blue triangles), p1 (up red triangles), and p2 (yellow squares) induced by
cycling of the readout transition. Note that p0 decays approximately exponentially in time
with a characteristic time scale ?cyc = 9.6 ?s, after which the population is transfered from
state |0? to states |1? and |2?. The black circles indicate p0 + p1 + p2 and the error bars
represent the standard deviation coming from the repetition of the measurements over two
days. (b) Free evolution of populations starting predominantly from state |2?. The data in
both (a) and (b) is adequately explained by an optical pumping model (solid lines) (see
text) where the dominant error mechanism is a non-radiative decay due to dielectric loss.
population
population
55
contrast with the phase sensitive amplifier which only amplifies one quadrature.
Before introducing JPA, we need to first understand why we need Josephson-junction-
based amplifiers. The general goal of any amplifiers is to fight against noise and extract
weak signals but one also need to realize that the noise at the input of the amplifier gets
magnified together with the signal. Starting from the photons in the cavity, the noise already
exists from Heisenberg uncertainty principle [98]. The electric field of a single-mode signal
can be written as
1
E(t) = (ae?i?t + a?ei?t). (3.15)
2
Its variance ??E2? = ?|?a|2?/2, where ?|?a|2? = ?|a|2? ? |?a?|2 is the symmetric variance
of a and |a|2 = (aa? + a?a)/2. The uncertainty principle puts an constraint on the variance
?|?a|2? ? 1/2. The lower bound is the half-quantum of the vacuum noise. When it is
reached, the noise is quantum limited.
The attenuators and the amplifiers on the measurement line not only modify the noise
at the input but also add extra noise [99]. Assuming the noise at the input is Nin in terms
of the photon number, the noise at the output is Nout = G(Nin + Nadded). Nadd is the
added noise and G is the gain of the component. G > 1 for an amplifier and G < 1 for an
attenuator. Because the signal also gets modified by the same factor Sout = GSin, the SNR
at the output is
Sout SNRin
SNRout = = . (3.16)
Nout 1 +Nadded/Nin
Therefore, any components, including amplifiers, can only degrade the SNR due to the
inevitable added noise. This may sound counter-intuitive and does not mean we should not
put amplifiers on the measurement line. To understand this paradox, we can first look at
the case for the attenuator. The added noise(is )
1
Nadded = ? 1 NT (3.17)
G
and NT is the thermal photon number
1
NT =
e?
. (3.18)
?/kBT ? 1
T is the physical temperature of the attenuator and give rise to the noise through Planck?s
blackbody radiation into the line. The prefactor in front of NT in Eq. 3.17 is crucial to make
56
the model self-consistent. Without this factor, one can imagine that by connecting a chain
of 0 dB attenuators at room temperature, the noise amplitude can be increased to infinity,
which is obviously unphysical. Using Eq. 3.17, it can be checked that two consecutive
attenuators with the same temperature T and loss factor L = 1/G are equivalent to one
attenuator with temperature T and loss factor L2.
The typical readout frequency in our experiment is 7.5 GHz, corresponding to a temper-
ature of 0.36 K. The thermal photon numbers at different stages of the dilution refrigerator
are listed in Table 3.1. The noise is rather small compared to the quantum noise of the pho-
ton in the cavity but starts to become significant at the pulse tube plate (3 K). Imagine there
is a loss as small as 3 dB, the prefactor 1/G? 1 ? 1, indicating an added noise equalling to
the thermal noise will be added to the input noise, which is already a significant amount.
From Eq. 3.16, it is straightforward that the ratio Nadded/Nin determines how much SNR is
reduced. Now we can see why amplifiers are necessary. In fact, the strategy is to increase
Nin and Sin simultaneously at the cost of a small portion of SNR, such that the SNR be-
comes resilient to the Nadded afterwards because of the small Nadded/Nin ratio. Otherwise,
the weak qubit signal on the order of 10 photons coming out from the cavity can be easily
buried into the thermal noise due to a small loss on the line. Due to the limit of gain on
a single amplifier, multiple amplifiers at different stages are required to transmit the signal
from the base plate inside the fridge all the way up to the room temperature acquisition
setup.
T 14 mK 80 mK 0.78 K 3 K 53 K 300 K
NT 7e-12 1e-2 1.7 7.8 147 833
Table 3.1: Thermal photon noise at different stages. Photon numbers are calculated
from Eq. 3.18 at 7.5 GHz which is the typical readout frequency.
As we mentioned earlier, there is no free lunch in nature. The amplifier itself is always
accompanied by an added noise. The Haus-Caves theorem puts a lower bound on the phase
insensitive amplifier?s added noise that is requir(ed by qu)antum mechanics
Namplifier
1 1
added ? 1? . (3.19)2 G
The amplifier close to the lower bound is called the quantum limited amplifier. When the
gain G ? 1, the amplifier adds at least half of a photon noise to the signal. Generally
speaking, the added noise is due to the uncontrolled degrees of freedoms. the added noise
can be introduced in the idler mode [100]. One can define an equivalent noise temperature
57
Te for an amplifier such that
Namplifier
1
added = e??/kBTe ? . (3.20)1
In the high temperature limit kBTe ? ??, or equivalently Namplifieradded ?, the noise tempera-
ture can be approximated as
Namplifier
T = added
??
e . (3.21)
kB
Another physical quantity to describe the added noise of an amplifier is the noise figure. It
is defined as the reduction of SNR when the input noise is from a resistor at T0 = 290 K,
SNRin Nadded Te
F = = 1 + ? 1 + . (3.22)
SNRout Nin T0
In this way, one can immediately know where is optimal to put an amplifier judging by its
noise temperature or noise figure.
Quantum efficiency ? is a commonly used concept to describe the added noise in the
measurement line. For a single amplifier, it is given by
1
? = . (3.23)
Nadded + 1/2
In some literature, the definition can be slightly different ?? = ?/2. If the amplifier is
quantum limited Nadded = 1/2, ? = 1, indicating this is the best case one can achieve.
For the case of an amplification chain with large gain, the total quantum efficiency can be
calculated by normalizing the total noise at the output with the total gain
1
?tot = . (3.24)
Ntot/Gtot + 1/2
Just like the property of SNR, adding extra amplifiers will only degrade the total quantum
efficiency. A good measurement setup should have the total quantum efficiency close to
that of the first stage amplifier, which should be close to 1 [101]?[103].
3.3.2 JPA and JTWPA
JPA is one of the simplest quantum limited amplifier. The circuit model is depicted in
Fig. 3.9. The incoming signal consists of a strong pump tone and a weak signal Iin = Ipump+
58
Figure 3.9: Circuit diagram of a JPA galvanically connected to a transmission line. Iin
and Iout are the incoming and outgoing current in the transmission line. The total current
at the end of the transmission line is I, which splits into the current going to the Josephson
junction (IJ) and the current going to the capacitor (IC).
Isignal. The phase difference across the junction is denoted as ? which can be also separated
into two parts corresponding to the pump and the signal ? = ?pump + ?signal. Because the
signal tone is much smaller than the pump tone. We first solve for the system?s response to
the pump and ignore the signal part. Therefore we assume Iin = Ipump = Ip cos(?pt+?) and
? = ?pump = ?a cos?pt. We are going to introduce the conclusions briefly in this section and
leave the derivation in appendix C.1. The relation between the phase oscillation amplitude
and the input curren(t amplitude is )2 ( )
?2 2
(?2 ? ?2)? ? 0 ?3 2Ip0 p a a + (2?? 2p?a) = , (3.25)8 C?0
where ?20 = I0/(C?0) and ? = 1/(2ZcC). We keep the lowest non-linear term for the
Josephson junction.
At small pump power (i.e. small Ip), the circuit behave like a harmonic oscillator and
the resonant frequency is ?0. The Purcell rate is ? = 2?. At large pump power, the phase
amplitude ?a?s response to the pump at different frequency starts to display non-linearity
in Fig. 3.10. When the power is above the critical power, bifurcation starts to occur where
two stable states can exist at the same time at one pump tone. In this regime, the system
is called Josephson bifurcation amplifier (JBA) [104] which can also have a performance
close to the quantum limit.
The system?s response to the weak signal can then be solved assuming ?pump is not
affected by the small signal. We can define cin(out) as the field operator for the input (output)
59
Figure 3.10: Phase amplitude ?a at different pump frequency. The power is close to
the critical power and the curve deviates from the linear Lorentzian response. The external
quality factor is 100.
signal and ?? = ? ? ?p is(the signal detuning from the pump. Then w)e have
? ? 2?(?i?
? ? i(?0 ? ?p) + ? ? i?0?/2)
cout(? ) = 1 +
?
(?i?? ? ? cin(? ) (3.26)+ ?0)( i? + ?1)
?e2i??0?/2
+ ? ?
(?i?? + ?0)(? ?
c (?? ), (3.27)
i? + ? in1)
=G(??)cin(??) +M(??)c?in(???). (3.28)
The parameters ?0/1 and ? are defined as
? ? = ?2a(/4, ) (3.29)?2?2? 0 ? ? ? ?0? 2?0/1 = ? ?0 ?p . (3.30)16 2
For simplicity, let?s substitute ?? with ?. The signal gain Gs and image gain Gi can be
defined as
Gs(?) ? |Gs(?)|2, (3.31)
G (?) ? |M (?)|2i s . (3.32)
The image tone (i.e. the idler tone) and the signal tone are symmetrically distributed on
two sides of the pump tone. When the signal only has one mode, the idler mode on the
other side of the pump becomes the source of the added noise that will get mixed with the
60
Figure 3.11: Signal gain Gs at different pump frequency and power. The signal fre-
quency is kept the same as the pump frequency in this simulation and ??p = ?pump ? ?0.
At small power (P = 0.5Pcritical), the signal gain is small (purple line). When the power
is close to the critical power (P = 0.99Pcritical), the gain peak is high and sharp (green
line). The pump frequency at the maximum gain is also further away from the low power
resonance frequency ?0.
signal mode at the output.
As shown in Fig. 3.11, when the pump and signal have the same frequency, the maxi-
mum signal gain Gs gets larger with increasing pump power and diverges at critical power
Pcritical. The gain peak shifts to the low frequency side, which is consistent with the bending
direction of the curve in Fig. 3.10. On the other hand, we can fix the pump frequency at the
maximum gain and change the signal power, as demonstrated in Fig. 3.11. When the signal
is at the same frequency as the pump (? = 0), the gain is maximized. The gain curve is
symmetric with respect to the pump frequency. The?half-maximum bandwidth B? is smaller
at higher pump power. One can prove that B? ? 1/ Gmax, which is quite common in many
amplifiers. At the same time, it is also worth mentioning that, in practice, it is more im-
portant to talk about the ?bandwidth? achieving over 20 dB rather than the half-maximum
bandwidth B?.
The bandwidth B? when Gmax reaches 20 dB is also called dynamic bandwidth. Typ-
ically it is on the order of few MHz [59]. From Fig. 3.12, we can see the bandwidth is
proportional to ?, which is linked to the external Q of the circuit. Because JPA requires a
high-Q system, one can not broaden the bandwidth by simply lower the quality factor. One
strategy is to fabricate a SQUID instead of a single junction which adds tunability to the
system. By operating at different flux point, one can achieve 20 dB gain over 500 MHz,
61
Figure 3.12: Signal gain Gs at different signal frequency and pump power. The pump
frequency is set to get the maximum gain. At higher power, the bandwidth is smaller, but
the frequency range where the gain is above 20 dB is actually broader.
which is called the static bandwidth.
In recent years, a new type of amplifier called Josephson traveling-wave parametric
amplifier was invented. The JTWPA used in our lab is from Lincoln Labs. It takes advantage
of a metamaterial transmission line made by Josephson junction chain. It can achieve 20 dB
gain over 3 GHz bandwidth by satifying the phase-matching condition. This is accomplished
by manipulating the dispersion relation using an array of subwavelength resonators. Apart
from the wide bandwidth, JTWPA can also tolerate higher input power than JPA. This is
quantified by 1 dB-compression point ? the input power when the gain is reduced by 1dB.
The typical 1 dB-compression point can be -120 dBm for JPA, while our JTWPA can reach
-105 to -95 dBm.
3.4 Initialization Schemes
In fluxonium experiments, fast initialization is instrumental to increase the averaged signal
contrast and hence speed up the measurement. It is also going to be indispensable in
the future when doing repetitive error corrections on logical qubits during a complicated
algorithm. We explored some initialization techniques in our lab and other schemes will
also be discussed in this section.
62
3.4.1 Preselection initialization
If the readout is quite QND, one can preselect the data to achieve an effective initialization
in the experiment. At the beginning of each sequence, an additional preselection readout
pulse is applied to perform single-shot readout. This means that the acquisition setup will
record two set of data in an interleaved way. One corresponds to the measurement result
before the sequence and the other is the usual measurement after the sequence.
The first readout pulse is QND, meaning that it does not change the ensamble probabil-
ity distribution of the mixed state. Therefore directly averaging the post-sequence measure-
ment data will give the same result as if there is no preselection pulse. Instead, one needs to
analyze the single-shot histogram of the pre-sequence measurement and select part of the
data. For example, as shown in Fig. 3.13, if we want to initialize the qubit to |10? state, we
select the sequence indices for which the pre-sequence data fall into the solid circle. Then
we pick out the data from the post-sequence measurement that have the same sequence
indices. The data after selection look like Fig. 3.13b. The population is concentrated into
the |10? blob. There is some residual population visible on |11? state. This is mainly because
the non-QNDness of the pre-sequence pulse moves the population to the |11? state.
This initialization method sacrifices the data collection efficiency and is not scalable. As
the number of qubit increases, the population of a specific state can decrease exponentially.
Because the contrast can only be enhanced after collecting a large amount of data and
post-processing, it is not as straightforward as other initialization schemes.
3.4.2 Cavity tone initialization
It is observed that for the devices used in our experiments, the readout can be highly non-
QND with a high readout power and long pulse duration. The population tend to be moved
to the exicted states, as shown in Fig. 3.14. The mechanism is not fully understood yet,
but there is evidence that the high levels are involved during the process. Typically a 10 ?s
delay need to be set between the initialization pulse and the following sequence in order
to wait for the high level population gets depleted and decays back to the computational
space.
We tried this scheme on two different two-qubit devices. We achieved 69 % and 82 %
excited state population on one device along with 86 % and 88 % on the other device.
63
a
b
Figure 3.13: Initialization by preselection. (a) Single-shot data from the heralding read-
out pulse. The populations extracted from the heights of the blobs are p00 = 0.17, p01 =
0.17, p10 = 0.37, p11 = 0.29. The experiments corresponding to the points within the solid
circle are selected in order to initialize the population to |10?. (b) Single-shot histograms
after preselection. The populations are p00 = 0.02, p01 = 0.02, p10 = 0.72, p11 = 0.25.
There is still a significant population at |11?, which is due to the readout non-QNDness.
64
Figure 3.14: Initialization using the cavity pulse Non-QNDness. A strong pump pulse at
the readout frequency is applied to pump the qubit population to the excited states. A 10 ?s
delay is placed after the initialization pulse in order to deplete the population outside the
compuational space. The population of the ee state can reach 0.6 after 10 ?s of pumping.
65
0.8
0.6 p0
p1
0.4 p2
0.2
0.0
0 5 10 15 20 25 30
time(?s)
Figure 3.15: Qubit population initialization by pumping |1?-|3?. Time-domain evolution
of populations during |1?-|3? pumping. 88% of the population can be prepared to |0? after
pumping for 5 ?s.
3.4.3 Initialization in the fluorescence experiment
With the ability to drive the even transitions at the flux point in the fluorescence experiment
(Sec. 3.1.2), we could prepare the ground state by driving the transition |1?-|3? as presented
in Fig. 3.15. By design of the readout transition, |3? decays to |0? much faster than to other
states. Therefore if thermal population gets pumped to |3?, most of it decays to |0?. We
checked that the qubit can be initialized with p0 ? 0.9 after 5 ?s pumping, limited by the
driving strength of the pumping transition |1? ? |3?. In theory, the maximal value of p0
approaches unity upon increasing T1 and reducing the error rate of decay from state |3? to
state |1?.
