ABSTRACT
Title of dissertation: COHERENT CONTROL OF
LOW ANHARMONICITY SYSTEMS
FOR SUPERCONDUCTING QUANTUM COMPUTING
Shavindra Premaratne, Doctor of Philosophy, 2018
Dissertation directed by: Professor Frederick C. Wellstood
Department of Physics
Dr. Benjamin S. Palmer
Laboratory for Physical Sciences
This dissertation describes research to coherently control quantum states of su-
perconducting devices. In the first project, the state of an 8 GHz 3D superconducting
Al cavity at 20 mK was manipulated to add a quantum of excitation. Preparing a
harmonic resonator in a state with a well-defined number of excitations (Fock states) is
not possible using one external classical drive. I generated Fock states by transferring
a single excitation from a 5.5 GHz transmon qubit to a cavity using Stimulated Raman
Adiabatic Passage (STIRAP). I also extended the STIRAP technique to put the cavity in
higher Fock states, superpositions of Fock states, and Bell states between the qubit and
the cavity. Master-equation simulations of the system’s density matrix were in good
agreement with the data, and I obtained estimated fidelities of 89%, 68% and 43% for
the first three Fock states, respectively.
The second project involved implementing an entangling gate between two Al/AlOx/Al
transmon qubits that were mounted in an Al cavity and cooled to 20 mK. Pertinent
system frequencies were as follows: one qubit was at 6.0 GHz, the other qubit at
6.8 GHz, the cavity at 7.7 GHz, and the qubit-qubit dispersive shift was −1 MHz. By
applying a specially-shaped pulse of duration τg = 907ns, I implemented a generalized
CNOT gate using an all-microwave technique known as Speeding up Waveforms by
Inducing Phases to Harmful Transitions (SWIPHT). Using quantum process tomography,
I found that the gate fidelity was 80%–82%, close to the 87% fidelity expected from
decoherence in the transmons during the gate time. Details of the device fabrication,
device characterization, measurement techniques, and extensive modeling of device
behavior are presented, along with χ-matrix characterization of single-qubit gates and
SWIPHT gates.
COHERENT CONTROL OF LOW ANHARMONICITY SYSTEMS FOR
SUPERCONDUCTING QUANTUM COMPUTING
by
Shavindra Priyanath Premaratne
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2018
Advisory Committee:
Professor Frederick Wellstood, Chair/Co-Advisor
Dr. Benjamin Palmer, Co-Advisor
Professor Andrew Childs, Dean’s Representative
Professor Christopher Jarzynski
Dr. Kevin Osborn
Professor James Williams
© Copyright by
Shavindra Priyanath Premaratne
2018
Dedicated to my father, my mother, my brother, my sister
and
my wife Kasuni
මෙග් අම්මාට, මෙග් අප්පච්චිට, මෙග් මල්ලිට, මෙග් නංගිට,
සහ
මෙග් බිරිඳ කසුනිට පුදමි ෴
ii
Acknowledgments
The past few years I spent at the University of Maryland and the Laboratory for
Physical Sciences have been some of the best in my academic life. My achievements
would not have been possible without the help I received from many people. I humbly
take this opportunity to thank them.
First of all, I thank Benjamin Palmer and Frederick Wellstood, who are amazing
physicists. It was my privilege to call them my advisors and mentors. Over the past
four years, I learned innumerable things from Ben. Ben’s persistence and perseverance
is unparalleled among all the people I have worked with. He has an amazing ability to
visualize a solution to a grand problem, and chip away at it until it is resolved. Working
with him both in the lab, and when developing a theory, I was amazed at how much can
be accomplished by steadily and methodically attacking a problem. Fred is an amazing
scientist with a wealth of knowledge. His inquisitive nature into minute details led
me to question many things that I sometimes took for granted, during the course of
modeling and measurements. Some of the perspectives Fred had were awe-inspiring,
and his ability to look at anything in multiple angles, is truly awesome.
I am grateful to Professors Chris Jarzynski, James Williams, and Kevin Osborn
for agreeing to be members of my dissertation committee. I thank Andrew Childs
for agreeing to serve as the Dean’s representative for my committee and for the
encouragement and support I received when he was the professor for CMSC858K.
Over the past four years, I received help from all of my labmates. Everything
was built on teamwork and I am especially grateful to Sergey Novikov and Baladitya
Suri, who helped me get started with my research work in Ben’s lab. I thank Jen-Hao
Yeh who arrived at Ben’s lab at about the same time I did. He was the lead postdoc
iii
Acknowledgments
in Ben’s lab, and I am glad we were able to collaborate on projects, and chat about
everyday things. I am grateful to Jay Lefebvre and Raymond Jachja, who were two
undergraduate students that helped me out in various ways, including getting me
acquainted with Autodesk Inventor and programming the Tektronix AWG. I thank Rui
Zhang and Tamin Tai for the many useful discussions we had over the past few years.
I thank Rangga Budoyo, who was the only person to send a warm reply to my email,
when I was still trying to decide if UMD would be my graduate school. After I came
to College Park, he was kind enough to show me his lab and his dilution refrigerator,
which was inoperable at the time. At that time, I thought I would never get into
research requiring low-temperature operations. It was a stroke of fate and serendipity
that I came to work in Ben’s superconducting quantum computing lab, with Fred (who
happened to be Rangga’s advisor) as my co-advisor. This was truly a fortuitous series
of events, that allowed me to work in one of the best labs, with some of the greatest
people around.
I thank my friends from LPS — Luke Robertson, Wan-Ting Liao, Joyce Coppock,
Yaniv Rosen, Nathan Siwak, Pavel Nagornykh, Neda Forouzani and Tim Kohler — for
discussions regarding important research matters, as well as trivial day-to-day things.
I appreciate the advice and support I received from Bruce Kane and Chris Richardson.
I thank the LPS cleanroom, research, and administrative staff members who were
always helpful with whatever issue I brought to them. I should particularly mention
Toby Olver, Steve Brown, Curt Walsh, Jim Fogleboch, Paul Hannah, Dan Hinkel, David
Gutierrez, Doug Ketchum, Warren Berk, Debtanu Basu, Dave Bowen, Jonas Wood,
Niquinn Fowler, Althia Kirlew, and Cynthia Evans who were simply amazing people to
work with. I consider myself lucky, to have had the opportunity to interact with people
who had such an amazing work ethic. The machinists in the LPS machine shop were
always ready to help out with any project we were working on. I thank Donald Crouse,
Ruben Brun, John Sugrue, and Bill Donaldson who fabricated many different parts for
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Acknowledgments
us. I am grateful to Greg Latini and Mark Thornton, for helping out whenever we had
issues with electronics or computers. Over the past few years, Greg and Mark probably
saved me several days of work and frustration by finding solutions to the problems
that I encountered.
I thank the members of our lunchtime group, Victor Yun, Donghun Park, Charlie
Krafft, Yongzhang Leng, and Mihaela Ballarotto with whom I had many interesting
conversations. It was never a dull moment when they were around. Victor’s enthusiasm
was infectious, and sometimes he was the sole reason some of us would go for a jog, to
be re-energized. Victor and Donghun invited me to accompany them on a skiing trip,
which was an unforgettable experience. Charlie was kind enough to show me thistle
sailing, though we didn’t get a chance to sail on that day. I consider myself truly lucky
to have worked at the Laboratory for Physical Sciences. The people I met here were
amazing, and I am grateful for all the support I received.
I thank Sophia Economou, Frederick Strauch, Timothy Sweeney, Xiu-Hao Deng,
and Edwin Barnes for useful discussions regarding my research projects.
I thank all my professors at University of Maryland, from whom I learned many
things. In particular, I am grateful to Alexey Gorshkov and Steven Anlage who en-
couraged and helped my work in many ways. I extend my great appreciation to the
UMD Physics department staff members — Pornshuda Rirksopa, Paulina Alejandro,
Margaret Lukomska, Linda Ohara, Jane Hessing, Amy Streets, and Jessica Crosby —
who ensured we had as little paperwork to deal with as possible. Their support enabled
me to focus completely on the work I was doing, and worry very little about other
matters.
It was an honor and a privilege to work at Laboratory for Physical Sciences and
University of Maryland.
There were people outside of UMD and LPS who helped me in innumerable ways.
Without them, the time I spent in the US as a graduate student would not have been
v
Acknowledgments
complete. These are the people who were our family away from home. Abeyratne
Bandara, Chandrika Bandara, and Sanjay Bandara were such a welcoming family to
us. Though we met only once or twice annually, we always enjoyed the little time we
had to spend with them. Suresh Jayasekara, Aruna Ramanayaka, Sajini Randeniya,
Shashikala Ratnayake, and Sameera Meegoda were our great friends, who helped us
in time of need, and developed an incredible bond with us. I also want to thank my
little buddies Nethuka Ramanayaka, Iruka Ramanayaka, and Dinith Jayasekara who
with their antics, were a constant source of delight. I am grateful to Dulith Abeykoon,
Vidumin Dahanayake, and Aishan Gonaduwa for their help in getting me set up in the
US, which would have been arduous task without them.
I thank my parents-in-law and my brother-in-law Geemal for their constant support
and encouragement.
I am grateful to my parents, my brother, and my sister for their immense love, care,
support and encouragement. Without them, I would not be where I am today.
Finally, I thank my wife and best friend, Kasuni, who supported me throughout my
time in the US. I am most fortunate to have met such a wonderful person, and for her
to be my wife is bliss.
vi
Table of Contents
Dedication ii
Acknowledgements iii
List of Tables xi
List of Figures xii
List of Abbreviations xv
List of Publications xvii
Chapter 1 Introduction 1
1.1 Superseding Classical Computation . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Coherent Control of Quantum Systems . . . . . . . . . . . . . . . . . . . . 3
1.3 Overview of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Chapter 2 Theory 8
2.1 Theory of the Transmon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.1 Physics of the Josephson Junction . . . . . . . . . . . . . . . . . . 8
2.1.2 Physics of the Transmon . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Circuit Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 The Quantum Harmonic Oscillator . . . . . . . . . . . . . . . . . 13
2.2.2 The Transmon Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 15
2.2.3 The Jaynes-Cummings Hamiltonian . . . . . . . . . . . . . . . . . 16
2.2.4 Hamiltonian of the Driven System . . . . . . . . . . . . . . . . . . 19
2.3 Density Matrix and Master Equation Formalism . . . . . . . . . . . . . . . 21
2.3.1 The Bloch Sphere Representation . . . . . . . . . . . . . . . . . . . 21
2.3.2 The Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.3 The Lindblad-Kossakowski Master Equation . . . . . . . . . . . . 24
2.4 Quantum Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.1 Quantum State Tomography . . . . . . . . . . . . . . . . . . . . . 26
2.4.2 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . 28
2.4.3 Quantum Process Tomography . . . . . . . . . . . . . . . . . . . . 30
Chapter 3 Device Design and Fabrication 36
3.1 Transmon Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Cavity Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Microwave Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
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TABLE OF CONTENTS
3.4 Black Box Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.5 Transmon Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.6 Cavity Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.6.1 Cavity Cleaning and Preparation . . . . . . . . . . . . . . . . . . . 52
3.6.2 Cavity Tuning and Packaging . . . . . . . . . . . . . . . . . . . . . 54
3.6.3 Details of Fabricated Cavities . . . . . . . . . . . . . . . . . . . . . 56
Chapter 4 Experimental Setup for Controlling and Measuring Qubits 58
4.1 The Dilution Refrigerator Setup . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Vector Network Analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3 Heterodyne Pulsed Measurement System . . . . . . . . . . . . . . . . . . 64
4.4 Microwave Signal Generators . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4.1 Synthesized Signal Generators . . . . . . . . . . . . . . . . . . . . 67
4.4.2 Arbitrary Waveform Generators . . . . . . . . . . . . . . . . . . . . 68
4.5 Device Control and Measurement . . . . . . . . . . . . . . . . . . . . . . . 69
Chapter 5 System Characterization 72
5.1 Cavity Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Transmon Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2.1 Dispersive Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2.2 The Non-linear High-power Readout . . . . . . . . . . . . . . . . 77
5.3 Transmon Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4 Rabi Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.4.1 Qubit Population Calibration . . . . . . . . . . . . . . . . . . . . . 87
5.4.2 Qubit Pulse Calibration . . . . . . . . . . . . . . . . . . . . . . . . 88
5.4.3 Off-resonant Rabi drives . . . . . . . . . . . . . . . . . . . . . . . . 90
5.4.4 The Rabi Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.5 Relaxation Time Measurements . . . . . . . . . . . . . . . . . . . . . . . . 92
5.6 Ramsey Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.6.1 Precise Determination of Qubit Frequency . . . . . . . . . . . . . 95
5.7 Spin-Echo Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.8 Photon Number Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.8.1 Determining the Cavity Coupling to the Input Drive . . . . . . . .101
5.9 Summary of Measured Parameters of the Devices . . . . . . . . . . . . . 103
Chapter 6 Fock State Generation using Stimulated Raman Adiabatic Pas-
sage 106
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.2 The Device and the Experimental Setup . . . . . . . . . . . . . . . . . . . 108
6.3 Modeling and Characterizing the System . . . . . . . . . . . . . . . . . . 110
6.4 Generating Fock States n= 1,2, 3 . . . . . . . . . . . . . . . . . . . . . . . 113
6.5 Generating Fock State Superpositions . . . . . . . . . . . . . . . . . . . . .121
6.6 STIRAP Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.7 Fractional STIRAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
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Chapter 7 A Generalized CNOT Gate 130
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.2 The Device and the Experimental Setup . . . . . . . . . . . . . . . . . . . 132
7.2.1 Details of the cQED System . . . . . . . . . . . . . . . . . . . . . . 132
7.2.2 The System Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 133
7.2.3 Joint Qubit State Readout . . . . . . . . . . . . . . . . . . . . . . . 135
7.3 The SWIPHT Gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .141
7.4 Quantum State Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.4.1 Constructing the Likelihood Function . . . . . . . . . . . . . . . . 145
7.4.2 QST During SWIPHT Evolution . . . . . . . . . . . . . . . . . . . . 147
7.4.3 Verification of Coherence and Phase Control . . . . . . . . . . . . 149
7.5 Quantum Process Tomography . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Chapter 8 Conclusion 158
8.1 Summary of Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8.1.1 Fock State Generation . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8.1.2 A Generalized CNOT Gate . . . . . . . . . . . . . . . . . . . . . . . 159
8.2 Possible Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .161
8.2.1 Improving Fock State Generation . . . . . . . . . . . . . . . . . . . .161
8.2.2 Improving the SWIPHT gate . . . . . . . . . . . . . . . . . . . . . . 162
Chapter A Experimental Determination of Quality Factors for a Bandpass
Cavity 163
A.1 Modeling the Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
A.2 ABCD Matrix Analysis for the Model . . . . . . . . . . . . . . . . . . . . . 164
A.3 Obtaining the S-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
A.4 Approximations for Calculation of Quality Factors . . . . . . . . . . . . . 166
A.4.1 Qout =Qin = 2QE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
A.4.2 ω=ω0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Appendix B Rabi Oscillations after FSG 168
Appendix C MATLAB Code for STIRAP Master Equation Simulations 173
C.1 The Main Program for Simulating FSG upto n= 3 . . . . . . . . . . . . . 173
C.2 Subprogram for the Lindblad Master Equation . . . . . . . . . . . . . . . 177
Appendix D Code for Quantum Tomography 179
D.1 Mathematica Program for Performing Quantum State Tomography . . 179
D.2 MATLAB Programs for Constructing the Likelihood Function . . . . . . 185
D.2.1 MATLAB Function to Import the Raw Data Files . . . . . . . . . . 185
D.2.2 MATLAB Function to Obtain the Density Matrix of a Ket . . . . . 185
D.2.3 MATLAB Function to Construct a Single Term of the Likelihood
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
D.2.4 MATLAB Program for Parallel Construction of All the Terms of
the Likelihood Function . . . . . . . . . . . . . . . . . . . . . . . . .191
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D.2.5 MATLAB Program to Construct the Expression for the Complete-
ness Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
D.3 Mathematica Program to Perform Quantum State Tomography . . . . . 195
D.4 Mathematica Program to Generate the Simulation Data for QPT . . . . 199
Appendix E QPT Process Matrices for Implementing SWIPHT 200
E.1 QPT Process Matrices for SWIPHT Gates . . . . . . . . . . . . . . . . . . . 200
E.2 QPT Process Matrices for Single-Qubit Gates . . . . . . . . . . . . . . . . 203
Bibliography 207
x
List of Tables
Table 2.1 Complete set of orthogonal tomography operators . . . . . . . . . . 31
Table 3.1 List of fabricated cavities . . . . . . . . . . . . . . . . . . . . . . . . . 57
Table 5.1 Summary of parameters for the devices . . . . . . . . . . . . . . . . 105
Table 6.1 Parameter values used in STIRAP pulses . . . . . . . . . . . . . . . 116
Table 6.2 Optimized cavity and qubit pulse parameters used in simulating
Fock state generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Table 6.3 Sensitivity of process fidelity to STIRAP parameters . . . . . . . . .121
Table 7.1 Device parameters for the two qubits coupled to the cavity . . . . 134
Table 7.2 Parameters for joint-qubit readout . . . . . . . . . . . . . . . . . . . 139
Table 7.3 List of tomographic maps . . . . . . . . . . . . . . . . . . . . . . . . 145
Table 7.4 QPT performance metrics for gates . . . . . . . . . . . . . . . . . . 154
xi
List of Figures
Fig. 2.1 Schematic of a Josephson junction . . . . . . . . . . . . . . . . . . . 9
Fig. 2.2 Eigenenergies of a charge qubit for various EJ/EC . . . . . . . . . 12
Fig. 2.3 Dispersive shift of the resonator . . . . . . . . . . . . . . . . . . . . 18
Fig. 2.4 The Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Fig. 2.5 Illustration of quantum process tomography . . . . . . . . . . . . . 32
Fig. 3.1 The qubit chip design for microwave simulations . . . . . . . . . . 38
Fig. 3.2 The cavity design for microwave simulations . . . . . . . . . . . . 39
Fig. 3.3 Mesh for the substrate and capacitor pads . . . . . . . . . . . . . . . 41
Fig. 3.4 Waveports in HFSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Fig. 3.5 Blackbox quantization results . . . . . . . . . . . . . . . . . . . . . . 44
Fig. 3.6 The transmon design for electron beam writing . . . . . . . . . . . 46
Fig. 3.7 Photomicrographs of the transmon after ZEP development . . . . 48
Fig. 3.8 Photomicrographs of the transmon after MMA development . . . 49
Fig. 3.9 Summary of the transmon fabrication process . . . . . . . . . . . . . 51
Fig. 3.10 The cavity design for machining . . . . . . . . . . . . . . . . . . . . 52
Fig. 3.11 Photograph of cavity 6D . . . . . . . . . . . . . . . . . . . . . . . . . 57
Fig. 4.1 Photograph of the CF-450 Dilution refrigerator . . . . . . . . . . . 60
Fig. 4.2 Input/Output microwave lines in the dilution refrigerator . . . . . 61
Fig. 4.3 Heterodyne measurement setup . . . . . . . . . . . . . . . . . . . . 66
Fig. 4.4 Schematic of circuit for controlling and measuring transmon qubits 71
Fig. 5.1 High-power and low-power cavity spectroscopy . . . . . . . . . . . 73
Fig. 5.2 Cavity spectroscopy versus drive power with transmon in |g〉 . . 75
Fig. 5.3 Cavity spectroscopy versus drive power with transmon in |e〉 . . 76
Fig. 5.4 Pulse sequence to obtain cavity switching curves . . . . . . . . . . 78
Fig. 5.5 Switching curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Fig. 5.6 Pulses for transmon spectroscopy . . . . . . . . . . . . . . . . . . . 82
Fig. 5.7 Transmon Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 83
Fig. 5.8 Two-photon process between |g〉 and |f〉 . . . . . . . . . . . . . . . 84
Fig. 5.9 Pulse sequence for Rabi oscillations . . . . . . . . . . . . . . . . . . 85
Fig. 5.10 Rabi Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Fig. 5.11 The π-pulse and the π/2-pulse . . . . . . . . . . . . . . . . . . . . . 89
Fig. 5.12 Rabi Coupling Measurement . . . . . . . . . . . . . . . . . . . . . . . 91
Fig. 5.13 Pulse sequence for T1 measurement . . . . . . . . . . . . . . . . . . 92
Fig. 5.14 T1 measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Fig. 5.15 Pulse sequence for Ramsey oscillation measurement . . . . . . . . 94
Fig. 5.16 Ramsey oscillations measurement . . . . . . . . . . . . . . . . . . . 95
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LIST OF FIGURES
Fig. 5.17 Drive frequency versus Ramsey frequency . . . . . . . . . . . . . . 96
Fig. 5.18 Pulse sequence for T2 measurement . . . . . . . . . . . . . . . . . . 97
Fig. 5.19 Spin-echo T2 measurement . . . . . . . . . . . . . . . . . . . . . . . 98
Fig. 5.20 Pulse sequence for photon number splitting . . . . . . . . . . . . . 99
Fig. 5.21 Measurement of photon number splitting spectrum . . . . . . . . 100
Fig. 5.22 Photon number splitting spectrum versus drive power . . . . . . . 102
Fig. 6.1 The transmon-cavity device . . . . . . . . . . . . . . . . . . . . . . . 109
Fig. 6.2 STIRAP in a transmon-cavity system . . . . . . . . . . . . . . . . . . .111
Fig. 6.3 Generation of the first three Fock states . . . . . . . . . . . . . . . . 115
Fig. 6.4 Verification of |g1〉 Fock state . . . . . . . . . . . . . . . . . . . . . . 118
Fig. 6.5 Generation of Fock state superpositions with STIRAP . . . . . . . 122
Fig. 6.6 Fock state superpositions’ variations with STIRAP phases . . . . . 123
Fig. 6.7 Reverse STIRAP and measurement of cavity decay rate . . . . . . 125
Fig. 6.8 Cavity-qubit entangled state preparation . . . . . . . . . . . . . . . 127
Fig. 7.1 Computer generated model of the device . . . . . . . . . . . . . . . 133
Fig. 7.2 Two-transmon S-curves . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Fig. 7.3 Cavity powers chosen for joint qubit readout . . . . . . . . . . . . 138
Fig. 7.4 Pulse train for joint qubit readout . . . . . . . . . . . . . . . . . . . 138
Fig. 7.5 SWIPHT pulse shape . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Fig. 7.6 Pulse train used during QST . . . . . . . . . . . . . . . . . . . . . . . 146
Fig. 7.7 Evolution of populations during SWIPHT gate . . . . . . . . . . . . 148
Fig. 7.8 Phase control of coherences . . . . . . . . . . . . . . . . . . . . . . . 149
Fig. 7.9 Pulse train for QPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Fig. 7.10 Process matrices from QPT . . . . . . . . . . . . . . . . . . . . . . . 153
Fig. A.1 Simplified effective circuit model for a bandpass resonator. . . . . 163
Fig. B.1 Rabi oscillations performed after |g1〉 generation . . . . . . . . . . 169
Fig. B.2 Rabi oscillations performed after |g1〉 generation under ωg1→e1
drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Fig. B.3 Rabi oscillations performed after |g1〉 generation under ωq drive .171
Fig. B.4 Rabi oscillations performed after |g2〉 generation under ωg2→e2
drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
Fig. E.1 χ-matrix for SWIPHT49π/36L . . . . . . . . . . . . . . . . . . . . . . . . .201
Fig. E.2 χ-matrix for SWIPHT0H . . . . . . . . . . . . . . . . . . . . . . . . . . .201
Fig. E.3 χ-matrix for SWIPHT7π/12H . . . . . . . . . . . . . . . . . . . . . . . . 202
Fig. E.4 χ-matrix for I ⊗ I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Fig. E.5 χ-matrix for I ⊗ Rπx . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Fig. E.6 χ-matrix for I ⊗ Rπy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Fig. E.7 χ-matrix for Rπx ⊗ I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Fig. E.8 χ-matrix for Rπy ⊗ I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Fig. E.9 χ-matrix for I ⊗ Rπ/2x . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Fig. E.10 χ-matrix for I ⊗ R−π/2x . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
xiii
LIST OF FIGURES
Fig. E.11 χ-matrix for I ⊗ Rπ/2y . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Fig. E.12 χ-matrix for I ⊗ R−π/2y . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Fig. E.13 χ-matrix for Rπ/2x ⊗ I . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Fig. E.14 χ-matrix for R−π/2x ⊗ I . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
Fig. E.15 χ-matrix for Rπ/2y ⊗ I . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
Fig. E.16 χ-matrix for R−π/2y ⊗ I . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
xiv
List of Abbreviations
AMO Atomic-Molecular-Optical
AWG Arbitrary Waveform Generator
BBQ Black Box Quantization
BNC Bayonet Neill-Concelman
CAD Computer Aided Design
CNOT Controlled NOT
CPB Cooper Pair Box
cQED Circuit Quantum Electrodynamics
CR Cross Resonance
DC Direct Current
DI De-ionized (water)
DUT Device Under Test
FSG Fock State Generation
f-STIRAP fractional Stimulated Raman Adiabatic Passage
HEMT High Electron Mobility Transistor
HFSS High Frequency Structure Simulator
HPF High Pass Filter
I/Q In-phase/Quadrature
IF Intermediate Frequency
IPA Iso-propyl Alcohol
IVC Inner Vacuum Chamber
JJ Josephson Junction
LO Local Oscillator
LPF Low Pass Filter
LPS Laboratory for Physical Sciences
OVC Outer Vacuum Chamber
QED Quantum Electrodynamics
QPT Quantum Process Tomography
QST Quantum State Tomography
RF Radio Frequency
RIP Resonator Induced Phase
rms root-mean-square
RO Read out
SEM Scanning Electron Microscope
SET Single Electron Transistor
SHO Simple Harmonic Oscillator
SMA SubMiniature version A
xv
List of Abbreviations
SNAP Selective Number dependent Arbitrary Phase
SPAM State Preparation And Measurement
SQUID Superconducting Quantum Interference Device
SSG Synthesized Signal Generator
STIRAP Stimulated Raman Adiabatic Passage
SWIPHT Speeding up Waveforms by Inducing Phases to Harmful Transitions
TE Transverse Electric
TLS Two Level System
VNA Vector Network Analyzer
xvi
List of Publications
[1] S. P. Premaratne, F. C. Wellstood, and B. S. Palmer, “Microwave photon Fock
state generation by stimulated Raman adiabatic passage,” Nat. Commun. 8,
14148 (2017).
[2] S. P. Premaratne, F. C. Wellstood, and B. S. Palmer, “Characterization of co-
herent population-trapped states in a circuit-QED Λ system,” Phys. Rev. A 96,
043858 (2017).
[3] S. P. Premaratne, J.-H. Yeh, F. C. Wellstood, and B. S. Palmer, “Implementation
of a generalized CNOT gate between fixed-frequency transmons,” In review,
2018.
[4] S. Novikov, T. Sweeney, J. E. Robinson, S. P. Premaratne, B. Suri, F. C. Well-
stood, and B. S. Palmer, “Raman coherence in a circuit quantum electrodynam-
ics lambda system,” Nat. Phys. 12, 75 (2016).
[5] J.-H. Yeh, J. LeFebvre, S. Premaratne, F. C. Wellstood, and B. S. Palmer, “Mi-
crowave attenuators for use with quantum devices below 100 mK,” J. Appl.
Phys. 121, 224501 (2017).
xvii
CHAPTER 1
Introduction
1.1 Superseding Classical Computation
In 1936–1937, Church [1] and Turing [2] independently formalized the notion
of computability. These ideas laid the foundation for the subsequent development
of classical computers in the twentieth century. The Strong Church–Turing Thesis
captures the essence of using an algorithm to compute a function [3, 4]: Every finitely
realizable physical system can be perfectly simulated by a universal Turing machine by
finite means. However, this definition was incompatible with the paradigm of classical
computation upon which virtually all modern computers are currently built.
Classical computing relies on an inherently discrete digital system. Since many
physical systems are continuous, and governed by quantum mechanics rather than
classical physics, it is not possible to efficiently simulate all realizable physical sys-
tems with classical computation. Quantum computation offers a potential solution.
Universal Quantum Computing is compatible with the Strong Church–Turing thesis
and also had properties not reproducible by any Turing machine [4]. Following the
1
1.1. SUPERSEDING CLASSICAL COMPUTATION
initial ideas of Feynman [5], Benioff [6], and Albert [7] in the 1980s, Deutsch [4]
introduced the concept of a fully quantum model for computation. This model had
the potential to supersede classical computation when solving certain classical and
many quantum problems. In principle, a quantum computer Q would be capable
of efficiently simulating every finite realizable physical system. With arbitrarily high
accuracy, it could simulate zero temperature systems, all instances of quantum com-
puters, and quantum simulators. Furthermore, unlike a Universal Turing machine T ,
Deutsch’s machine Q could simulate any finite classical discrete stochastic process (i.e.
randomized algorithms) also efficiently [4].
In 1994, Shor [8] described a quantum algorithm that was vastly superior to any
known classical algorithm for resolving a composite number into its prime factors.
This caused a burst of excitement and stimulated activity that continues to this day
to realize actual working quantum computers. Following demonstrations of primitive
quantum computing systems, DiVincenzo [9] put forth a set of criteria that a universal
quantum computer needed to satisfy,
• A scalable physical system with well-characterized qubits
• Ability to initialize the state of the qubits to a simple fiducial state
• Qubit coherence times that are much longer than gate operation times
• A universal set of quantum gates
• A reliable technique to measure the state of the qubits with high fidelity
Since Shor’s proposal, researchers worldwide have pursued building quantum
2
1.2. COHERENT CONTROL OF QUANTUM SYSTEMS
computers using many different types of quantum systems. Initially, well-understood
naturally ocurring systems such as atoms [10], ions [11], and nuclear spins [12] showed
the most promise as qubits. However, as interest in quantum computing spread, more
sophisticated, carefully engineered, and unconventional solid-state quantum systems
such as quantum dots [13] and superconducting devices [14]were pursued. In addition
to these examples, so many other systems have been used to demonstrate quantum
coherence and quantum operations over the past 20 years, that a comprehensive review
would be far too long for this dissertation.
1.2 Coherent Control of Quantum Systems
A formidable difficulty in building a functioning and useful quantum computer
is the problem of decoherence. Universal Quantum computation requires quantum
entanglement and the use of superpositions of different states. However, excited states
eventually relax back to the ground state and superposition states dephase; both effects
can be described as a loss of information to the environment. While decoherence can
be suppressed by reducing the coupling of a quantum system to the environment, it
is impossible to completely isolate a system and eliminate decoherence. In particular,
any deliberate operations we perform on the system will necessitate some type of
interaction with the system, and this will result in the introduction of decay channels
[15]. Imperfections in carrying out quantum operations is another difficulty in building
quantum computers. In general, quantum operations span a continuous space (e.g.
3
1.2. COHERENT CONTROL OF QUANTUM SYSTEMS
rotations by arbitrary angles on the Bloch sphere). In the real world, carrying out
operations to arbitrary precision is impossible – and operations will inevitably have
errors associated with them [15]. These are serious issues and a major breakthrough
was the development of quantum error correcting codes and related techniques [16].
Much of the foundation for superconducting quantum computing was actually
developed in the 1980s. Key work included discovery of macroscopic quantum tun-
neling [17, 18] and quantum energy levels in Josephson devices [19], as well as the
development of single-electron devices [20], the Cooper-pair box [21], and the search
for macroscopic quantum coherent oscillations in the rf superconducting quantum in-
terference device (SQUID) [22, 23]. The field of superconducting quantum computing
truly began with the seminal work by Nakamura et al. [14] in which a Cooper-pair-box
(CPB) [24] was found to undergo coherent quantum oscillations. A CPB works on the
principle that the excess number of Cooper pairs on a tiny superconducting island is
a sensitive function of the reduced gate charge [24]. The state of the device can be
measured using a sensitive single electron transistor (SET) or by coupling the device
to a resonator. Unfortunately, charge qubits are sensitive to quasiparticles and 1/ f
charge noise [25], and severely limited the coherence time of the quantum states [26].
The direct susceptibility of a CPB to charge noise was “remedied” by Koch et al.
[27], by making the electrostatic charging energy E = e2C /2CΣ much smaller than the
Josephson energy EJ. This remedy also removed the sensitivity of the device to applied
gate voltage, and pushed it out of the charge qubit limit and into the phase qubit limit.
The reduction in EC was accomplished by adding a shunting capacitance across the
junction to increase the total capacitance to CΣ. The new device was christened a
4
1.2. COHERENT CONTROL OF QUANTUM SYSTEMS
“transmon,” which stood for transmission line plasma oscillation qubit [27]; it was not
quite a phase qubit, since it lacks a bias current and required a specific range of EC to
achieve a useful anharmonicity.
At present, superconducting qubits are leading candidates for building scalable
quantum computers. They now have reasonably high coherence times, fast gate times,
and are relatively easy to fabricate in large numbers using fairly standard lithographic
techniques. However, fabricating qubits with precisely chosen frequencies is still a
major challenge for scaling up to systems with a large number of superconducting
qubits. The largest superconducting quantum computer at present appears to be a
20-qubit system at IBM [28]. Ongoing work to develop systems with 50 or more qubits
has also been reported [29–31]. As many government and private entities continue
to invest in building a useful quantum computer, I expect many more systems will be
reported that satisfy all five of DiVincenzo’s criteria. A major goal of such efforts in
the next few years is the demonstration of quantum supremacy [32], i.e. showing a
quantum computer can outperform the fastest classical computer, for some specific
computation.
Quantum computing is a computing paradigm that has some significant poten-
tial advantages over classical computing for certain problems. While the power and
advantages of quantum computing have been proven theoretically, realizing an effi-
cient, accurate, and large quantum computer remains a daunting challenge. Many
difficulties must be overcome and novel techniques developed before efficient quantum
computation becomes a practical alternative to classical computers.
5
1.3. OVERVIEW OF THE DISSERTATION
1.3 Overview of the Dissertation
In this dissertation, I discuss how I used superconducting qubits to realize coherent
control over a three-dimensional microwave resonator by generating Fock states, and
to generate entanglement between two qubits using a generalized CNOT gate. The
qubits I fabricated and measured were fixed-frequency transmons. The transmons
were mounted in three-dimensional superconducting cavities to allow measurement of
the qubit state and to provide isolation of the qubit from the surrounding environment.
In Chapter 2, I describe the theory behind the transmon and circuit quantum-
electrodynamics (cQED), which paved the way for many key developments in super-
conducting quantum computing. I also describe the master equation formalism that
I used to simulate quantum interactions and model the behavior of my devices. In
Chapter 3, I discuss the design and fabrication of the cavities and transmons. This
chapter explains how I went from a set of experimental design parameters to an actual
device.
Chapter 4 describes the experimental setup I used to acquire data. I discuss the
instruments I used and present circuit schematics for the different measurement setups.
In Chapter 5, I discuss the process I used to systematically determine the cavity and
qubit characteristics. I also elaborate on why the qubit calibration is not a sequential
process, but rather an iterative process that required multiple runs to fine-tune the
parameters.
Chapter 6 describes my work on using Stimulated Raman Adiabatic Passage (STI-
6
1.3. OVERVIEW OF THE DISSERTATION
RAP) to transfer an excitation from a transmon to a superconducting microwave cavity
[33]. I found good agreement between my results and simulations for the generation
of the first three Fock states, and I verified the generation of the first two Fock states
using Rabi oscillations. The coherence of the technique was also demonstrated by
performing STIRAP on various prepared superposition states. I discuss the distinct
interference patterns I observed and compare them to master equation simulations.
In Chapter 7, I describe my implementation of a generalized CNOT gate between
two superconducting qubits using a technique known as Speeding up Waveforms
by Inducing Phases to Harmful Transitions (SWIPHT). For this project, I repackaged
two transmons into a single cavity to create a two-qubit device that was suitable for
implementing SWIPHT. My gating results were in good agreement with master-equation
simulations. I also verified the performance of the gate by using quantum process
tomography, which I describe in detail. I conclude the dissertation in Chapter 8, where
I summarize the main results from my research and discuss possible avenues for further
investigation.
7
CHAPTER 2
Theory
2.1 Theory of the Transmon
2.1.1 Physics of the Josephson Junction
The Josephson effect is one of the most remarkable phenomena associated with
superconductors [34]. In a seminal article, Josephson [35] predicted that two super-
conductors separated by a sufficiently thin barrier (insulator or a superconducting
weak link) would permit a current to flow with a zero voltage difference between
the two superconducting electrodes (see Fig. 2.1). This “supercurrent” is due to the
quantum tunneling of Cooper pairs [36]. Furthermore, if a constant voltage difference
V is maintained between the two electrodes, the difference in the phases of the two
order parameters associated with the two superconducting regions was predicted to
oscillate at a well-defined frequency f = 2eV/h [34, 35, 37].
The phenomenon of a supercurrent Is at zero voltage bias is called the DC Josephson
effect, and the phenomenon of an alternating superconductor phase resulting from
8
2.1. THEORY OF THE TRANSMON
a fixed voltage bias is called the AC Josephson effect. The equations describing the
effect are
Is = Ic sin∆φ (2.1)
d(∆φ) 2eV
= J (2.2)
dt ħh
where Ic is the “critical current” or the maximum supercurrent that can flow through
the junction, ∆φ and VJ are the difference in the superconducting phase and voltage,
between the superconducting electrodes, respectively [34, 37]. Combining the two
Josephson relations (Eq. (2.1) and Eq. (2.2)), one finds that a Josephson junction
consisting of two superconducting electrodes separated by a thin insulating barrier
obeys:
‚ Œ
ħh 1 dI
VJ = ±
s
Æ . (2.3)
2e I2c − I2 dts
For an inductor, the voltage drop across the inductor is related to the rate of change
of the current by VJ = L
dIs
J dt . Comparing this to Eq. (2.3) one sees that a Josephson
S I S
𝜙0 𝜙0 + Δ𝜙
Fig. 2.1 Schematic of a Josephson junction. Two superconducting electrodes (yellow) are
separated by a thin insulating barrier (striped red). The superconducting phase difference
between the two electrodes is ∆φ.
9
2.1. THEORY OF THE TRANSMON
junction behaves as a non-linear inductor with the Josephson inductance LJ given by,
ħh 1 Φ
L = ± = 0J Æ (2.4)2e I2 − I2 2πI cos∆φc s c
where Φ0 = h/2e is the flux quantum. By integrating IsVJ with respect to time, the
total work done on the junction U can be shown to be
U = −EJ cosφ, (2.5)
where the Josephson energy EJ is,
I Φ
EJ ≡
c 0 . (2.6)
2π
2.1.2 Physics of the Transmon
The Hamiltonian for the transmon is the same as that for an un-biased phase qubit,
and can be obtained from that of a charge qubit as
 
2
HCPB = 4EC n̂− ng − EJ cos φ̂, (2.7)
where n̂ is the operator for the excess number of Cooper pairs that tunnels onto the
CPB island, φ̂ is the operator for the Josephson phase difference across the junction,
ng = CgVg/e is the reduced gate charge, and Vg is the gate voltage applied across the
gate capacitor Cg to the island [26]. Since n̂ and φ̂ are conjugate operators, Eq. (2.7)
10
2.1. THEORY OF THE TRANSMON
can be written [26] in either the charge basis as
• ˜
∑
 
2 E
Hcq = 4EC n̂− ng |n〉 〈n| −
J (|n〉 〈n+ 1|+ |n+ 1〉 〈n|) , (2.8)
n 2
or in the phase basis as
 2
1 ∂
Hcq = 4EC − ng − EJ cos φ̂. (2.9)i ∂ φ̂
In the phase basis, the Hamiltonian (Eq. (2.9)) has exact solutions [38] that can be
written in terms of the Mathieu functions [27], and the eigenenergies are given by
 ‹
E
Em(ng) = ECa
J
2[n − , (2.10)g+k(m,ng)] 2EC
 

where aν(q) is the characteristic value for the νth Mathieu cosine function and k m, ng
is a sorting function for eigenvalues. The sorting function k is given by,
•  ‹ ˜
∑
 
 l   
 	
k m, ng = int 2ng + mod 2 int ng + l(−1)m [(m+ 1) div 2] (2.11)2
l=±1
where int(x) is a function rounding x to the nearest integer, and (x div y) gives the
integer quotient of x and y [27].
By solving Eq. (2.10) for various EJ/EC ratios, we can see that higher values of EJ/EC
flatten the energy levels as a function of ng, at the expense of reduced anharmonicity
(see Fig. 2.2). Nevertheless, one finds that for EJ > EC, the anharmonicity only
decreases as a weak power law of EJ/EC, while the charge dispersion (variation in Em
11
2.2. CIRCUIT QUANTUM ELECTRODYNAMICS
(a) 𝐸J/𝐸C = 1 (b) 𝐸J/𝐸C = 10 (c) 𝐸J/𝐸C = 100
25 4
20 3
3
15
2
2
10
1 1
5
0 0 0
-2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2
𝑛g 𝑛g 𝑛g
Fig. 2.2 (a) Eigenenergies of a charge qubit versus reduced gate charge ng for energy levels
m= 0 (black), m= 1 (blue), m= 2 (red), m= 3 (green), and m= 4 (orange) with EJ/EC = 1
(the Cooper-pair-box regime), (b) EJ/EC = 10, and (c) EJ/EC = 100 (the transmon regime).
For all the plots, the zero of energy has been defined as the value for E0(0). Furthermore,
the level structure has been normalized with the quantity E01 = E1(0.5)− E0(0.5) for ease in
comparison.
with ng) is suppressed exponentially with EJ/EC [27], as expected for a device that
is in the phase regime. By setting EJ/EC ∼ 100 excellent charge noise suppression is
obtained, along with an anharmonicity equal to ∼ EC, which can be made sufficiently
large to allow for reasonably-fast qubit operations, without driving upper levels of the
system.
2.2 Circuit Quantum Electrodynamics
Cavity quantum electrodynamics (QED) involves the interaction between atoms and
photons in high-quality-factor cavities, often in the few photon limit where quantum
12
𝐸𝑚 𝑛g /𝐸01 0.5
2.2. CIRCUIT QUANTUM ELECTRODYNAMICS
effects prevail [39]. Cavity QED studies have long been carried out in both the optical
regime and the microwave regime, and many striking results have been obtained
[39–41].
Both the ideas of cavity QED and the work in the field of microwave kinetic induc-
tance detectors (MKIDs) eventually led to circuit quantum electrodynamics (cQED)
[42–45]. In cQED, superconducting devices are coupled to a microwave resonator.
This simple arrangement produces several benefits [42],
• The resonator can be used to isolate the qubit from the external electromagnetic
environment
• The resonator enables a quantum non-demolition readout method for the state
of the qubit
• Coupling to the qubit enables techniques for manipulating the microwave cavity
states
• The resonator can act as a quantum bus [46] for entangling qubits
In this section, I describe the theory for the components in my cQED experiments and
the Hamiltonian for the combined qubit-cavity system.
2.2.1 The Quantum Harmonic Oscillator
The simple harmonic oscillator (SHO) is one of the most well understood quantum
systems in physics. Some physicists even still claim that the SHO might be the only
13
2.2. CIRCUIT QUANTUM ELECTRODYNAMICS
thing that is understood, or that most of the physical world can be constructed from
approximations that are based on the SHO. Although both claims are incorrect, under-
standing the harmonic oscillator is critical for understanding cQED and many other
systems [47].
The classical equation of motion for an undriven LC SHO is given by CΦ̈+Φ/L = 0,
where Φ is the flux, C is the capacitance, and L is the inductance [48]. Comparing this
equation to that of a SHO, one can see the Φ will oscillate freely at resonance frequency
p
ωr = 1/ LC . The Hamiltonian for the system can then be written as
Q2 Φ2
H = + , (2.12)
2C 2L
where Q is the charge on one plate of the capacitor. This equation from classical physics
can be directly converted to the Hamiltonian for a quantum harmonic oscillator (QHO),
Q̂2 Φ̂2
Ĥ = + (2.13)
2C 2L
where Q̂ and Φ̂ are Hermitian operators. The non-Hermitian raising and lowering
operators can be constructed as
√
 
√Cω iQ̂
â = r Φ̂+ (2.14)
2ħh Cωr
√
 
√
â†
Cω
= r
iQ̂
Φ̂− , (2.15)
2ħh Cωr
where â is the “annihilation” operator and ↠is the “creation” operator. The Hamilto-
14
2.2. CIRCUIT QUANTUM ELECTRODYNAMICS
nian for the harmonic oscillator (or resonator) can then be written as
 ‹  ‹
1 1
Hres = ħhω †r â â+ = ħhωr N̂ + , (2.16)2 2
where N̂ = â†â is the photon number operator [47].
2.2.2 The Transmon Hamiltonian
In Sec. 2.1.2 I gave exact expressions for the energy of the nth transmon level in
terms of characteristic values of the Mathieu functions. These expressions are not
transparent. Instead, perturbation theory can be used to find simpler approximate
results when EC/EJ 1. The general asymptotic expressions for transmon eigenen-
ergies Em, absolute anharmonicity αabs = (Em+2 − Em+1) − (Em+1 − Em), and relative
anharmonicity αrel = αabs/E0 can then be obtained as [27, 49],
 ‹
p 1 E  

Em = −E + 8E E m+ −
C 2m2J J C + 2m+ 1 (2.17)2 4
αabs = −EC (2.18)
√
√ E
αrel = −
C . (2.19)
8EJ
In the limit EJ  EC, the transmon Hamiltonian reduces to a slightly anharmonic
oscillator,
 ‹
† 1 EC  
H † † †trans = ħhωp b b+ − 2b bb b+ 2b b+ 1 − EJ, (2.20)2 4
15
2.2. CIRCUIT QUANTUM ELECTRODYNAMICS
p
where ωp = 8EJEC/ħh is the plasma frequency of the transmon, while b and b† are
annhilation and creation operators for the transmon, analogous to those of a resonator.
Alternatively, if we define the transition frequency for the mth level as ωm = Em/ħh (see
Eq. (2.17)), then the transmon Hamiltonian can be written simply in terms of the
eigenkets |m〉 as,
∞
∑
Htrans = ħhωm |m〉 〈m| . (2.21)
m=0
2.2.3 The Jaynes-Cummings Hamiltonian
When a qubit or a two-level system (TLS) with transition frequency ωq is coupled
to a single-mode cavity, the total system may be described by the Jaynes-Cummings
Hamiltonian [48–50],
 ‹
1 1  

HJC = ħhω †r a a+ + ħhωqσz +ħhg aσ+ + a†σ− . (2.22)2 2
Here, ħhg quantifies the coupling strength between the qubit and the resonator. In
particular, if ωq =ωr and one excitation is placed in the system, then the excitation
will be exchanged between the qubit and the cavity at a rate g.
Equation (2.22) can be extended to describe a resonator interacting with a multi-
level system. The resulting generalized Jaynes-Cummings Hamiltonian can be written
16
2.2. CIRCUIT QUANTUM ELECTRODYNAMICS
as
∑
H (gen)
 

JC = ħhωra
†a+ ħhω j | j〉 〈 j|+ħhg ab†ge + a† b (2.23)
j
∑ ∑
 

= ħhω a†r a+ ħhω j | j〉 〈 j|+ ħhg j, j+1 a | j + 1〉 〈 j|+ a† | j〉 〈 j + 1| , (2.24)
j j
p
where j ≥ 2. Here, I use the notation of Bishop [49] and define gn,n+1 ' gge n as
the coupling strength for the n-photon manifold in the transmon-cavity system. In
Eq. (2.23) and Eq. (2.24), the zero-point energy for the cavity and multi-level system
was eliminated. To clearly distinguish the cavity levels from the levels of the transmon,
throughout the dissertation, I will use {g, e, f, h, . . . } to indicate transmon levels.
When the transition frequency ωge =ωe −ωg is far detuned from the resonance of
the cavity ωr compared to the coupling strength gge, i.e. gge |ωr −ωge|, the system
is said to be in the “dispersive regime”. In this regime, Eq. (2.24) can be approximately
diagonalized and the transmon truncated to the lowest two levels (qubit levels) [48,
49]. For convenience, I can define the transition frequencies ω j, j+1, detunings ∆ j, j+1,
and dispersive shifts χ j, j+1 as
ω j, j+1 ≡ω j+1 −ω j (2.25)
∆ j, j+1 ≡ω j, j+1 −ωr (2.26)
g2
≡ j, j+1χ j, j+1 . (2.27)
∆ j, j+1
Truncating Eq. (2.24) to the two lowest transmon levels (|g〉 and |e〉) and taking
the dispersive limit then yields a Jaynes-Cummings Hamiltonian with the following
17
2.2. CIRCUIT QUANTUM ELECTRODYNAMICS
−𝜒ge
−𝜒
−12𝜒ef
𝜔r 𝜔෥
𝜔
r
Fig. 2.3 Dispersive shift of the resonator.
convenient form [49, 51],
H (disp)
1
JC = ħhω̃ a
†
r a+ ħhω̃geσz +ħhχa†aσz, (2.28)2
where χ = χge −χef/2, ω̃ge =ωge +χge, ω̃r =ωr −χef/2, and χge, χef are as defined
by Eq. (2.27) (see Table 5.1 for detailed device parameters).
If we assume ωge <ωr it is easy to see that χge < 0 and χef < 0. Then for typical
couplings and detunings between the resonator and the transmon, the dispersive shifts
of the resonator are qualitatively shown in Fig. 2.3. In particular, when the qubit is in
the ground state, the bare cavity frequency ωr (solid black) shifts higher to the dressed
frequency ω̃r (dashed black). When the transmon is occupying |g〉, the resonator is
shifted to ω̃(|g〉)r = ω̃r −χ =ωr −χge (red), and when the transmon is occupying |e〉,
the resonator is shifted to ω̃(|e〉)r = ω̃r +χ =ωr +χge −χef (blue).
18
2.2. CIRCUIT QUANTUM ELECTRODYNAMICS
2.2.4 Hamiltonian of the Driven System
Using Eq. (2.28) to describe the undriven Hamiltonian for the dressed cavity qubit
system, I next consider adding a term to the Hamiltonian to describe the microwave
drive. Following the method of Steck [52], I consider the interaction term between a
microwave electric field and a charged TLS. The electric field is given by
1  

E~(t) = "~E cosω t ≡ "~E eiωd t0 d 0 + e−iωd t , (2.29)2
where "~ is the polarization vector for the field, E0 is the amplitude of the electric field
of the drive, and ωd is the drive frequency.
I now assume that the TLS has an electric dipole moment operator d~, which can be
rewritten in the following form [52],
 

d~̂ = 〈g| d~ |e〉 (|g〉 〈e|+ |e〉 〈g|)≡ 〈g| d~ |e〉 σ− +σ+ . (2.30)
The drive Hamiltonian can then be written as
H ~̂ ~int = −d · E (2.31)
E0  
  
= 〈g|"~ · d~ |e〉 σ− +σ+ eiωd t + e−iωd t (2.32)
2
E0  
' 〈g|"~ · d~ |e〉 σ−eiωd t +σ+e−iωd t , (2.33)
2
where in going from Eq. (2.32) to Eq. (2.33) the rotating wave approximation (RWA)
has been applied, allowing the fast, counter-rotating, terms to be eliminated [52]. The
19
2.2. CIRCUIT QUANTUM ELECTRODYNAMICS
Rabi frequency for the atom interacting with the field can then be defined as
E
Ω = − 0q 〈g|"~ · d~ |e〉 . (2.34)ħh
With this definition, the interaction Hamiltonian for a charged TLS interacting with a
classical electromagnetic drive can be written as,
ħhΩ
H q
 

= σ−eiωd t +σ+e−iωd tint . (2.35)2
The drive Hamiltonian for a resonator can be obtained by replacing the qubit
operators in Eq. (2.35) with the corresponding ladder operators for a resonator [49]:
(res) ħhΩr  
H = aeiωd t + a†e−iωd tint , (2.36)2
where Ωr is the effective Rabi drive frequency for the resonator. The driven dispersive
Jaynes-Cummings HamiltonianH =H (disp)JC +Hint can then be written as,
ħh  

H̃ = ħhω̃ †ra a+ ω̃ge + 2χa†a σ2 z
ħhΩq  
 ħhΩr  
+ σ−eiωd t +σ+e−iωd t + aeiωd t + a†e−iωd t . (2.37)
2 2
Following the method of Bishop [49], one can apply an appropriate unitary transfor-
mation to remove the time dependence in Eq. (2.37) and move into the rotating frame
of the drive. This yields the following time-independent driven Jaynes-Cummings
20
2.3. DENSITY MATRIX AND MASTER EQUATION FORMALISM
Hamiltonian [48, 49],
ħh  
 ħhΩ
H ħ † † q
 
 ħhΩ  

= h∆ a a+ ∆ − +r ge + 2χa a σz + σ +σ +
r a+ a† , (2.38)
2 2 2
where ∆r = ω̃r −ωd and ∆ge = ω̃ge −ωd.
2.3 Density Matrix and Master Equation Formalism
2.3.1 The Bloch Sphere Representation
Using Dirac notation, any physically distinct pure quantum state of a qubit can be
written in the form
θ
|ψ〉= cos |g〉+ eiφ
θ
sin |e〉 . (2.39)
2 2
One implication of Eq. (2.39) is that any superposition state of a single qubit can be
specified using two independent variables θ and φ.
As I discuss next in Sec. 2.3.2 on the density operator formalism, one needs three
independent parameters to describe the mixed states of an ensemble of qubits. One
way to visualize the state of a qubit ensemble is by using the three expectation values