3.4.4 Other fluxonium initialization schemes
Similar to the fluorescence experiment, the cavity also provides a lossy transition with large
radiative decay rate. Here we use |?n? for the joint state with qubit at state ? and n photons
in the cavity. The transition |g1? ? |g0? and |e1? ? |e0? decay at a rate of ? that is on the
order of tens of MHz and can be utilized to initialize the qubit (Fig. 3.16).
One way to do this is to use the sideband transition |e0?? |g1? [105], [106]. Away from
the half flux quanta, one can directly pump this transition just like pumping the |1?-|3? tran-
sition in the fluorescence experiment. At the half flux quanta, the sideband transition is a
population
66
a b
Figure 3.16: Other initialization diagrams. (a) Sideband cooling. The system population
is pumped to |g1? using the two-photon transition of the red sideband transition |e0? ? |g1?
and then decays back to |g0? rapidly. (b) Initialization using the |h0? state. Two microwave
drives are applied to the system to move the population from |g0? to |e1? through |h0?. The
population then decays back to |e0?.
forbidden even transition and can not be directly driven. Instead, by applying a strong drive
at half of the sideband transition, the population can be pumped up through two-photon
process. A initialization fidelity of 99 % has been reported in Ref. [106]. Alternatively, if a
fast flux line is available on the chip, one can apply a flux pulse to bias the fluxonium away
from the sweet spot temporarily to open the forbidden channel and go back to sweet spot
after the initialization pulse [85].
Another approach is to use a intermediate state, for example, the |h0? state [49]. Be-
cause |g0? ? |h0? and |h0? ? |e1? both have large matrix elements, one can drive these two
transitions simultaneously to pump the population to |e1? state. A population of 95 % can
be initialized to the excited state in this way.
67
Chapter 4
Microwave Gate Experiments with
Fluxonium
Entanglement is a critical resource in quantum information. It makes a quantum computer
fundamentally distinct from a classical computer. One might be able to describe superposi-
tion of quantum states with classical probabilities. But this type of attempt will fail in the
presence of quantum entanglement as it breaks the Bell?s inequality. The quantum magics
such as quantum teleportation need to be performed at the cost of quantum entanglement,
which is also the case for the exponential speedup of the Shor?s algorithms. Entangle-
ment is generated with gates on the qubits in a quantum processor and the essential part
is two-qubit gates. For superconducting qubits, the gate can be activated with flux [49],
[85], [107]?[114] or microwave pulses [115]?[124]. The invention of flux-tunable cou-
plers makes it possible to turn the interaction between qubits on and off on-demand while
the microwave toolbox are also being expanded for the fixed frequency qubits. Generally
speaking, flux gates have shorter gate times due to its ability to turn on a large interac-
tion. But the qubit coherence time can be degraded away from the sweet spot, especially
when the qubit transition crosses two-level defects on the spectrum. On the contrary, the
microwave gate are performed with better coherence and does not require a fast flux line,
which is simpler in terms of the hardware. However, its gate time is usually longer. The
incoherent error of a gate coming from deco(herence c)an be estimated with [125]
tgate 1 1
? = + . (4.1)
3 T1 T?
Therefore the gate time and the coherence both need to be taken into account to get an
optimal gate fidelity. Recent experiments have exhibited two-qubit gate fidelity approaching
68
99.9 % for both types of architecture.
Even though fluxonium is flux-tunable by nature, we focus on microwave-activated gates
in this chapter. There are efforts exploiting flux gates on fluxoniums including theoretical
proposals and experimental demonstrations [49], [85], [126]?[128]. We believe that mi-
crowave gates implemented at flux sweet spot can make the most of the high coherence of
fluxonium qubits. We are going to first introduce the tools for characterizing gates including
process tomography and randomized benchmarking based techniques. We will discuss our
single-qubit gate measurement and then describe different types of two-qubit microwave
gates that we explored for fluxoniums. The last section will briefly present our routine for
parameter optimization.
4.1 Characterization techniques
We used quantum process tomography (QPT) and various randomized benchmarking based
methods to characterize the gates. QPT is a universal technique to identity a quantum
operation and can be implemented in most cases. However, it can be limited by state-
preparation-and-measurement (SPAM) errors [129]. This concern is addressed in random-
ized benchmarking techniques by repeating the gates and fitting for the decay constant.
There are some other techniques reported in literature but will not be discussed in this
section, such as QPT with Pauli transfer matrix representation [130] and the gate set to-
mography (GST) [131].
4.1.1 Quantum process tomography
Just like quantum state tomography (QST) can depict a full picture of a quantum state,
QPT includes all the information of a quantum operation. In our experiments, we perform
a process tomography of the gate operation using a standard procedure described in [132],
[133]. The quantum process to be characterized is interleaved between one preparation
pulse and one tomography pulse chosen in Table 4.1. The process matrix ? characterizes
fully the quantum process E . We prepare 16 input states {?j} and measure the output states
through QST. Using the information from the 16 measurements, we can extract the process
matrix ?.
State tomography
For a n qubit system, the Hilbert space dimension is d = 2n. There are 4n = d2 correspond-
ing Pauli matrices ?v1 ??v2 ?? ? ???vn , where vi = 0, 1, 2, 3. They are also d?d matrices and
69
Preparation pulses Tomography Pulses
II, IX?/2, II, X?I, X?/2I, X??/2I,
IY?/2, IX??/2, Y?/2I, Y??/2I, IX?, X?X?,
X?I, X?X?, X?/2X?, X??/2X?, Y?/2X?, Y??/2X?,
X?Y?/2, X?X??/2, IX?/2, X?X?/2, X?/2X?/2, X??/2X?/2,
Y?/2I, Y?/2X?, Y?/2Y?/2, Y??/2Y?/2, IX??/2, X?X??/2,
Y?/2Y?/2, Y?/2X??/2, X?/2X??/2, X??/2X??/2,
X??/2I, X??/2X?, Y?/2X??/2, Y??/2X??/2,
X??/2Y?/2, X??/2X??/2 IY?/2, X?Y?/2, X?/2Y?/2, X??/2Y?/2,
Y?/2Y?/2, Y??/2Y?/2, IY??/2, X?Y??/2,
X?/2Y??/2, X??/2Y??/2,
Y?/2Y??/2, Y??/2Y??/2
Table 4.1: Preparation and tomography pulses used for two-qubit quantum process
tomography. The first letter refers to the rotation axis and the subscript is the rotation
angle. I is the identity. We prepare d2 = 16 input states (where d = 4 is the dimension
of the Hilbert space). An overcomplete set of 36 pulses is used for the state tomography ?
combined with a maximum likelihood estimation ? to reduce the sensitivity of our tomog-
raphy to pulse imperfections.
are linearly independent of each other. Therefore the Pauli matrices form the basis for any
complex d ? d matrices. This naturally applies to any density state ? and it can be written
as a linear superposition of the Pauli matrices
1 ?
? = mv1,v2???vn(?v1 ? ?v2 ? ? ? ? ? ?vn) (4.2)2n
with coefficients mv1,v2???vn = tr(?v1 ? ?v2 ? ? ? ? ? ?vn?). For example, a single qubit density
matrix is
1
? = (I + tr(X?)X + tr(Y ?)Y + tr(Z?)Z). (4.3)
2
Here we used the fact that ? is Hermitian with trace one and there are only three inde-
pendent real parameters, corresponding to the three Bloch vector coordinates. Similarly,
a two-qubit density matrix can be described by 15 independent real parameters. One can
definitely write it in terms of the summation of 16 Pauli matrices. Alternatively, we parame-
terize the density matrix with 16 real parameters as ? = T ?T /Tr(T ?T ), where T is a lower
70
triangular matrix given by ? ?
?? t? 1
0 0 0
T = ?? t5 + it6 t2 0 0??
?
. (4.4)
t11 + it12 t7 + it
?
8 t3 0?
t15 + it16 t13 + it14 t9 + it10 t4
This Cholesky decomposition ensures that ? corresponds to a physical quantum state (see
Appendix F of Ref. [134]). Even though we introduced an extra redundant degree of
freedom for a more concise representation, it turns out that they can still be handled well
in numerical optimizations.
To do QST on a two-qubit joint state, the density matrix of is reconstructed by maximiz-
ing the likelihood of a list of 36 measurements obtained by applying the tomography pulses
given in Table 4.1. The Maximum Likelihood Estimation (MLE) technique [135] searches
the space of density matrices to find the one that is the most likely to reproduce the ob-
servations. To calculate the expected output signal for an arbitrary density matrix, we first
calibrate the measurement operator
M = ?II?I ? ?I + ?IZ?I ? ?z + ?ZI?z ? ?I + ?ZZ?z ? ?z (4.5)
where ?ij are complex coefficients. ?ij are measured by applying the pulses {I,X ?2?} =
{II, IX?, X?I,X?X?} to the initial state before measuring the output signal, taking into
account the initial state effective temperature from the single-shot measurement. After
calibrating the measurement operator, we can predict the output signal for a state ? after
any of the 36 tomography pulses. The state ? is then obtained by maximizing the likelihood
of the 36 output measurements over the 16 parameters of T . Here we did not do single-shot
readout to directly measure the population in each of the 36 measurements and measured
the averaged value tr(M?) instead, simply because fore our data acquisition setup, it is
faster than demodulating each recorded trace and fitting the single-shot histograms.
Process matrix
The ? matrix representation is defined as follows. For any quantum process E acting on a
density matrix ?, the final density matrix can be written as a superposition of basis operators
E?n acting on ? ?
? ?final = E(?) = E?m?E?n?mn. (4.6)
mn
71
? ? ? ?
The operator basis we choose is {?I/ 2, ?x/ 2, ?y/ 2, ?z/ 2}?n. ? is a d2 ? d2 com-
plex matrix containing d4 ? d2 independent real parameters required by the density matrix
properties. To calculate ? matrix, we first prepare d2 linearly independent density matrices
that automatically form the basis for any density matrix. When we apply E to one of these
density matrix ?j , the final state can be writte?n with the basis
E(?j) = ?jk?k. (4.7)
k
The density matrix basis is prepared by applying the preparation pulses in Table 4.1. In
the experiment, the final state E(?j) is obtained with QST. It is worth mentioning that the
density matrices are not orthogonal to each other (and actually they can be mixed states).
The coefficients ?jk can not be directly calculated with tr(E(?j)?k) like what we do with the
Pauli matrix basis. One need to fi?rst calculate the Gram matr? ?
ix
tr(?1?1) . . . tr(?d2?1)
. . ?
G = ?? .. . . . .. ?? . (4.8)
tr(?1?d2) . . . tr(?d2?d2)
The coefficients can be calcul?ated by? ? ? ?? ?j1 ?? tr(E(?j)?1)? .. ? = G? ??? . ?1. .. ?? . (4.9)
?jd2 tr(E(?j)?d2)
On the other hand, the right hand side of Eq. 4?.6 can be written as
E? ? E?? mnm j n = ?jk ?k, (4.10)
k
which can be calculated theoreti?cally. Combining?Eq. 4.7 and Eq. 4.10, we have
?mn?
mn
jk ?k = ?jk?k. (4.11)
kmn k
Because the expansion on the basis {??k} is unique, the coefficients should all equal
?mnjk ?mn = ?jk. (4.12)
mn
72
a
b
Figure 4.1: ? matrix in quantum process tomography of a CZ gate. (a) The ? matrix of
a CZ gate in theory. The height of the histogram indicates the magnitude of the matrix ele-
ment and the color indicates the phase. The non-zero elements only exist for II, IZ, ZI, ZZ
indexes.(b) The measured ? matrix for the CZ gate introduced in Sec. 4.3.1. The fidelity
calculated by comparing the ? matrix is 0.959.
73
Therefore the ? matrix is directly related to the measurement observables through the ?
matrix. For convenience, we can reshape the matrix ? and ? to vectors and the tensor ? to
d4 ? d4 matrix
??? = ??. (4.13)
Then it is straightforward to calculate the vector ?? = ??? with ? = ??1 and reshape ?? back
to its matrix form. In the case that ??1 does not exist, one can always find a generalized
inverse ? such that ??? = ?.
The ideal quantum process is described by a unitary operator
Eideal(?) = U?U ?. (4.14)
We can write the unitary operator U in terms of the operat?or ba?sis {E?n}
? ?? e1e
U = e E? = (E? , E? , ? ? ? , E? 2)??? 2. ?
?
n n 1 2 d
..n ??
?
. (4.15)
ed2
We can define e? = (e1, e2, . . . , e Td2) and the ideal process matrix is ? Tideal = e?e? from
Eq. 4.6. The gate fidelity of the measured ? matrix [136] compared to ?ideal is F (?, ?ideal) =
(tr(???ideal)d+ 1)/(d+ 1). Fig. 4.1 shows the ? matrix histogram of a CZ gate in Sec. 4.3.1
with fidelity 0.959. Additional process matrices ? of CPhase gates for control phases
? = ?/16, ?/8, and ?/4 are also measured. We found the QPT fidelity saturates around
0.978 due to SPAM errors and it is always lower than the fidelity from the cross-entropy
benchmarking experiment.
4.1.2 Randomized benchmarking
Randomized benchmarking (RB) [137]?[140] is a standard technique to quantify the aver-
age fidelity of the gate inside the Clifford group [141]. Inspired by this idea, many other
similar techniques are invented to focus on different types of fidelity measurements. As
will introduced later, interleaved randomized benchmarking (IRB) is used to measure the
fidelity of a specific Clifford gate. Purity benchmarking is to measure the average incoher-
ent error of the Clifford group. Cross-entropy benchmarking is used to measure the fidelity
of a non-Clifford gate.
74
There are 24 unique Clifford gates (up to a overall phase factor) in the single-qubit
Clifford group. These gates describe any possible rotations between the six axial states
(eigenvectors of the Pauli matrices) on the Bloch sphere. Depending on the pulses used in
a experiment, the Clifford gate representation can be different as shown in Table 4.2. One
can write Clifford gates using only X and Y rotations (including I). Or equivalently, each
Clifford gate can be constructed with one X or Y rotations at most with Z rotations. A
Clifford gate can have a different number of pulses (i.e. I,X, Y , or Z gates). On average,
one Clifford gate has 1.875 pulses In the first representation and 1.833 pulses in the second
representation. Two-qubit Clifford group is much larger and has 11520 gates. They fall into
four categories: single-qubit based gates, CNOT-like gates, iSWAP-like gates, and SWAP-like
gates. Likewise, they can have different representations. For example, one can use CZ gate
and single-qubit X and Y rotations to construct the Clifford gates. In this case, it requires
on average 1.5 CZ gates and 8.25 single qubit gates to build a two-qubit Clifford gate.
The pulse sequence for a standard RB is depicted as Fig. 4.2a. A series of randomly
chosen Clifford gates Ci are applied to the qubit (the random number seed also changes
with the sequence length m). From the mathematical properties of a group, we know that
multiplying Clifford gates will still end up with a gate in the Clifford group and its inverse
also exists in the Clifford group. Therefore we can apply a recovery gate Cr at the end to go
back to the initial state. The whole process is equivalent to an identity gate ideally. Due to
the gate errors accumulated in the sequence, the final state always deviates from the initial
state. We can measure the final state population and average it over many different random
Clifford sequence with the same sequence length. The sequence fidelity (sometimes people
directly use the qubit population) follows the curve
Fseq(m) = Ap
m +B. (4.16)
A and B absorb the SPAM errors and the error of the recovery gate. p is the depolarizing
parameter that is related to the average error of a Clifford gate r = (d? 1)(1? p)/d, where
d = 2n is the Hilbert space dimension. The average error for each pulse in the Clifford gate
is therefore rpulse = r/1.875 for the left representation in Table 4.2. In the experiment, we
fit the exponential decay curve with Eq. 4.16 and use the fitting error to calculate the error
bar on the Clifford gate fidelity F = 1 ? r (or 1 ? rpulse for the fidelity of the pulse). It is
obvious that by definition the above formalism can be directly applied to multi-qubit case.