 
〈σ̂x〉, σ̂y , and 〈σ̂z〉 to specify the components of a vector. The resulting “Bloch vector”
21
2.3. DENSITY MATRIX AND MASTER EQUATION FORMALISM
can be written as [52]

 
σBloch = 〈σ̂x〉 î+ σ̂y ĵ+ 〈σ̂z〉 k̂. (2.40)
If the ensemble is in a pure state, the density matrix will be represented by |ψ〉 〈ψ| and
only two parameters are needed to fully specify the state of the ensemble; in this case,
σBloch will be a unit vector. When the ensemble is in a mixed state, the length of the
vector is less than 1, and it will be proportional to the degree of purity. In all cases,
σBloch will lie within a unit sphere, which is known as the “Bloch sphere” (see Fig. 2.4).
The angles θ and φ in Eq. (2.39) represent the angular spherical coordinates of the
qubit state on the Bloch sphere.
2.3.2 The Density Matrix
The density operator ρ̂ is very useful for understanding quantum mechanical
behavior of real systems [47]. The density operator naturally allows one to treat a
statistical ensemble of quantum systems that may be in different states – a so-called
“mixed ensemble”, with the “density matrix” representing the “mixed state” of the
system. A simple example is when we have an ensemble of systems each in a pure
∑
state |αi〉 with a fractional population pi such that i pi = 1. In this case, the density
operator ρ̂ can be defined as
∑
ρ̂ ≡ pi |αi〉 〈αi| . (2.41)
i
22
2.3. DENSITY MATRIX AND MASTER EQUATION FORMALISM
|g〉
1 1g + e g + 𝑖 e
2 2
|e〉
Fig. 2.4 The Bloch sphere with different states labelled. Points on the equator correspond to
equal superpositions between |g〉 and |e〉 with different phases. Points on the surface of the
sphere correspond to pure states of the qubit, while points within the sphere indicate mixed
states of the qubit.
If the operator Ô is an observable of the system, then the “ensemble average” of Ô is,

   

Ô = tr ρ̂Ô . (2.42)
In general, the density operator ρ̂ must satisfy the following properties:
• ρ = ρ† (Hermiticity)
• ρ j j ≥ 0 (non-negative diagonal elements)
• tr (ρ) = 1 (unit trace)
Given that the elements of an ensemble are not interacting with each other, it is
possible to obtain an equation of motion for the density matrix. For an isolated system,
23
2.3. DENSITY MATRIX AND MASTER EQUATION FORMALISM
the state will evolve in time according to the Schrödinger equation [47], and one can
show that the evolution of ρ is then given by
∂ ρ
iħh = − [ρ,H ] =H ρ −ρH (2.43)
∂ t
where [ρ,H ] is the commutator of the density matrix ρ and the HamiltonianH .
2.3.3 The Lindblad-Kossakowski Master Equation
Equation (2.43) describes the evolution of the density matrix of an isolated system
under a unitary operation. However, real systems are not completely isolated from
their environment. Dissipation and decoherence are inevitable in non-isolated systems
and these processes must be included for realistic simulations of a quantum system.
The quantum master equation approach involves modifying Eq. (2.43) to include terms
corresponding to decoherence channels, which cause non-unitary evolution of the
states. The idea is to treat the system of interest as a “small system” that is coupled to
a “large reservoir”. If the reservoir is sufficiently large, it should undergo little change
due to its interaction with the small system and can be assumed to remain in thermal
equilibrium.
To formulate a master equation for our cQED system, I will assume that the reservoir
is Markovian – i.e. the reservoir has no memory of its interactions with the small system.
In this case, the reservoir component of the density matrix can be traced out to obtain
the reduced density matrix for the small system [53]. Since I am only interested in the
24
2.3. DENSITY MATRIX AND MASTER EQUATION FORMALISM
evolution of the small system, and not the reservoir, this master equation approach is
appropriate.
The above set of assumptions lead to the Lindblad-Kossakowski master equation
[54, 55]:
∂ ρ
iħh = − [ρ,H ] +L [ρ] , (2.44)
∂ t
where L [ρ] is the Liouvillian describing the non-Hermitian evolution of the small
system [53]. The Liouvillian is given by
 ‹
∑ ∑
L [ρ] = Γ D [A ]ρ = Γ A ρA †
1 1
i i i i i − A
†
i Aiρ − ρA
†
i Ai (2.45)2 2
i i
where D are dissipators, Γi is the decoherence rate, andAi is the ‘Lindblad operator’
or ‘jump operator’ corresponding to the ith decoherence channel [49, 53].
The master equation for the coupled transmon-cavity system is thus,
∂ ρ   γ
iħh = − [ρ,H ] + κ D [a]ρ + Γ D σ− ρ + D [σz]ρ (2.46)
∂ t 2
where H is the driven transmon-cavity Hamiltonian (Eq. (2.38)), κ is the total re-
laxation rate of the cavity, Γ is the relaxation rate of the transmon and γ is the pure
dephasing rate of the transmon [49]. Here, I have only included three well-known
decoherence channels: relaxation of the cavity, relaxation of the resonator, and de-
phasing of the transmon. On the other hand, I have assumed that the system is at
temperature T = 0K, so there is not rate for thermal excitations to be generated in the
25
2.4. QUANTUM TOMOGRAPHY
qubit or the cavity [56, 57].
2.4 Quantum Tomography
2.4.1 Quantum State Tomography
The cQED readout method I used yields 〈σ̂z〉 for a qubit state (see Sec. 5.2.1 and
Sec. 5.2.2). However, as I discussed in Sec. 2.3.1, three quantities need to be determined
to fully characterize a qubit that is in a mixed state. It should be emphasized that is
not possible to completely determine the state of a qubit when given just a single copy
[3, 58, 59]. However, when multiple copies of the same unknown qubit state can be
generated, it is possible to reconstruct the density matrix for the system [3, 58, 59].
This method is referred to as Quantum State Tomography (QST). Tomos is Greek for
“to cut” or “slice”, and the method involves reconstructing the qubit state from multiple
slices or projections. In some sense, the purpose of quantum state tomography (QST) is
to determine 〈σ̂x〉 and 〈σ̂y〉, which can then be used with 〈σ̂z〉 to construct the density
matrix for the qubit [60].
At this point, it is helpful to understand the precise relationship between ρ and
 	
〈σ̂x〉 , 〈σ̂y〉, 〈σ̂z〉 . The relationship is closely related to orthographic projections.
Orthographic projection is commonly used in engineering drawings to render a 3D
object on a 2D surface. Typically three such orthogonal projections are used to represent
a 3D object, corresponding to visualizing the object from three different viewpoints.
26
2.4. QUANTUM TOMOGRAPHY
This is analogous to QST, in that the Bloch sphere is measured in three orthogonal
directions. However, unlike in technical drawings, in QST the Bloch vector is rotated in
three particular directions and the average result obtained from repeated application
of the same projective measurement. This is equivalent to taking three orthogonal
measurements with the Bloch vector held fixed. For single qubit tomography, the three
measurements are [60],
• A projective measurement yielding the probability pz for the system to be in the
excited state |e〉,
• a Rπ/2x operation on the qubit followed by a projection yielding a probability px
that the system was in the state |g〉p+i|e〉 ,
2
• a Rπ/2y operation on the qubit followed by a projection yielding a probability py
that the system was in the state |g〉p+|e〉 .
2
These three measurements can be inverted to obtain the Stokes parameters [61], which
are the {x , y, z} components of the Bloch vector [60]:
〈σ̂x〉= 2py − 1 (2.47)

 
σ̂y = 2px − 1 (2.48)
〈σ̂z〉= 1− 2pz, (2.49)
27
2.4. QUANTUM TOMOGRAPHY
and the density matrix for the qubit [60],
   
ρ 1g,g ρg,e 1− pz py − ipx − 2(1− i)
ρ = =  . (2.50)
   
ρ ρ p 1e,g e,e y + ipx − 2(1+ i) pz
2.4.2 Maximum Likelihood Estimation
The method described in Sec. 2.4.1 for reconstructing the density matrix is conve-
nient and relatively straightforward. However, it is prone to errors when applied to
real data, i.e. results obtained from experimental measurements. In particular, errors
associated with the measurements or state preparation, including rotations of the Bloch
vector during QST, will induce errors in the values of px , py , and pz. The resulting
density matrix can even be unphysical, as manifested by the loss of one or more of
the density matrix properties listed in Sec. 2.3.2. To prevent the reconstruction of an
unphysical density matrix, I used a Maximum Likelihood Estimation (MLE) technique
restricted to physically constrained matrices [62].
The method I describe below is based on the method described by James et al. [63],
and I adapt it to perform quantum state tomography on our two-qubit system. I start
with an explicitly “physical” matrix which will be constrained to have unit trace, be
Hermitian and have positive eigenvalues. It can be proven that any matrix ρ̂ of the
form
T̂ † T̂
ρ̂ =  
 (2.51)
Tr T̂ † T̂
28
2.4. QUANTUM TOMOGRAPHY
will satisfy all the requirements for a density matrix [63]. Since an arbitrary density
matrix for a two-qubit system will have 15 independent parameters, it is convenient to
choose a tridiagonal form for T̂ [63],
 
t1 0 0 0
 
 
 
 t 5 + i t6 t2 0 0
 
T̂ =  . (2.52)
 
 t11 + i t12 t7 + i t8 t3 0 
 
 
t15 + i t16 t13 + i t14 t9 + i t10 t4
Using this form for T̂ , I write the density matrix using Eq. (2.51) and parameterize it
with the 16 real variables t i (the unit trace condition will set the number of independent
variables to 15).
For clarity, I assume that this system is measured using a joint-qubit readout scheme
(see Chapter 7). In such a scheme, the measurement signal mi will have weighted
contributions from populations from all four eigenstates of the two-qubit system. If
the system’s density matrix is given by ρ, then the outcome of the measurement is
4
∑
m= ciρii (2.53)
i=1
where ci indicates the output if the system was in the i
th eigenstate. Each measurement
will have an associated uncertainty which can be written in terms of the uncertainties
εi in a corresponding measurement of ci. Now, following James et al. [63], I define the
29
2.4. QUANTUM TOMOGRAPHY
likelihood function
” € Š —
∑ 2
N m − 4∑ j i=1 ci Ĝ jρ̂Ĝ
†
j
L (t ii1, t2, . . . , t16) = 2 (2.54)” € Š —∑4
j= †1
i=1 εi Ĝ jρ̂Ĝ j ii
where the Ĝ j are two-qubit operators which form a complete set of orthogonal tomo-
graphic operations when N = 15. The numerator of each term in L quantifies the
closeness of the matrix ρ̂ to the measurements. The sum in the denominator is the
estimated uncertainty in the measurement m j. For N > 15, one has an overcomplete
set of tomographic operations, and this allows the optimization routine to find the
most likely physical state with higher precision. An example of a complete set of
tomographic operations for two-qubit state tomography is given in Table 2.1 [64].
By simultaneously minimizing the likelihood function L with respect to all the
parameters tk, it is possible to determine the best estimate for {t1, t2, . . . t16}. These
values can then be used to reconstruct the most likely density matrix for the system,
by substituting into Eq. (2.51). In practice, to expedite the search for the minimum,
it can be helpful to initialize the variables with the expected ideal values for tk. The
ideal values for tk can be obtained by performing a Cholesky decomposition on the
ideal density matrix, expected from either theory or simulations.
2.4.3 Quantum Process Tomography
The QST analysis I discussed in Sec. 2.4.1 can be used for the reconstruction of
a density matrix for an unknown state. Although this gives the most likely density
30
2.4. QUANTUM TOMOGRAPHY
Table 2.1 A complete set of orthogonal tomography operators [65, 66] used for two-qubit
QST.
Operator Two-qubit gate representation
Ĝ1 I ⊗ I
Ĝ I ⊗ Rπ/22 x
Ĝ3 I ⊗ Rπ/2y
Ĝ4 I ⊗ Rπx
Ĝ5 R
π/2
x ⊗ I
Ĝ π/26 Rx ⊗ R
π/2
x
Ĝ Rπ/27 x ⊗ R
π/2
y
Ĝ Rπ/28 x ⊗ R
π
x
Ĝ π/29 R y ⊗ I
Ĝ π/210 R y ⊗ R
π/2
x
Ĝ π/211 R y ⊗ R
π/2
y
Ĝ12 R
π/2 π
y ⊗ Rx
Ĝ Rπ13 x ⊗ I
Ĝ Rπ π/214 x ⊗ Rx
Ĝ Rπ ⊗ Rπ/215 x y
matrix for a specific quantum operation on a specific starting state, it does not provide
complete information regarding the operation itself. To fully characterize the operation,
we need to perform the same operation on a complete set of initial states [3], and
then perform QST on the resulting output states. This technique for characterizing a
quantum operation or process is known as quantum process tomography (QPT). The
essence of QPT is to prepare a complete set [66] of initial states, perform the desired
operation on each state, and then do QST on all the resulting states to quantify the
performance of the gate [3] (see Fig. 2.5).
A quantum operation or quantum process can be thought of as a mapping between
input and output density matrices. Given an input state described byρin, and a quantum
31
2.4. QUANTUM TOMOGRAPHY
process E , the density matrix ρout for the output state is
ρout = E (ρin) . (2.55)
Given a set of ρin and ρout, the goal is to find an explicit form for E . Thus, the objective
e
e
Unknown 
Quantum 
QST
Process
ℰ
g + e
g
2
g + 𝑖 e
⋮ e
2
Fig. 2.5 Illustration of quantum process tomography (QPT). The quantum operation is repre-
sented as a black box. A complete set of initial states for a two-qubit system are sent in, one at
a time. Each output state is then analyzed with QST. By studying the input state to output state
transformation, a complete description of the operation (E) can be determined.
32
2.4. QUANTUM TOMOGRAPHY
is to find a set of operators Ôi, such that
∑
ρout = E (ρ
†
in) = ÔiρinÔi (2.56)
i
is satisfied for an every possible input density matrix ρin [3].
To proceed, we write the constituent operators Ôi in terms of the chosen basis
operators  j as
∑
Ôi = qi j j, (2.57)
j
where qi j are complex numbers representing the relative weights of basis operators.
We can then write Eq. (2.56) using the basis operators as
∑
E (ρ ) = χ Â †in jk jρinÂk, (2.58)
j,k
∑
where χ = q q∗jk m mj mk are matrix elements of the “process matrix” or “χ-matrix” which
completely and uniquely describes the quantum operation E [3]. The dimensionality of
the process matrix will be d4× d4 where d is the number of qubits. Since the χ-matrix
needs to satisfy positivity and Hermiticity [3, 67], I can use Cholesky decomposition as
described in Sec. 2.4.1 and Sec. 2.4.2 to constrain the matrix and write:
χ = T̂ † T̂ , (2.59)
where T̂ has the appropriate tridiagonal form with a dimensionality of d4. In addition,
33
2.4. QUANTUM TOMOGRAPHY
any real process has to be trace-preserving [3, 67], and this implies:
∑ ∑
Î = Ô Ô† = χ Â †i i jk jÂk. (2.60)
i j,k
I enforce this completeness condition by introducing a cost function
∑¦
 2  2©
Ξ= Re(Z jk)− I jk + Im(Z jk) , (2.61)
j,k
∑
where Z = j,k χ jk jÂ
†
k.
To apply this method to measurement, I follow O’Brien et al. [67], and implicitly
define the quantum process as
∑
ρ(σ) = Â (σ)out χ
†
jk jρin Âk, (2.62)
j,k
where (σ)ρ is the input density matrix and ρ(σ)in out is the output density matrix corre-
sponding to the initial state σ. By using a complete set of initial states, one can use
a maximum likelihood estimation technique to extract the best estimation for χ. For
QPT, I used the following likelihood function:
 ∑4  
 ( ) † 2m σ∑∑ σ,τ − i=1 ci Ĝτρout ĜL (t τ iii,λ) = +λΞ, (2.63)∑4   ( ) 
 σ † 2
σ τ i=1 εi Ĝτρout Ĝτ ii
where τ is the index for the tomographic operators Ĝτ, λ is a Lagrange multiplier
[67], and the final term in Eq. (2.63) is used to enforce the completeness relation
in Eq. (2.61). By minimizing the likelihood function with respect to the variables t i
34
2.4. QUANTUM TOMOGRAPHY
and λ, the most likely quantum process matrix χ giving rise to the mapping will be
determined.
As in QST, by using an overcomplete set of tomographic operations and initial states,
the accuracy of this technique can be improved. For example, in Chapter 7 I describe
QPT on a two-qubit system where I used an overcomplete set of 36 initial states.
35
CHAPTER 3
Device Design and Fabrication
Proper design of the quantum system is essential to obtain reliable and repeatable
results in many solid state quantum computing experiments. The original proposal for
a circuit quantum electrodynamics (cQED) platform was based on planar resonators
and Cooper-pair box qubits [42, 43, 68]. Using 3D cavities instead of lumped-element
or distributed-element resonators decreased the electric field energy stored in the
interfaces of the materials [69]. The first use of such cavities in superconducting
quantum computing resulted in significantly longer transmon lifetimes and larger
internal quality factors of resonators. Since then, fabrication methods and designs
for planar geometries have improved and lifetimes for 2D and 3D structures are now
comparable [70, 71]. Nevertheless, I chose to use a 3D cavity for its simplicity and ease
of use, which allowed me to focus on improving the control of the qubit and cavity.
In this chapter I will discuss how I designed the transmon and the cavity following
the groundwork laid by Dr. Sergey Novikov [72] in our research group. I will then
explain how I used finite-element-method microwave simulations in conjunction with
Black Box Quantization to determine suitable system parameters, depending on the
36
3.1. TRANSMON DESIGN
experimental requirements. I will conclude the chapter by explaining how the trans-
mon and cavity were fabricated, and how the device was packaged and prepared for
cooldown.
3.1 Transmon Design
The concept of the transmon [27] is quite simple – fabricate a small Josephson
junction with an appropriate shunting capacitor. However, there are a few important
design considerations when incorporating such a device in a 3D cavity so that it can
perform quantum operations. In addition to decreasing the charging energy EC, the
pads of the capacitor also serve as an antenna and set the transmon-cavity coupling.
Wang et al. [73] examined different physical designs to obtain optimal performance
for the 3D transmons and studied the effect of various interfaces. They found that
certain interfaces resulting from processing and fabrication contributed the most to
dissipation.
My design for the transmon was largely based on a design by Novikov [72], which
is very similar to the original 3D transmon design of Paik et al. [69]. In my transmon
design (see Fig. 3.1), the shunting capacitor had two 650µm × 500µm pads on a
sapphire substrate. The two pads were adjacent to each other on their shorter side,
with a separation of 150µm. Although the electric field of the capacitor is dispersed in
the substrate, vacuum, and interfaces, with this design most (∼ 90%) of the energy
stored in the electric field is in the substrate, which has low dielectric loss and relatively
37
3.2. CAVITY DESIGN
(a) (b)
2 mm 0.5 mm
Fig. 3.1 (a) Front view and (b) 3D perspective wireframe view of the qubit chip for 3D
microwave simulations. The capacitor was designed in Inventor along with the leads that contact
the Josephson junction. The junction itself was not part of the 3D microwave simulations.
high permittivity.
3.2 Cavity Design
The cavities for my cQED experiments were also based on cavities used by Novikov
[72]. All of my cavities were designed to have a TE101 mode frequency close to
8 GHz. The 1 GHz to 10 GHz frequency range is well suited for experiments due to
the availability of compatible commercial instruments. This frequency range is also
well suited for QC experiments since thermal populations can be suppressed at the
milliKelvin temperatures achievable using a dilution refrigerator. In particular, this
allowed me to initialize the system in a well-defined ground state by simply waiting
long enough for the system to dissipate energy. I also chose the cavity resonance to
be at a higher frequency than the qubit, as this enhances the lifetime of the qubit, by
38
3.2. CAVITY DESIGN
(a) (b)
FRONT RIGHT
2 cm
BOTTOM
Fig. 3.2 The cavity design for microwave simulations. (a) Orthographic projections and (b)
3D perspective wireframe view of the designed cavity with the transmon shunting capacitor.
reducing contributions to loss from the Purcell effect from higher modes [74].
For microwave simulations, I drew a rectangular cavity within Autodesk Inventor,
with length l = 35 mm, width w = 20.5 mm, and depth h = 5mm (see Fig. 3.2 for a
schematic and Fig. 3.11 for a photograph). I also included a 5 mm× 6.5mm× 450µm
sapphire chip and the transmon shunting capacitor (see Fig. 3.1) to mimic a transmon.
The sapphire chip decreases the fundamental mode frequency of the cavity to just below
8 GHz. The input and output coupling ports were included as perfectly conducting
pins that protruded into the cavity. I then imported the CAD design into the microwave
simulation software Ansys HFSS (High Frequency Structure Simulator) as a SAT file
[75]. Although HFSS has its own design capability, I preferred laying out the cavity
geometry using Inventor because it was easier to use and had superior capabilities.
39
3.3. MICROWAVE SIMULATIONS
3.3 Microwave Simulations
After importing the CAD layout for the chip and cavity into HFSS, I next designated
the materials for the different regions. The bulk of the cavity and the coupling pins
were set to be perfect electrical conductors, the empty space was set as vacuum and the
chip material was set to be sapphire with relative permittivity "r = 10. The material
for pads of the shunting capacitor and the leads to the capacitor pads were also set to
be perfect electrical conductors.
Since HFSS is a finite-element solver [75], it is important to have sufficiently fine
meshing on the surfaces in order to obtain accurate solutions. This means that I had to
restrict the maximum length of meshing elements (see Fig. 3.3). For initial simulations,
I usually set the maximum length to be 5–10% of the shortest non-zero dimension of
the object. For example, the capacitor pads had an initial maximum meshing restriction
of 50µm. Of course this ignored the much smaller thickness (∼ 80 nm) of the pads.
Since my main interest was about the behavior of the transmon, I manually set the
meshing for the leads to the capacitor pads, the capacitor pads, and the sapphire chip.
The remaining regions used an automatically generated mesh.
To perform an HFSS microwave simulation in the “driven modal” method, I set up
ports #1 and #2 to couple microwaves in to and out of the cavity. To represent the
ports, I drew two circles each starting from the center of the coupling pin and ending on
the surface of cavity, as shown in green and yellow in Fig. 3.4. I assigned each circle on
the cavity wall as Waveports with an impedance of 50Ω (see Fig. 3.4). The integration
40
3.3. MICROWAVE SIMULATIONS
y
z
x
1 mm
Fig. 3.3 Mesh for the substrate and capacitor pads after restricting the maximum length of
elements. Picture taken from HFSS.
line for the excitation was radially directed from the center of the circle (the pin) to
the edge of the circle (the cavity wall). This ensured that the ports were generating
the microwave signals correctly. To represent the junction, I drew a rectangle between
the two capacitor leads. This rectangle was designated as a lumped port (#3) with an
impedance of 10 GΩ. I discuss how I used this lumped port in Sec. 3.4, to extract the
cavity transmission simulation results and to determine the required parameters for
the junction. After setting up the ports, I set the frequency range for the sweep and
ran HFSS to find the transmission S21 through the cavity to confirm that the resonance
was close to 8 GHz and the waveports were set up correctly.
41
3.4. BLACK BOX QUANTIZATION
y
z x
1 cm
Fig. 3.4 Waveports for coupling microwaves in (green port) and out (yellow port) of the
microwave cavity, as depicted in HFSS. Both waveports are set to have an impedance of 50Ω
and a radial excitation.
3.4 Black Box Quantization
Black Box Quantization (BBQ) is a semi-classical method that was developed to
determine the approximate energy level structure of coupled transmon-cavity systems
[76]. In this method, microwave simulations are used to model the electromagnetic
environment around the JJ, including the capacitor pads, chip and cavity.
To perform BBQ analysis of my device, I extracted the self admittance versus
frequency data Y33(ω) for the lumped port (#3) from the HFSS output results (see
Sec. 3.3). The discrete data of Y33(ω) was interpolated using Mathematica to obtain a
smooth analytic function [72]. From the 100 nm × 125 nm area of the JJ, I estimated
the capacitance of the junction by itself to be Cj = 4 fF [77]. Since Cj is much less than
42
3.4. BLACK BOX QUANTIZATION
the shunting capacitance from the pads, which dominates the charging energy EC, this
rough estimate for the junction capacitance is sufficient for design purposes. As part
of the BBQ analysis, the JJ is treated as a simple linear inductor with inductance LJ0.
Accordingly, I set LJ0 between 3 nH to 15 nH depending on the target characteristic
frequency for the transmon. I then analytically combined the admittance contribution
from the JJ capacitance and inductance, with Y33(ω) from the simulation to obtain the
total admittance:
1
Ytot(ω) = Y33(ω) + + iωCj. (3.1)iωLJ0
From Im [Ytot(ω)], I used BBQ to obtain the transmon characteristic frequency, the
expected dispersive shifts, the anharmonicity in the system, and other pertinent param-
eters [76]. An example for BBQ analysis is shown in Fig. 3.5 where Y33(ω) extracted
from HFSS results and Ytot(ω) calculated through Mathematica are shown. The two
poles of Ytot represent the characteristic frequencies of the qubit and cavity. When I
set LJ0 = 4.5nH, the analysis predited the qubit frequency to be at ωq ≈ 5.9 GHz (see
Fig. 3.5).
From Re [Y (ω)], I note that I also used the BBQ analysis to estimate the lifetimes
of the transmons from Purcell coupling to the external 50Ω environment or bulk
dielectric loss of the substrate. My simulations did not include lossy surface layers
or lossy residue that could be left from the fabrication process, due to the very small
meshing that would be required. BBQ also cannot incorporate other loss mechanisms
such as non-equilibrium quasiparticles. Thus, it was only possible to obtain an upper
43
3.4. BLACK BOX QUANTIZATION
15
10 𝑌33 𝜔
5
𝑌tot 𝜔
0
qubit cavity
-5
-10
5 6 7 8 9
𝜔
(GHz)
2𝜋
Fig. 3.5 Blackbox quantization results for simulating the transmon SPP-v1-Q7. I set LJ0 =
4.5nH to obtain the qubit frequency ωq ≈ 5.9 GHz and the cavity frequency ωr ≈ 7.6 GHz.
bound for the lifetime.
It should be noted that Solgun et al. [78] have developed a more advanced and ac-
curate method to perform quantization of superconducting quantum circuits. However,
the relative simplicity of BBQ makes it particularly useful for design work, and I found
that the resulting simulated parameters for my single-transmon and two-transmon
devices were accurate to within a few percent. For these reasons, I continued using
the original BBQ method developed by Nigg et al. [76].
44
𝑌 10−3 Ω−1
3.5. TRANSMON FABRICATION
3.5 Transmon Fabrication
My transmons were fabricated in the LPS cleanroom using electon-beam lithography,
and double-angle evaporation, with both the Josephson junction and the shunting
capacitor pads deposited at the same time. I first cleaned a 3” c–plane oriented polished
sapphire wafer [79] using acetone, methanol, and isopropyl alcohol (IPA). I then spun
on a 900 nm layer of MMA(8.5)MAA EL11 [80] at 1000 rpm for 60 s, and baked the
wafer in an oven for 5 min at 180 ◦C. After letting the wafer cool down, I spun a
100 nm layer of ZEP520A DR2.3 [81], and baked the wafer for the second time in
an oven for 5 min at 180 ◦C. The wafer was then hard baked for 30 min at 180 ◦C. I
then used a thermal evaporator to deposit a 10 nm thick Al layer over the ZEP, to act
as an anti-charging layer during e-beam writing. Here, I used a thermal evaporator,
to ensure that there was no stray ultraviolet or x-ray exposure of the e-beam resist
layers, as one would get with an electron-beam evaporator. I next spun on a 5µm thick
coating of FSC-M resist [82] at 2000 rpm for 60 s, to protect the lower layers of the
wafer during dicing. The wafer was finally baked at 120 ◦C for 3.5 min to evaporate
any remaining solvents. After the protective layer was applied, I diced the wafer into
5 mm × 7 mm chips using a Dicing Blade Technology Type CA-010-270-080H blade
[83] in a Disco DAD3220 dicing saw [84]. Individual chips were cleaned, patterned
and processed when I had to fabricate a device. This completed the initial processing
of the sapphire wafer, and I note that this process was nominally identical to that used
by Suri [48] and Novikov [72].
45
3.5. TRANSMON FABRICATION
100 μm 10 μm 5 μm
Fig. 3.6 The computer aided design (CAD) of the transmon for the electron beam writing.
Three different magnifications are depicted to successively show the details from the shunting
capacitor pads to the Josephson junction region.
Prior to e-beam patterning of a chip, I removed the protective FSC-M layer using
successive 40 s dips in Acetone, Methanol and IPA. I then used a JEOL JSM-6500F
scanning electron microscope (SEM) at LPS to write the pattern (see Fig. 3.6). The
blue and red regions in Fig. 3.6 were written at a magnfication of 500x while the green
capacitor pads were written at a 70x magnification. For the fine red regions, I used a
dosage of 175µC cm−2, while the green/blue regions were written with a dosage of
100µC cm−2. The horizontal red line had a pattern width of 100 nm and the vertical
red line had a pattern width of 125 nm. The two red regions were separated by 200 nm
to form the suspended bridge [85] for the double-angle evaporation of the JJ. I note
46
3.5. TRANSMON FABRICATION
that the smaller features were written at a beam current of ∼ 30pA while the pads
were written at a current of ∼ 2.5nA.
After the e-beam writing, I removed the anti-charging layer by first agitating the
chip in a beaker of OPD 4262 [86] for 60 s, then agitating the chip in a beaker of
de-ionized water for 60 s, followed by a quick 3 s dip in IPA, and finally blow drying
with nitrogen.
I next developed the ZEP by agitating the chip in a beaker of Amyl Acetate [87]
for 2 min, followed by a 1 min agitation in a beaker of IPA, and completed this step
with blow drying using nitrogen. After developing the ZEP, I imaged the chip with a
Keyence confocal laser microscope to obtain optical micrographs as shown in Fig. 3.7.
After imaging the chip, I developed the MMA by first agitating the chip in a 5:1 volume
ratio solution of IPA:DI for 260 s, followed by a 60 s agitation in de-ionized water, and
finally blow drying. Following MMA development, I imaged the chip again with the
Keyence microscope to obtain micrographs as shown in Fig. 3.8.
I performed the two imaging steps after development to ensure that there were
no problems with the developed patterns. When the process was successful, the
micrographs after ZEP development clearly showed the existence of the bridge region
(see Fig. 3.7), and the micrographs obtained after MMA development allowed me to
confirm that the undercut was as expected (see Fig. 3.8).
After development, the chip was placed in a thermal evaporator dedicated for Al
deposition. The evaporator had a custom-built chamber with commercial vacuum
components. This evaporator was pumped to < 1× 10−6 torr over 12 h using a Varian
turbomolecular pump. Aluminum was deposited at two angles from the normal to
47
3.5. TRANSMON FABRICATION
(a)
200 μm
(b)
10 μm
Fig. 3.7 (a) 10x and (b) 150x magnification photomicrographs of the resist layers after
developing ZEP.
the chip, following the double-angle evaporation technique introduced by Dolan [85].
The first Al layer had a thickness of 30 nm and was deposited at an angle 12.5° to
the normal. I then oxidized the Al surface to create the insulating barrier for the JJ;
48
3.5. TRANSMON FABRICATION
(a)
200 μm
(b)
10 μm
Fig. 3.8 (a) 10x and (b) 150x magnification photomicrographs of the resist layers after
developing MMA.
typically at an O2 pressure of ∼ 16 Pa for ∼ 5 min. The chamber was evacuated again,
and I deposited the second Al layer with a thickness of 50 nm at an angle −12.5°.
Finally a passivation step was performed by oxidizing the device for ∼ 20min at an O2
49
3.5. TRANSMON FABRICATION
pressure of ∼ 33Pa.
When fabricating the JJ, the duration of the oxidation was the primary factor
I used to control the tunnel barrier thickness. To determine a good value for the
duration, I built multiple test junctions simultaneously and measured their resistance,
to obtain statistics for the JJ characteristics. These resistances were compared to the
target value for RJ, that I found using my designed value for EJ (see Sec. 3.4) and the
Ambegaokar-Baratoff formula [34, 88],
h ∆
R = AlJ . (3.2)e2 8EJ
I used∆Al/e ≈ 210µeV as the superconducting energy gap for thin-film Al. Bulk Al has
a superconducting energy gap of 170µeV [89], but the value used here is slightly larger
because thin superconducting films of Al are well-known to have a larger gap [89–94].
Since I typically wanted EC/h ≈ 200MHz and EJ/h ≈ 20 GHz so that EJ/EC ≈ 100,
Eq. (3.2) yielded a target value of RJ ≈ 8kΩ.
To complete the fabrication of a transmon, I had to lift-off the remaining unexposed
resist. I used a modified version of the lift-off recipes developed by Novikov [72] and
Suri [48]. First, I set a hotplate to 125 ◦C, and used it to heat two crystallizing beakers
each with 125 ml of Microposit-1165 for ∼ 1h; the liquids reached a steady-state
temperature of ≈ 80 ◦C. The processed chip was then placed in the first beaker for
60 min and agitated every 8 min. Next, I transferred the chip to the second beaker and
agitated every 3 min. After 15 min, the chip was thoroughly cleaned using IPA and
DI water, and blown dry. I placed completed chips in a plastic container which itself
50
3.6. CAVITY FABRICATION
(a) (b) (c) e-Beam
Al
ZEP
MMA
Sapphire
(d) (e) Al (f) O
𝜃 2𝑖
(g) (h) Al (i)𝜃𝑓
Fig. 3.9 Summary of the transmon fabrication process. (a) Spinning the dual-resist layer
on sapphire, (b) depositing the Al anti-charging layer and dicing the wafer into chips, (c)
electron-beam exposure in SEM, (d) development of the resists, (e) 30 nm deposition of Al at
θi = 12.5° to normal, (f) oxidation of Al, (g) completed oxidation (h) 50 nm deposition of Al
at θ f = −12.5° to normal, and (i) lift-off for MMA resist.
was encased in a static shielding bag, to prevent destruction when handling. The full
fabrication process is depicted in Fig. 3.9.
3.6 Cavity Fabrication
Using the electromagnetic simulations discussed in Sec. 3.3, I designed a cavity
with suitable dimensions (see Fig. 3.10), and had the LPS machine shop build several
51
3.6. CAVITY FABRICATION
(a) (b)
TOP RIGHT TOP RIGHT
1 cm 1 cm
FRONT
FRONT
Fig. 3.10 (a) Orthographic projections and 3D perspective view for base and (b) lid of the
microwave cavity.
from 6063 Al alloy. This is a corrosion-resistant, non-magnetic, hardened alloy. The
cavities were made of superconducting Aluminum to minimize resistive losses from
the cavity surfaces.
3.6.1 Cavity Cleaning and Preparation
After machining, I had to do some additional preparation work on the cavity to
achieve a relatively high internal quality factor Q i. First, it was necessary to clean
debris from the small scratches and crevices that were created during machining. In
particular, the blind tapped holes would tend to collect loose metal fragments and
lubricants from the machining. I used dental picks and tweezers to dislodge, break
apart, and extract the debris. Although the tools might seem unconventional in a
52
3.6. CAVITY FABRICATION
physics laboratory, their strength, reach, and maneuverability made them well-suited
for my purpose. Of course these tools were designed to work with cavities, although
not those made of Aluminum.
After the debris was removed, I thoroughly degreased, cleaned, and etched the two
pieces of the cavity according to the following steps:
1. Degreasing by ultrasonication in successive baths of acetone, methanol, IPA, and
de-ionized (DI) water for ≈ 10min each.
2. Cleaning in a 50 ◦C heated Alconox detergent bath for 10 min with a magnetic
stirrer.
3. Etching in two successive baths of Avantor Aluminum Etch 80-15-3-2 at 50 ◦C
for 60 min each. A magnetic stirrer was used for agitation, along with periodic
rotation of pieces to remove attached gas bubbles.
4. Thorough rinsing of pieces with DI water spray.
5. Cleaning in a 50 ◦C heated Alconox detergent bath for 1.5 h with a magnetic
stirrer to neutralize any remaining acid.
6. Thorough rinsing with DI water spray, followed by immediate drying with com-
pressed nitrogen.
7. Drying the cavity pieces on a hotplate at 150 ◦C for 1.5 min to evaporate residual
chemicals.
Etching removes material, and makes holes and slots larger. This effect is more
53
3.6. CAVITY FABRICATION
pronounced for smaller features, and it is necessary to take this into account during
the design phase.
3.6.2 Cavity Tuning and Packaging
I coupled microwaves into and out of the cavity using SubMiniature version A
(SMA) feedthroughs. The flange of the connector was secured to the outer surface
of the cavity and a pin connected to the inner conductor protruded into the cavity
space. The coupling of the pin to a cavity mode will depend on the physical position
of the pin, the length of the pin, and the mode. For example, with pins located as in
Fig. 3.2, we would expect the coupling to the TE101 mode to be lower than for the
TE102 mode since the pins are further from the single anti-node of the electric field of
the TE101 mode than they are from the antinodes of the TE102 mode; the proximity
of the coupling pins to the anti-nodes of the electric field determine the strength of
capacitive coupling to the pins. Conversely, the proximity of the coupling pins to the
anti-nodes of the magnetic field would determine the strength of inductive coupling to
a coil shaped termination of the line.
The strength of coupling between the coaxial line and the cavity for a given cavity
mode is primarily adjusted by changing the length of the center conductor pin of
the SMA jack. In general, as the pin protrudes further into the cavity the coupling
will increase. However, it should be noted that beyond a certain critical length, the
coupling decreases. This is due to the metal in the pin causing a large distortion of the
cavity modes, which can essentially be shorted out by the pin. This critical length for
54
3.6. CAVITY FABRICATION
the protrusion of the pin was observed to be ≈ 1 mm when measured from the inner
surface. My procedure for tuning the SMA port couplings of the cavity was as follows:
1. I placed a sapphire chip with a test device in the cavity lid slot for the qubit.
Then, using an Indium gasket around the lip of the lid, sealed the cavity and
secured it using screws.
2. I attached two identical unmodified non-magnetic EZ Form #705615-801 SMA
feedthroughs to the cavity. Obtained the parameter S21(ω) and fit it using the
method described in Appendix A. From this fit, I determined the coupling quality
factors (Qin and Qout) for the input and output generic ports (Q
6
in =Qout ∼ 10 ).
3. I removed the generic SMA output feedthrough. The remaining input port had
known coupling Qin. To achieve the best results, I input the desired coupling
values and calculate the target values for S22 and S21 at resonance as well as the
total Q for the desired cavity mode.
4. I next made a modified output port by extending the center pin of another SMA
connector. To do this, I used a short piece of Coax Company SC-219/50-SC
low-loss semi-rigid coaxial cable with a silver-plated copper center conductor.
I removed the outer conductor and the teflon jacket to obtain just the center
conductor. After snipping off a≈ 6 mm piece, I soldered it to the center conductor
of a generic SMA feedthrough. This ensured that the pin was reaching well into
the cavity. I obtained VNA measurements of S22 and S21 versus frequency and fit
this data, to determine the coupling Qout for the modified port. If the coupling
55
3.6. CAVITY FABRICATION
Q was too low, I removed the connector, trimmed the pin’s length, reattached,
and re-measured the coupling Q. The procedure was repeated until the desired
coupling Q was obtained for the given port. Typically I wanted Qin ∼ 105 and
Qout ∼ 104. The actual cavity Q values used in experiments are listed in Sec. 5.9.
5. After the first pin was adjusted as desired, I replaced the connector with another
SMA jack with an elongated center pin. The trimming and fitting process was
repeated until that port was properly set up as well. This method was suitable
for setting up couplers with Q ∼ 103–106. After both pins were set, I removed
the control port and replaced it with the first modified port.
6. I unsealed the cavity, replaced the test chip with the actual transmon chip, placed
In pieces between the chip and the slot, and secured the chip. I next applied a
fresh In seal around the lip of the cavity lid, and sealed the cavity using McMaster-
Carr non-magnetic Al screws. Finally, I note that I had to take proper precautions
against electrostatic discharge while mounting the transmon chip in the cavity.
3.6.3 Details of Fabricated Cavities
For my experiments, I had the LPS machine shop build four nominally identical
cavities based on my design, using 6063 Al from the supplier McMaster-Carr. After
receiving the cavities, I prepared them as described in Sec. 3.6.1. A few details on the
cavities are given in Table 3.1. Figure 3.11 shows a photograph of the cavity 6D. A
summary of parameters for microwave cavities used in experiments is listed in Sec. 5.9.
56
3.6. CAVITY FABRICATION
Table 3.1 List of fabricated cavities
Fabrication date Cavity Name Note
7/1/2014 6C Initial Machining
7/1/2014 6D Initial Machining
4/8/2015 6F Initial Machining
4/8/2015 6G Initial Machining
9/26/2017 6G Modification to havetwo chip slots
10 mm
Fig. 3.11 Photograph of cavity 6D
57
CHAPTER 4
Experimental Setup for Controlling and Measuring
Qubits
Superconducting quantum integrated circuits are currently one of the leading
candidates for building a scalable quantum computer. Given that superconducting
circuits can be fabricated with a wide range of parameters, there is a very large phase
space of possible device parameters. This space is greatly reduced due to technological
and practical considerations [95]. For superconducting qubits, I will assume that we
need to satisfy:
kBT  ħhωq 2∆ (4.1)
where T is the operating temperature of the environment that is coupled to the qubit,
ωq/2π is the qubit transition frequency, and ∆ is the energy gap of the superconductor
[95]. The first inequality in Eq. (4.1) allows a qubit to be initialized in its ground
state by simply waiting much longer than the relaxation time T1, and it also minimizes
thermal excitation. The second inequality in Eq. (4.1) must be satisfied to minimize
58
4.1. THE DILUTION REFRIGERATOR SETUP
thermal generation of quasiparticles by breaking Cooper pairs. Both requirements
can be satisfied by using a superconductor with Tc > 1 K and keeping characteristic
transition frequencies in the 5 GHz to 10 GHz range [95]. The qubits I fabricated were
made from thin films of Al (Tc ' 1.2 K) and had ωq/2π in the range 5 GHz to 6 GHz,
while the Al cavity had a resonance frequency ωr/2π≈ 8 GHz.
4.1 The Dilution Refrigerator Setup
Since I designed my devices to have characteristic frequencies between 5 GHz and
10 GHz, I needed to cool them down to well below 1 K. In practice, for thin film
Al devices we find loss from thermally generated quasiparticles to be significant for
T ≥ 150mK. Keeping T ∼ 20 mK also ensures that the Boltzmann factor e−ħhωq/kT <
10−6, resulting in minimal thermal excitations of the qubit or cavity. The most popular
method for cooling superconducting qubits is the 3He-4He dilution refrigerator [96].
Dilution refrigerators offer continuous operation at temperatures as low as a few mK,
which is ideal for satisfying Eq. (4.1) [97].
All my experiments were conducted in a Leiden CF-450 cryogen-free dilution
refrigerator [98]. This apparatus had a nominal cooling power of ∼ 450µW at a
mixing chamber temperature of 120 mK. The first two stages of the Leiden refrigerator
were cooled by a two-stage pulse tube cryocooler to 50 K and 3 K, respectively. The
qubit or qubits were mounted in an Al cavity that was thermally anchored to the
mixing chamber stage, which had a base temperature of 15 mK. The operation of this
59
4.1. THE DILUTION REFRIGERATOR SETUP
Fig. 4.1 Photograph of the CF-450 Dilution refrigerator with two Cu mounting posts. The
thermal shields and cryoperm shields have been removed.
60
4.1. THE DILUTION REFRIGERATOR SETUP
300 K
50 K
LPS Custom Attenuator
3 K 20 dB
Midwest Attenuator
1 K
K&L Cryogenic Low Pass Filter
80 mK 30 dB
DC Block
Quinstar/Pamtech Isolator
Caltech CITCRYO 4-12A 
HEMT Amplifier 20 dB
SC-219/50-CN-CN coax
SC-219/50-Nb-Nb coax
UT-85C-LL coax 15 mK
UT-085C-Form-LL coax
Direct connection or 
DUT
Huber+Suhner Sucoflex 102 coax
Cryoperm Shield
Copper Shielded Room
Fig. 4.2 Input/Output microwave lines in the dilution refrigerator.
refrigerator, and details of the apparatus, are described extensively by Novikov [72].
Figure 4.1 shows a photograph of our CF-450 dilution refrigerator.
While the refrigerator cools to 15 mK, it is very challenging to ensure that the
microwave signals propagating to the device are well-thermalized to a low temperature.
Thermal noise on the input or output lines can produce unwanted excitation of the
qubit and cavity, and lead to relaxation and dephasing [56, 57, 99, 100]. To block
external RF interference, and to ensure thermalization of noise on the input signal
61
4.1. THE DILUTION REFRIGERATOR SETUP
line to the device-under-test (DUT), broadband attenuators are placed strategically