75
a
b
c
d
Figure 4.2: Different benchmarking techniques based on random Clifford gates. (a)
Standard randomized benchmarking (RB) sequence. A random sequence of Clifford gates
{C1, C2, ? ? ? , Cm} is applied to the qubit and a recovery gate Cr is applied at the end to
convert the qubit back to the initial state. (b) Interleaved randomized benchmarking (IRB)
sequence. The test Clifford gate G is inserted after each random Clifford gate in the RB
sequence. (c) Single-qubit purity benchmarking (PB) sequence. State tomography pulses
are applied at the end of the RB sequence in order to measure the final state purity. (d)
Two-qubit cross-entropy benchmarking (XEB) sequence. The sequence consists of m cycles,
during which a random single qubit Clifford gate [CAi , C
B
i ] is applied to both qubits, fol-
lowed by the test two-qubit gate G. Unlike IRB, the test gate G in XEB can be a non-Clifford
gate.
76
Interleaved randomized benchmarking
If we insert a specific Clifford gate G after each Clifford gate Ci in the RB sequence (see
Fig. 4.2b, now Cr needs to undo the sequence including G), we can measure the fidelity of
G by comparing the depolarizing parameters from the RB measurement [140]. Let?s denote
the depolarizing parameter of the IRB measurement by pG (the one of the RB measurement
by p). The gate error of G is given by
(d? 1)(1? pG/p)
rG = , (4.17)
d
and the gate fidelity is defined as FG = 1 ? rG. One can also insert n repetitions of G into
the sequence (Gn is still a Clifford gate). The total incoherent error increases linearly while
the unitary error increases in a non-linear way [142]. We use this to demonstrate the CZ
gate in Sec. 4.3.2 is limited by decoherence.
Purity benchmarking
To evaluate the average incoherent error of a Clifford gate, one can add tomography pulses
at the end of the RB sequence (Fig. 4.2c) to measure the purity of the final states, which
is called purity benchmarking (PB) [125], [143]. Gate errors such as misrotations do not
change the state purity while decoherence errors do. The purity is defined by P = tr(?2). It
can be normalized so that it ranges from 0 to 1 ( )
d 1
Pnormalized = ? P ? . (4.18)d 1 d
For a single qubit, P 2 2 2normalized = ??x? + ??y? + ??z? , which can be directly measured
with the three tomography pulses in Fig. 4.2c. One can repeat the same RB sequence with
different tomography pulses, calculate the purity, and then average the purity for different
random sequences. The purity decays as
P mnormalized(m) = Au +B. (4.19)
u is the unitarity. The incoherent error of a Clifford gate on average is
d? 1 ?
rincoherent = (1? u). (4.20)
d
77
Cross-entropy benchmarking
Non-Clifford two-qubit gate is a helpful tool to construct a universal gate set for arbitrary
unitary operations. Non-Clifford gates can not be evaluated with traditional IRB techniques
because it is hard to build a recovery gate for every random sequence. In recent years,
cross-entropy benchmarking technique [3], [109], [144], [145] (XEB) was implemented as
a universal approach to benchmark non-Clifford gates. Fig. 4.2d shows the pulse sequence
for a two-qubit XEB sequence. It is similar to the IRB sequence in the sense that it consists
of m cycles. In each cycle, we apply random single-qubit Clifford gates to the two qubits
individually, followed by the non-Clifford gate G. At the end of the sequence, instead
of applying a recovery gate like what we do in IRB, we directly measure the population
distribution of the two qubits {p00,i, p01,i, p10,i, p11,i}, where i = 1, 2, ? ? ? , l is the random
sequence index. Combining the l distributions into one, we have
{ } {p00,1 p01,1 p10,1 p11,1 p00,2 p01,2 p10,2 p11,2Pmeasured = pj = , , , , , , , ,
l l l l l l l l
? ? ? p00,l p01,l p10,l p11,l, , , , }, (4.21)
? l l l l
and 4lj pj = 1. On the other hand, we can calculate the probability distribution Pexpected =
{qj} in theory with the knowledge of the composition of each random sequence. We can
use cross-entropy to quantify the ?distance? between these two distributions
?4l
S(Pmeasured, Pexpected) = ? pj log qj (4.22)
j
The cross-entropy is minimized when the two distributions are identical. The sequence
fidelity can be therefore defined as
S(Pincoherent, Pexpected)? S(Pmeasured, Pexpected)
FXEB = ? , (4.23)S(Pincoherent, Pexpected) S(Pexpected, Pexpected)
where Pincoherent = {1/(4l), 1/(4l), ? ? ? , 1/(4l)} is the maximally incoherent mixture. When
the measured distribution agrees with the expected one, FXEB = 1. When the final states
are maximally mixed states, FXEB = 0. By fitting the sequence fidelity versus number of
cycles with FXEB = Apm +B, the Pauli error per cycle rPcycle is given by
d? 1
rcycle = (1? p) (4.24)
d
P d+ 1rcycle = rcycle (4.25)d
78
Then we extract the non-Clifford gate Pauli error rPG from the equation (1 ? rPcycle) = (1 ?
rP )(1? rP )(1? rP ), where rP , rPA B G A B are the average single-qubit Pauli error measured from
the simultaneous RB experiment on the two qubits. The gate fidelity of G is calculated by
converting Pauli error back to the gate error rG. Even though XEB is claimed to be able to
evade the SPAM errors, a readout error on the order of 10 % coming from non-QNDness still
needs to be calibrated out in order to get a reasonable single exponential decaying curve of
FXEB.
4.2 Single-qubit Gates
In this section, we are going to show how to perform single-qubit gates, i.e. the X,Y, and
Z gates, with microwaves. We will discuss two different scenarios ? on-resonance driving
and off-resonance driving. In our experiment, we also studied qubit control crosstalk which
can be a notable issue for single-qubit gates across a multi-qubit device.
4.2.1 X and Y gates
We can consider a simple spin model in the lab frame
H lab = ??q ??z + cos(?dt+ ?d)?x. (4.26)
2 2
The second time-dependent term in the Hamiltonian is a classical driving term. For the
0 ? 1 transition of a fluxonium, the drive strength ? is proportional to the electric field
amplitude at the qubit and the fluxonium matrix element | ?0|n |1? |. In order to obtain an
analytical solution, we go into a rotating frame with frequency ?d and initial phase ?d for a
time-independent Hamiltonian. If we denote the state in the lab frame by |?lab?, it becomes
|?RWA? = U ? |?lab? in the rotating frame. The transformation operator U = exp(?iOt),
where O = ??d?z/2 is on resonance with the drive. In the rotating frame, the Hamiltonian
transforms as
HRWA = U ?H labU ? iU ??tU. (4.27)
Then we can apply the rotating wave approximation (RWA) to keep the co-rotating term
and neglect the counter-rotating term. One can think of the counter-rotating term as a
drive with a large detuning ?d + ?q ? ?. As we will demonstrate later, large detuning
drive barely changes the qubit population but it can shift the qubit frequency. With RWA,
79
the Hamiltonian is now time-independent
HRWA
? ? ?
= ? ?z + cos?d?x ? sin?d?y. (4.28)
2 2 2
? = ?q ? ?d is the detuning of the drive. Because there is a degree of freedom on choosing
which direction of the x axis on a 2D plane, we can offset the qubit phase by applying a
unitary operator U? = exp(i??z/2) to the system, so that |?RWA? ? U ? RWA? |? ? and the
Hamiltonian becomes
? ? ?
HRWA = ? ?z + cos(?d ? ?)?x ? sin(?d ? ?)?y. (4.29)
2 2 2
We can choose ? = ?d and the drive only involves the ?x term
HRWA = ?? ??z + ?x. (4.30)
2 2
When the drive is on resonance, ? = 0. The Bloch sphere does not spin anymore and the
qubit evolution is purely an X rotation that is described by
UX = exp(?
?X
i ?x), (4.31)
2
where ?X = ?t. For example, the gate X? corresponds to ?X = ? (note that it is not
?X/2 = ?), which is a ? pulse on the qubit. X2? is equivalent to an identity gate in qubit
systems because it only brings an overall phase factor to the state. One can realize a Y
rotation by changing the phase of the drive to ?d ? ?/2 while using the same phase offset
? = ?d in Eq. 4.29.
In a typical experiment, the pulses sequences are sent to the device continuously and
each sequence contains its own initialization pulse and readout pulse. It is more conve-
nient to treat each sequence as an independent measurement and set the time origin at the
beginning of that sequence. However, the microwave generated from the RF source is con-
tinuous, meaning that the microwave phase offset ?d may be different for two successive
sequences. For example, let?s assume the sequence length is T and the phase offset of the
first sequence is ?d,0. Hence the phase offset of the second sequence is ?d,1 = ?d,0 + ?dT .
To prevent an X gate in one sequence becomes a Y gate in another sequence, we need to
keep in mind that we need to re-define the x axis of the Bloch sphere in each sequence, so
that ? = ?d is satisfied in Eq. 4.29. This will be important when we come to Sec. 4.3.3.
80
4.2.2 ac Stark shift and virtual Z gates
The Z rotation can be trickier than the X and Y rotations. We have to keep in mind that
we eventually need to work in the rotating frame on resonance with the qubit. We can not
just do nothing and wait because the qubit does not spin around the z axis in this frame.
The Z gates can be straightforward with the use of a flux pulse which can change the qubit
frequency for some time to acquire a Z phase. But this is at the cost of leaving the flux
sweet spot which is not the optimal solution. In fact, there are multiple ways to implement
Z gates using only microwaves. In practice, the virtual Z gate [146] works the best in our
experiment. But before introducing this technique, let?s look at the other approaches which
are helpful to understand qubit dynamics.
One way to realize a Z rotation is through off-resonance driving. The qubit subjected to
Eq. 4.30 rotates on the Bloch sphere around the axis n? = (??/2,?/2, 0) and the effective
Rabi frequency is ?
? 2eff = ? +?2, (4.32)
meaning that after t2? = 2?/?eff , the qubit finishes one cycle on the Bloch sphere and
returns to its starting point with an overall phase. Therefore the evolution operator in this
frame is an identity UI .
Now we can transform UI to the on-resonance rotating frame. First, let?s define U?(t) =
exp(i?dt?z/2) is the transformation operator from the lab frame to the off-resonance frame
and U0(t) = exp(i?qt?z/2) is from the lab frame to the on-resonance frame. Assuming the
evolution begins at t1 and ends at t2 = t1+t2? So the evolution operator in the on-resonance
frame is
?Z
U = U ?0(t2)U
?
?(t2)UIU? (t1)U0(t1) = exp(?i ?z), (4.33)2
where ?Z = ?t2? is picked up due to the different rotating speeds of the two frames.
If we fix the drive strength and increase the detuning such that ? ? ? > 0, the rotation
axis of the Rabi oscillation n? becomes close to the ?z axis. We can consider that one cycle
acquires a Z phase of ?2? (the sign is crucial here). Because the effective Rabi frequency
is fast, we can wait for n cycles to accumulate the Z phase ?Z = n?t2? ? 2n?. The average
accumulation rate is
? ?Z
??Z = = ? ? ?2 +?2. (4.34)
nt2?
81
Actually in this large detuning regime, it is not necessary to completely go back to the
starting point in the off-resonance frame because it will only lead to a tiny error due to a
little bit of X rotation. In this case, the external microwave drive can keep adding Z phase
to the qubit as if the qubit frequency is shifted. Indeed, this can be understood with the
dressed state approach and the energy difference between the dressed states and the bare
states is the ac Stark shift.
We can diagonalize the Hamiltonian Eq. 4.30
?
?2 +?2 ? +??
Hdressed = ?sign(?) ?z = ? ?z. (4.35)
2 2
?
The ac Stark shift (i.e. the qubit frequency change) is ?? = sign(?) ?2 +?2? ?. When the
drive is red-detuned ? = ?q??d > 0, ?? > 0 indicating the |0? state energy is pushed down
and the |1? state energy is pushed up by |??|/2, i.e. the transition is ?distended?. When the
drive is blue-detuned, the energy levels go to the opposite direction and the transition is
?squeezed?. We denote the dressed eigenstates by |0?? and |1??. One can prove that they are
actually located on the rotating axis n?.
When we apply the far-detuned drive |?| ? ?, we can approximate the original qubit
state |?? = ? |0?+ ? |1? as |?? = ? |0??+ ? |1?? in the off-resonance frame, because |i?? ? |i?.
That?s why we can use the qubit frequency shift to describe the unitary evolution
? ???UZ = exp( i( )?z), (4.36)
2
which is consistent with our previous calculation for the evolution of multiple cycles on the
Bloch sphere.
The ac Stark shift picture is straightforward when we are thinking of the accumulated
Z rotations with an off-resonant drive. However, it seems it only works at a large detun-
ing, because otherwise we can not have |i?? ? |i? and the bare states |i? will keep rotating
around the axis n?. Fortunately, this issue can be addressed with adiabatic pulses. A process
is adiabatic when the Hamiltonian changes slow enough so that the bare eigenstates can fol-
low the evolution of the instantaneous dressed eigenstates. That is, if H(t) changes slowly
with time, the eigenstates |i(0)? of H(0) evolve as |i(t)? up to a phase factor, where |i(t)?
is the instantaneous eigenstates of H(t). This idea was implemented in flux pulses [107]
which tune the repulsion between two coupled states. We can use the same Hamiltonian of
82
Eq. 4.30 to describe the two levels
H(t) = ??(t) ??z + ?x (4.37)
2 2
and assume ? = ?(t) changes with flux. The population that does not follow the dressed
state is
? ??
2
Plost = exp( | |), (4.38)2 ?t?
which is called the Landau-Zener tunnel probability [147]. The adiabatic condition is sat-
isfied when Plost ? 1, indicating that |?t?| ? ?2. For microwave pulses, we can fix ?
and change the pulse amplitude over time ? = ?(t). Similarly, we can have the adiabatic
condition for microwave pulses is
|?t?| ? ?2. (4.39)
As long as Eq. 4.39 is satisfied, the initial eigenstates will follow the rotating axis n? as it
slowly travels away from the z axis with increasing pulse amplitude and comes back when
the pulse is slowly turned off. In this case, we can safely use ac Stark shift to describe this
process even if |?| ? ? without worrying about the ?x error. The adiabatic pulse can be
understood in the frequency domain. We can take a Gaussian pulse for example. Assume
the pulse has a width of 2? in time domain, its Fourier transform is also Gaussian with a
width of 1/(2?). If the pulse changes slowly with time, ? is large and its Fourier spectrum is
sharp so that there is less frequency component on the qubit transition that has a detuning
? away from the Gaussian center.
One can immediately point out the disadvantage of an adiabatic pulse which is that it
requires a long time to finish such an operation. Again We can take the Gaussian pulse for
example to do a rough estimation. Assume the pulse has a width of 2? and amplitude of ?m
that is on the same order of magnitude with the detuning ?m ? ?. Therefore ?t?(t) ? ?m/?
and Eq. 4.39 leads to ? ? ? /?2m ? 1/?. If ?/(2?) is on the order of 10 MHz, the width
of the pulse needs to be much larger than 100 ns, which is definitely too long for a single
qubit gate. To mitigate this problem, we can leverage the pulse-shaping techniques such
as the Derivative Removal by Adiabatic Gate (DRAG) [148]?[150]. The basic idea is to
add an auxiliary pulse on the other quadrature (i.e. through ?y) with an amplitude that is
proportional to ?t?(t)/?, which reduce the Fourier spectrum near the transition. With this
technique, the pulse can be faster while maintaining part of the adiabaticity. In fact, DRAG
83
was originally proposed to suppress the leakage to the higher levels in transmon systems.
This is because the anharmonicity of transmon is on the order of 100 MHz. If a pulse with
? = 50 MHz with a duration of 10 ns is applied to the 0? 1 transition, the 1? 2 transition
will also be off-resonantly driven, which is inevitable. Therefore adiabaticity is required to
make sure the population can come back to the computational states.