at various cooling stages in the dilution refrigerator (see Fig. 4.2). Although these
attenuators filter out high-frequency blackbody noise emanating from the 300 K stage
electronics, they also generate blackbody radiation corresponding to their operating
temperature. The refrigerator I used had an input line with 70 dB of attenuation
from lumped element components and further attenuation contributed by cupro-nickel
coaxial cables used for some sections (see Fig. 4.2). This yielded a noise temperature
≤ 50mK for signals reaching the DUT [57]; this was significantly higher than the base
temperature of the refrigerator, for reasons that are still not well-understood. The
30 dB and 20 dB attenuators at the cold plate and mixing chamber stages (see Fig. 4.2)
were custom-made by Dr. Jen-Hao Yeh to have high cooling-power and low noise
temperature at cryogenic temperatures [57].
The output line from the cavity had 60 dB of directional isolation provided by
two Pamtech CWJ1015-K3 isolators and one Quinstar QCI-075900X000 isolator (see
Fig. 4.2). The input and the output lines to the device also had K&L 11L250-12000/T20000
low-pass filters (LPFs) with a 12 GHz cutoff to block noise that could cause excitations
of higher modes of the cavity. The output signal from the cavity was amplified at 3 K by
a Caltech CITCRYO 4-12A high-electron-mobility-transistor (HEMT) amplifier which
had a nominal noise temperature Tn < 5 K [101].
The DUT was enclosed in two Amuneal Manufacturing Amumetal 4K cryoperm
shields [102] to protect the device from low-frequency magnetic fields. These shields
were attached to the mixing chamber stage. The mixing chamber stage (at 15 mK)
had no dedicated shield, but both the cold plate stage (at 80 mK) and the still stage
62
4.2. VECTOR NETWORK ANALYZER
(at < 1 K) had Au-plated Cu shields. The 3 K, 50 K, and 300 K stages all had Al shields.
The 3 K and 50 K shields were wrapped in blankets of Aluminized Mylar. The 300 K
and 50 K stages composed the outer vacuum chamber (OVC), while the stages at 3 K
and below composed the inner vacuum chamber (IVC). There were no other magnetic
shields used at room temperature. The entire dilution refrigerator was housed in a
grounded Cu shielded room to reduce electromagnetic interference.
4.2 Vector Network Analyzer
A vector network analyzer (VNA) is a powerful instrument for characterizing a
microwave network based on its scattering parameters (S-parameters) [103]. The
S-parameters are obtained by measuring the amplitude and the phase of reflected and
transmitted signal components with respect to a DUT. The Agilent 5071C VNA I used
has an internal microwave source that can sweep the frequency while simultaneously
measuring the two-port S-parameters, making it very useful for microwave spectroscopy
of qubits and cavities. I used the VNA to accurately measure the loaded quality factor
of the cavity at 15 mK, and at room temperature to tune the cavity coupling ports as
described in Sec. 3.6.2.
I mainly used the Agilent E5071C VNA for preliminary characterization of our
devices. After a device was cooled to 15 mK, I first measured transmission through
the cavity as a function of frequency. Although the cavity was designed to have pre-
determined spectra, due to imperfections during machining, chemical processing and
63
4.3. HETERODYNE PULSED MEASUREMENT SYSTEM
variations in mounting the qubit chip, its frequency would inevitably vary slightly from
the design value. Furthermore, due to thermal contraction and changes in the London
penetration depth, the frequency changes when cooling the device from 300 K to 15 mK;
the room temperature value was a good starting guess. This simple measurement also
allowed me to determine if there was a serious problem with the qubit.
It is known that the cavity transmission at high power (average cavity photon
number n̄ 6cav ∼ 10 ), is insensitive to the state of the qubit and the cavity resonance
frequency will be at the bare frequency ω(bare)r [104]. However, when a low power is
used (n̄cav ≤ 10), the dressed cavity resonance ω̃r depends on the state of the qubit
(see Fig. 2.3). By sweeping the output signal power from the VNA during spectroscopy,
I examined these two regimes and checked that the qubit had a well-behaved response
(see Fig. 5.2). Furthermore, measuring the shift χ = ω(bare) − ω̃(|g〉)ge r r allowed me to
estimate the dispersive shift between the cavity and the transmon qubit.
4.3 Heterodyne Pulsed Measurement System
Most of the qubit gate operations I used were accomplished by applying one or
more discrete microwave pulses. Since qubits have a limited lifetime, it was also
necessary to perform qubit measurement using a pulsed measurement system. Most of
my measurements and control operations were carried out using a pulsed control and
measurement setup (see Chapter 5). Although the VNA offered exquisite measurement
capability for continuous-wave applications, for gating and qubit state measurements,
64
4.3. HETERODYNE PULSED MEASUREMENT SYSTEM
I used the customized setup depicted in Fig. 4.3. The qubit state was interrogated by
measuring the transmission of pulsed signals through the cavity. The apparatus was
set up for signal detection at either the bare or dressed cavity frequency, depending on
whether I used a high-power or low-power readout technique (see Sec. 5.2).
To perform digital signal processing on the data, an analog-to-digital converter
was used to convert the high frequency microwave signals from the cavity output
line to a digital record (see Fig. 4.3). The two most commonly used methods for
radio-frequency (RF) signal down-conversion are “homodyne” and “heterodyne” [105].
Homodyne detection is also known as direct-conversion, since the original RF signal
is mixed with a local oscillator (LO) signal which is operated at the same frequency,
i.e. ωLO =ωRF, to yield a zero-frequency measurement signal. By using an IQ mixer
which mixes the sine and cosine components separately, it is possible to perform a
quadrature measurement of the RF signal. Although homodyne detection has fewer
components such as synthesized signal generators, it can be difficult to correct and
compensate for non-idealities present in the IQ mixer. These non-idealities can depend
on the frequency and amplitude of the LO, harmonic distortions, as well as DC offsets
[105].
As an alternative, heterodyne detection involves down-conversion to an interme-
diate frequency (IF) using a heterodyne receiver. With this type of receiver ωIF =
ωLO −ωRF, and an IF frequency ωIF ∼ 1 MHz to 10 MHz was suitable for my cQED
applications. The in-phase and quadrature (I/Q) components of the RF signal were ex-
tracted from the measured amplitude and phase of the IF signal. For high-performance
detection, it is also possible to use a superheterodyne receiver which performs double
65
4.3. HETERODYNE PULSED MEASUREMENT SYSTEM
RF
C320520 VHF-2700A
R L LO
I
To 
SHP-1000+
DUT
UT-085C-Form-LL coax
Mini-Circuits
BNC Cable SLP-21.4+ SHP-1000+
L
Mini-Circuits ZFL-400LN+ Amplifier I
Picosecond R
5915
Mini-Circuits High Pass Filter Output
From 
Low Pass Filter DUT
MAC Technology Directional Coupler
Agilent U1082A 
R L Marki M1-0310LA Double-balanced Mixer Ref Digitizer Signal
I
Fig. 4.3 Heterodyne setup for measuring reference signal and the output signal from the DUT.
down-conversion with two different LO and IF frequencies. Double-conversion or an
image reject mixer can be used to prevent image noise fold-over [106, 107]. Although
a heterodyne detection system has more complex circuitry and also requires a separate
LO signal generator, it is insensitive to harmonic distortion and DC offsets, and does
not require calibration for accurate I/Q measurements [105].
All my pulsed qubit measurements were performed with a single-conversion het-
erodyne detector shown schematically in Fig. 4.3. The input signal to the DUT was
separated into two branches using a 20 dB directional coupler and the LO signal was
separated into two branches using a 6 dB directional coupler. One branch of the input
signal was mixed with the filtered LO signal and used as the reference waveform for
the digitizer. The other branch of the input signal was sent to the DUT. The output
signal from the DUT was mixed with the second branch of the LO signal, filtered, and
66
C32056
4.4. MICROWAVE SIGNAL GENERATORS
amplified, before being sent to the second input channel on the digitizer. The I/Q
demodulation was performed using software in LabVIEW. In my measurements, I set
IF = 10 MHz and used 1000 onboard averages of the voltage using the Agilent Acqiris
1082A digitizer board to perform measurements efficiently.
To ensure phase matching during acquisition and averaging, the RF signal generator
and the digitizer were triggered by the same source, which was an Agilent 33250A
arbitrary waveform generator locked to a Rb atomic clock (see Fig. 4.4).
4.4 Microwave Signal Generators
4.4.1 Synthesized Signal Generators
For signal generation, I used Agilent E8257D/E8267D synthesized signal generators
(SSGs) that output a highly coherent continuous analog tone at a set frequency (300 kHz
to 20 GHz). By triggering the SSG output signal, I generated pulsed wave trains with
high accuracy and precision. The time between the trigger reception and pulse output
had a minimum duration of 70 ns for the Agilent SSGs. When timing pulses from
different SSGs with respect to each other, this delay had to be properly accounted
for. Unfortunately, it is not possible to have in-phase and quadrature (I/Q) control
solely using a triggered SSG. Hence, I used these instruments for preliminary qubit
characterization, and to perform measurements which did not require I/Q control such
as Rabi oscillations and Ramsey oscillations (see Chapter 5).
67
4.4. MICROWAVE SIGNAL GENERATORS
4.4.2 Arbitrary Waveform Generators
I used arbitrary waveform generators (AWGs) capable of digitally generating arbi-
trary waveforms, as signal sources for my experiments. With these complex instruments,
a user can program any desired waveform as a series of voltage values versus time
(waypoints), and the AWG will generate the specific waveform, within its bandwidth
and voltage resolution limitations. The output of AWGs are limited in their frequency
range by the sampling rate, in their voltage resolution, and the maximum length of
pulses.
The Tektronix AWG70002A that I used for device control had a sampling rate of
25 GSa/s, while the microwave signals used in our experiments had frequencies in
the range 4 GHz to 8 GHz. Based on the sampling theorem attributed to numerous
scientists and engineers [108–110], real-world AWGs can faithfully generate analog
signals up to 40% of its sampling rate [111]. Since the device frequencies in our system
were < 10GHz, the Tektronix AWG70002A had the capability to directly generate the
required pulses for qubit and cavity control. For my experiments, I fixed the sampling
rate at 25 GSa/s and made any change in signal frequency directly by adjusting the
waypoints.
For all programmed waveforms, the instrument had a minimum waveform length
of 2400 samples and a maximum waveform length of 8× 109 samples per channel,
limited by the AWG memory of 16 GB. At 25 GSa/s, the shortest pulse corresponded
to a minimum length of 96 ns and the longest pulse corresponded to a maximum
length of 320 ms. Although most of the waveforms I programmed into the AWG had
68
4.5. DEVICE CONTROL AND MEASUREMENT
lengths > 96ns, to simplify programming, all the waveforms I created had a 100 ns
length zero-voltage buffer-zone at the beginning. Each pulse train was designed in
LabVIEW and uploaded as a text file to the AWG via an ethernet connection. The
text file consisted of a discrete set of voltage values at the maximum sampling rate of
the instrument. Following upload, each text file was read using the AWG’s internal
software and stored in its native format in the AWG memory. The experiments were
then carried out by loading the waveforms sequentially into the AWG70002A and using
an Agilent 33250A AWG to trigger qubit or cavity pulses (see Fig. 4.4).
4.5 Device Control and Measurement
The essential components of the device control and measurement network have
been described in previous sections. Here, I take a holistic approach and trace the path
of microwave signals from generation to detection (see Fig. 4.4).
For preliminary characterization, all the signal generators shown in Fig. 4.4 are
disconnected, and the E5071C VNA is used both as the signal source and the detector.
In this configuration, the output signal from the top of the refrigerator is amplified by
a single MITEQ amplifier before being fed into the VNA input.
For pulsed qubit control and heterodyne measurements, the Agilent 33250A cavity
AWG was set up with the proper timing, to trigger its counterpart Agilent 33250A
qubit AWG, and the cavity SSG E8267D. The qubit AWG was in turn used to trigger the
qubit signal generator (either Tektronix AWG70002A or Agilent E8257D). A third SSG
69
4.5. DEVICE CONTROL AND MEASUREMENT
E8257D was used to apply a third tone, if necessary. The latter SSG could either be set
in continuous mode or triggered mode. The signals from the three signal generators
are routed through a Weinschel programmable atttenuator and combined using two
6 dB MAC attenuators (see Fig. 4.4). The combined microwave drive signal passed
through a DC block at the wall of the shielded enclosure and entered the input line at
the top of the dilution refrigerator.
As the drive signal passed through the cold stages, it was progressively attenuated in
the cupro-nickel cables (from 300 K to 0.08 K) and lumped element attenuators (70 dB
total attenuation). A Nb cable was used at the 80 mK stage to limit the conduction
of heat from higher temperature stages. The heavily attenuated signal was further
filtered with a LPF at the mixing chamber stage before being fed to the input pin of the
cavity.
The output signal from the DUT then passes through another LPF and three iso-
lators. The isolators and the LPF work in conjunction to prevent thermal noise from
propagating down the output line from the 3 K stage to the cavity output pin. The
output signal was amplified at 3 K with a Caltech HEMT amplifier [101] and further
amplified at room temperature with a MITEQ low-noise amplifier [112]. This signal
was then mixed to a lower IF using the heterodyne system before being digitized by
the Agilent 1082A digitizer. All the signal generators were synchronized with a single
Stanford Research Systems FS725/2 Rubidium frequency standard, to prevent drift.
70
4.5. DEVICE CONTROL AND MEASUREMENT
Agilent 33250A Cavity
Trigger E8267D
E83650B
C320520
E8257D VNA R L HP HPI
out
Mini-Circuits
SLP-21.4+ LP Mini-Circuits HP SHP-1000+
Tektronix ZFL-400LN+
AWG70002A HP SHP-1000+
LP
C32056 C32056 Picosecond 
L
I
Qubit 5915 R
Trigger E8257D
Ref Signal
SRS FS725/2 Rb Atomic Clock
Agilent U1082A 
LPS Custom Attenuator Trig Digitizer
VNA
Midwest Attenuator in
DC Block
Quinstar/Pamtech Isolator MITEQ AMF-3F
-0400800-07-10P
Caltech CITCRYO 4-12A HEMT Amplifier
Amplifier
MAC Technology Directional Coupler
300 K
50 K
Weinschel Programmable Attenuator 
3 K 20 dB
VNA Agilent E5071C Vector Network Analyzer
1 K
Agilent Synthesized Signal Generator
80 mK 30 dB
Arbitrary Waveform Generator
R L Marki M1-0310LA Double-balanced Mixer
I
LP K&L Cryogenic Low Pass Filter 20 dB
HP Mini-Circuits High Pass Filter
LP Low Pass Filter LP LP
15 mK
SC-219/50-CN-CN coax
SC-219/50-Nb-Nb coax
UT-85C-LL coax
DUT
UT-085C-Form-LL coax
BNC Cable Cryoperm Shield
Direct connection or 
Huber+Suhner Sucoflex 102 coax
Clock Synchronization Signals
Trigger Signals Copper Shielded Room
Fig. 4.4 Schematic of circuit for controlling and measuring transmon qubits.
71
VHF-2700A
VHF-2700A
C32056
CHAPTER 5
System Characterization
Device characterization is an essential component in quantum computing, especially
so in superconducting qubits. Due to the propensity for the transition frequencies,
couplings, and other qubit parameters to vary somewhat from one cooldown to the next,
it is important to properly characterize the devices at the beginning of each experiment
run. Parameters can also drift slightly during a run, and occasional calibration checks
are a good idea. This helps to ensure that quantum gating operations are performed
accurately. The basic physics of Rabi oscillations, relaxation time measurements,
Ramsey oscillations (Ramsey fringes), and spin-echo measurements are described in
detail by Meystre and Sargent III [53] and in many other places. In this chapter, I will
focus on the experimental approach to characterize my transmons.
72
5.1. CAVITY SPECTROSCOPY
5.1 Cavity Spectroscopy
I began the characterization process by performing spectroscopy near the cavity
frequency versus drive power. This revealed the quality factor of the cavity and let me
determine a good pulse length for reading out the transmon state. The spectroscopic
measurements also allowed me to determine a preliminary power bias point for the
high-power qubit readout. As mentioned in Sec. 4.2, it was particularly convenient to
perform this spectroscopy using the VNA.
Figure 5.1 shows an example where I did cavity spectroscopy at high power and
low power. The two traces show |S21| versus frequency f for input powers of −67 dBm
(black) and −117 dBm (red). The resonance at high-power occurs at the bare cavity
resonance frequency, and the resonance at low-power occurs at the dressed cavity
6×10−4
5×10−4
4×10−4
3×10−4
2×10−4
1×10−4
0
7.78 7.80 7.82 7.84 7.86 7.88
f (GHz)
Fig. 5.1 High-power (−67 dBm, black) and low-power (−117 dBm, red) spectroscopy showing
|S21| versus frequency f of cavity 6D during Measurement Run #1.
73
𝑆21
5.1. CAVITY SPECTROSCOPY
resonance frequency, where average stored photon number n̄cav ∼ 1. It is the dressed
cavity resonance frequency that depends on the state of the qubit (see Sec. 2.2.3). In
this case, since the qubit was not excited, the resonance at ωr/2π = 7.7968GHz is
the bare resonance, while the dressed resonance is at ω̃(|g〉)r /2π = 7.8524 GHz, and
this yields a shift of χge/2π= −56 MHz. I note that the effective quality factor of the
resonance also varied, as indicated by the different peak heights and widths.
Seeing the shift of the resonance as in Fig. 5.1 gave me the confirmation that the
qubit had survived the mounting and cooldown. I next completed a more compre-
hensive measurement of the spectrum as a function of power. An example of such
a measurement is shown in Fig. 5.2. This false color plot shows |S21|
2 of the cavity
resonance versus frequency f and drive power P. Note in particular the rich structure
in the intermediate powers. Examination of this structure is beyond the scope of this
work, but Mavrogordatos et al. [113] conducted an extensive analysis of the behavior.
Figure 5.2 shows the steady-state measured |S 221| versus applied frequency and
power; that allowed me to pick an initial bias point for transmon readout as discussed
next in Sec. 5.2. Using this spectroscopy, I picked the initial bias point suitable for
the non-linear high-power readout (see Sec. 5.2.2) at frequency ωr/2π = 7.9227 GHz
with an applied power of P = −57 dBm. This was a good preliminary bias point since
at powers > −57 dBm the cavity exhibits good transmission and at powers < −57 dBm
the cavity exhibits low transmission. By biasing the cavity at −57 dBm I expected the
cavity transmission to increase dramatically when the qubit was in an excited state
(see Sec. 5.2.2 for further details).
Following preliminary qubit characterization, I revisited cavity spectroscopy to
74
5.1. CAVITY SPECTROSCOPY
obtain the cavity transmission with the transmon in state |e〉 (see Fig. 5.3(a)). The
difference in the cavity transmission between state |e〉 and |g〉 is shown in Fig. 5.3(b).
The latter plot allowed me to verify the optimum position for biasing the cavity in
order to read out the transmon state efficiently. As is evident from Fig. 5.3(b), the
optimum bias point was at the bare cavity frequency with a bias power of −58 dBm.
𝑓init
𝑃init
-60
-20
-70
-80
-90
-100
-115
-110
7.91 7.92 7.93 7.94 7.95
f (GHz)
Fig. 5.2 |S21|
2 versus frequency f and drive power P for the cavity 6D, during Measurement
Run #5. Bare cavity resonanceωr/2π = 7.9227GHz and the dressed cavity resonance with the
transmon in |g〉 is at (|g〉)ω̃r /2π= 7.9430 GHz. The preliminary qubit readout point is depicted
by ( finit, Pinit)
75
P (dBm)
𝑆 221 (dB)
5.2. TRANSMON READOUT
𝑆 221 (dB) Δ 𝑆
2
21 (dB)
(a) -115 -20 (b) -40 +40
-60 -60
-70 -70
-80 -80
-90 -90
-100 -100
-110 -110
7.91 7.92 7.93 7.94 7.95 7.91 7.92 7.93 7.94 7.95
f (GHz) f (GHz)
Fig. 5.3 (a) |S 221| versus frequency f and drive power P of the cavity 6D, during Measurement
Run #5. Color corresponds to the transmission |S 221| in dB. The dressed cavity resonance
with the transmon in |e〉 is at (|g〉)ω̃r /2π = 7.9373GHz. (b) The difference ∆ |S
2
21| between
transmission |S 221| when the transmon state changes from |e〉 to |g〉.
5.2 Transmon Readout
5.2.1 Dispersive Readout
Dispersive readout was initially proposed by Blais et al. [42] and demonstrated by
Wallraff et al. [43] for reading out Cooper-pair box qubits in a circuit QED architecture.
The specific readout method they proposed involved probing the transmission properties
of the cavity with a low-powered pulse, typically at the dressed cavity frequency
ω̃(|g〉)r = ω̃r−χ with the qubit in state |g〉. From Eq. (2.28), we see that high transmission
of the microwave pulse is indicative that the transmon was in the ground state, and low
transmission is indicative that the transmon was in an excited state. For example, the
76
P (dBm)
P (dBm)
5.2. TRANSMON READOUT
dispersive readout could have been used in device SPP-v2-Q2 by probing the 6D cavity
at a power of −117 dBm (see Fig. 5.1). A “low” |S21|
2 signal would have indicated
the transmon was in an excited state, while a “high” |S21|
2 would indicate that the
transmon was in an excited state.
This low-power method had the advantage of being a quantum non-demolition
readout and being a weak measurement of the system [42]. However, due to the low
power used during the measurement, the output signal was weak as well. Without the
use of a quantum-limited amplifier, the signal must be averaged over many shots in
order to obtain a good signal-to-noise ratio.
In fact, most of my devices showed a low-power resonance (see Fig. 5.2), that
was somewhat distorted and weak in strength compared to the bare resonance at
high-power. Hence, I chose to employ a high-power readout technique, as described in
the next section.
5.2.2 The Non-linear High-power Readout
An alternative method of reading out the state of a transmon was demonstrated by
Reed et al. [104]. This method realized a projective strong measurement by applying a
high-power microwave pulse at the bare cavity frequency ωr. By mapping the cavity
transmission S21 as a function of the pulse power P and the transmon state, suitable
bias powers can be found which give good discrimination between the transmon
|g〉 and |e〉 states [104]. In general, occupying higher transmon levels will yield a
higher transmission at a lower pulse power [33]. This non-linear high-power readout
77
5.2. TRANSMON READOUT
technique was used exclusively in all my measurements since it showed better stability
and signal-to-noise ratio, compared to the dispersive readout for my devices.
The non-linear high-power readout was calibrated by measuring the cavity trans-
mission S21 at the bare cavity frequency as a function of pulse power P. I started this
calibration process by choosing the length of the cavity transmission measurement
pulse. Most of my cavities had lifetimes of 50 ns to 100 ns. For such cavities, I chose
the length of the cavity pulse to be 1µs so that the cavity had ample time to ring-up.
For my Fock state measurements [33], the cavity had a lifetime of ∼ 2µs and, in that
case, I chose the pulse length to be 3µs.
After fixing the pulse length, I applied the pulse sequence shown in Fig. 5.4 1000
times repeatedly, and averaged the amplitude of the transmitted signal separately in
two batches using the digitizer’s onboard averaging capability. The first set of mea-
surements (background) consisted of applying no operations for an idle time tidle ∼
100µs to 400µs, followed by performing a cavity transmission S21 measurement with
a pulse of duration tRO ∼ 1µs to 3µs. The second set of measurements (qubit) con-
sisted of applying the desired set of transmon operations and then applying the cavity
𝑡idle 𝑡idle 𝑡RO
Cavity 𝑆21 Transmon Cavity 𝑆21
Measurement Operations Measurement
𝑇
bg meas
Fig. 5.4 Pulse sequence to obtain cavity switching curves. tidle is the time between cavity
measurement pulses, used to ensure the system returns to a ground state. tRO is the length
of the readout pulse, sufficiently long to calculate the relative transmission depending on
the transmon state. The background measurement (bg) and signal measurement (meas) are
obtained without and with transmon operations, respectively.
78
5.2. TRANSMON READOUT
measurement pulse. I chose tidle to be over five times the lifetime of the longest-lived
component, to ensure the system always started in the ground state. The background
and qubit measurement pulses were interleaved, to allow me to detect random vari-
ations in the system, or systematic effects such as heating; these would show up as
variations in the background measurements. Given these measurement signals, I then
constructed (∆V/V )meas and the background signal (∆V/V )bg, where ∆V and V are
the demodulated in-phase voltage components at the “Signal” and “Reference” inputs
to the digitizer in Fig. 4.3, respectively. In most of my data plots, I actually report the
fractional difference signal
 ‹  ‹  ‹
∆V ∆V ∆V
= − . (5.1)
V V meas V bg
The phase of the reference pulse after digitization, was also adjusted through LabVIEW
software, in order to report all the signal information in the in-phase (I) component
for each Measurement Run. This meant that the information relating to the qubit state
was strictly visible as a difference in the detected amplitude and not in phase.
I initialized the transmon in various states using microwave drives and performed
the pulse sequence represented in Fig. 5.4 to obtain the “switching curves,” (∆V/V )
versus P, for the transmon-cavity system with the qubit prepared in different initial
states. These switching curves served as the roadmap that I used to determine the
optimum power P0 for the cavity transmission measurement pulse for the high-power
readout. Unfortunately, there were a few devices where the cavity measurement when
the transmon state was in |e〉 was not easy to distinguish, from that produced when
79
5.2. TRANSMON READOUT
0.10 |𝑔〉
|𝑒〉
0.08 |𝑓〉
|ℎ〉
0.06
0.04
0.02
0.00
-77 -73 -69 -65 -61 -57 -53
P (dBm)
Fig. 5.5 Switching curves of the cavity 6D during Measurement Run #2. The black curve
was obtained using the background measurement for different input powers P of the cavity
measurement pulse. The red, blue, and green curves were obtained in similar fashion after
initializing the transmon in the states |e〉, |f〉, and |h〉 respectively.
the transmon was in the state |g〉.
For example, Fig. 5.5 shows switching curves I obtained for device SPP-v1-Q7
prepared in states |g〉, |e〉, |f〉, and |h〉. It is evident that there is relatively poor discrim-
ination between the black (|g〉) and red (|e〉) curves for any power from −69 dBm to
−56 dBm. In contrast, the maximum possible signal value of ∆V/V ≈ 0.1, is found
at the peak of the green curve. The peak of the blue curve near −64 dBm also shows
excellent contrast between the |g〉 state and |f〉 state signals. In a situation such as
this, I found it highly advantageous to map the population |e〉 ⇒ |f〉 to increase the
contrast between the states during a measurement. This technique allowed me to
efficiently measure the state of the transmon with fewer false counts. Of course there
was a higher chance for the transmon state to decay during measurement, but this
was a price that was worth paying. After determining a good cavity bias power for the
80
DV/V
5.3. TRANSMON SPECTROSCOPY
high-power readout, I proceeded with characterizing the transmon, as described in the
next sections.
5.3 Transmon Spectroscopy
I used transmon spectroscopy to find the allowed transition frequencies of the
transmon. In all the transmons I measured, I did not know the Josephson energy EJ
prior to the cooldown. In fact, to minimize the possibility of destroying the junction
through electrostatic discharge, I deliberately did not measure the room temperature
resistance of live devices. Although I had a good idea of the transmon capacitance CΣ,
not knowing EJ meant I did not know the characteristic frequency of the transmon
until I measured it through spectroscopy.
I performed transmon spectroscopy after first selecting a preliminary cavity bias
power. For example, for the map shown in Fig. 5.2, suitable preliminary bias points
are at {7.9227 GHz,−57dBm}, or at {7.9430GHz,−100 dBm}.
After setting a preliminary cavity power bias point, I performed the pulsed mea-
surement sequence depicted in Fig. 5.4 using a long pulse to excite the transmon. The
frequency of the pulse was incremented to perform spectroscopy, as shown in Fig. 5.6.
Since I was not initially sure of the power and length of a pulse required to excite the
transmon (i.e. power and length of a π-pulse), I started by using a high-power, long
saturation pulse to locate the transition frequency of the transmon. The saturation
pulse was typically ∼ 100µs for my transmons which had lifetimes ∼ 20µs.
81
5.3. TRANSMON SPECTROSCOPY
Although the preliminary cavity power bias point found from visual inspection
was not optimal, it usually had sufficient contrast to allow me to find the transmon’s
g ↔ e transition frequency. After finding this frequency, I was able to find a better
cavity power bias point and iteratively optimize the system for qubit measurements.
Figure 5.7 shows an example of a spectrum for device SPP-v1-Q6, after all measurement
parameters have been optimized; I used a saturation pulse of power −111dBm and
length tsat = 200µs to obtain the transmon spectrum.
A transmon transition spectrum such as Fig. 5.7 reveals a wealth of information
about the device. For example, the |g〉 → |e〉 transition frequency is ωge/2π = f1 =
6.3312GHz. The peaks at f2 and f3 are due to two-photon and three-photon transitions,
respectively, from the ground state to the second and third excited states; from there
I obtain ωgf/4π = f2 = 6.2335GHz, and ωgh/6π = f3 = 6.1295 GHz. From these
frequencies of the higher excited states, I determined the charging energy using EC/h≈
2( f1− f2) = 195MHz. However, as the higher energy levels are prone to Stark shifting
and power broadening, this value is close to but only approximately equal to EC.
When there was a doubt regarding the identity of a transition, I was able to check
that a spectral peak corresponded to single-photon, or multi-photon transitions by
𝑡sat 𝑡RO
Transmon Transition Cavity 
Saturation Measurement
𝑇
Fig. 5.6 Pulses for transmon spectroscopy.
82
5.4. RABI OSCILLATIONS
𝑓3 𝑓2 𝑓1
0.4
0.3
0.2
0.1
0.0
6.10 6.15 6.20 6.25 6.30 6.35 6.40
f (GHz)
Fig. 5.7 Transmon spectroscopy for transmon SPP-v1-Q6 during Measurement Run #6. The
three peaks have frequencies at f1 = 6.3312 GHz, f2 = 6.2335GHz, and f3 = 6.1295GHz.
The saturation pulses had length tsat = 200µs and an input power of −111 dBm. The cavity
measurement pulse had a power −81.5 dBm and a length tRO = 3µs.
measuring the width and amplitude of the spectroscopy versus power and examining
the power broadening that occurs in systems with anharmonicity [114]. At each
power, I fit the peak to a Lorentzian distribution and then plotted the widths versus
power [115]. The power dependence of the widths reveals the order of the transition.
Figure 5.8 shows an example for a two-photon transition between the |g〉 and |f〉 levels
of the transmon.
5.4 Rabi Oscillations
Rabi oscillations are central to many implementations of gates in quantum com-
puting. A Rabi oscillation describes the behavior of the state of a two-level system
83
DV/V
5.4. RABI OSCILLATIONS
(a)
-75 ∆V/V
0.08
-80 0.06
0.04
-85
0.02
0
-90
5.860 5.861 5.862 5.863 5.864
f (GHz)
(b) 12.0
11.5
11.0
10.5
10.0
9.5
9.0
-5.4 -5.2 -5.0 -4.8 -4.6 -4.4 -4.2
log10(Vrms)
Fig. 5.8 (a) The |g〉↔ |f〉 two-photon process spectroscopy for transmon SPP-v1-Q7 during
Measurement Run #5. ∆V/V was measured versus frequency f and applied power P. (b) Plot
of full-width at half-maximum log10(FWHM) versus rms applied voltage log10 (Vrms). The fit
(red line) to data (black points) has a gradient of 1.93 confirming that this is a two-photon
process.
84
log10(FWHM) P (dBm)
5.4. RABI OSCILLATIONS
that is driven for a time t by a sinusoidal oscillating applied field [47, 53]. When the
two-level system is driven on-resonance, it is possible to obtain a simple analytical
solution using the rotating wave approximation. Ignoring dissipation and dephasing,
driving a qubit at its |g〉↔ |e〉 transition frequency will cause the state of the system
to oscillate sinusoidally between the two levels [53]. These oscillations are known as
Rabi oscillations after Isidor Rabi who studied them in the field of nuclear magnetic
resonance [116].
In cQED, Rabi oscillations are generated by driving the transmon with a microwave
signal on resonance at the |g〉 → |e〉 transition frequency (see Fig. 5.9). With the
system passively cooled, the transmon starts in state |g〉. Applying a microwave drive
at frequency ωge induces stimulated transition oscillations between levels |g〉 and
|e〉. It is not possible to observe the full oscillation in a single experiment; instead
these measurements were performed by applying a Rabi drive for a time tRabi, before
measuring the qubit state using a high-powered cavity measurement. I typically
repeated this procedure 1000 times to obtain a single averaged data point ∆V/V , and
used this to find the population at time t. I repeated this averaged measurement with
different Rabi lengths tRabi. Figure 5.10 shows an example of a Rabi oscillation in
𝑡Rabi
Cavity 
𝜔ge Measurement
𝑇
Fig. 5.9 Pulse sequence for Rabi oscillations. The time tRabi is varied for the measurement.
85
5.4. RABI OSCILLATIONS
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0 200 400 600 800 1000
𝑡Rabi (ns)
Fig. 5.10 Rabi Oscillations in transmon SPP-v1-Q7 during Measurement Run #5. Microwaves at
5.96339 GHz were delivered to the cavity input with power−70 dBm to induce Rabi oscillations
of frequency fRabi ≈ 11.7MHz. Data (black points) are overlaid with a decaying sinusoidal fit
(red curve).
device SPP-v1-Q7.
In an ideal but unrealistic system, the Rabi oscillations would persist indefinitely, as
long as the drive was applied. However, real qubits do not exist in isolation. Dissipation
and dephasing are always present and will cause the amplitude of oscillations to decay,
leaving the system in a mixed state between the |g〉 and |e〉 states. This condition is
referred to as saturation. Assuming a frequency independent loss, the decay time of
the Rabi envelope T ′, is related to the relaxation time T1 and the coherence time T2 of
the qubit by [48]
1 1 1
= + . (5.2)
T ′ 2T1 2T2
In the special case where there is no dephasing, it turns out that T2 = 2T1 and Eq. (5.2)
86
DV/V
5.4. RABI OSCILLATIONS
then gives T ′ = 4T1/3. The experimental method that I used to determine T1 and T2
are discussed below.
5.4.1 Qubit Population Calibration
By measuring a complete Rabi oscillation (see Fig. 5.10) I was able to extract a few
important parameters and calibrate the qubit drive. The fitting function I used for the
Rabi oscillations was,
 ‹
∆V ′
= Ae−t/T sin(Ωt +φ) + B (5.3)
V Rabi
where A is the amplitude, T ′ is the envelope decay time, Ω/2π is the Rabi frequency,
φ ≈ −π/2 is the phase offset, and B is an overall offset. Note that this is not the
correct functional form for a Rabi oscillation, but a convenient form that was similar
enough at high drive power. Since a Rabi oscillation causes the state of the qubit to
vary sinusoidally, and I assume the system starts in the ground state, I can now relate
the maximum excited population to the detected signal. When the system is prepared
in the ground state |g〉 at t = 0, Eq. (5.3) will give
 ‹
∆V
= Asinφ + B ' −A+ B. (5.4)
V g
87
5.4. RABI OSCILLATIONS
On the other hand, if the system is prepared in the excited state at time t = 0, then
one finds
 ‹
∆V
= Asin(π+φ) + B ' A+ B. (5.5)
V e
Thus the general expression for converting a detected signal to the corresponding qubit
excited population Pe is,
 