Another way to realize a Z gate is to use a combination of X and Y gates. As demon-
strated in Table 4.2, the Clifford group can be constructed using X and Y rotations. For
example,
Y? ?X? = Z?, (4.40)
X??/2 ? Y?/2 ?X?/2 = Z??/2, (4.41)
X??/2 ? Y??/2 ?X?/2 = Z?/2. (4.42)
The errors of these Z gates are the summation of the corresponding X and Y gate errors.
The consequence is that the Z gate is always much worse than the X and Y gates.
Finally, we are going to introduce the optimal realization of Z gates for our experiments
? the virtual Z gate. When we want to perform a Z gate, intuitively we will rotate the Bloch
sphere around the z axis by applying certain pulses. One just need to realize that rotating
the Bloch sphere is equivalent to rotating the x and y axes in the opposite direction and the
solution is going to be much simpler. Whenever we perform a Z gate, we only need to re-
define the directions of the x and y axes conceptually without changing the qubit physically.
That is why it is called the virtual Z gate, as it happens instantly on the software level. As
we derived in Sec. 4.2.1, the X gate is defined after we choose the direction of the x axis
by setting ? = ?d in Eq. 4.29. Hence the same microwave pulse can be a Y rotation if we
set ? = ?d ? ?/2, and a Y pulse becomes an X pulse in with the new definition.
For example, let?s say we want to apply a sequence as X? ?Z?/2?X? ?Y?. The first X? pulse
is applied with the drive phase ?d (meaning that ? = ?d). The virtual Z?/2 gate changes the
axis definition and now ? = ?d??/2, which is memorized by the pulse sequence compiling
program. Therefore when we apply the second X? pulse, we need to change the drive
phase to be ?d + ?/2, which is a Y?? pulse in the original definition. For the last Y? pulse,
the drive phase should be ?d, which is a X? pulse in the original definition. In the end, the
sequence will just look like X? ?Y?? ?X? in the original definition. Other arbitrary virtual Z
rotation angles can be implemented in the same way. Unlike a physical Z pulse, the virtual
Z gate affects the definition of all the X and Y types of gate (including the two-qubit gates
with X or Y terms) after it in the sequence, while the Z type of gates are not affected
(such as CPhase gates). Thus the program needs to keep track of the virtual rotation angles
84
a b
Figure 4.3: Single-qubit gate RB for a two-qubit device. (a) Ground state population
of qubit A and (b) population of qubit B as a function of the sequence length. A list of 51
randomly chosen sequence of Clifford gates are applied to the individually or simultane-
ously to the qubits before applying a recovery gate. The average single qubit fidelity for
qubit A (resp. qubit B) for a gate duration of 45 ns (resp. 26 ns) is (99.69 ? 0.01)% (resp.
(99.91? 0.003)%) in the individual case and (99.63? 0.01)% (resp. (99.57? 0.01)%) in the
simultaneous case. The duration of a simultaneous operation is given by the longest gate
time (gate time of qubit A or gate time of qubit B if a zero-duration pulse is applied to A).
Taken this into account, the residue error on qubit B comes from cross-talk
throughout the sequence in order to assign the correct phase to the microwave pulses.
Because the virtual Z gate has zero duration, it is almost immune to decoherence just
like identity gate, except for the case where qubit A is applied with an X or Y gate and qubit
B applied with a virtual Z gate needs to wait for A to finish. The major error for the virtual Z
gate comes from the imperfect microwave IQ mixing, which is negligible compared to other
gate errors as long as the IQ mixer is working at the correct parameter range. So the virtual
Z gate is almost as perfect as the identity gate in our experiment with fidelity close to 1.
What?s more, the virtual Z gate is highly flexible with arbitrary phases and they are equally
simple to implement. It outperforms all the other realizations of Z gates in every aspect and
becomes our optimal option. This is the reason that we chose the second representation of
single-qubit Clifford group in Table 4.2 when benchmarking the gates. However, this does
not mean the ac Stark shift is not an important effect. As we will introduce in the rest of
this chapter, it can still play a major role in cross-talk and two-qubit gates.
85
4.2.3 Cross-talk
The cross-talk between single-qubit gates on different qubits can be classified as quantum
cross-talk and classical cross-talk [123], [151]?[153]. Quantum cross-talk refers to the
entangling operators between the qubits. It can be the static ZZ interaction or the cross-
resonance ZX terms in the presence of the drive. We will come back to them in Sec. 4.3.
Classical cross-talk, on the contrary, is manifested as IX or IZ types of terms that do not
create entanglement. This usually happens when the control signal on one qubit drive
line leaks to other qubit drive lines through the mutual capacitance or inductance. In this
case, two remote qubits can be affected by each other?s control signal if their control lines
are too close somewhere on the device. The classical cross-talk can also happen when the
electromagnetic field generated by the drive line on one qubit is not localized enough in
space, such that it is sensed by other nearby qubits. The latter case is especially significant
in a 3D architecture. This type of leakage is small, but can contribute to an error on the
order of 0.001 to the single-qubit gates. One can partly solve this problem by increase the
detuning between the qubits, such that the X types of misrotations are converted to the
Z type of error described by the ac Stark shift, which can be also suppressed with large
detuning. But as the number of qubit increases, this can potentially lead to the problem of
frequency. One can also apply corrections to the single-qubit pulses. For example, when
applying a X gate on qubit A, we can apply a weak Z rotation simultaneously on qubit B to
cancel out the ac Stark shift (this does not work with the virtual Z gate if qubit B is applied
with X or Y rotations at the same time). This approach can be tedious with a large number
of qubits. A clever way to address the classical cross-talk is to eliminate it on the hardware
level through ground plane shielding with airbridges [3], [154], [155].
In our experiment, we characterized cross-talk between two qubits by performing single-
qubit RB on each qubit ? individually and simultaneously. Z rotations are performed with
virtual Z gates. Therefore, our Z rotations and identity gates have a zero duration. In the
individual single-qubit RB sequence, one Clifford gate contains on average 0.83 physical
pulses. As shown in Fig. 4.3, the average individual single-qubit fidelity is (99.69 ? 0.01)%
for qubit A with 45 ns-long pulses and (99.91? 0.003)% for qubit B with 26 ns-long pulses.
In our simultaneous RB sequence, there is a possible idle time for one qubit when the
other qubit is rotated on the Bloch sphere. So the simultaneous RB sequence is longer
than the individual RB sequences with the same number of Clifford gates. The change of
fidelity when operating the qubits simultaneously cannot be directly associated to cross-
talks but can be thought of as an upper bound of the cross-talk. With such sequences, the
simultaneous single qubit fidelity is (99.63?0.01)% for qubit A and (99.57?0.01)% for qubit
86
B. Because qubit A has longer pulses than qubit B, the change of qubit A fidelity should be
closer to the cross-talk error, which is still quite small compared to the other errors such as
incoherent error.
By assigning the same sequence length to the individual RB and simultaneous RB, one
can quantify the cross-talk (or addressability) following the procedure described in [156].
Even though this condition is not satisfied in our measurement, we can still measure the
correlations in the errors by examining ?p = p12 ? p1|2p2|1. The depolarizing parameter p1|2
is obtained by fitting qubit A population p10 + p11 in the simultaneous RB and p12 is from
fitting p00 + p11. The correlation is ?p = (0.11? 0.05)% from our measurements.
4.3 Two-qubit Gates
As we mentioned at the beginning of this chapter, entanglement is essential for quantum
speedup. However, it is not a sufficient condition. According to Gottesman?Knill theorem,
a circuit consisting of only Clifford gates can be perfectly simulated within polynomial time
on a classical computer. Therefore a truely powerful quantum computer should be able to
execute a universal gate set, which can construct any unitary operations. This means even-
tually we need to be able to perform all kinds of non-Clifford gate with the finite number of
gates that we can physically implemented. For example, a standard set of universal gates
contains the Hadamard gate, phase gate, CNOT gate, and the ?/8 gates. One would need
two or more of these gates to construct a specific two-qubit gate needed in an algorithm,
such as a CPhase gate required in the Shor?s algorithm. This type of composite gate will
have an error worse than its components. Therefore, it will be quite beneficial if we have
direct access to different types of two-qubit gate from the hardware level. In this way, we
can expand the toolbox of the fundamental gates and create shortcuts in a real algorithm.
In this section, we are going to discuss the realization of several microwave two-qubit
gate schemes for capacitively coupled fluxoniums, including non-Clifford gate sets. As we
will illustrate, entanglement can be either created with the high energy transitions (off-
resonance CPhase gate and the near-resonance CZ gate) or using only computational states
(off-resonance CPhase gate, bSWAP gate, and the cross-resonance gate).
4.3.1 Off-resonance CPhase gate
An arbitrary control-phase (CPhase) gate can be realized by inducing ZZ interaction with
an external microwave drive. Here we are going to introduce two experiments. In one
experiment, we leveraged the high energy transitions. In the other experiment, we only
87
utilized the computatonal states.
For the first experiment, qubit A (B) has a frequency fA = 217.2 MHz (fB = 488.9 MHz).
While the qubit lifetimes are above 100 ?s, the Ramsey coherence times are in the 10?14 ?s
interval, limited by insufficient thermalization of the measurement lines. The relevant part
of the measured two-qubit spectrum is shown in Fig. 4.4a. The simulated quantity ?kl?k?l?
attached to every transition is the Rabi frequency that would be induced by a resonant drive
with the same amplitude for all transitions. The computational states |00?, |10?, |01?, and
|11? are separated in energy from non-computational states by at least a few GHz, which
exceeds the qubit frequencies by almost an order of magnitude. Within the computational
subspace, the two-qubit dynamics obeys the Hamiltonian
ZI IZ ZZ
H = fA + fB + ?ZZ , (4.43)
2 2 4
where ? staticZZ = ?ZZ = ?357 kHz is the ZZ-interaction strength. The quantity ?staticZZ has a
negative sign because the non-computational levels push stronger on level |11? than on the
other computational levels.
Tuning the magnitude of ?ZZ by an externally applied microwave drive can be under-
stood as follows. First, let us note that the capacitive coupling leads to much stronger inter-
action between the non-computational states, splitting the otherwise degenerate transitions
|10??|20? and |11??|21? by ? = 8.47 MHz ? |?staticZZ |, which provides us the resource to dy-
namically modify ?ZZ . The same is true for other transition pairs, e.g. |11??|12?, |01??|02?,
or |00? ? |03?, |10? ? |13?, although the splitting may vary. Consider driving the circuit at
the frequency fd, blue-detuned from the |10? ? |20? resonance by an amount ? ? ? (see
Fig. 4.4b). Except for the paired transition |11? ? |21?, there are no other circuit transitions
in the GHz-vicinity (see Fig. 4.4a) due to the large anharmonicity. Therefore, the effect of
the drive reduces to the attraction between levels in pairs {|10? , |20?} and {|11? , |21?}, while
other transitions are not driven. Sp?ecifically, if qubit B is in the ground state, there is a pos-
itive ac Stark shift ?fStark(?, ?) = ( ?2 + ?2 ? ?)/2 on the qubit A frequency, which can be
thought of as pulling level |10? towards level |20? by the ?-detuned drive (see Sec. 4.2.2).
However, this shift is larger when qubit B is in the excited state, because the detuning
(? ? ?) is smaller and the effective drive amplitude ? is larger. Therefore, the qubit ac-
quires a differential ac Stark shift, which is equivalent to modifying the ZZ-term in Eq. 4.43
as ? = ?static + ?driveZZ ZZ ZZ , where ?
drive Stark
ZZ = ?f (? , ? ??) ? ?fStark11?21 (?10?20, ?). Because
?static < 0 and ?driveZZ ZZ > 0 for typical fluxonium parameters, the total qubit-qubit interaction
?ZZ in Eq. 4.43 can be tuned through zero or increased by about an order of magnitude
compared to the static value by adjusting the drive frequency and amplitude (see Fig. 4.4c
88
a
b c
21
20
10 11
00 01
Figure 4.4: Differential ac Stark shift in a two-fluxonium circuit. (a) Spectrum of al-
lowed transitions of the two-fluxonium circuit extracted from spectroscopy data. The sim-
ulated quantity ?kl?k?l? defines an on-resonance Rabi frequency that a drive field would
induce, assuming the field amplitude is frequency-independent. Note that only transitions
between states with different parity are allowed. The green arrow indicates the drive fre-
quency fd used to induce the differential ac Stark shift. (b) Schematic of the energy level
diagram for qubit B in the ground state (left) and qubit B in the excited state (right). A
drive at frequency fd pulls the energy level |11? more than the energy level |10?, without
affecting levels |00? and |01?, which is equivalent to a differential ac Stark shift ?driveZZ > 0.
(c) Calculated total ZZ-interaction rate ? = ?|?static| + ?driveZZ ZZ ZZ for a fixed drive amplitude
(?11?21 = 52 MHz). Note, at an appropriate detuning ?, the total qubit-qubit interaction is
switched off, we get ?ZZ = 0.
qubit A energy
89
and Fig. 4.5c).
We verify that the qubit-qubit interaction indeed takes the form of Eq. 4.43 using a
tomography protocol depicted in Fig. 4.5a [157]. The pulse sequence shown results in
the observation of Ramsey-type fringes oscillating at the frequency ?ZZ . Fig. 4.5b shows
the measured oscillations by sweeping the driving frequency around fd ? 4.5 GHz and
fixing the amplitude such that ?11?21 = 52 MHz. As the drive frequency approaches either
the |10? ? |20? or |11? ? |21? transitions (marked by the black dashed lines at 4.488 GHz
and 4.496 GHz, respectively), the ZZ-interaction rate reaches about 6 MHz before the off-
resonant ac Stark shift picture breaks down (Fig. 4.5b). Beyond that point, the drive field
and the two non-computational transitions undergo a coherent energy exchange, witnessed
by the detection of rapidly oscillating ripple features in Fig. 4.5b. The ZZ-interaction can
also be controlled by off-resonantly driving the pair of doublets of transitions |00? ? |03?,
|10? ? |13?, |01? ? |31?, |00? ? |30?, near the frequency f ?d ? 6.5 GHz, where we obtained
?ZZ > 10 MHz (Fig. 4.6).
The ZZ-term in Eq. 4.43 can be switched on and off on a time scale of about 10 ns using
Gaussian-edge pulses supplemented with the commonly used derivative removal (DRAG)
distortion. During the time interval of ?ZZ(t) ?= 0, the states |10? and |11? accumulate
different phases ?10 and ?11, which is equivalent to the action of a unitary evolution oper-
ator U = diag(1, e?i?10 , 1, e?i?11) in the computational subspace. Using virtual Z rotations,
the accumulated phase can be entirely assigned to any state such as |11? to implement a
controlled-phase operation UCP(?) = diag(1, 1, 1, e?i?) where ? = ?11 ? ?10. In principle,
one can modulate both the drive frequency and amplitude during the gate pulse, but for
simplicity we fixed the drive frequency and tried two different values, fd = 4.545 GHz (de-
tuned by 49 MHz from transition |11?? |21?) and f ?d = 6.665 GHz (detuned by 55 MHz from
|00? ? |30?).
At ? = ? the CP gate belongs to the Clifford group, i.e. it becomes the controlled-Z
gate. Hence, it can be characterized using randomized benchmarking (RB), which evades
SPAM limitations. Using procedures similar to those described in Ref. [120], we optimized
the gate pulses and obtained a CZ gate fidelity of 0.989 ? 0.001 at fd = 4.545 GHz and
0.991 ? 0.001 at f ?d = 6.665 GHz (Fig. 4.7). Next, we characterize the CP gate at ? = ?
using the cross-entropy benchmarking technique. The XEB procedure applied to the CZ
gate at fd = 4.545 GHz yields a gate fidelity of 0.988 ? 0.001 which agrees with the results
of randomized benchmarking and hence validates the use of XEB for other values ? of the
controlled phase.
Finally, we apply the XEB procedure to a family of CP gates with the value of ? equally
spaced by ?/16. The extracted gate error grows approximately linearly in ? with a slope
90
a
b c
d
Figure 4.5: Tuning the ZZ-interaction. (a) Pulse sequence used to measure the interac-
tion rate ?ZZ . The qubits evolve under Uac that corresponds to the ZZ-term in Eq. 4.43. We
use a refocusing pulse on each qubit in the middle of the sequence to cancel single-qubit
Z rotations. (b) Induced ZZ-interactions (a two-qubit Ramsey-type fringe) as a function
of drive frequency around the |10? ? |20? and |11? ? |21? transitions. The color scale is
proportional to ?ZI?, which is related to qubit A population. The oscillation rate expect-
edly increases on approaching a resonance condition. (c) The induced interaction rate at
different drive frequency and drive strength. It can reach about ?ZZ = 5 MHz on-demand.
(d) Cancellation of the qubit-qubit interaction. When applying a drive at 4.65 GHz, the ZZ-
oscillations slow down and speed back up as the drive amplitude passes though the value
?11?21 = 30.4 MHz. At this point, we can extract that ?ZZ < 20 kHz.
91
a b
Figure 4.6: ZZ-interactions induced by driving around 6.5 GHz. (a) ZZ-oscillations mea-
sured with the pulse protocol depicted in Fig. 4.5(a). Within this frequency range, the four
transitions |00??|03?, |10??|13?, |01??|31?, and |00??|30? contribute to the ac Stark shift in
the computational subspace. (b) The induced interaction rate at different drive frequency
and drive strength. We achieve interaction rates exceeding 10 MHz.
of about 3 ? 10?3 per radian (Fig. 4.8(d), circular markers). For ? = ?/16, the CP gate
error reaches 2 ? 10?3, which is close to the experimental resolution limit. In order to
understand the origin of the gate error, we performed detailed master-equation simulations
of the driven two-fluxonium system with the measured coherence time in Table 4.3 [121].
Our numerical model closely reproduces the data while relying only on experimentally
measured parameters (Fig. 4.8(d), square markers). According to the model, the error is
entirely due to incoherent processes, while the coherent error is absent down to the 10?4
level thanks to the strong anharmonicity of fluxonium?s non-computational transitions.
The second experiment is perform on another two-qubit device with fA = 498 MHz and
fB = 853 MHz. Compared to the device used in the first experiment, the qubit frequencies
are higher and the charge matrix elements are larger leading to a stronger effective cou-
pling between the computational states with the same mutual capacitance between the two
fluxonium circuits. This can be supported with the measured static ZZ interaction of 0.7
MHz. Such a strong effective coupling makes it possible to induce substantial ZZ interaction
while keeping the population inside the high-coherence computational space.
Similar effects have been explored with transmon systems [122], [123] where the mi-
crowave drive needs to be applied to the two qubits simultaneously. This can be understood
as an off-resonant effect of the ZX term of the drive (see Sec. 2.4). A ZX term alone can
not generate different ac Stark shift because qubit B experiences the same drive strength
with a ? phase difference depending on the state of qubit A, indicating equal ac Stark shifts
92
a
b
Figure 4.7: Interleaved randomized benchmarking (IRB) for CZ gates. (a) IRB of a
145 ns CZ gate using the 1 ? 2 manifold. The drive frequency is fd = 4.545 GHz (detuned
by 49 MHz from transition |11? ? |21?) and the drive amplitude is ?11?21 = 68 MHz. The
gate fidelity is 0.989 ? 0.001 from IRB, which is consistent with the XEB fidelity on the
same gate. (b) IRB of a 115 ns CZ gate using the 0 ? 3 manifold. The drive frequency is
f ?d = 6.665 GHz (detuned by 55 MHz from transition |00? ? |30?) and the drive amplitude
is ?00?30 = 91 MHz. The gate fidelity is 0.991? 0.001.
93
a
b
Figure 4.8: Cross-entropy benchmarking for arbitrary CPhase gates. (a) XEB for the
phases ? = ?/2 and ?. We extract the Pauli and gate errors from the decay of the XEB
sequence fidelity with the number of cycles. The gate fidelity at ? = ?/2 reaches (99.2 ?
0.1)%. (b) Fidelity of the controlled-phase gate family. The green circles are obtained
by cross-entropy benchmarking and the magenta squares are obtained by master equation
simulations using the average T E1 and T2 shown in Table 4.3. The gate fidelity averaged
over the complete family exceeds 99.2%.
94
Figure 4.9: Measured ZZ rates when applying the drive near the qubit transitions.
The drive frequency is 650 MHz and the drive amplitude ?B is the equivalent Rabi rate
when driving qubit B on resonance. The data point at 0 pulse amplitude shows the static
ZZ interaction of the system is 0.7 MHz. The total ZZ rate gets stronger with larger drive
amplitude and then saturates around 7 MHz which is mainly because the rotating wave
approximation starts to break down.
95
in both cases. The symmetry can be broken when an addition IX term is created by ap-
plying the drive with the same frequency at qubit B. The non-local input field of the 3D
architecture fulfills this requirement by nature, enabling us to induce differential ac Stark
shift with only one port.
The ZZ interaction is induced by driving at 650 MHz in Fig. 4.9. We quantify the drive
strength with the Rabi frequency of qubit B when applying the same power on resonance.
The induced ZZ rate can be as high as 7 MHz, suggesting that a potential fast CZ gate. The
ZZ rate saturates at large drive strength. This is mainly because the driving strength starts
to be comparable with the qubit frequency and the rotating wave approximation starts to
break down. We optimized a CZ gate at ?B = 133 MHz. The gate duration is 138 ns
and the measured fidelity is (99.0 ? 0.1)%, which we believe is limited by driving induced
decoherence [152], [158] and pulse distortion [106].
4.3.2 Near-resonance CZ gate
A CZ gate can be implemented by Rabi drive |10? ? |20? and |11? ? |21? at a frequency
in between. Let us start by considering the gate transitions |10? ? |20? and |11? ? |21?
as two-level systems. Our gate exploits the geometric-phase accumulations during round
trips in these two systems. The accumulated phase can be divided into two parts: the
dynamical phase, which is proportional to the evolution time and the energy of the system,
and the geometric phase, which depends only on the trajectory followed in the Hilbert
space. Applying a microwave tone drives the system with the Hamiltonian
Hdrive = (?AnA + ?BnB) cos(2?fdt). (4.44)
When the drive frequency fd is nearly resonant with one of the two gate transitions, we
observe Rabi oscillations (Fig. 4.10a) with the resonance Rabi frequencies
?10?20 = |?10|Hdrive|20?|, (4.45)
?11?21 = |?11|Hdrive|21?|, (4.46)
respectively. A strong hybridization of the |12? and |21? states creates an imbalance between
the rotation speeds given by the ratio r = ?11?21/?10?20 ? 1.36.
In order to eliminate leakage to higher states, one needs to synchronize off-resonance
Rabi oscillations determined by the two transitions to ensure that the state vector always
comes back to the computational subspace. This is achieved by matching the generalized
96
(a) (b)
20
15
10
(c)
10-0
5 10-1
10-2
10-3
0 10-4
5.169 5.179 5.189 5.199 5.209 30 40 50 60 70 80
drive frequency (GHz) gate time (ns)
Figure 4.10: Gate principle of the near-resonance CZ gate. (a) Rabi oscillations near
the |10? ? |20? and |11? ? |21? transitions versus driving frequency with a 330 ns-long pulse.
The two transitions display different resonance Rabi frequencies characterized by the ratio
r = ?11?21/?10?20 ? 1.36. The drive frequency fd indicated in brown given by Eq. 4.48
is used to synchronize the oscillations on the two transitions. (b) Bloch sphere representa-
tions of the trajectories in the {|10 ?, |20?} and {|11?, |21?} manifolds in the frame rotating
at the drive frequency. The quantum state follows a closed path in the Hilbert space leading
to a relative phase accumulation given by the difference of the solid angles spanned by the
two trajectories. (c) Gate error versus gate duration simulated using the Hamiltonian in the
presence of a Gaussian flat-topped pulse and without decoherence. For every gate dura-
tion, drive frequency and amplitude are optimized to minimize the coherent leakage error
(dashed line). The dots represent the gate error caused by incorrect phase accumulation
?? = ?11 ? ?10 ?= ?. With the optimal pulse duration around ? 62 ns, the infidelity due to
coherent errors is well below 10?3.
pulse amplitude (mV)
demodulated signal (?V)
-20 -10 0 10
gate error
97
Rabi frequencies ? ?
? = ?2 + ?2 = ?2 211?21 10?20 + (? ??) , (4.47)
where ? = f11?21 ? fd is the detuning between the |11? ? |21? transition and the drive fre-
quency (Fig. 4.10a). A full rotation is then performed in the shortest gate time tgate = 1/?.
During the gate operation, the state vector trajectory can be depicted in a Bloch sphere
representation (Fig. 4.10b) when the system starts in |10? or |11?. These circular trajec-
tories, which are traveled in opposite directions with respect to the centers of the Bloch
spheres, define two cones inside the spheres. The cones and directions of travel define
the solid angles ?10 = 2?[1 ? (? ? ?)/?] and ?11 = 2?(1 + ?/?), which correspond to
a geometric phase accumulation ?ij = ??ij/2 on state |ij?. Our gate thus implements
a unitary operation U = diag(1, 1, ei?10 , ei?11). Using virtual Z rotations, the phase dif-
ference can be assigned to any computational state such as |11? to realize a controlled-
phase operation U = diag(1, 1, 1, ei??) with ?? = ?11 ? ?10. A CZ gate is obtained when
?? = ?(?11 ? ?10)/2 = ???/? = ??. Using this condition in Eq. 4.47 we obtain the
optimal drive frequency (brown arrow?on Fig. 4.10a)
? r2 ? (r2 ? 1)2 + r2
= ? ? 0.29 . (4.48)? r2 1
For ? given by Eq. 4.48 and ? = ?, a CZ gate with zero leakage is achieved in time exactly
tgate = 1/?.
Our understanding of the gate process is validated by simulating the Hamiltonian in
the presence of the drive term, given by Eq. 4.44. The full Hamiltonian takes into account
the dynamical phase ? = 2?? t ? 10?2dyn ZZ gate ? ? which is negligible after one gate
operation thanks to the small ZZ-interaction term. Starting with the driving frequency
given by Eq. 4.48, we minimize the final populations leakage out of the computational
subspace by adjusting the drive amplitude and frequency for every gate duration. Because
of the rising and lowering edges of the pulse in the simulation, we find a slightly longer
optimal gate duration ? 62 ns (compared to 1/? = 45.5 ns) for which coherent errors on
the gate fall below < 10?3, see Fig. 4.10c. Note, even at the error level of 10?4, the optimal
point does not require an excessive fine-tuning of the gate pulse parameters: it corresponds
to a fraction of a percent variation in terms of the gate time, gate pulse amplitude, and
frequency.
We assess the quality of the optimized CZ gate by iterative interleaved randomized
benchmarking [159]. We obtain a reference fidelity FClifford2 = (96.0? 0.1)%. Interleaving
the gate CZn, obtained by concatenating n pulses corresponding to a single CZ gate, yields
98
(a)
0.8
0.7
...
0.6
0.5
0.4 ...
0.3
0.20 5 10 15 20 25 30
(b)
99
98
97
96
95
94
93
92
91 error rate 8 x 10-3 /gate
1 2 3 4 5 6 7 8 9 10
Figure 4.11: IRB of different number of CZ gates. (a) Average probability p(|11?) as
a function of sequence length. For each sequence length, we average over 100 random
sequences repeated 1500 times. We insert a variable number n = 0, 1, ..., 10 (encoded in the
color) of CZ gates between the randomly chosen Clifford gates. (b) Fidelity of CZn versus
n. The gate error grows linearly with n indicating that the errors are incoherent.
the CZ gate fidelity of F (CZ) = (99.2 ? 0.1)% for n = 1 (yellow curve in Fig. 4.11a). The
error grows linearly with n (we tried n = 2, 3, ..., 10) (Fig. 4.11b), which is a solid evidence
that the gate error is due to incoherent processes [142].
4.3.3 bSWAP gate
A bSWAP gate can be realized through driving the two-photon transition of the 00 ? 11
transition. We can start from the spin model introduced in Sec. 2.4 and define the total
Hamiltonian Htot = H? + F? cos(?dt + ?3) using Eq. 2.37 and Eq. 2.40. To understand the
origin of the two-photon effect, we need to first go to a rotating frame with transformation
Fidelity(CZn) (%)
99
operator U2 = exp(?iO2t) and
?? ????d 0 0 0??? 0 0 0 0O = ?2 ?
?
?? , (4.49)0 0 0 0
0 0 0 ?d
which we called the bSWAP gate frame. In this rotating frame, state |00? and |11? become
almost degenerate and form a low-energy manifold. We can define ? = ?(??B ? ??A)/2 and
? = ?d? (??A+ ??B)/2. If we can treat the drive term as a perturbation ?A,?B < ?, we can
apply Schrieffer-Wolff transformation [160] (see Sec. C.2) to arrive at a Hamiltonian Heff
with the effective coupling between |00? and |11? as well as the coupling between |01? and
|10?
?IZ ?ZI ?ZZ ?ER
Heff = IZ + ZI + ZZ + (XX ? Y Y )
2 2 2 4
?EI ?O
+ (XY + Y X) + (XX + Y Y ), (4.50)
4 4
where ?ER = ?E cos 2?3, ?EI = ?E sin 2?3 and
?
?E = ? (?
2
A sin 2(?? ? ?+) + ?2B sin 2(?2 2 ? + ?+)) (4.51)2(? ? )
?
? = (?2 2O ? A sin 2(?? ? ?+)? ?B sin 2(?? + ?+)). (4.52)2(?2 ?2)
The IZ, ZI, and ZZ terms are the computational state energies modified by the ac Stark
shifts in the rotating frame. Let?s ignore these terms for now and focus on the unitary
evolution of th?e last three terms in Eq. 4.50? ?? cos ?Et 0 0 ?iei(2?3) sin ?Et? 2 2??
?
0 cos ?Ot2 ?i sin
?Ot
2 0 ?U ?XY = ? t . (4.53)0 ?i sin O cos ?Ot2 2 0 ??
?ie?i(2?3) sin ?Et 0 0 cos ?Et2 2
It is clear from Eq. 4.53 that XX?Y Y and XY +Y X terms are the X-like and Y -like Rabi
rotations for the 00 ? 11 transition, corresponding to the bSWAP process. ?E is the Rabi
frequency with a quadratic dependence on the drive amplitude ?A,?B and 2? determines
the direction of the rotation axis. The XX + Y Y term is the X-like Rabi rotation for the
01 ? 10 transition, corresponding to the iSWAP process with Rabi frequency ?O, which
100
Figure 4.12: Two-tone spectroscopy of the |00? ? |11? two-photon transition under
different drive strength. The drive amplitude ?B is the equivalent Rabi rate when driving
qubit B on resonance. At low power, the transition frequency is at 677 MHz. The two-
photon transition shifts significantly at higher power due to the unequal ac stark shifts on
the two qubits.
should be avoided if we only want a bSWAP gate.
Next, we can include the ac Stark shift terms. Because the ZZ term commutes with
all the other terms in Eq. 4.50, we can separate it from the unitary evolution of the other
operators UH = UXY ZUZZ , where UXY Z contains the IZ, ZI, and the X or Y types of
terms. For simplicity, we rewrite the single-qubit Z terms as
?IZ ?ZI ?+ ??
IZ + ZI = (IZ + ZI) + (IZ ? ZI), (4.54)
2 2 4 4
and focus on?the off-diagonal terms in UXY Z .? ? i(2? ) ?E ?? t?? u 0 0 ie 3 sin E00? ?
? 2
? E? ? t ?? 0 u
?O
01 ?i ? sin O 0 ?? 2 ?
UXY Z = ? ?? O? ?O Ot ? .0 i ? sin 2 u10 0?O ???
?ie?i(2?3) ?E ?sin Et 0 0 u
?? 2 11E
? ? (4.55)
?? (??E O) are the effective Rabi frequency defined as ?
?
E = ?
2 +?2+ E (?
?
O = ?
2
? +?
2
O).