  

∆V − ∆VV V
P ge =  
  
 . (5.6)∆V ∆V
V −e V g
Note that this depends crucially on the assumption that the system was initialized in
the ground state. In reality, this cannot be done perfectly, and application of Eq. (5.6)
will yield some systematic errors.
5.4.2 Qubit Pulse Calibration
With the signal calibrated in terms of the qubit population, I can now find the pulse
lengths corresponding to two basic operations: the bit-flip and the Hadamard gate.
When I program the AWG to apply the drive tone for a Rabi oscillation, I explicitly
set the phase of the drive as φd = 0°. This sets the reference x-axis with respect to the
Bloch sphere (see Chapter 2). The bit-flip operation which I perform is equivalent to
the Rπx gate, which yields a half-cycle-rotation of the Bloch vector around the x-axis. We
refer to this pulse as a π-pulse or a Xπ-pulse. I determine the duration tπ of this pulse
by applying the microwave drive on resonance and finding the time which maximizes
88
5.4. RABI OSCILLATIONS
the output signal∆V/V (see Fig. 5.11). I note that this bit-flip operation can be written
equivalently as
 
0 1
Rπx ≡ σx = |0〉 〈 |+ | 〉 〈 |=
 
1 1 0 (5.7)
 
1 0
The Hadamard gate can be described in terms of two rotations on the Bloch sphere:
a π-rotation around the x-axis followed by a π/2-rotation around the y-axis. The
length for the π/2 rotation pulse is tπ/2, which is the time required to perform a
quarter-cycle rotation on resonance. We refer to such a pulse as a π/2-pulse. The
𝑡𝜋/2 𝑡𝜋
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0 50 100 150 200 250 300 350 400
𝑡Rabi (ns)
Fig. 5.11 Rabi Oscillations for the transmon SPP-v1-Q7 during Measurement Run #5. The
π-pulse length was tπ ' 122 ns and the π/2-pulse length was tπ/2 ' 61ns. The transmon
|g〉 → |e〉 transition was driven on resonance with a power −79 dBm to obtain fRabi = 4.1MHz.
89
DV/V
5.4. RABI OSCILLATIONS
Dirac representation of the Hadamard gate H is
 
 ‹  ‹
|0〉+ |1〉 |0〉 − |1〉 1 1 1
H= p 〈0|+ p 〈1|= p   (5.8) 
2 2 2
1 −1
Figure 5.11 shows an example for transmon SPP-v1-Q7 when I extracted the tπ
and tπ/2 times from a Rabi oscillation curve; with a microwave drive power giving
fRabi = 4.1MHz Rabi oscillations, I obtained tπ ' 122 ns and tπ/2 ' 61 ns.
5.4.3 Off-resonant Rabi drives
If the microwave drive is applied with a frequency detuning ∆ from the |g〉↔ |e〉
transition frequency, the two-level system will still undergo oscillations with a smaller
amplitude and larger Rabi frequency. These are known as generalized Rabi oscillations.
In the presence of relaxation and dephasing, there is no analytical form for the resulting
Rabi oscillations. Nevertheless, the generalized Rabi frequency and amplitude are
approximately given by [52]
p
Ω 2 2gen ' Ω +∆ (5.9)
Ω2
Agen ' , (5.10)
Ω2 +∆2
where Ω is the Rabi frequency observed on resonance. Note that Agen is less than
unity for ∆ 6= 0, indicating that it is not possible to fully excite the system with an
off-resonant drive.
90
5.4. RABI OSCILLATIONS
5.4.4 The Rabi Coupling
Figure 5.10 shows Rabi oscillations in a transmon using microwave power set to
a single value. I repeated this measurement at different powers to obtain a series of
Rabi oscillations with different Rabi frequencies. As I discussed in Chapter 2, the Rabi
frequency fRabi ≡ Ω/2π at sufficiently high drive power is proportional to the amplitude
of the drive. By measuring the variation of fRabi with the applied qubit drive voltage, I
obtained the coupling strength Υ (in units HzV−1) of the drive to the transmon.
25
20
15
10
5
0
0 100 200 300 400 500 600
𝑉𝑉in (μVrms)
Fig. 5.12 Rabi coupling measurement for transmon SPP-v1-Q7 during Measurement Run
#5. Microwaves at a frequency 5.96339 GHz were delivered to the cavity input with power
−84 dBm to −64 dBm to induce Rabi oscillations. The resulting data was fitted using Eq. (5.3)
to extract fRabi for each input voltage Vin (black points), and plotted versus the input rms
voltage. A linear fit to the data (red line) yields the slope or the Rabi coupling strength to be
Υ = 37kHz µV−1 for the device.
91
𝑓𝑓Rabi (MHz)
5.5. RELAXATION TIME MEASUREMENTS
5.5 Relaxation Time Measurements
The relaxation time T1 determines the lifetime of the excited state |e〉 of the qubit.
To measure T1, I applied a calibrated π-pulse as described in Sec. 5.4.2. I then waited a
time trel and performed a cavity measurement which I used to obtain the probability of
being in the excited statePe at time trel. Averaging 1000 such measurements completed
a single averaged measurement at time trel. These steps were repeated for varied trel, as
illustrated by the pulse sequence in Fig. 5.13. I fitted the obtained data to the formula
P = Ae−trel/T1e + B (5.11)
where A ' 1 and B ' 0. Figure 5.14 shows an an example T1 measurement in
transmon SPP-v1-Q6. This device had a stable lifetime of 6.71µs as verified by multiple
measurements on successive days.
I note that it was important to know T1 for many reasons. For example, for
properly calibrated quantum operations and measurements, the reset time tidle between
𝑡𝜋
𝑡rel
𝑅𝜋
Cavity 
𝑥 Measurement
𝑇
Fig. 5.13 Pulse sequence for T1 measurement. The time trel is varied for the measurement.
92
5.6. RAMSEY OSCILLATIONS
1.0
𝑒0
𝑒−1
0.8
𝑒−2
−3
0.6 𝑒
𝑒−4
𝑒−50.4
0 10 20 30 40
𝑡rel (μs)
0.2
0.0
0 10 20 30 40
𝑡rel (μs)
Fig. 5.14 Relaxation time measurement of transmon SPP-v1-Q6 during Measurement Run
#6. Data (black points) are overlaid with an exponential decay fit (red curve). (inset) T1
measurement on a semilog plot.
measurements must be long enough to ensure that the qubit has sufficient time to
relax back to the ground state. I ensured this by setting tidle > 5T1, which will give
Pe < 0.7% if the qubit started in state |e〉. This step is one of many, required to optimize
and fine-tune the measurement process described in Sec. 5.2.
5.6 Ramsey Oscillations
The Ramsey method is a crucial technique in atomic timekeeping and provides
interesting information about a qubit [52]. Norman Ramsey was a former student
of Isidor Rabi who extended molecular beam methods to develop a highly accurate
method of atomic interferometry [117]. Ramsey’s method (see Fig. 5.15) involved
first applying a π/2-pulse to put an atom into an equal superposition of the |g〉 and
93
𝒫e
𝒫e
5.6. RAMSEY OSCILLATIONS
𝑡𝜋/2 𝑡𝜋/2
𝜋 𝜋
𝑅2 𝑡 𝑅2 Cavity 𝑥 Ramsey 𝑥 Measurement
𝑇
Fig. 5.15 Pulse sequence for Ramsey oscillation measurement. The time tRamsey is varied for
the measurement.
|e〉 states. The atom was then allowed to evolve freely for a time tRamsey. Another
π/2-pulse was then applied and the state of the system measured.
Depending on the magnitude of the accumulated phase of the superposition state,
the final measured population would differ. As tRamsey is varied in a given experiment,
this accumulated phase can vary from 0 to π to 2π resulting in complete oscillations.
For ωd =6 ωge, the frequency of these Ramsey oscillations depends on the detuning ∆
between the applied drive frequency ωd and the qubit transition frequency ωge.
Steck [52] provides an informative analogy for the Ramsey interference fringes: the
Ramsey method is similar to Young’s double-slit experiment where the two π/2-pulses
act as two slits separated in the time domain, resulting in interference fringes which
appear as a function of the frequency or detuning. Ramsey interferometry samples
the integrated noise that impacts the transmon, and hence is a sensitive metric of the
coherence.
94
5.6. RAMSEY OSCILLATIONS
1.0
0.8
0.6
0.4
0.2
0.0
0 200 400 600 800 1000 1200 1400
𝑡Ramsey (ns)
Fig. 5.16 Ramsey oscillations for transmon SPP-v1-Q6 during Measurement Run #7. I obtained
the two Ramsey oscillation traces by applying the π/2-pulses at frequencies of 6.7520 GHz
(black), and 6.7527 GHz (red). The resulting Ramsey frequencies fRamsey were 2.247 MHz and
1.549 MHz, respectively.
5.6.1 Precise Determination of Qubit Frequency
Figure 5.16 shows two examples of Ramsey oscillations I observed in transmon
SPP-v1-Q6. I fit those Ramsey oscillations to damped sinusoidal functions with an
average decay time of T ∗2 = 6.1µs and frequency which depends on the detuning. For
a frequency independent loss mechanism, Ramsey decay time T ∗2 obeys
1 1 1 1 1 1
∗ = + + = + , (5.12)T2 2T1 Tφ T † T2 T †
where Tφ is the pure dephasing time and T
† is a time constant characterizing inhomo-
geneous broadening (see next section).
During Measurement Run #3, I was able to obtain Ramsey oscillations with detun-
95
𝒫e
5.7. SPIN-ECHO MEASUREMENTS
5.540
5.535
5.530
5.525
5.520
-10 -5 0 5 10
𝑓Ramsey (MHz)
Fig. 5.17 Plot of drive frequencyωd/2π versus Ramsey frequency fRamsey for transmon SPP-v1-
Q7 during Measurement Run #3. The drive frequency of π/2-pulses was varied from 5.52 GHz
to 5.54 GHz. The data (black) are overlaid with the linear fit (red), and the y-intercept value
from the fit is taken to be the precise qubit frequency.
ings ∆ in the range −10 MHz to 10 MHz for transmon SPP-v1-Q7. By plotting the drive
frequency ωd versus the resulting Ramsey frequencies fRamsey, I obtained the |g〉 → |e〉
transition frequency with a high precision (see Fig. 5.17). By using the y-intercept of
a linear fit to the data, I extracted the qubit frequency precisely to be 5.529 72 GHz.
It should be noted that when ∆ ≈ 0, long traces have to be obtained to precisely
determine the frequency. Given the relatively short coherence times this is often not
possible, as was evident with the two data points for ωd ≈ 5.529GHz in Fig. 5.17.
96
𝜔d/2𝜋 (GHz)
5.7. SPIN-ECHO MEASUREMENTS
5.7 Spin-Echo Measurements
The method of spin-echo was developed by Erwin Hahn [118] to decrease, to first
order, effects of inhomogeneous broadening when studying the coherence of a spin
ensemble. Inhomogeneous broadening means different atoms in an ensemble have
slightly different transition frequencies due to local effects such as variations in the local
magnetic field, or a spread in velocities for a train of atoms/ions [52]. When a Ramsey
experiment is performed on such an ensemble, some atoms will be accumulating phases
faster than others, after the first π/2-pulse. When the second π/2-pulse is applied
and the ensemble is measured, this differential phase accumulation will reduce the
resulting coherent signal, much like pure dephasing. A spin-echo measurement (Hahn
echo) involves applying a π-pulse around an orthogonal axis, halfway between the
two π/2-pulses as illustrated in Fig. 5.18. The π-pulse causes the spins to evolve in a
time-reversed fashion, allowing the slow parts and fast parts to refocus. The spin-echo
technique has become commonplace in other applications as well, and can be used to
𝑡𝜋/2 𝑡𝜋 𝑡𝜋/2
𝜋
𝜋 1𝑡 𝜋 1 22 𝑅 𝑡 𝑅 Cavity 𝑅 2 echo𝑥 𝑦 2 echo
𝑥
Measurement
𝑇
Fig. 5.18 Pulse sequence for spin-echo measurement. The time techo is varied for the mea-
surement.
97
5.7. SPIN-ECHO MEASUREMENTS
1.0
0.8
0.6
0.4
0.2
0.0
0 4 8 12 16 20 24 28
𝑡echo (μs)
Fig. 5.19 Spin-echo measurement showing probabilityPe to be in the excited state versus time
techo for transmon SPP-v1-Q6 during Measurement Run #7. Data (black points) are overlaid
with fit to an exponential decay (red curve) with time constant 6.23µs and offset ≈ 0.5.
eliminate the effects of unwanted interactions by applying corrective pulses [119].
Note that if the local effects do not vary in time, at least during the timescale of
evolution, the ensemble constituents will be refocused just prior to the application of
the second π/2-pulse. Thus the echo removes effects of inhomogeneous broadening.
Of course, this does not remove effects from Tφ and T1, and the resulting echo decays
exponentially. When this technique was applied in optical systems [120], the system
itself would emit a light pulse akin to the original excitation, producing an optical
“echo” pulse.
Figure 5.19 shows an example of a spin-echo measurement I took on transmon
SPP-v1-Q6. By fitting the data to an exponential decay with an offset as in Eq. (5.11),
I obtained the T2 coherence time; this device had a relaxation time of T1 = 3.47µs
and a coherence time of T2 = 6.23µs. In this case, the coherence time T2 was close to
98
𝒫e
5.8. PHOTON NUMBER SPLITTING
2T1 = 6.94µs, implying that the dephasing time was much longer than T1. In general,
for frequency independent loss mechanisms, T2 is related to T1 by the relation,
1 1 1
= + . (5.13)
T2 2T1 Tφ
5.8 Photon Number Splitting
The cQED systems I examined were composed of a cavity as well as one or two
transmons. I needed to characterize certain aspects of the cavity as well, as it interacted
with the qubits. Here I discuss the coupling of an applied field to the cavity. Unlike
the case of the Rabi coupling to a qubit (see Sec. 5.4), it is not so straightforward to
measure this coupling.
In the dispersive limit, at sufficiently low photon numbers, the dressed cavity is
a slightly anharmonic system, due to the high-order Kerr nonlinearity arising from
the interaction with the transmon [121]. Thus, a sufficiently strong drive field will
yield a coherent state with multiple photons in the cavity, rather than causing coherent
𝑡sat 𝑡RO
Transmon Transition 
Cavity 
Saturation Measurement
Cavity saturation (weak drive)
𝑇
Fig. 5.20 Pulse sequence for photon number splitting. The long pulses are applied to ensure
the steady-state of the cavity and the transmon.
99
5.8. PHOTON NUMBER SPLITTING
0.04
0.03
0.02
0.01
0.00
6.18 6.19 6.20 6.21 6.22
f (GHz)
Fig. 5.21 Measurement of photon number splitting spectrum for transmon SPP-v1-Q7 in the
cavity 6D during Measurement Run #2. Data (black points) are overlaid with a fit to five
Lorentzians (red curve). The drive power at the cavity input was Pcav ≈ −112 dBm and the
calculated mean photon number was 〈n〉 ≈ 1.6. The qubit spectroscopic peaks correspond to
different cavity photon numbers, as labeled.
oscillations between the two lowest levels. Depending on the intensity of the drive and
the decay time of the resonator, the driven cavity will reach a steady coherent state
with a photon population following a Poisson distribution [122].
The strength of the coupling of the cavity to the input drive line can be found by
measuring the average photon number corresponding to a given drive strength. In
my cQED system, I used the transmon as a sensitive detector of the average photon
occupation in the cavity; in the low photon number limit, each additional photon in
the cavity causes a shift in the qubit frequency an amount 2χ (see Sec. 2.2.3).
Figure 5.20 shows an example where I measured the transition spectrum while
applying a weak continuous cavity tone. The cavity photon distribution is “imprinted”
100
∆V/V
𝑛 = 4
𝑛 = 3
𝑛 = 2
𝑛 = 1
𝑛 = 0
5.8. PHOTON NUMBER SPLITTING
as a series of peaks in the qubit spectrum, with each peak corresponding to the qubit
transition with a specific number of photons in the cavity; this technique is known
as measuring the photon number splitting spectrum [123]. By extracting the relative
heights of the peaks, I determined the average number of photons in the cavity [124],
and the strength of the coupling between the input pin and the cavity. I note that
this approach only works well if the system is in the strong dispersive regime where
2χ > {κ, 1/T2} (κ is the decay rate for the cavity and T2 is the coherence time for the
qubit), and if the qubit and the cavity are driven lightly to ensure that the individual
photon number peaks are well-resolved.
5.8.1 Determining the Cavity Coupling to the Input Drive
By measuring the number splitting spectrum as a function of the microwave input
drive strength, I obtained the average cavity photon occupation 〈n〉 versus the drive
voltage Vrms. Naively, this relationship should be quadratic in the low-power limit, i.e.
〈n〉 ∝ V 2rms. It is also possible to restate this relationship in terms of the effective cavity
drive strength Ωcav [48, 123, 125],
Ω2cav QE 1〈n〉= , (5.14)
2π QL (κ/2)2 +χ2
where QL and QE are the loaded and external coupling quality factors of the cavity,
respectively.
Figure 5.22 shows an example where I measured the photon number splitting
101
5.8. PHOTON NUMBER SPLITTING
-105
∆V/V
-115
0.45
-125
-135
-145
-155
0.00
-165
-175
6.15 6.17 6.19 6.21
f (GHz)
Fig. 5.22 Photon number splitting spectrum versus drive power for transmon SPP-v1-Q7 in the
cavity 6D during Measurement Run #2. The transmon spectrum was measured while applying
a weak cavity drive at frequency 7.94683 GHz with different powers Pcav.
spectrum versus cavity drive power. At low powers Pcav < −145dBm, only the peak
corresponding to zero cavity photons is visible. Above −145 dBm other cavity photon
number peaks appear, following a Poisson distribution. For powers Pcav > −115 dBm
the series of peaks shift to the left, indicating significant Stark shifting. In this case I
found Ωcav/2π > 3MHz.
102
𝑃cav (dBm)
5.9. SUMMARY OF MEASURED PARAMETERS OF THE DEVICES
5.9 Summary of Measured Parameters of the Devices
Table 5.1 gives a summary of the parameters for the devices I fabricated and
measured for this dissertation. I determined the input and output quality factors for
the cavity (Qinput and Qoutput) by measuring S21, S22, and S11 at room temperature. The
loaded quality factor of the cavity Qloaded was measured after cooling the cavity to
20 mK. I obtained the cavity lifetime Tcav either using a ringdown measurement (Tcav =
Tringdown/2) or from the loaded quality factor (Tcav =Qloaded/ωr). The charging energy
EC and the Josephson energy EJ of the transmons were extracted from spectroscopic
measurements. I used the effective dispersive shift χeff to calculate the effective qubit-
p
cavity coupling geff = χeff∆ge. The Rabi coupling Υ was measured as described
in Sec. 5.4.4. The attenuation values at the qubit frequency for the input line were
estimated based on S21 measurements on the line at room temperature.
All SPP-v1 transmons were fabricated on sapphire substrates. The SPP-v2 transmon
was fabricated on a silicon substrated that was not treated with HF. Both cavities 6D
and 6G were machined from 6063 Al. Except for the final cooldown in 2017, all
other devices listed were packaged in 2015. SPP-v2-Q2 was replaced with SPP-v1-Q7
on June 22, 2015. Subsequent to this repackaging, the two halves of the cavities
were not opened during the rest of 2015 and 2016; nevertheless the coupling quality
factors were adjusted by simply removing and reattaching modified SMA ports. For the
cooldown in 2017, the cavity 6G was opened, machined to have two slots equidistant
from the center, and repackaged with both qubits SPP-v1-Q6 and SPP-v1-Q7 inside. The
103
5.9. SUMMARY OF MEASURED PARAMETERS OF THE DEVICES
original center slot was closed off with Indium to ensure boundary integrity.
104
5.9. SUMMARY OF MEASURED PARAMETERS OF THE DEVICES
105
Table 5.1 Summary of parameters for the devices I fabricated and measured.
Measurement Run #1 #2 #3 #4 #5 #6 #7
Cooldown Date 5/8/2015 6/22/2015 9/12/2015 9/12/2015 2/25/2016 2/25/2016 9/28/2017
Cavity 6D 6D 6D 6G 6D 6G 6G
Transmon Name SPP-v2-Q2 SPP-v1-Q7 SPP-v1-Q7 SPP-v1-Q6 SPP-v1-Q7 SPP-v1-Q6 SPP-v1-Q6 SPP-v1-Q7
Qoutput 350000 309000 207000 3000 207000 3000 3000
Qinput 730000 682000 420000 80000 420000 — 80000
Qloaded 42130 69180 119464 2887 124450 1493 2425
fbare (GHz) 7.796813 7.921156 7.9222 7.9222 7.92272 7.92241 7.6804375
f (g)dress (GHz) 7.8524 7.94683 7.936997 7.9416 7.9430 7.95021 7.7463
f (e)dress (GHz) 7.8354 7.93887 7.932949 7.9358 7.9373 7.94164 7.732 7.7398
Tcav (µs) 0.86 1.39 2.4 0.058 2.5 0.03 0.1075 0.1075
fge (GHz) 6.78133 6.22097 5.52972 5.93986 5.96339 6.33121 6.75427 6.07135
fgf/2 (GHz) 6.7084 6.12236 5.42422 5.83758 5.862 6.2335 6.6654 5.9718
fgh/3 (GHz) 6.6304 6.01735 — 5.729 — 6.1295 6.5686 —
EC/h (MHz) 146 206 211 204 203 200 180 199
EJ (GHz) 41 25 19 23 23.4 26.7 31.9 23.18
EJ/EC 280 121 90 112 115 134 177 116
χge/2π (MHz) −55.6 −25.7 −14.8 −19.4 −20.3 −27.8 −65.9 −65.9
χef/4π (MHz) −47.1 −21.7 −12.8 −16.5 −17.4 −23.5 −58.7 −62.6
χeff/2π (MHz) −8.5 −3.98 −2.02 −2.9 −2.85 −4.29 −8.35 −3.0
gge/2π (MHz) 238 209 188 196 199 210 208 225
gef/2π (MHz) 331 287 258 269 275 290 293 318
geff/2π (MHz) 95 83 70 76 75 83 84 74
T1 (µs) 0.425 50 24 20 18 6.71 3.47 9.04
T ∗2 (µs) 0.567 9.1 22 20 20 10.5 6.2 12.9
T ′ (µs) 0.43 14.4 20 20 15 8.37 4.24 12.3
T2 (µs) — — — — 25 11.7 6.2 14.6
Υ (MHz µV−1) 0.29 0.15 0.030 0.21 0.037 2.75 0.32 0.11
Attenuation (dB) — 77 77 83 88 83 90 90
CHAPTER 6
Fock State Generation using Stimulated Raman
Adiabatic Passage
6.1 Introduction
Quantum states with a well defined number of quanta, or Fock states [126], have
many roles in quantum information processing including quantum key distribution
[127], quantum memory [128], generation of identical photons for remote entangle-
ment [129] and universal control of quantum elements [130]. Following proposals
for generating Fock states in AMO-based cavity QED systems [131, 132], the first
experimental demonstration involved exchanging an excitation between a Rydberg
atom and a cavity [133].
Fock states were first generated in a circuit quantum electrodynamics (cQED) system
by coupling a tunable superconducting qubit to a microwave resonator [134, 135].
In those experiments, the excited state of the superconducting qubit was tuned to be
resonant with the resonator and the excitation from the qubit was then transferred to
106
6.1. INTRODUCTION
the resonator by vacuum Rabi oscillations. An advantage of this technique was that
the transfer could be performed in a relatively fast time of 25ns; sufficiently short that
dephasing was not an issue. More recently, arbitrary Fock states were generated using
selective number dependent arbitrary phase (SNAP) gates on a fixed-frequency qubit-
cavity system [130, 136]. The SNAP approach took advantage of a highly coherent
qubit-cavity system to generate the desired states in a time of 1µs.
STIRAP involves the transfer of population from an initial state |0〉 to a final state
|2〉 via an intermediate state |1〉 by applying two Gaussian coherent Raman pulses that
overlap in time. The pulse sequence is counter-intuitive, in that the initial pulse is on
the |1〉 to |2〉 transition, and the second pulse is on the |0〉 to |1〉 transition [137, 138].
STIRAP is an interesting technique because it can overcome a forbidden transition (for
example |0〉 to |2〉 to first order), while never occupying the intermediate state, and is
robust against slight variations in the drive parameters [137]. Previous proposals and
demonstrations of STIRAP using superconducting quantum devices involved transfer
of populations between the three lowest levels of the device [139–141]. Such qutrit
levels can have a relatively high anharmonicity and this simplifies the implementation.
In contrast, I employed STIRAP to operate on composite cavity-qubit states, a system
which has a relatively small anharmonicity. Using a STIRAP protocol with the third
excited state of the qubit-cavity system as the intermediate state, I transferred the first
excited state population of the qubit to the cavity, and thereby created a single photon
Fock state.
In this chapter, I describe the use of a two-photon Stimulated Raman Adiabatic
Passage (STIRAP) protocol, to transfer the excitation from a fixed-frequency transmon
107
6.2. THE DEVICE AND THE EXPERIMENTAL SETUP
qubit to a microwave cavity Fock state in 354 ns. By performing driven coherent manip-
ulations on the generated state, and comparing the results to multi-level simulations, I
determined the fidelity for the n= 1 Fock state to be ≥ 85%, a value similar to other
initial demonstrations of Fock states [134–136]. In Sec. 6.2, I describe the device I
used, and the experimental setup. Section 6.3 will describe how I modeled the system,
and how I implemented the master-equation simulations. In Sec. 6.4, I describe how
I extended the process to create the second and third Fock states, with simulated
fidelities of 68% and 43%, respectively. These fidelities were limited by the decay rate
of the cavity relative to the adiabatic process duration. In Sec. 6.5, I demonstrate how
this protocol can be used to create a superposition of Fock states. In Sec. 6.6, I describe
reversal of STIRAP. I describe how I generated entanglement between the cavity and
the qubit in Sec. 6.7. I will conclude this chapter by summarizing the main results in
Sec. 6.8.
6.2 The Device and the Experimental Setup
The cQED system used in my experiment consisted of the transmon SPP-v1-Q7
embedded in the three-dimensional superconducting Al cavity 6D (see Fig. 6.1(a))
[27, 69]. The cavity-transmon system was attached to the mixing chamber of a Leiden
Cryogenics CF-450 dilution refrigerator that operated at a temperature of 15 mK. The
transmon [27] had a transition frequency ofωq/2π = 5.5297 GHz between the ground
and excited state. I fabricated the transmon on a sapphire substrate (see Chapter 3)
108
6.2. THE DEVICE AND THE EXPERIMENTAL SETUP
(a) ((bb))
10 mm 0.5 mm
Fig. 6.1 The transmon-cavity system. (a) Photograph of transmon chip in the top of opened
3D Al cavity. (b) Optical micrograph of transmon SPP-v1-Q7 on sapphire chip.
and it had a single Al/AlOx/Al Josephson junction capacitively shunted by two large
500µm×650µm Al pads which reduced its charging energy (see Fig. 6.1(b)). The large
pads also provided coupling with strength g/2π = 69.8 MHz to the TE101 fundamental
mode of the cavity with a resonance frequency ωc/2π = 7.9350GHz. All the results in
this chapter were obtained during Measurement Run #3 (see Sec. 5.9).
The input line to the cavity had an attenuation of 77 dB and the output had more
than 60 dB of directional isolation at the cavity and qubit frequencies (see Sec. 4.1). To
drive the transmon-cavity system, microwaves were delivered to the input port of the
cavity with a coupling Qin ∼ 400, 000 (adjusted and measured at room temperature).
The state of the system was interrogated using a high-power cavity transmission
measurement that was sensitive to the transmon state [104]. The output port had
coupling Qout ∼ 200, 000 (adjusted and measured at room temperature) and sent the
109
6.3. MODELING AND CHARACTERIZING THE SYSTEM
transmitted microwaves to a low noise HEMT amplifier at 3 K [101]. The output signal
was further amplified at room temperature with a low-noise amplifier (see Fig. 4.4).
The signal was then mixed down to an intermediate frequency of 10 MHz, digitized
and analyzed. The experimental setup is described in detail in Chapter 4 (see Fig. 4.4)
and is similar to that described by Novikov et al. [142, 143].
The generation of Gaussian pulses was carried out by programming a Tektronix
AWG70002A arbitrary waveform generator (AWG). The maximum sampling rate for
the instrument was 25 GSa/s with 10 bits of resolution for voltage. The output from
the AWG was amplified at room temperature with a Mini-Circuits ZVA-183-S+ amplifier
before being sent into the dilution refrigerator.
6.3 Modeling and Characterizing the System
In my setup, the coupling strength g ∼ 70MHz is much smaller than the qubit-
cavity detuning ∆ =ωc−ωq = 2.4 GHz. In this limit, the undriven qubit-cavity system
can be well described by the dispersive Jaynes-Cummings Hamiltonian [50]:
1
HJC = ħhω2 q
σz +ħh(ωc +χσz)N̂ (6.1)
where N̂ is the photon number operator for the cavity, σz is the Pauli-Z operator,
and χ/2π = −2.0 MHz is the effective frequency dispersive shift resulting from the
cavity-qubit coupling.
110
6.3. MODELING AND CHARACTERIZING THE SYSTEM
(a) (b)
⋮ |e2〉 𝛿1 |e1〉
|g2〉
𝜔s1, Ωs 𝜔p1, Ωp
|e1〉
𝜔q + 2𝜒 |g1〉 |e0〉
𝜔c + 2𝜒|g1〉
(c) Amplitude
|e0〉
𝜔c
𝜔q
e0 → |e1〉
|g0〉 g1 → |e1〉 Time
(ns)
0 100 200 300
Fig. 6.2 STIRAP in a transmon-cavity system. (a) Level structure (not to scale) for the
transmon-cavity system. Letters and numbers within kets denote qubit and cavity photon-like
states, respectively. (b) Levels used for the generation of |g1〉 Fock state with the Stokes (red)
and pump (blue) tones with frequencies {ωs1, ωp1} and drive strengths {Ωs, Ωp}, respectively.
δ1 is the common detuning for both drives from |e1〉. (c) Pulse envelopes for the Stokes (red)
and pump (blue) tones.
The Hamiltonian HJC gives an energy level structure that has two ladders. One
ladder is for all states that have the qubit in the ground state |g〉, while the second
ladder is for the excited state |e〉 (see Fig. 6.2(a)). The rungs of each ladder correspond
to a different integer number of cavity photons. My state measurement technique
allowed me to find the total probability Pe to find the system in the |e〉 ladder, by
averaging repeated measurements.
The transmon had a characteristic relaxation time of T1 = 24µs and a Ramsey
decay time T ∗ = 22µs. The cavity had a lifetime κ−12 = 2.5µs, set by Qout to allow for
relatively fast measurements. I note that my device operated in the strong dispersive
¦ ©
regime of circuit QED since 2χ  κ, 1 1T , T ∗ [124].1 2
111
6.3. MODELING AND CHARACTERIZING THE SYSTEM
For my numerical simulations, the master equation for the density matrix ρ was
solved in the time domain. The Jaynes-Cummings ladders were truncated to a max-
imum of 11 levels in the |g〉 ladder and 10 levels in the |e〉 ladder. Denoting the
cavity and qubit microwave drive strengths Ωc(t) and Ωq(t), respectively, the drive
Hamiltonian can be written as (see Sec. 2.2.4)
Hint = ħhΩc(t)(a+ a†) +ħhΩq(t) (|g〉 〈e|+ |e〉 〈g|) (6.2)
where a and a† represent the annihilation and creation operators for the cavity, respec-
tively.
Cavity and qubit dissipation was incorporated through the Lindblad-Kossakowski
formalism [54, 55]. The driven system Hamiltonian can then be written as, H =
HJC +Hint and the resulting master equation for the density matrix ρ can be written
as,
 ‹
∂ ρ 1 ∑ 1 1
= [H ,ρ] + Γ A ρA† − A†A ρ − ρA†A . (6.3)
∂ t iħh j j j 2 j j 2 j j
j
Here, the index j runs through the various decoherence channels (cavity relaxation,
qubit relaxation, qubit dephasing) and Γ j represents the decoherence rate for each
channel. A j are the Lindblad operators corresponding to each channel. The qubit and
cavity relaxation Lindblad operators can be given in the form A −qdec = σ and Acdec = a,
respectively, and the qubit dephasing can be represented by the Lindblad operator
Aqdph = σz.
112
6.4. GENERATING FOCK STATES N = 1,2, 3
Fidelity between any two states (for example realized state φ and target state ψ)
described by density matrices ρ and σ, can be defined as
Æp p
F(ρ,σ) = Tr ρσ ρ. (6.4)
The above definition simplifies considerably when one of the states is a pure state (e.g.
|g1〉, or |g0〉+ |g2〉) [3]. As I was trying to generate pure states, I instead used
F (ρ, |ψ〉) = [F(ρ, |ψ〉)]2 = 〈ψ|ρ|ψ〉 (6.5)
where |ψ〉 is the target pure state and ρ is the density matrix for the generated state.
Equation (6.5) is more intuitive than Eq. (6.4), since it is just the probability of finding
the system in state ψ.
6.4 Generating Fock States n= 1, 2,3
The process I used to generate the single-photon Fock state |g1〉 involved use of
the four lowest levels in the system: |g0〉, |e0〉, |g1〉, and |e1〉 (see Fig. 6.2(a)). After
passively cooling the system to the |g0〉 state, a 32 ns duration πg0→e0 pulse at the qubit
frequency ωq was sent to add one excitation to the qubit and obtain the state |e0〉.
To transfer population from |e0〉 to |g1〉, which is a first order forbidden transition,
STIRAP was used via the intermediate level |e1〉 (Fig. 6.2(b)). In this protocol, I first
applied a Stokes tone at frequency ωs1 =ωq+ 2χ −δ1, slightly detuned by an amount
113
6.4. GENERATING FOCK STATES N = 1,2, 3
δ1 from the |g1〉 to |e1〉 transition frequency and with a Gaussian envelope for the
amplitude. I then applied a slightly delayed Gaussian-shaped pump tone pulse that
overlapped with the Stokes pulse. The pump pulse had a frequencyωp1 =ωc+2χ−δ1,
which included the same detuning δ1 as the Stokes tone.
For a conventional three-level system undergoing STIRAP, process fidelity does not
depend heavily on the detuning from the intermediate level. However, my system has
a multi-level transmon coupled to a multi-level cavity. I found that a negative detuning
from level |e1〉 (i.e. −δ1 < 0) was beneficial to minimize leakage to higher levels, and
I optimized δ1 through simulations. The counter-intuitive profile for the Stokes and
pump pulses was
   