101
a b
Figure 4.13: Rabi oscillations of the |00? ? |11? two-photon transition. (a) chevron
pattern of Rabi oscillations at different drive frequency. The drive strength ?B = 222 MHz.
The pattern is not symmetric with respect to the resonance frequency. (b) Rabi oscillation
on resonance. The plateau for a ? pulse is 161 ns.
Interestingly, ?+ (??) serves as the detuning of the 00?11 (10?01) two-photon transition.
This originates from the fact that IZ +ZI does not commute with XX ? Y Y or XY + Y X
and IZ ? ZI does not commute with XX + Y Y . Because ?+ is strongly dependent on
ac Stark shifts, the two-photon transition can drift significantly at different drive power, as
illustrated in Fig. 4.12. One needs to search for the on-resonance frequency whenever the
power is changed to make sure ?+ = 0. Because the ac Stark shifts can also depend on the
drive frequency, the Rabi chevron pattern is not symmetric in Fig. 4.13 as in a typical Rabi
measurement. However, the oscillation contrast is still maximized on resonance. At this
frequency, the Rabi oscillation of 10? 01 is small due to the large detuning of ??.
An vital thing to notice is that, up to now, we are working in the rotating frame of O2
defined in Eq. 4.49. We still need to transform the unitary operator to the single-qubit gate
frame of ? ?
????? /2 0 0 0
O = ???
+
? ?0 ???/2 0 0 ??
1 ?? , (4.56)0 0 ???/2 0
0 0 0 ??+/2
and U1 = exp(?iO1t). For the sake of convenience, we only consider the bSWAP process in
102
Eq. 4.53 and set ?Et = ? for a full SWAP between |00? and |11?. The unitary operator in
the bSWAP gate frame is
?? ?? 0 0 0 ?iei(2?3)?
bSWAP ??? 0 1 0 0 ?U = ?2 . (4.57)0 0 1 0 ??
?ie?i(2?3) 0 0 0
We can assume the bSWAP gate starts at t = t1 and ends at t = t2. Following a similar
frame transformation procedure mentioned earlier in Sec. 4.2.2 and illustrated in Fig. 4.14,
we have the unitary operator in the single-qubit gate frame
UbSWAP = U ?(t )U (t )UbSWAPU ?1 1 2 2 2 2 2(t1)U1(t1). (4.58)
Of course, we need to consider the phase offset of the single-qubit control microwave ?1
and ?2, such that UbSWAP1 ??U
? U ? UbSWAP?1 ?2 1 U?2U?1 . Therefore in prac?tice, the operator is
?? 0 0 0 ?iei?E?? ?0 e?i?(t1?t2)bSWAP ? 0 0U1 = 0 0 ei?(t1?t2) 0 ???? , (4.59)
?ie?i?E 0 0 0
where ?E = ?(t1 + t2)??1 ??2 +2?3. The diagonal terms in Eq. 4.59 are trivial. They are
nothing but fixed single-qubit Z phases. However, the phase on the off-diagonal elements
are non-trivial because it depends on the position of the pulse in the sequence through
?(t1 + t2)! It originates from the fact that a two-qubit system only has two degrees of
freedom on choosing the phase offset, therefore a third microwave drive at a different
frequency can not maintain a relatively fixed phase over the whole sequence.
To prove that ?E changes with the pulse position, we performed a Ramsey-type of ex-
periment illustrated in Fig. 4.15. If the unitary operator of the bSWAP pulse takes the form
of ?
??
?
0 0 0 ?iei?E??
UbSWAP ??? 0 1 0 0= ??? , (4.60)0 0 1 0
?ie?i?E 0 0 0
the observable ?ZI? = ? sin?E = ? sin(2?f?t + ?0). By fitting the Ramsey data in
103
Figure 4.14: Rotating frame transformation for bSWAP gate. U1(t) is the unitary trans-
formation between the bSWAP gate frame and the lab frame. U2(t) is the unitary transfor-
mation between the single-qubit gate frame and the lab frame. The goal is to obtain the
evolution operator in the single-qubit frame UbSWAP1 from the bSWAP gate frame.
104
Fig. 4.15b, we found |f?| = 158 MHz. We can cancel out this effect by assigning a position-
dependent phase on the bSWAP pulse such that ? = f t + ?offset3 pulse 3 . The Ramsey fringes
disappear when fpulse = 79 MHz. These measurements are all consistent with the fact that
|?/(2?)| = |fd ? (f?A + f?B)/2| = 79 MHz in our setup.
What?s more, because we need to repeat the sequence for many times, ?E is required to
be stable across different consecutive sequences, otherwise the off-diagonal terms will look
random over many repetitions and be eventually averaged out. Assume the total duration
of a sequence is T . This converts to the condition of (2?d ? ??A ? ??B)T = 2n?, where n
is an integer. If three independent RF sources are used for the single-qubit drives and the
two-photon drive, they need to be synchronized to the same clock and their relative phase
fluctuation should also be small. We monitored the phase difference fluctuations over 30
hours between two RF sources (Rohde&Schwarz SGS100A) in our lab. The relative phase
can drift on the order of 0.03, which can be translated into an error of 2e-4 by calculating
the fidelity of two unitaries [161]
( ? ) ??? ( )??2Tr U U + Tr U ?expU ?
F = . (4.61)
20
Unfortunately, we found a significant ZZ interaction (i.e. ?ZZ in Eq. 4.50) is induced
as we apply the two-photon drive which is on the same order of the pure bSWAP gate
rate. The associated ZZ term can undermine the entangling power of the whole unitary
[?162]. We noticed ?that this issue can be addressed by constructing a composite gate as
bSWAP ? X?I ? bSWAP. This composite gate echos out the ZZ interaction and has
the same entangling power with a CNOT gate.
4.3.4 Cross-resonance gate
Similar to the transmon system, we can perform cross-resonance (CR) [115] gate on fluxo-
nium qubits as well [106]. The cross-resonance effect can be understood with the ZX terms
in Eq. 2.40. when the drive field applied on qubit A is on resonance with qubit B, the qubit
B feels a different drive phase based on the state of qubit A. In practice, we worked with
fluxoniums in a 3D architecture. The ZX and IX terms can be generated simultaneously.
Tuning the relative strength between these two terms can be done by leveraging the two
input ports of a 3D cavity. The microwave drive can be applied to the two ports simulta-
neously (at the same frequency). By adjusting the drive strength and the phase difference
of the two ports, one can achieve selective darkening of a transition in the computational
105
a
b c
Figure 4.15: bSWAP Ramsey measurements. (a) Pulse sequence to measure the accu-
mulated phase due to different rotating frame. UbSWAP brings an additional phase that
depends on the position of the pulse in the sequence. (b) bSWAP Ramsey oscillation.
The measured oscillation at |f?| = 158 MHz is proportional to ?ZI?, which is related to
qubit A population. (c) bSWAP Ramsey oscillation rate |f?| at different pulse phase fre-
quency. The bSWAP pulse phase is changed depending on the position of the pulse cen-
ter ?bSWAP = 2?fpulsetcenter to cancel out the additional phase from frame transforma-
tion. |f?| vanishes at fpulse = 79 MHz, which is consistent with the frequency difference
|f?|/2 = |fd ? (f?A + f?B)/2|. The red region indicates the noise floor of |f?| from fitting due
to finite sequence length.
106
Figure 4.16: Relative fluctuation between two different RF source. A Ramsey exper-
iment is carried out to monitor the phase difference ?? between two independent Ro-
hde&Schwarz SGS100A RF sources. ?? can have a drift of 0.03 rad over 30 hours, which
can contribute an error of 2e-4 estimated from Eq. 4.61.
space. For example, when ?00|Hdrive |01? = 0 as demonstrated in Fig. 4.17, a CNOT gate
can be implemented.
As we discussed in Sec. 4.2.3, the side effect of cross-resonance is the quantum cross-
talk as we perform single-qubit gates. This is more severe for the non-local input field of
a 3D cavity. One can apply the same trick in the CNOT gate and utilize the two input
ports. Likewise, ?00|Hdrive |01? = ?10|Hdrive |11? can be achieved by adjusting the drive
parameters on the two ports, leading to a full cancellation of a ZX term as we execute the
single-qubit gate on B.
4.4 Parameter Optimization
Here is the routine when we optimize the two-qubit gate. We start from optimizing the
single-qubit gates. We run Rabi and Ramsey experiments to find the single-qubit pulse du-
ration (amplitude) and frequency. We perform a parameter search optimizing the sequence
fidelity of fixed length RB, also known as ?Optimized Randomized Benchmarking for Im-
mediate Tune-up? (ORBIT), with a numerical Nelder-Mead method. This is to fine tune the
pulse parameters including the DRAG coefficient.
107
a
b
Figure 4.17: Gate principle for a cross-resonance gate. Two microwave drives are ap-
plied to the two qubits with different frequency ?A and ?B when (a) qubit A is at |0? state
and (b) qubit A is at |1? state. The drives have the same frequency but different amplitude
and phase. The drive at qubit A generates a ZX term. Combined with the IX term of
the drive at qubit B, one can achieve selective darkening of the |00? ? |01? transition by
adjusting the relative amplitude and phase of the drives. Qubit B can only be driven when
qubit A is at |1? state.
108
Next we perform some basic experiments to determine the parameters of the two-qubit
gate. For example, we run Ramsey-type of measurements to measure the ZZ interaction and
the single-qubit ac Stark shifts of a CZ gate. Then we confirm the two-qubit unitary with
QPT and correct any erroneous Z rotations with virtual Z gates and adjust the controlled
phase with pulse duration. In the end, we execute ORBIT on the two-qubit gate to fine tune
the duration, DRAG coefficient and the virtual Z rotations. For non-Clifford gates, we used
the sequence fidelity of the XEB measurement instead.
109
X & Y rotations only with Z rotations
I I
X? X?
Y? X?
Y?, X? Z?
X?/2, Y?/2 Y?/2, Z?/2
X?/2, Y??/2 Y??/2, Z??/2
X??/2, Y?/2 Y?/2, Z??/2
X??/2, Y??/2 Y??/2, Z?/2
Y?/2, X?/2 X?/2, Z??/2
Y?/2, X??/2 X??/2, Z?/2
Y??/2, X?/2 X?/2, Z?/2
Y??/2, X??/2 X??/2, Z??/2
X?/2 X?/2
X??/2 X??/2
Y?/2 Y?/2
Y??/2 Y??/2
X??/2, Y?/2, X?/2 Z??/2
X??/2, Y??/2, X?/2 Z?/2
X?, Y?/2 Z?/2, X?/2, Z?/2
X?, Y??/2 Z?/2, X??/2, Z?/2
Y?, X?/2 Z??/2, Y?/2, Z??/2
Y?, X??/2 Z??/2, Y??/2, Z??/2
X?/2, Y?/2, X?/2 Z??/2, X?, Z??
X??/2, Y?/2, X??/2 Z?/2, X?, Z??
Table 4.2: Single-qubit Clifford gates. The Clifford gates can have different representa-
tions based on the pulse used in the experiment.
T1 (?s) T
R
2 (?s) T
E
2 (?s)
|00? ? |10? 158? 207 10? 12 14? 15
|00? ? |01? 116? 141 13 20? 25
|11? ? |21? 4.9? 6.2 2.6? 2.8 3.3
Table 4.3: Energy relaxation time T1, Ramsey coherence time TR2 , and spin echo
coherence time TE2 . The ranges corresponds to time fluctuations during the experiment.
110
Chapter 5
Summary
5.1 Fluorescence readout
We showed that a single macroscopic degree of freedom ? the superconducting phase-
difference ? can combine a highly coherent qubit and its QND readout via conditional
fluorescence, in full analogy with Dehmelt?s electron shelving scheme. This first imple-
mentation of conditional readout by fluorescence in superconducting circuits relies on the
high anharmonicity of the fluxonium circuit and does not require an additional degree of
freedom like other implementations of V and lamda-shaped artificial atoms in circuit-QED
[163]?[165]. The present experimental setup can be supplemented with quantum-limited
linear amplifiers [166] or recently developed microwave photon counters [167]?[169] for
a single-shot operation. Indeed, we measured readout histograms separated by a third
of the standard deviation, a quantity that can be increased by a factor 10 by adding a
quantum-limited amplifier, leading to measurement distributions overlapping by 1 %. With
the current experimental parameters and the fluorescence integration time ?m = 500 ns, the
resulting single-shot measurement would see 5 % of population leaving the ground state
during readout. This amount can be reduced below the percent by a moderate increase of
the QND-fidelity. Moreover a further increase of the capacitive coupling of the circuit to the
waveguide will lead to larger readout coupling rates without affecting the qubit coherence
time, which is limited by dielectric loss. Such improvements would make readouts based on
conditional fluorescence comparable to state-of-the-art circuit-QED readouts using cavities.
Our approach offers the most direct interface between superconducting qubits and prop-
agating microwave photons while maintaining near state-of-the-art coherence times. These
properties make it a useful resource for quantum optics in the microwave range, with ap-
plications such as single photon emission and detection and photon routing controlled by
111
superposition of states [170], [171]. The possibility to route photons in a superposed way
is particularly interesting and has not been demonstrated so far, as previous implemen-
tations rely on classical controls. From a waveguide QED point of view, the addition of
a long-coherence memory transition paves the way to the preparation of highly-entangled
state over a quantum network [172] and its use for quantum computation protocols such as
teleported gates [173]. Finally our approach is compatible with a 2D implementation and
offers high advantages over traditional cQED in terms of scalability, as readout resonators
takes the most space due to their mode structure.
5.2 Two-qubit gates
The demonstrated all-microwave control over the ZZ-interaction is technologically attrac-
tive for scaling-up superconducting quantum processors. Our scheme has the following
advantages. First, it does not require close arrangement of qubit frequencies, in contrast to
the case of the cross-resonance (CR) gate [152], [174] for transmons, which would miti-
gate the spectral crowding issues [152], [175], allows tolerance to fabrication variability,
and enables multiplexing of the qubit control. For example, here we used only one input
port for the entire experiment. The qubit frequency could be as low as 100 MHz, as in our
earlier demonstration of the CZ-gate [120], but it can also be around 200 ? 500 MHz (the
main device here) or even 700 ? 1300 MHz (the second device here). Second, fluxonium?s
highly anharmonic spectra allow great flexibility of choosing drive frequencies and to oper-
ate the gate in the far-detuned regime, in which higher states are almost unoccupied during
the gate operation, reducing leakage error and error due to decoherence of higher states.
The required drive power is also comparable to single-qubit rotations [175]. This makes
our method highly scalable. Third, our scheme does not require any additional circuitry as
in schemes to cancel ZZ coupling using tunable couplers.
Among the previously explored two-qubit gates with a high degree of flexibility, our
off-resonance CPhase gate scheme is reminiscent of the resonator-induced phase (RIP) gate
[176]?[178] for transmons. The RIP gate populates an auxiliary bus mode with off-resonant
photons to induce a differential Stark shift. In comparison, our CP gate requires no aux-
iliary modes or complex pulse-shaping, has closely-confined driven dynamics, and already
enables a higher fidelity in devices with very sub-optimal coherence, all thanks to the strong
anharmonicity of fluxoniums.
Dressing qubits with microwave drives to control their interaction opens up a new route
to implement on-demand interaction in a qubit register. Similar methods for transmon
qubits have been very recently reported by two groups [122], [123]. In both cases, the
112
ZZ coupling is controlled by driving off-resonantly computational transitions of supercon-
ducting circuits. Moreover, Ref. [123] successfully applied this technique to a multiqubit
system, demonstrating the scalability of this approach.
Although the benchmarked fidelity of the two-qubit gates is already high in the experi-
ments involving the high energy transition, the current experimental setup contains a num-
ber of imperfections, most of which can be eliminated in the next generation experiments.
Let us start the discussion by briefly summarising these imperfections.
From the error estimation taking into account the Ramsey coherence times of transitions
|10?? |20? and |11?? |21?, they contribute to a significant portion both for the off-resonance
CPhase gate and the near-resonance CZ gate. Even though the off-resonance CPhase gate
can mitigate the decoherence from the higher levels by reducing the drive strength or in-
creasing the detuning, such that the Stark-shifted dressed states have less high level com-
ponents, this is at the cost of a longer gate time. We believe that the limited coherence
time of transitions involving the second excited state is resulted from the relatively short
T1 due to suboptimal fabrication conditions. Importantly, fluxonium?s |1? ? |2? transition is
very similar in term of decoherence mechanisms to a transmon qubit transition [50] and we
therefore expect no fundamental obstacles in reaching coherence times around 50? 100 ?s
in the future work.
In the computational subspace, coherence times can potentially exceed 500 ?s, given the
reported coherece times in Ref. [50]. However, here the qubit coherence time is only several
tens of microseconds, even at their flux sweet-spots, which can be explained by the presence
of about 5 ? 10?3 thermal photons, on average, in the readout resonator. This dephasing
source is generic to superconducting circuit experiments and it can be mitigated in the
future with improved cryogenic filtering of the measurement lines [179], [180]. Currently
the qubit coherence is also susceptible to the first order sensitivity to external flux noise
caused by the sweet spot misalignment. The flux difference in the loop of two fluxoniums
mainly originates from the magnetic defects in the environment and the magnetic field
inhomogeneity of the coil. The sweet spot misalignment should be close to 0.1 % of the flux
quanta in order to achieve optimal coherence simultaneously on the two qubits. For the 3D
cavity architecture in our experiment, this problem can be solved by tilting the cavity, such
that the flux difference coming from the coil can be adjusted to cancel out that from the
defects. Another better solution would be to design the two-qubit closer to each other in
space. It can be more robust in different magnetic environments. Of course, in the presence
of the individual on-chip flux bias line for each qubit, this problem will disappear naturally.
With improved qubit coherence, the two-qubit gates only relying on computational states
should have much better performance. Our simulation shows that coherent gate errors
113
in the low 10?4-range are possible for our device parameters and without sophisticated
pulse optimization. Because a large drive strength is required for these type of gates, more
investigation is needed to confirm whether the coherence could be degraded by the strong
drive. Once we achieve such low gate errors, it is nor far from the large scale universal
quantum computation.
Turning to our microwave packaging choice, we used a 3D cavity to perform all the gates
for the sake of technical simplicity. However, it also generates significant classical cross-talk.
The ground-shielding technique in the 2D architecture can enable selective addressing of
each qubit and hence improve significantly the single qubit gates fidelity. We believe that
scaling of our gate scheme can be done by moving to the 2D-circuit designs, in complete
analogy with processors based on capacitively-coupled transmons [6], [181], [182]. Note,
our experiment does not benefit from the usually high quality factors of 3D-resonators, and
hence it is compatible with a traditional 2D-circuit technology without conceptual modifi-
cations.
The single-shot readout in our two-qubit gate experiment with 3D cavities has noticeable
non-QND effect, which has not been fully understood yet. Evidence has shown that it
is related to the high energy states in the system. Readout error is not less important
than the gate error for a logical qubit in QEC. There has been studies [183] carried out
to demonstrate that it is possible to perform QND readout with high fidelity on fluxonium
qubits. Fully understanding the non-QND mechanism in the fluxonium-cavity system is
necessary to achieve a fast and accurate readout operation.
114
Appendices
115
Appendix A
Simulation
A.1 Transmission simulation
We use Ansys HFSS to simulate transmission and reflection for a 3D enclosure (cavity,
waveguide, etc.) using the driven modal solution. For example, Fig. A.1 shows the design
of a waveguide. We set up a vacuum region with perfect conductor boundary condition.
We assign wave port excitation conditions at the bottom of the ports, which have a ring
shape interface. The S parameters are then simulated by calculating the electromagnetic
field inside the box with finite element methods. The simulated S parameters at GHz range
are given in Fig. A.2 for the waveguide. In our simulation, we optimize the reflection over
various box dimensions and the launcher dimensions to make sure that S11 can be below
-20 dB over the desired frequency range.
Similar simulation was carried out in a bandpass filter project. We call it a domino
bandpass filter because it has many post-shape resonators sitting inside a tunnel (Fig. A.3).
These resonators share similar resonant frequencies such that the modes can hybridize to
form an energy band to let photons pass through. There are similar launchers at the en-
trance/exit to couple the travelling photons in the transmission line with the collective
modes in the tunnel. The posts are not identical and their dimensions are optimized with
a numerical algorithm. The external python script runs the simulation file, acquires the S
parameters, and iterates the model dimensions. The optimal design can have GHz range
flat pass-band. We machine the filter with the optimized parameters and measure the trans-
mission and reflection with a VNA, as presented in Fig. A.4. The simulation agrees well with
the measurements, demonstrating the accuracy of HFSS microwave simulation.
The purpose of this domino bandpass filter is to build a cryogenic broad band Purcell
filter made of pure copper for good thermalization. On the other hand, the posts and the
116
Figure A.1: Waveguide model in HFSS. Wave port excitation conditions are assigned at
the bottom of the two ports. The other boundaries and the launchers (the yellow parts) are
set to be perfect conductors.
a b
Figure A.2: S parameters simulation of a waveguide. (a) The reflection coefficient |S11|.
By optimizing the waveguide and the launcher dimensions, it can have a 5 GHz passband
where the reflection is below -20 dB. (b) The transmission coefficient |S12|. The transmis-
sion is flat and approaches 1 within the passband.
117
a b
Figure A.3: Domino bandpass filter model in HFSS. (a) Side-view of the filter. Each
post is a resonator and coupled to its neighbor. Together they form collective modes and
a energy band. Within the band, the photon can travel freely from one post to the next
just like dominoes. (b) Top-view of the filter. The launchers are designed to increase the
coupling between the post and the transmission line. The dimensions of each post are the
parameters to optimize the filter performance. They can be slightly different in order to
have a flat passband.
a b
Figure A.4: Domino bandpass filter S parameters. (a) The reflection coefficients. Most
resonances can be predicted by the simulation. The parameters are optimized such that
the first energy band (around 7 GHz) has a large density of states. Within this passband,
the measurements show low reflection coefficients from both ports. (b) Transmission coef-
ficients. The transmission can be flat in GHz range and the simulation agrees well with the
measurements.
118
meandering tunnel are expected to block possible high energy photons in the transmission
line that can break the cooper pair and create quasiparticles on the device. However, this
component still need more tests and we has not install it in any serious fluxonium experi-
ments yet.
A.2 Coupling simulation
Besides, HFSS can simulate the capacitance and coupling between a lumped element and a
resonant mode or the transmission line. For this purpose, we use the eigenmode solution.
The qubit is treated as a harmonic oscillator by a lumped inductance connected with a pair
of metallic pads (3D objects) as illustrated in Fig. A.5. The lumped inductance is a 2D square
sheet assigned with a boundary condition with a specified L. Because the qubit structure is
much smaller than the whole box, the mesh needs to be finer around the qubit area for a
more accurate solution. The simulation gives the resonant modes in the system, including
the 3D modes of the box and the qubit mode. The simulated mode is a complex number
with the real part representing the resonant frequency and the imaginary part representing
the decay rate. For a perfect conductor boundary condition, the imaginary part is zero. A
resistive boundary condition can be assigned to simulate the quality factor of the box mode
or the radiative decay rate of the qubit mode. The mode frequencies can be calculated with
different lumped inductance L (see Fig. A.6). The mode that changes with L is the qubit
mode and can be used to calculate the capacitance of the qubit antenna which gives the EC
in the qubit circuit. The qubit anti-crosses with the box mode that does not change with L.
By subtracting the two mode frequencies, one can find the coupling strength between the
qubit mode and the box mode.
One can assign a lumped 50 ? resistance boundary at the ports which is equivalent to
a infinitely long transmission line. The imaginary part suggests the radiative decay rate
of the qubit. We simulate this in the waveguide experiment and find that it is consistent
with the pass-band simulation from the S parameters. In addition, if the waveguide is a
reflectional setup (i.e. with only one port open), standing waves are formed inside the box
and at certain frequency the electromagnetic field vanishes near the qubit region (indicating
a node at the qubit). In this case, the qubit radiation is suppressed (Fig. A.7).
119
Figure A.5: Lumped inductance in HFSS. A lumped inductance boundary condition is
assigned to the surface connecting two metallic antenna in the HFSS model to simulate the
qubit. The mesh need to be finer here due to the small dimension compared to the copper
box.
a b
Figure A.6: Simulation of qubit capacitance and coupling strength to the cavity. (a)
Simulated mode frequency versus the lumped inductance. A resonant qubit mode is formed
with the capacitance from the antenna and the lumped inductance. This mode anti-crosses
with the cavity mode which does not change with the lumped inductance. The antenna ca-
pacitance can be calculated with the qubit mode frequency away from the anti-crossing. (b)
Frequency difference of the simulated two modes. At the anti-crossing point, the frequency
difference is at its minimal and can be used to calculate the coupling strength g.
120
Figure A.7: Simulation of radiative decay rate of a qubit inside a waveguide. The
radiative decay rate can be simulated by assigning a 50 ? lumped resistance boundary
condition at the port which is equivalent to a infinitely long transmission line. The simulated
modes are complex numbers. The real part is the mode frequency and the imaginary part
is the decay rate.
121
A.3 Fluxonium simulation
We use Qutip python package to simulate fluxonium spectrum and dynamics. We start by
constructing the basic quantum operator. The phase and charge operators are expressed as
the position and momentum operator in the harmonic oscillator basis. We first diagonalize
the single-fluxonium Hamiltonian using a relatively large Hilbert space and use part of the
eigenstates with low energy for the coupling Hamiltonian. We use the sesolve method to
simulate the unitary evolution for the four computational states in a two-qubit system.
122
Appendix B
Experiment techniques
B.1 Fabrication
The fabrication process starts from dicing the wafer. We use the high resistivity silicon wafer
or sapphire wafer for the device. For silicon samples, we dip in inside the water-ammonium
fluoride-hydrofluoric acid solution (Buffer HF Improved from Transene, Inc.) for 2 min to
remove the native oxide layer on the surface. The silicon chip treated with this buffer oxide
etch (BOE) is expected to have lower dielectric loss tangents. Then we rinse the silicon chip
with DI water.
Before spin coating the ebeam resists, we sonicate the chip inside acetone and IPA for 3
min each and blow dry it with nitrogen gas. We apply a layer of MMA EL 13 by spinning
at 5000 rpm for 1 min and bake the chip on a hot plate at 180 ?C for 1 min. We then
apply a layer of 950 PMMA A3 by spinning at 4000 rpm for 1 min and bake it at the same
temperature for 30 min.
The fluxoniums are fabricated with the Dolan bridge technique. We do ebeam lithog-
raphy to write patterns on the resists. For sapphire wafer, this requires an anti-charging
layer to remove the accumulated charge during the ebeam writing because sapphire is an
insulator. The anti-charging layer is a 11 nm aluminum layer deposited in Plassys. After
writing, it is removed by etching the aluminium in potassium hydroxide (KOH) solution.
We use Elionix to expose the resist with 100 kV electron beam. Most of the structures are
exposed with 1 nA current, except that the long couplers are exposed with 20 nA or 100 nA
current.
We develop the mask with a IPA and DI water solution which has 3:1 (IPA:water) volume
ratio. The developer is daily kept inside a thermostat at 6 ?C. When developing, we create
a simple water bath environment for the beaker by adding ice to 200 mL water to keep it
123
around 6 ?C. The chip is dipped inside the developer for 2 min and then visually checked
under the microscope.
Then we put the chip into Plassys for deposition and oxidation. The first layer is 20
nm Al deposited at 1 nm/s at an angle of 23.83 ?C. Next, oxygen flows into the chamber
and the Al layer gets oxidized at 100 mbar for 10 min. After that, oxygen is pumped out.
A second layer of Al with 40 nm is deposited at 1 nm/s at an angle of -23.83 ?C. Finally
we take out the device and start the liftoff process. The chip is submerged in acetone and
baked at 60 ?C for 3 hr.
Because the optimal ebeam doses change with the machine status or device design. We
perform dose tests when necessary. This can be done with different sizes of junctions and
doses of the critical region. We also find that the plasma frequency of our junction can drift
by a factor of two throughout the year. It is suspected this is related to the change of the
ambient humidity. After a device is fabricated, it can age at the atmosphere. The plasma
frequency goes down and 0?1 transition frequency at half flux quanta goes up. It is possible
that qubit frequency increase by 10 % in a month. To target the qubit frequency within the
desirable range, apart from carefully controlling the fabrication condition to reach consis-
tency, the laser annealing technique attracts more interests recently [184], [185].
B.2 3D cavity
We run HFSS simulation introduced in Appendix A to determine the dimension of the 3D
cavity. The cavity is manufactured with CNC machining using copper 101. The cavity is
polished with sand paper and Dremel tool. We also tried electropolishing but did not see
significant improvement on qubit coherence yet. We typically use non-magnetic connectors
at the input and output ports if available. A thin copper wire is soldered with the pin
of the connector. It reaches inside the box to couple to the cavity mode. By adjusting
the length of the wire, we can approximately have the wanted external quality factor by
fitting the reflection with Eq. 3.1. Because we use indium wire to seal the connectors, the
measured external quality factor can change with or without the indium seal due to 1 mm
level of change on the length of the wire inside the box. Therefore, it is better to seal the
connectors first and adjusting the length of the wire by cutting it if it is too long. The input
port and output port are on different parts of the cavity. When we measure the external
quality factor of one port on one part, we use the other part from a spare cavity without
ports to prevent the cavity mode from being modified by a long pin. After preparing the
ports, we mount the chip in the cavity by sandwiching small pieces of indium wire for better
thermal contact and also securing the chip in place. Finally the cavity is sealed with indium
124
and clamped with a vice to make sure the indium has a tight contact with other surfaces.
B.3 Cryogenic setup
The wiring diagram for the two-qubit experiment is presented in Fig. B.1. The experiment
was carried out in Bluefors LD 250 cryogen-free dilution refrigerator (DR). As explained
in Sec. 3.3, we need multi-stage amplification on the readout line and multiple cryogenic
circulators are installed to prevent the noise propagating in the reverse direction. Ideally,
the attenuators on the driving line should be all placed at the base plate for minimum added
noise. But due to the cooling power of each plate, the attenuators are also distributed on
multiple stages. We can simulate the thermal photon numbers of the setup depicted in
Fig. B.1 with the formalism introduced in Sec. 3.3. The dashed curves with different color
in Fig. B.2a show the attenuated thermal photons coming from the attenuators at different
temperature. Even though the thermal photons from room temperature and 3 K plate
are attenuated heavily, they are still the dominant contributors of the noise. At 7.5 GHz,
which is the typical cavity frequency, the total photon number is 1.7e-3, corresponding to
an effective temperature of 56 mK. However, this is only the ideal case because the real
temperature of the attenuators should be higher than that of the fridge plates. Additional
attenuation at base temperature is required for lower thermal photon number and hence
higher qubit coherence. At the same time, one also need to make sure the attenuators are
well thermalized and driving power will not heat up the fridge.
We use homemade Eccosorb filters (see Fig. B.3) on the driving line to absorb high
energy photons that can leads to quasiparticle loss on the qubit. Eccosorb CR 110 has low
loss at GHz range and starts heavy attenuation above 100 GHz [186]. The 4.8 mm diameter
through-hole of the copper body is designed to match the 50 ? impedance based on the
relative permittivity of Eccosorb CR 110 [187]. The connectors are sealed with indium.
We pot Eccosorb in the through-hole and bake it at around 100 ?C for lower viscosity and
shorter cure time.