(t − t )2 (t − t )2
Ω(t) = Ωs exp −
s − psin [ω t +φ ] + Ω exp sin [ω t +φ ],
2σ2 s1 s p 2 2 p1 pσ
(6.6)
where Ωs and Ωp were the peak amplitudes of the Stokes and pump tones respectively,
σ was the width of the Gaussian envelope for both pulses, ts and tp were the times
when the Stokes and pump envelopes were at their maximum, and φs and φp were
offset phases in the Stokes and pump carrier tones respectively.
To determine the initial choices for STIRAP parameters, density matrix simulations
were performed at the outset of the experiment. I solved the master equation in the
laboratory frame, while applying the time dependent STIRAP protocol. Following
initial experiments, the model was refined using a total of 11 cavity levels in the
|g〉-ladder and 10 cavity levels in the |e〉-ladder. The programmed waveforms used for
114
6.4. GENERATING FOCK STATES N = 1,2, 3
the experiment were used in the numerical simulations (see Table 6.1).
Figure 6.3 shows the measured probabilities Pe (filled points) of the qubit being
in its excited state versus time t during generation of the first, second and third Fock
states using the STIRAP protocol. For comparison, results from the simulations are
(a) 𝜋 STIRAP 𝜋 STIRAP 𝜋 STIRAP
g0→e0 e0→g1 g1→e1 e1→g2 g2→e2 e2→g3
t
(b) 1.0
Exp
Sim
g0
0.8 g1
g2
g3
0.6
0.4
0.2
0.0
0 200        400 600 800 1000 1200
𝑡 (ns)
Fig. 6.3 Generation of the first three Fock states. (a) Pulse sequence for generating Fock states
|g1〉 , |g2〉 and |g3〉. The π-pulses (yellow) transfer |g0〉 → |e0〉 , |g1〉 → |e1〉 and |g2〉 → |e2〉.
The Stokes (red) and pump (blue) pulses transfer |e0〉 → |g1〉 , |e1〉 → |g2〉 and |e2〉 → |g3〉.
(b) State probabilities P versus time t for |g1〉 , |g2〉 and |g3〉 Fock state generation protocol
shown in (a). Measured data are total qubit excited state populationsPe (black dots). Solid red
curve shows the correspondingPe from density matrix simulations. Dashed/dotted curves show
simulated evolution of the populations of Fock states |g0〉 , |g1〉 , |g2〉 and |g3〉 corresponding
to 0, 1, 2, and 3 photons in the cavity respectively.
115
𝒫
Amplitude
6.4. GENERATING FOCK STATES N = 1,2, 3
Table 6.1 Parameter values used in STIRAP pulses. The drive phases φp1 and φs1 had no
effect on the generation of the Fock states and for simplicity were set to zero.
Parameter Symbol Value
Stokes drive strength (qubit-like) Ωs/2π 9.6 MHz
Pump drive strength (cavity-like) Ωp/2π 26.2 MHz
Stokes-pump pulse separation tsep 14 ns
Standard deviation of Gaussian pulses σ 70.8 ns
Total length of each truncated Gaussian pulse tlen 340 ns
Common detuning for Stokes/pump drives from |e1〉 δ1 8.1 MHz
Common detuning for Stokes/pump drives from |e2〉 δ2 9.7 MHz
Common detuning for Stokes/pump drives from |e3〉 δ3 11.3 MHz
|g1〉 → |e1〉 Stokes frequency ωs1/2π 5.5176 GHz
|e0〉 → |e1〉 Pump frequency ωp1/2π 7.9248 GHz
|g2〉 → |e2〉 Stokes frequency ωs2/2π 5.5119 GHz
|e1〉 → |e2〉 Pump frequency ωp2/2π 7.9233 GHz
|g3〉 → |e3〉 Stokes frequency ωs3/2π 5.5063 GHz
|e2〉 → |e3〉 Pump frequency ωp3/2π 7.9217 GHz
shown as curves. To collect each point at time t in this plot, execution of the protocol
was halted at time t, and a high-power cavity pulse applied to destructively measure
the probability of the qubit being in the excited state. The protocol was started again
from the beginning and repeated 1000 times to obtain the average probability to be
in the excited state Pe. For the first 32ns, a πg0→e0 was sent to the device and upon
completion Pe > 96% (confidence level obtained from evaluating false counts during
single shot measurements). After placing the system in |e0〉, the Stokes and pump
STIRAP pulses were sent to the device using the parameters listed in Table 6.1. The
total duration to complete STIRAP was 354ns. Comparing the data to the simulations,
I found that the fidelity upon completion was Fg1 = 89%, limited by photon leakage
out of the cavity.
Due to the relatively small photon anharmonicity and the relatively short lifetime of
116
6.4. GENERATING FOCK STATES N = 1,2, 3
photons in the cavity, Wigner tomography [144] could not be performed in my system.
Instead, I examined the fidelity of the Fock state by performing Rabi oscillations
immediately following the STIRAP protocol (see Appendix B). The Rabi oscillations
were performed by driving at both the |g1〉 → |e1〉 transition frequency ωg1→e1, and
at the |g0〉 → |e0〉 transition frequency ωq, for a range of drive amplitudes. For small
drive amplitudes Vdrive, the oscillations when driven at ωq (Fig. 6.4(a)) show a larger
Rabi frequency fRabi and lower Pe amplitude, consistent with the drive frequency being
detuned from the |g1〉 → |e1〉 transition. Each Rabi oscillation was fit to the sum of
two oscillatory functions to extract the frequency fRabi and the amplitude ARabi of each
component. The amplitude of the resonant component is a metric for the fidelity with
which the STIRAP process generated the desired state |g1〉.
Figure 6.4(b) shows the four extracted Rabi frequencies when the system was
driven at the two drive frequencies ωg1→e1 and ωq. Two of the data sets are linear with
drive voltage and extrapolate to a zero Rabi frequency at zero drive voltages, consistent
with a resonant drive process. The other pair of Rabi frequencies extrapolates, at small
drive voltages, to a frequency of approximately 4 MHz, consistent with a non-resonant
process with detuning 2χ (Fig. 6.2(a)). The fidelity of the |g1〉 Fock state can be
determined from the amplitude of the Rabi oscillations, as shown in Fig. 6.4(c). At
small drive voltages, the amplitude for the resonant component when driving atωg1→e1
yielded a fidelity of Fg1 ≥ 85%.
Following generation of |g1〉, the second (|g2〉) and third (|g3〉) Fock states were
produced in a similar manner (see Fig. 6.3). When generating the higher Fock states, I
tuned the parameters through simulations, to optimize the fidelity of the desired state
117
6.4. GENERATING FOCK STATES N = 1,2, 3
(a)1.0
0.5
0.0
0 200 400 600
t (ns)
(b)
12
8
Resonant 
4 𝜔g1→e1
Off-res 
Resonant 
𝜔
Off-res q
0
0 100 200 300 400
𝑉drive (μVrms)
(c) 1.0
0.8
0.6
Resonant 
𝜔g1→e1
Off-res 
0.4
Resonant 
𝜔
Off-res q
0.2
0.0
0 100 200 300 400
𝑉drive (μVrms)
Fig. 6.4 Verification of |g1〉 Fock state: (a) Driven Rabi oscillations at ωg1→e1 (blue) and ωq
(red) frequencies, at a relatively small drive amplitude of Vdrive = 53 µVrms. (b) Extracted Rabi
frequencies fRabi versus drive amplitude Vdrive following generation of |g1〉 through STIRAP.
Dashed and dotted lines depict expected frequencies for a two-level system subjected to an on-
resonant and 4.04 MHz detuned off-resonant drives, respectively. (c) Extracted Rabi amplitude
oscillations. Dashed and dotted lines depict expected behavior for a two-level system after
setting up the cavity in a Fock state with Fg1 = 89%.
118
𝐴Rabi 𝑓Rabi (MHz) 𝒫e
6.4. GENERATING FOCK STATES N = 1,2, 3
Table 6.2 Optimized cavity and qubit pulse parameters used in simulating Fock state genera-
tion.
Parameter Symbol Value
Number of transmon levels nq 2
Maximum number of cavity excitations nex 10
Non-dressed cavity frequency ω̃c/2π 7.934 97 GHz
Non-dressed qubit frequency ω̃q/2π 5.531 75 GHz
Effective cavity-qubit coupling g/2π 69.8 MHz
|g0〉 → |e0〉 π-pulse frequency ωg0→e0/2π 5.5297 GHz
|g1〉 → |e1〉 π-pulse frequency ωg1→e1/2π 5.5257 GHz
|g2〉 → |e2〉 π-pulse frequency ωg2→e2/2π 5.5216 GHz
|g0〉 → |e0〉 π-pulse strength Ωg0→e0/2π 17.0 MHz
|g1〉 → |e1〉 π-pulse strength Ωg1→e1/2π 15.4 MHz
|g2〉 → |e2〉 π-pulse strength Ωg2→e2/2π 14.7 MHz
|g0〉 → |e0〉 π-pulse time tg0→e0 32 ns
|g1〉 → |e1〉 π-pulse time tg1→e1 34 ns
|g2〉 → |e2〉 π-pulse time tg2→e2 32 ns
(see Table 6.2 for the general system parameters used in simulations). |g2〉 was created
by adding another excitation to the system: a πg1→e1 pulse was followed by another
STIRAP protocol taking the system to |g2〉, with a comparison to the simulations
yielding a fidelity Fg2 = 68% (see Fig. B.4). A similar process was implemented to
generate |g3〉, with a comparison to the simulations yielding a fidelity Fg3 = 43%. As
with |g1〉, simulations revealed that Fg2 and Fg3 were limited by κ.
For comparison, the red curve in Fig. 6.3 shows the simulated results for the excited
state probability using the optimized parameters in Table 6.1 and Table 6.2. Except
for some small, fine-scale oscillatory behaviour found in the simulation during the
|g2〉 and |g3〉 STIRAP protocol, there is good agreement with the experiment. The
dashed and dotted curves show the simulation of the expected populations of the
119
6.4. GENERATING FOCK STATES N = 1,2, 3
various Fock states in the cavity, which I could not measure directly. The rotating wave
approximation was not employed in the simulations, and the 21 level simulation for
|g1〉 generation was also checked against a simulation performed with 51 levels in the
two-ladder system, which did not yield significant differences.
I note that Bergmann et al. have derived both local and global criteria for adiabatic
following in a system, to verify that a system is truly undergoing a STIRAP process [137].
The local adiabaticity criterion addresses the instantaneous non-adiabatic coupling
possible with various pulse shapes. As my experiment employed smooth Gaussian
pulses, the local adiabaticity criterion was naturally satisfied. The global adiabaticity
criterion addresses adiabaticity for the entire process, and for STIRAP is given by
Ωeff∆τ > 10 (6.7)
q
where Ωeff = Ω2p +Ω2s is the root-mean-square (rms) Rabi frequency and ∆τ is the
total overlap time for the two pulses [137]. Using Eq. (6.6) and my parameters for
∫
generating |g1〉, I found Ωeff dt ∼ 30, satisfying global adiabaticity Eq. (6.7). The
relative insensitivity of the STIRAP protocol to parameter fluctuations was verified
through simulations (see Table 6.3). Examining the table, one sees that a ±10%
variation in the parameters resulted in significantly smaller reductions in fidelity, with
the exception of increase in Ωp or decrease in δ.
120
6.5. GENERATING FOCK STATE SUPERPOSITIONS
Table 6.3 Sensitivity of process fidelity to STIRAP parameters: Simulated loss of fidelity in
|g1〉 generation due to varying the STIRAP parameters by 10% from the predetermined optimal
values.
Loss of fidelity from parameter adjustment
Process Parameter
Decrease of 10% Increase of 10%
Stokes drive strength (Ωs) 3.2% 6.4%
Pump drive strength (Ωp) 2.9% 10.3%
Pulse truncate length (tlen) 1.2% 3.5%
Standard deviation of pulse (σ) 0.4% 5.6%
Separation between pulses (tsep) 0.5% 0.5%
Common detuning from |e1〉 (δ) 16.9% 1.8%
π-pulse length (tg→e) 2.3% 2.5%
6.5 Generating Fock State Superpositions
STIRAP can also be used to create superpositions of Fock states. To generate a
superposition of |g0〉 and |g1〉, I replaced the initial πg0→e0 pulse for the Fock state
p
creation with a 16 ns (π/2)g0→e0 pulse to create the superposition state (|g0〉+|e0〉)/ 2.
Next I used the |e0〉 → |g1〉 STIRAP protocol to coherently transfer the population in
|e0〉 to |g1〉 (see Fig. 6.5(a)). Following the 370ns total execution time, I observed
Pe ≈ 0 and simulations showed that the state was an equal superposition of |g0〉 and
|g1〉.
During this STIRAP process, I found that Pe exceeded 0.5 and displayed an oscilla-
tory pattern (see Fig. 6.5(a)). To understand this aspect of the data, I measured the
behavior when the phase of each microwave drive was independently varied. I found
that the oscillations depended strongly on the phase φπ/2 of the initial π/2 pulse and
the phase φs of the Stokes tone but not the phase φp of the pump tone (see Fig. 6.6).
121
6.5. GENERATING FOCK STATE SUPERPOSITIONS
(a) 1.0
Exp
Sim
0.5
0.0
0 100 200 300
t (ns)
(b) 1.0
0.5
0.0
0 100 200 300
t' (ns)
Fig. 6.5 Generation of Fock state superpositions with STIRAP. (a) Equal superposition between
|g0〉 and |g1〉 generated using STIRAP after a π/2 initialization pulse. All phases for the drives
and the initialization were set to zero (φπ/2 = φs = φp = 0). (b) STIRAP performed on the
system after driving to a mixed steady state with a 150µs long pulse at frequency ωq. t
′ = 0
corresponds to the end of the saturation pulse. Experimental data points (black circles) and
simulated pooled populations (solid red curve) are shown.
122
𝒫 𝒫e e
6.5. GENERATING FOCK STATE SUPERPOSITIONS
(a) 1.0
𝜙𝜋/2
0.8
0
0.6 90
180
0.4 270
0.2
0.0
0 100 200 300
t (ns)
(b) 1.0
0.8 𝜙𝑠
0
0.6 90
180
0.4
270
0.2
0.0
0 100 200 300
t (ns)
(c) 1.0
𝜙𝑝
0.8
0
90
0.6
180
0.4 270
0.2
0.0
0 100 200 300
t (ns)
Fig. 6.6 Fock state superpositions’ variations with STIRAP phases. Equal superpositions
between |g0〉 and |g1〉 are generated using a π/2 initialization pulse. (a) The phases φ of
π/2 initialization, (b) Stokes and (c) pump pulses were changed while the other phases were
held constant at zero. The experimental data points for phases 0◦ (black), 90◦ (red), 180◦
(magenta) and 270◦ (blue) as well as the respective simulated evolutions of Pe (solid curves)
are shown. The phase of the initialization and Stokes pulses exhibit complepmentary behavior
in modifying the oscillatory features during the generation of (|g0〉+ |g1〉)/ 2. The phase φp
of the pump pulse has no impact on the oscillatory features.
123
𝒫 𝒫e e 𝒫e
6.6. STIRAP REVERSAL
This phase dependence, and the fact that Pe exceeds 0.5, during execution suggests
that the population in |g0〉 in the initial state plays a pivotal role when turning on the
Stokes tone. Comparing the data to the simulations again revealed good agreement
(see solid curves in Fig. 6.6). The simulations also implied that the final state was
prepared with a fidelity of approximately 93%, limited by the decay rate κ of the cavity.
To further examine the dependence on the initial state, STIRAP was performed
after the qubit was driven to a mixed steady-state, removing qubit phase coherence.
This resulted in the disappearance of oscillations observed previously (see Fig. 6.5(b)).
I inferred that the oscillations were the direct result of coherent interference within
the system when undergoing STIRAP. I believe the four lowest levels |g0〉 , |e0〉 , |g1〉
and |e1〉 set up a suitable system for observing quantum interference between differ-
ence excitation pathways [145], though confirming the source of this interference
was beyond the scope of this work. The existence of oscillations during the STIRAP
process demonstrates the coherent nature of the behavior as well as the preservation
of coherence during execution.
6.6 STIRAP Reversal
I discussed in Sec. 6.4 and Sec. 6.5, the use of STIRAP for the transfer |e0〉 → |g1〉.
The reverse process |g1〉 → |e0〉 can also be accomplished simply by reversing the order
of pulses. When performing the reversal immediately after |g1〉 Fock state generation,
the recovered population is ≈ 80%; consistent with the significantly longer time to
124
6.6. STIRAP REVERSAL
(a) STIRAP STIRAP
e0→g1 g1→e0
𝜋g0→e0
Δ𝑡
𝑡
(b) 1.0
Exp
Sim
g0
g1
0.5
g2
e0
e1
e2
0.0
0 200 400 600
t (ns)
(c) 0.8
Expt
0.6
Fit
0.4
0.2
0.0
0 1 2 3 4 5 6
Δ𝑡 (μs)
Fig. 6.7 Reverse STIRAP and measurement of cavity decay rate. (a) Pulse sequence to generate
|g1〉 and map back with a time-delay ∆t. (b) Data from reversing STIRAP (black points) and
simulated evolution (solid red curve) for ∆t = 0. Dashed lines show simulated evolution of
lowest Jaynes-Cummings levels. (c) Recovered populations from reversals performed with
various ∆t time-delays (black points) along with the exponential-decay fit (solid red) with
decay constant τ= 2.51µs.
125
(rec)
𝒫 𝒫e e
Amplitude
6.7. FRACTIONAL STIRAP
complete this two step process (see Fig. 6.7(a),(b)).
A delay can be introduced prior to the reversal pulses to investigate the |g1〉 decay
which occurs with rate κ. By varying the delay time ∆t and plotting versus the re-
covered population P (rec)e , I was able to obtain the cavity decay time (see Fig. 6.7(c)).
For longer delays, a lower population is available for the reverse mapping, and even-
tually (∆t = 6µs) most of the population has decayed to |g0〉, and there is little
population to be transferred back to |e0〉. Fitting yielded an exponential decay time of
τcav = κ−1 = 2.51µs, which was in excellent agreement with independent measure-
ments of cavity lifetime.
6.7 Fractional STIRAP
In the previous sections, STIRAP was employed as a SWAP operation to transfer
any population in the excited state of the qubit to the first excited state of the cavity,
or vice versa. In contrast, Vitanov et al. [146] described a technique by which it
was possible to create superpositions of states by performing a partial transfer, which
they termed f-STIRAP (fractional STIRAP). They achieved this by allowing the Stokes
pulse to precede the pump pulse as in conventional STIRAP, but letting them vanish
simultaneously at the end. By adjusting the synchronization time, the superposition
components could be adjusted [146].
In my system, I implemented f-STIRAP by decreasing the Stokes and pump ampli-
126
6.7. FRACTIONAL STIRAP
(a)
1.0
2
1.9
0.8 1.8
1.7
0.6 1.6
1.5
1.4
0.4
1.3
1.2
0.2 1.1
1
0.0
0 50 100 150 200 250 300 350 400
t (ns)
(b) 1.0
0.8
Exp
0.6 Sim
g1
0.4 |e0〉
|g0〉
0.2
0.0
0 100 200 300
t (ns)
Fig. 6.8 Cavity-qubit entangled state preparation. (a) Plot showing Pe versus t for different
Stokes and pump drive attenuation factors β . The data points and pooled excited populations
from simulations (solid curves) with colors corresponding to the value of β are shown. (b)
Generation of the coherent superposition between |g1〉 (orange dashed) and |e0〉 (blue dash-
dotted) occurs for β = 1.5. Data points (black) and simulated excited population (red curve)
are shown.
127
𝒫 𝒫e
6.8. SUMMARY
tudes by the same attenuation factor β from the optimal STIRAP values so that
Ω′s = Ωs/β
(6.8)
Ω′p = Ωp/β .
By varying β , partial transfers were performed from the qubit to the cavity, as verified
by the residual population in the qubit |e〉 state (see Fig. 6.8(a)). When β = 1.5,
comparing the simulations to the data gave final populations for |g1〉 and |e0〉 as
0.43 and 0.52, respectively (see Fig. 6.8(b)). The resultant state was approximately
0.723 |e0〉+ (0.563− 0.324i) |g1〉 with a 6% probability for being in other states. This
was consistent with the generation of a highly entangled state between the qubit and
the cavity.
6.8 Summary
In summary, a STIRAP-based protocol for coherently transferring the population
from a superconducting qubit to n = 1, 2, and 3 Fock states in a superconducting
microwave cavity was demonstrated. The protocol required 3n drives to move the
system from the ground state, to the nth Fock state.
Based on my simulations, increasing the lifetime of the cavity by a factor of ten
[147] would improve the fidelity for generation of the first Fock state from the present
value of 89% to 95%. An increased lifetime would also allow the application of more
sophisticated measurement analysis techniques on the generated states [136], which
128
6.8. SUMMARY
were not viable in the present system due to the relatively low anharmonicity. Further
improvements in the Fock state fidelity would require increasing the width of the
STIRAP Gaussian pulses (σ). Alternatively, increasing the dispersive shift χ would
enable the use of stronger drives and shorter STIRAP times, as increased anharmonicity
reduces leakage to other states. Implementing adiabatic shortcuts [148–150] is another
approach which could potentially speed up the transfer process. I note that if the cavity
lifetime is much shorter than the qubit lifetime, STIRAP can be used to reset the qubit
by transferring population from the |e〉 ladder to the |g〉 ladder and using the short
cavity lifetime to initialize to |g0〉.
I note that the STIRAP approach also works in the reverse direction; if the order of
the Stokes and probe pulses are reversed, then the population will be transferred from
the cavity to the qubit. In fact, I used this technique to measure the lifetime of the first
Fock state and found it to be 2.51µs, in good agreement with the decay rate of the
cavity. This protocol could be used for single-shot detection of itinerant microwave
photons with noise limited only by the fidelity of the qubit readout. Such a high-fidelity
single-microwave-photon detector may allow measurements of the Bell state between
the qubit and the cavity, including violation of Bell’s inequality [151].
129
CHAPTER 7
A Generalized CNOT Gate
7.1 Introduction
Entanglement is at the core of quantum computing, as can be seen from the require-
ment that we must be able to do arbitrary single-qubit rotations and an entangling
two-qubit gate in order to form a universal quantum gate set [3, 152]. Different types
of entangling gates have performance that is facilitated or hindered depending on the
computer architecture, and a wide variety of such gates have been examined using
trapped ions [11], semiconductor quantum dots [153], and nitrogen-vacancy centers
in diamond [154]. For superconducting qubits, entangling gates can be broadly cate-
gorized into those based on flux-tunability & parametric control, and all-microwave
control [155, 156]. Understanding the trade-offs and performance of gates for quan-
tum computing applications is an active area of research, with scalability, ease of use,
gate speed and gate fidelity being important factors in determining the suitability of a
gate for a particular quantum system.
In cQED [42] based systems, flux-tunable elements allow fast and efficient gates
130
7.1. INTRODUCTION
at the expense of more control lines and shorter coherence times, mostly attributed
to flux-noise and operating the qubits off the sweet spot [157]. Examples of such
gates include the dynamically tuned C-Phase gate [158–160], the iSWAP gate [161],
p
and the direct-resonance iSWAP gate [162, 163]. Gates utilizing all-microwave
controls are slower in general, require fewer control lines but enable longer qubit
lifetimes. Examples include the cross-resonance (CR) gate [164, 165], resonator-
p
sideband-induced iSWAP gate [166], the microwave-activated C-Phase gate [167],
and the resonator-induced-phase (RIP) gate [46, 168]. A major limitation of the CR
gate is the requirement of positioning qubit frequencies within a narrow window, which
proves cumbersome when scaling to many qubits, due to frequency crowding issues
[169]. The CR gate also requires the qubits to be addressed through individual input
drive lines, which is a potential limitation. On the other hand, the RIP gate is able to
operate with a common qubit drive through a bus cavity and with a large frequency
detuning. However, the interaction between the qubits is directly mediated through
the cavity and exposes the system to significant losses via the cavity, especially if one
wants to use a strongly coupled cavity for high-speed readout.
In this chapter, I present my results from an experimental implemention of an
all-microwave generalized CNOT gate, using the technique Speeding up Waveforms by
Inducing Phases to Harmful Transitions (SWIPHT) [170, 171]. The SWIPHT gate has a
few potential advantages compared to the CR gate, including that the detuning between
qubits is not as critical for gate performance, and individual qubit addressability is not
required. Furthermore, as the interaction between qubits is indirectly mediated by the
cavity, the SWIPHT gate is not as susceptible to decoherence from the cavity as the RIP
131
7.2. THE DEVICE AND THE EXPERIMENTAL SETUP
gate is. Our cavity was very short-lived by-design to allow a rapid readout, but this had
no apparent effect on gate performance, as verified by master equation simulations.
As I discuss below, I found that my results were in good agreement with the master
equation simulations and that the gate fidelity was primarily limited by the gate time,
the lifetime of the qubits, and SPAM errors.
7.2 The Device and the Experimental Setup
7.2.1 Details of the cQED System
The cQED system I used in this experiment consisted of the two transmons SPP-v1-Q6
and SPP-v1-Q7 on separate chips mounted in the single three-dimensional supercon-
ducting Al cavity 6G. The device was cooled on the mixing chamber of the Leiden
CF-450 dilution refrigerator, nominally to a base temperature of 20 mK [27, 69]. The
cavity 6G, which originally only had a single slot in the center of the lid, was modified
by machining two slots for the transmons, with each slot being 3 mm equidistant from
the center of the lid (see Fig. 7.1). Each transmon had a single Al/AlOx/Al Josephson
junction and two large 500µm× 650µm Al pads capacitively shunting the junction.
All the results I report in this chapter were obtained during Measurement Run #7.
The transition frequencies between the two lowest levels (|g〉 and |e〉) of the trans-
mons were ωL/2π = 6.07135 GHz for qubit L (QL), and ωH/2π = 6.754 27GHz for
qubit H (QH). The direct capacitive coupling between the transmons, combined with
132
7.2. THE DEVICE AND THE EXPERIMENTAL SETUP
L
H
2 cm
Fig. 7.1 Computer generated model of two transmons and the two sections of the 3D Al cavity.
the indirect coupling through the 7.7463 GHz cavity resonance, resulted in a qubit-qubit
dispersive shift [172] of 2χqq/2π= −1.04MHz. The detailed system parameters are
given in Table 7.1. I determined the system parameters such as the bare frequencies of
the components, and the couplings between components by diagonalizing the coupled
Hamiltonian for the system (see next Section).
7.2.2 The System Hamiltonian
The effective Hamiltonian for the coupled system can be written as,
– ( j) ™
∑ E € Š € Š
H̃0 = ħh
C
ω̃ a† † † † †j j a j + a j a j a j a j − 1 + g j aRa2 j
+ aRa j
j=L , H
 