We wind long superconducting (SC) wires T48B-SW-M using a drill around a brass cylin-
der body to make the coil. The wire diameter is 0.102 mm and the total room temperature
resistance is 7.85 k?. The incoming wire and the outgoing wire are anchored onto the coil
body with stycast 2850 and are twisted together where they leave the coil to cancel out
stray magnetic field. We use manganin wire as the guiding wire because manganin wire is
thicker and more visible while the SC wire is very fragile. The wires are thermalized onto
the posts (screws) at the 100 mK and 4K plates using GE varnish. Above the 4 K plate, the
wires are soldered to a connector which connects to normal wires. On our setup, we notice
125
RT
Pasternack
20 dB coupler
DC 4-12.4 GHz
TWPA pump DC
R&S SMW200A
Yokogawa Minicircuits
DC GS200 MITEQ BLK-18W-S++40dB
53 K
3 K
LNC4_16B
LNF
+40dB 4-16 GHz
780 mK
80 mK
LP 0.25 -10 GHz
14 mK
Eccosorb
 LPF
50 4-8 GHz
50 4-8 GHz
JTWPA
coil 50 4-8 GHz
50 4-8 GHz
50 4-8 GHz
in out
copper cavity
cryoperm + aluminum foil
Figure B.1: Low temperature wiring diagram of the two qubit experiment. The thermal
photons are attenuated by multi-stage attenuators. The readout signal is amplified by multi-
stage amplifiers. The circulators are necessary to keep out the amplifier back amplification
and the thermal noise.
-20dB -20dB -20dB
-20dB -10dB -20dB
126
a b
Figure B.2: Thermal photon number simulation. (a) Thermal photon numbers from dif-
ferent stages. Each stage attenuates the thermal photons from the upper stages and also
radiates the photons at its own temperature. The thermal photon number (black line) seen
by the cavity is a summation of all the stages. The dotted vertical line indicates the cav-
ity frequency at 7.5 GHz and the photon number at this frequency is 1.7e-3. (b) Effective
temperature calculated from the total thermal photon numbers. Even though the effective
temperature is higher at the high frequency, the actual photon number is lower. The effec-
tive temperature at 7.5 GHz is 56 mK.
Figure B.3: Homemade Eccosorb filter. The homemade Eccosorb filter is assembled with
a copper body, two stainless steel SMA connectors on each side and a stainless steel center
pin. After mounting one connector and the center pin, Eccosorb CR110 is potted around
the pin. When fully cured, the other connector can be installed. There are four free-fit 4-40
screw holes on the copper body for being mounted onto the cold fingers and thermalized.
127
the fridge temperature will have noticeable increase when the coil current is above 30 mA.
B.4 Room temperature setup
We used two different room temperature heterodyne setups as described in Fig. B.4. In
Fig. B.4a, we use a single-sideband-mixer (SSB) to mix down the RF source frequency fLO
to fr = fLO ? fmod before the signal goes into the fridge. On the other hand, the same
RF source acts as the local oscillator and mixes with the readout signal coming from the
fridge at the IQ mixer. Both I and Q channels are demodulated and their combination can
cancel out the noise at fr = fLO + fmod. To prove this, we can assume the signal coming
out of the RF source is SLO = ALO cos(2?fLOt+ ?1), where ?1 is an arbitrary phase offset.
The readout signal coming out of the fridge is Sout = Aout cos(2?frt + ?1 + ??), where
?? is dependent on the measurement line and the device. Then let?s consider the noise
components at frequency fLO ? fmod, as they will go into the demodulated signal. So we
write the total signal as
Sout = Aout cos(2?frt+ ?1 +??) +N? cos(2?(fLO ? fmod)t+ ??)
+N+ cos(2?(fLO + fmod)t+ ?+). (B.1)
After the IQ mixer, the I and Q channels can be written, without loss of generality, as
SI = Bout cos(2?fmodt???) +M? cos(2?fmodt+ ?1 ? ??)
+M+ cos(?2?fmodt+ ?1 ? ?+), (B.2)
SQ = Bout sin(2?fmodt???) +M? sin(2?fmodt+ ?1 ? ??)
+M+ sin(?2?fmodt+ ?1 ? ?+). (B.3)
We can demodulated the signal after digitizing the data and use phasor to represent the
terms on the IQ plane.
s = b e?i?? +m ei(?1???) +m e?i(?? 1
??+)
I out + , (B.4)
s = ib e?i?? + im ei(?1???) ? im e?i(?? 1??+)Q out + . (B.5)
Therefore we can keep s = sI ? isQ = 2b e?i??out + 2m ei(?? 1???) to cancel out the noise
component at fLO ? fmod.
The other setup in Fig. B.4b, however, have two separate RF sources, one as the LO and
the other as the readout tone. The readout RF source itself is connected to an AWG and is
128
a b
ACQ
CH1B1 CH2
IF 50 50 LO
Splitter
RF Polyphase RF
SSB mixer cavity cavity
+ mod freq LO + mod freq LO
ACQ I Q ACQ ACQ
RF source CH1 CH2 RF source mixer CH1
IQ mixer RF RF Low pass
Fridge In Fridge Out Fridge In Fridge Out
Figure B.4: Room temperature readout signal demodulation schemes. (a) The first
scheme. The output signal from the RF source is cavity frequency plus the modulation
frequency. It gets splitted with the SMA T-adapter. One way goes to the single-sideband
(SSB) mixer and gets down-converted to the cavity frequency. The other way goes to the
LO port of the IQ mixer and is used to mix down the signal from the fridge. (b) The second
scheme. The left RF source is at cavity frequency. It gets splitted with a power splitter
(Minicircuits ZFRSC-183-s+). One way directly goes to the fridge. The other way gets
mixed with the LO as the phase reference channel.
IQ (and pulse) modulated. The two channels of the acquisition card are assigned as a signal
channel and a reference channel to cancel out the relative phase fluctuation between the
two RF sources. Compared to the setup above, this setup has more phase stability on the
final demodulated value because the relative phase fluctuation on the AWG can be cancelled
as well. This setup requires more RF sources but because we use the built-in IQ modulation
on the readout RF, the output power can be much larger than the first setup.
Because we need to combine multiple microwave tones with large power in the fluxo-
nium experiment, we used different microwave components to combine the signals. Mini-
circuits ZFRSC-183-S+ is a two-port power combiner and the total loss is about 7 dB.
ZN4PD1-63HP-S+ is a four-port combiner with a similar loss. The SMA T adapter can be
also used as a two-port power combiner with lower loss but the isolation is worse than the
minicircuits one. The directional coupler is used as a combiner when one port can tolerate
high attenuation (20 dB). In this way, the other port can send the signal almost without
loss.
Splitter
mixer cavity
RF source
129
Appendix C
Derivations
C.1 Signal Gain for JPA
In this section, we are going to derive the signal gain for a JPA presented in Sec. 3.3.2 in a
classical way. A quantum mechanical description can be found in Ref. [59]. The first step
is to find the equation of motion (EOM) for the phase difference ? in Fig. 3.9. The total
current is the sum of the current in the two branches I = IC + IJ . Using the Josephson
equations, we have
d(CV ) d2?
IC = = C?0 , (C.1)
dt dt2
IJ = I0 sin ?. (C.2)
On the other hand, for the propagating waves in a transmission line, the voltage and the
current satisfy the following relation
I = Iin ? Iout, (C.3)
V = Vin + Vout = (Iin + Iout)Zc. (C.4)
Combining Eq. C.3 and Eq. C.4,
I = 2Iin ?
?0 d?
V/Zc = 2Iin ? . (C.5)
Zc dt
130
Substituting Eq. C.1 and Eq. C.2 in to Eq. C.5, we have the EOM of the phase ?
d2?(t) ?0 d?(t)
C?0 + + I0 sin ? = 2I2 in(t). (C.6)dt Zc dt
Let?s keep the lowest non-linear term sin ? ? ? ? ?3/6 and define ?20 = I0/(C?0), ? =
1/(2ZcC). The EOM can be simplified as
d2?(t) d?(t) 32 ? ? (t) 2Iin(t)+ 2? + ?0(?(t) ) = (C.7)dt2 dt 6 C?0
The incoming field can be separated into two parts ? a large pump and a weak signal.
Iin = Ipump+ Isignal. We can first neglect the signal and solve for the phase amplitude in the
presence of the pump. Assuming Iin = Ipump = Ip cos(?pt+ ?) and ? = ?pump = ?a cos?pt,
Eq. C.7 will lead to E(q. 3.25 in the main te)xt ( )
?2
2 2
(?2 ? ?2)? ? 0 ?3 + (2?? ? )2 2Ip0 p a a p a = . (C.8)8 C?0
Here we neglect the fast rotating term and simplify ?3 as
3 3
?3
? ?
= ?3a cos
3 ?pt =
a (3 cos?pt+ cos 3?pt) ? a 3 cos?pt (C.9)
4 4
Next we can deal with the weak signal part assuming it does not affect the pump. We can
write
Iin = Ip cos(?pt+ ?) + Isignal, (C.10)
? = ?a cos?pt+ ?signal. (C.11)
Similarly from Eq. C.7, the EOM for the signal tone is
d2?signal(t) d?signal(t)
+ 2? + ?2
2
1(t)?signal(t) = Isignal(t). (C.12)dt2 dt C?0
?1(t) is defined as ( )
2 2
?2(t) = ?2 ? ?a ?1 + a1 0 cos(2?pt? 2?) (C.13)4 4
131
The coefficient of the differential equation Eq. C.12 is time dependent. We can solve it in
the frequency domain, or equival?ently, b?y introducing the field operators.
? ? d??
I (t) = ? ?a (?)e?i?tsignal in + h.c., (C.14)2Zc 0 2?? d?
?signal(t) = ?
?i?t
s(?)e + h.c.. (C.15)
0 2?
In this way, Eq. C.12 becomes
? ?
2
2 ? ? ? 4? Zc ??(?0(1 ?)? ?2 ? i2??)?s(?)? 0 e2i???s(2?p ? ?) = ain(?), (C.16)2 ?0 2
where ? = ?2a/4. We can define ?
? = ? ? ?p as the detuning between the signal mode and
the pump tone. The operator of the inter?nal field in the JPA circuit is
2EJ
c(??) = ? (? ?
? s p
+ ? ). (C.17)
?0
The field operators of the input and output signal are c ? ?in(out)(? ) = ain(out)(?p + ? ). With
the new field operators, Eq. C.16 yields
?0? ?0? ?
(? ? ? ? ? 2i? ? ? ?0 p ? ? ? ? i?)c(? )? e c (?? ) = 2?cin(? ). (C.18)
2 4
From the input-outpu?t theory (or the boundary condition Eq. C.4 equivalently), we have
c (??)+c (??) = ?i 2?c(??). Then we can get rid of the internal field operator c(??out in ) and
express cout in terms of c(in ? )
? ? 2?(?i? ? i(?0 ? ?p) + ? ? i?0?/2)cout(? ) = 1 + ?
(?i?? + ? )(?i?? cin(? ) (C.19)0 + ?1)
?e2i??0?/2
+ ? ? ? ? c
? (???in ), (C.20)( i? + ?0)( i? + ?1)
=G(??)c ? ? ? ?in(? ) +M(? )cin(?? ), (C.21)
which is Eq. C.19 in the main text. Th?e parameter ?0/1 is
?2?2 ( ? ?)0 2
? 00/1 = ? ? ? ?0 ? ?p ? . (C.22)16 2
132
C.2 Schrieffer-Wolf Transformation
Imagine a Hamiltonian with a low-energy manifold and a high-energy manifold. If there is
an inter-manifold coupling, we can apply Schrieffer-Wolf transformation to find an effective
intra-manifold coupling. To clarify this point, we take the 00 ? 11 two-photon transition
introduced in Sec. 4.3.3 for an examp?le. We denote the u?nperturbed Hamiltonian by H0???00 0 0 0
H ?0 = ??
?
0 ?01 0 0 ???? . (C.23)0 0 ?10 0
0 0 0 ?11
Inside the bSWAP gate frame (see Eq. 4.49), |00? and |11? become almost degenerate
(|?| << |?|) ? ?
??? 0 0 0???0 ? 0 0 ??
?
H0 = . (C.24)
0 0 ?? 0 ??
0 0 0 ??
We can rearrange the order of the basi?s? ??? 0 0 0???0 ?? 0 0 ?H0 = ???
?
, (C.25)
0 0 ? 0
0 0 0 ??
such that the upper left block is the low-energy manifold and the lower right block is the
high-energy manifold. Therefore, we ca(n write the)tota(l Hamilton)ian in the form of
H0l 0 0 Vlh
H = H0 + V = + , (C.26)
0 H0h Vhl Vhh
where V is the driving term. Because V = V ?, we can know Vhl = V
?
lh. The drive near the
two-photon transition frequency (? (??A + ??B)/2) provides a coupling term connecting the
two manifolds. It can not exchange population between them because of large detuning.
However, this coupling effectively links |00? and |11? in the low-energy manifold. The dy-
namics can be solved by treating the driving term as a perturbation and block-diagonalizing
133
H. We can assume ( )
Heff 0
Heff = exp(iS)(H0 + V ) exp(?iS) = l , (C.27)
0 Heffh
and (1) (2) 3 . Similarly, (1) (2)S = S + S + O(V ) Heff = H 30 +Heff +Heff + O(V ). Neglecting the
high order terms,
1 ( )2 1 ( )2
Heff = [1 + iS
(1) + iS(2) ? S(1) ](H0 + V )[1? iS(1) ? iS(2) ? S(1) ] + ? ? ? .
2 2
(C.28)
Therefore
( ) (1)H (1)eff = i[S (, H0)] + V (C.29)
(2) 2 2
H = i[S(2)
1 1
eff , H0]? S(1) H ? H S(1) + S(1)0 0 H (1) (1)0S + i[S , V ]. (C.30)2 2
We assume S(1) takes the form of ( )
(1)
S(1)
0 S
= lh(1) . (C.31)
Shl 0
The first order te(rm of the effective Hamiltonian is )
(1) (1)
(1) 0 i(Slh H0h ?H0lSlh ) + VlhHeff = (1) (1) . (C.32)
i(Shl H0l ?H0hShl ) + Vhl Vhh
Block-diagonalization requires( )
( (1)i Slh H0h ? (1)H0lSlh )+ Vlh = 0, (C.33)
(1)
i Shl H0l ?
(1)
H0hShl + Vhl = 0. (C.34)
These conditions can be used to determine (1) and (1)Slh Shl . In our example,
(1) (1)
Slh and Shl are
2?2 matrices. Assume |p? and |q? are the low and high energy eigenstates of H0 in Eq. C.25
with energy Ep and Eq. The matrix elements of
(1)
Slh can be calculated with
?p| (1) | ?p|Vlh |q?Slh q? = ?i ? , (C.35)Ep Eq
134
( )
and (1) (1)
?
Shl = Slh . After obtaining S
(1), we can check the second order term (2)Heff . Here
we list all the terms in Eq. C.30 exp?licitly. ( )?
(2) (2)
0 i S
(2) ? ( ) lh H0h ?H0lSlhi[S ,H0] = ? ,(2) (2)
i Shl H0l ?H0hShl 0
( ) ( ) (C.36)2 2
?1 S(1) 1H ? H S(1)0 0 (=2 2 )
(1) (1) (1) (1)
?1 Slh Shl H0l +H0lSlh Shl 0(1) (1) (1) (1) ,2 0 Shl Slh H0h +H0hShl Slh
( ) (C.37)
(1) (1)
(1) (1) 1 Slh H0hS 0S H S = hl0 ? ( (1) (1) , (C.38)2 0 Shl )H0lSlh? ?(1) (1) (1)i Slh Vhl ? VlhShl ( iSlh V(1) hhi[S , V ] = )? . (C.39)
? (1) (1) (1)iVhhShl i Shl Vlh ? VhlSlh
Similarly, block-diagonalizat(ion requires )
( (2) ? (2)) (1)i Slh H0h H0lSlh + iSlh Vhh = 0, (C.40)
(2)
i Shl H0l ?
(2)
H0hShl ?
(1)
iVhhShl = 0. (C.41)
S(2) can be determined from the above equations. Fortunately we don?t need to solve them
in our case because the diagonal terms does not depend on S(2). Additionally, Vhh = 0 in
our example. This makes(the p)roblem simpl(er an)d we have
(2) 1 2 1 2
Heff = ? S(1) H ? H (1) (1) (1) (1)0 0 S + S H0S + i[S , V ]. (C.42)2 2
The last step is to change the basis order back and we can arrive at Eq. 4.50 in Sec. 4.3.3.
135
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