+ħhω̃ †RaRaR +ħhJ a
†
LaH + a a
†
L H , (7.1)
133
7.2. THE DEVICE AND THE EXPERIMENTAL SETUP
Table 7.1 Device parameters for the two qubits coupled to the cavity. Values were obtained
by extracting eigenstates for the Hamiltonian Eq. (7.1) and comparing the resulting transition
frequencies to the measured spectrum.
System parameter Symbol Value
Bare frequency of transmon L ωL/2π 6.10322 GHz
Bare frequency of transmon H ωH/2π 6.79943 GHz
Bare frequency of cavity ωR/2π 7.66927 GHz
Charging energy of transmon L E(L)C /h 206.5 MHz
Charging energy of transmon H E(H)C /h 192.6 MHz
Transmon-transmon direct coupling J/2π 14.3 MHz
Cavity-transmon coupling (L) gL/2π 224.6 MHz
Cavity-transmon coupling (H) gH/2π 207.5 MHz
Cavity-transmon dispersive shift (L) χL/2π −3.3 MHz
Cavity-transmon dispersive shift (H) χH/2π −7.2 MHz
Qubit-qubit dispersive shift χqq/2π −1.04 MHz
Cavity lifetime 1/κ 50 ns
Relaxation time of transmon L T1L 9.0µs
Relaxation time of transmon H T1H 3.5µs
Spin-echo time of transmon L T2L 14.6µs
Spin-echo time of transmon H T2H 6.2µs
where a and a† are the annihilation and creation operators, and the subscripts L, H,
or R identifies whether the operation is on QL, QH, or the resonator state. Since the
gates I perform do not involve occupation of excited states of the cavity or transmon
levels higher than |e〉 (i.e. |f〉 , |h〉 , . . . ), for simplicity, I restrict consideration to the
computational two-qubit subspace of the system spanned by {|gg〉 , |ge〉 , |eg〉 , |ee〉},
where the labels within kets represent the states of QL and QH, respectively. In this
case, the undriven system can be described by the Hamiltonian:
ħh  
 ħh  
 ħhχ
H (L) qq0 = ωL +χqq σz + ω +χ σ
(H) + σ(L)σ(H)
2 2 H qq z 2 z z
, (7.2)
134
7.2. THE DEVICE AND THE EXPERIMENTAL SETUP
where σ(L)z and σ
(H)
z are the Pauli-Z operators for the qubits L and H, respectively.
The two-qubit system was driven using the SWIPHT protocol (described in Sec. 7.3),
with qubit L as the control, and qubit H as the target [170]. The drive Hamiltonian
can be expressed as:
ħhΩ(t)  
H − i(ωd t+φd) + −i(ωd t+φd)d = σH e +σH e , (7.3)2
where Ω(t) is the envelope of the drive pulse, σ+H is the raising operator for qubit H,
σ−H is the lowering operator for qubit H, φd is the drive phase and ωd is the drive
frequency for the gate pulse. The full system Hamiltonian for the two-qubit subspace
can then be written as,
H =H0 +Hd. (7.4)
7.2.3 Joint Qubit State Readout
Since the two qubits were coupled to a single readout cavity, I had to develop
a method to read out the state of both qubits. This was not the standard situation
where a single transmon was coupled to a cavity, and the cavity can be probed at
its bare frequency with a high-power pulse to destructively measure the qubit state
[104]. For the single qubit case, optimization of the pulse power allows maximum
discrimination between the |g〉 and |e〉 states of the qubit, and pumping to higher
states of the transmon can be used to achieve a higher degree of discrimination [33]. I
135
7.2. THE DEVICE AND THE EXPERIMENTAL SETUP
0.25
0.20  |gg›
 |eg›
 |ge›
0.15
 |fg›
 |ee›
0.10  |gf›
 |fe›
 |ef›
0.05
 |gh›
 |eh›
0.00
-80 -78 -76 -74 -72 -70 -68 -66
Cavity Bias( PdoBwme)r (dBm)
Fig. 7.2 Two-transmon S-curves. Cavity S21 measurements of ∆V/V versus Pbias averaged
over 1000 shots taken for 10 different two-qubit state initializations.
extended this standard approach to perform joint-state-readout of the two-transmon
system by judicious choice of measurement power and mappings to higher transmon
levels.
To set up the joint-readout, I first initialized the two-transmon system in each of
the nine states |gg〉 , |eg〉 , |ge〉 , |fg〉 , |ee〉 , |gf〉 , |fe〉 , |ef〉 , |gh〉 , |eh〉. After each
state initialization, a 2µs long pulse was applied on resonance with the bare cavity
frequency, and the resulting transmission ∆V/V averaged over 1000 shots. I did these
measurements as a function of cavity power Pcav, to obtain the switching curves ∆V/V
versus Pcav for each of the transmon states (see Fig. 7.2).
The vertical dashed lines in Fig. 7.3 show the seven bias powers P1–P7 that I selected
for joint qubit-state readout. I chose |e〉 → |f〉 or |e〉 → |f〉 → |h〉 mappings based on
136
Cavity Signal ∆EnVh/Vancement (V/V)
7.2. THE DEVICE AND THE EXPERIMENTAL SETUP
their ability to discriminate between the eigenstates |gg〉 , |ge〉 , |eg〉 , and |ee〉. The
bias powers were hand-picked to maximize the amount of information extracted using
a minimal number of measurements. For example, at power P4, the |gh〉 and |eh〉
signals were nearly identical (∼ 0.21), and the |gg〉 and |eg〉 signals were also close to
each other (∼ 0.01). Thus, if the excited state of qubit H was mapped |e〉 → |f〉 → |h〉,
P4 would be an ideal bias point to measure the state of transmon H, irrespective of the
state of transmon L. In contrast, if qubit H were only mapped |e〉 → |f〉, the resulting
signals for |gf〉 and |ef〉 would be further apart. Similarly, if qubit L was mapped
|e〉 → |f〉, the resulting signal is not as large (0.08–0.12) and provides less contrast
(see Fig. 7.3). Furthermore, I preferred to use points on Fig. 7.2 where the curves had
smaller slopes, to reduce the uncertainty in measured values. Table 7.2 summarizes
the resulting bias points and mappings that I used.
Figure 7.4 shows the joint measurement procedure; after gating operations on
the qubits were finished, one or two state maps were applied, followed by a cavity
measurement pulse. For a given initial state, the application of a mapping β followed
by a measurement pulse at cavity bias power Pβ resulted in an average transmission
signal
Mβ =C (β)P +C (β)P +C (β)gg gg eg eg ge Pge +C
(β)
ee Pee, (7.5)
wherePi j is the probability of finding the system in the state |i j〉, andC (β) is the outputν
when the system is in state ν and power Pβ is applied (see Eq. (7.6)). To determine the
Pi j for an unknown state, I determined the average values ofMβ for β = {1, . . . , 7} by
137
7.2. THE DEVICE AND THE EXPERIMENTAL SETUP
0.25
 |gg›
0.20  |eg›
 |ge›
0.15  |fg›
 |gf›
0.10  |fe›
 |ef›
0.05  |gh›
0.00  |eh›
-76 -75 -74 -73 -72 -71 -70 -69
Pbias (dBm)
Fig. 7.3 Cavity powers chosen for joint qubit readout. Plot of pulsed cavity transmission
measurements ∆V/V , averaged over 1000 shots, versus cavity bias power Pbias for different
initial two-qubit states. The measurement was performed at the bare cavity frequencyωR/2π =
7.680438 GHz. Dashed lines show powers P1–P7 chosen for joint-qubit readout. Each power
Pi has four associated symbols, corresponding to the system being prepared in |gg〉 (black),
|eg〉 (red), |ge〉 (blue), or |ee〉 (green), and then mapped to a higher level in some cases (see
Table 7.2) before being read out.
Mapping 𝛽
Qubit Cavity Measurement at 
Operations power 𝑃𝛽
𝑡
Fig. 7.4 Pulse train for joint qubit readout. The pulses used for the joint qubit readout are
depicted. Following qubit manipulation (red), one or two mapping pulses (yellow and gold)
are applied, and then the cavity measurement pulse (black) is applied at the corresponding
power Pβ (as listed in Table 7.2).
completing ∼ 1000 repeated preparations, mappings, and measurements of the same
 	
state. Finally, to find the best fit values for the probabilities P 2gg, Pge, Peg, Pee , χ
138
∆V/V
Map 1
Map 2
7.2. THE DEVICE AND THE EXPERIMENTAL SETUP
Table 7.2 Parameters for joint-qubit readout shown in Fig. 7.4. Seven different cavity input
bias powers Pβ between −75.4dBm and −69.5 dBm were used for joint qubit measurements.
Note that in all cases, the |e〉 state of transmon L or transmon H was mapped to the |f〉 or |h〉
state before applying the cavity readout pulse. Also Map 2 was only used for β = 4, 6, and 7,
to map |f〉 → |h〉 for QH.
Map 1 Map 2
β Pβ (dBm)
QL QH QH
1 −75.4 — e→ f —
2 −73.9 — e→ f —
3 −72.7 e→ f — —
4 −72.4 — e→ f f→ h
5 −71.3 e→ f — —
6 −70.5 — e→ f f→ h
7 −69.5 — e→ f f→ h
minimization was performed on Eq. (7.5) as discussed further in Sec. 7.4.
I did separate measurements to determine the calibration data matrix C and the
corresponding error matrix E for the seven cavity powers Pβ , for each of the initialized
eigenstates. The C (β) values can be directly read off of Fig. 7.3. Expressed as a matrix,
ν
I found:
 
0.554 3.69 0.724 11.6
 0.657 16.35 0.943 19.86 
 
 0.836 5.88 9.62 21.35 
C = 10−2 1.05 21.40 2.13 22.46   . (7.6)
 2.4 18.06 21.23 23.405 
 
 4.02 23.42 18.24 24.003 
15.06 23.959 23.44 24.367
139
7.2. THE DEVICE AND THE EXPERIMENTAL SETUP
The corresponding matrix of uncertainties for C was found to be
 
0.033 0.60 0.065 2.6
 0.052 0.92 0.099 0.25 
 
 0.082 0.41 1.80 0.20 
E = 10−2 0.11 0.22 0.18 0.16   . (7.7)
 0.2 0.51 0.24 0.086 
 
 0.27 0.13 0.45 0.065 
0.42 0.081 0.14 0.025
I obtained these statistics by repeating the 1000 shot measurements 1000 times, for
each of the data points in Fig. 7.3. For each set of 1000 shots, I found the mean value;
let C (β)ν, j be the mean for the j
th set. I then found C (β) by averaging together the 1000
ν
values of C (β)ν, j , and took the standard deviation of the C
(β)
ν, j to get E
(β). Thus E (β) is
ν ν
the uncertainty in C (β).
ν
The expression for Mβ given in Eq. (7.5) corresponds to a measurement that
returns an average value C (β) when the system is in eigenstate ν and power P
ν β
is
applied. A more complete model takes into account the noise in the readout electronics
and the bifurcating nature of the non-linear readout [104]. In this case, each single-
shot measurement is randomly drawn from a distribution that I will assume has the
following simple form [173]
  2  2€ Š € Š
 
 (β ,ν)  
(β) (β) V − V (β) (β ,ν)dp  1−K 
ν ν low K V − Vν high= K exp− + exp−  ,
dV (β ,ν)p € Š p € Šσ 2π 2 (β ,ν)
2 (β ,ν) 2

low σ σ 2π
(β ,ν)

low high 2 σhigh
(7.8)
 
dp(β)
where ν dV is the probability of a single-shot measurement producing an output
dV
between V and V + δV , where V ≡ ∆V/V , and K is a normalizing constant; here
140
7.3. THE SWIPHT GATE
∫  
  

K = 1 since dp dV = 1. K (β) and 1−K (β)dV are probabilities that the cavityν ν
bifurcates to the “high” and “low” transmission states when measured, respectively.
V (β ,ν) (β ,ν)low and σlow are the mean and standard deviation of the cavity signal ∆V/V when
it bifurcates to the “low” transmission state, respectively. V (β ,ν)high and
(β ,ν)
σhigh are the mean
and standard deviation of the cavity signal ∆V/V when it bifurcates to the “high”
transmission state, respectively. Note that the underlying distributions corresponding
to “high” and “low” transmission states are assumed to be normal distributions as
one would expect for simple added noise, and all the preceding listed quantities are
expressed with the measurement performed at a power Pβ with the system prepared
in the eigenstate ν. Full analysis of the results using this model of measurement is in
progress.
7.3 The SWIPHT Gate
The SWIPHT gate is based on a theoretical method that implements a desired
evolution on a driven two-level system [174, 175]. The method provides a general
framework to implement a multitude of operations on a two-level system. Economou
and Barnes [170] applied this method to a system of two transmons coupled to a single
cavity. In particular, they provided an analytical solution to implement a generalized
CNOT gate as well as a generalized CZ gate. I used their SWIPHT technique to implement
a generalized CNOT gate which I will call the SWIPHT gate.
An interesting feature of Economou and Barnes [170] implementation of the
141
7.3. THE SWIPHT GATE
SWIPHT gate is that the control pulse shape is entirely determined by the qubit-qubit
dispersive shift χqq. The duration of the pulse is
5.87
τg =   , (7.9)
2 χ qq
and the pulse shape can be written as
γ̈ q
Ω(t) = − 2 χ2q qq − γ̇2 cot (2γ), (7.10)
χ2 2qq − γ̇
where
 4 4
t t π
γ(t) = 138.9 1− + . (7.11)
τg τg 4
From Eq. (7.9) and Eq. (7.10) it can be shown that the maximum amplitude of the
pulse is
 
Ω  max = 0.887× 2 χqq . (7.12)
The SWIPHT gate operates on the principle of applying a π-rotation on a desired
transition (bit-flip), and simultaneously applying a 2π-rotation (no bit-flip) on an
undesired transition within the computational subspace. The resonator plays a minimal
role, allowing the use of a cavity with a strongly coupled output port for a fast qubit
readout. The generalized CNOT gate implemented by the SWIPHT gate can be written
142
7.3. THE SWIPHT GATE
as
 
0 eiϕd 0 0
e−iϕd 0 0 0
SWIPHT= 0 0 eiξ 0 , (7.13) 
0 0 0 eiζ
with ϕd = 0, ξ ≈ 1.16 rad, and ζ ≈ 1.98 rad (note ξ+ ζ = π). As with a standard
CNOT operation, the target qubit is flipped depending on the control qubit. However,
this is a generalized CNOT gate since the control qubit also acquires extra z-rotations
(represented by ξ and ζ) due to the non-trivial 2π-rotation imposed on the harmful
transition, as discussed next.
The numerical values in Eqs. 7.9 and 7.11 are determined by the boundary con-
ditions for the desired |gg〉 ↔ |ge〉 transition at frequency ωH to complete a π-
rotation, and the undesired |eg〉↔ |ee〉 transition at frequency ωH + 2χqq to complete
a 2π-rotation. For our experiment, ωd/2π = 6.75427 GHz, Ωmax/2π = 913kHz,
τg = 907ns, and the resulting control pulse envelope is shown in Fig. 7.5. The re-
quired SWIPHT pulse was generated using a 25 GSa/s Tektronix AWG70002A arbitrary
waveform generator. Since an AWG performs digital signal processing, it has a limited
sampling rate and voltage resolution, resulting in minor distortions of the output
waveform compared to an ideal waveform.
143
7.4. QUANTUM STATE TOMOGRAPHY
800
600
400
200
0
0 200 400 600 800
t (ns)
Fig. 7.5 SWIPHT pulse shape Ω/2π versus time t. The analytically derived SWIPHT pulse
envelope Ω(t) used in the experiment has a peak amplitude of Ωmax = 913 kHz and a duration
of τp = 907 ns.
7.4 Quantum State Tomography
I used the method outlined in Sec. 7.2.3 to determine the diagonal elements
(i.e. populations) ρii of the density matrix. In this section, I discuss how I also
obtained the off-diagonal elements (i.e. coherences) of the density matrix by using
quantum state tomography (QST). This involved applying rotations to the system just
prior to measurement [64, 176]. By performing these rotations before measurement,
partial information of the state was obtained. The partial information was then used
to reconstruct the complete density matrix using a maximum likelihood estimation
technique [63] as described next.
144
(kHz)
7.4. QUANTUM STATE TOMOGRAPHY
Table 7.3 List of tomographic maps. Overcomplete set of tomographic operations applied
prior to the measurement protocol.
Gate Tomographic Gate Tomographic
name map name map
G1 I ⊗ I G I ⊗ R−π/210 x
G π/2 −π/22 I ⊗ Rx G11 I ⊗ R y
G3 I ⊗ Rπ/2y G R−π/212 x ⊗ I
G Rπ/2 ⊗ I G R−π/24 x 13 y ⊗ I
G Rπ/2 ⊗ I G R−π/2 ⊗ R−π/25 y 14 x x
G Rπ/2 ⊗ Rπ/2 G R−π/2 ⊗ R−π/26 x x 15 x y
G Rπ/2 ⊗ Rπ/2 G R−π/2 −π/27 x y 16 y ⊗ Rx
G Rπ/2 ⊗ Rπ/28 y x G17 R−π/2y ⊗ R
−π/2
y
G Rπ/2 ⊗ Rπ/29 y y
7.4.1 Constructing the Likelihood Function
To determine the density matrix from my tomographic data, I constructed a likeli-
hood function L that characterized the closeness of the density matrix to the data. To
ensure that I extracted a physically meaningful density matrix, even if there were state
preparation and measurement (SPAM) errors, I used an explicitly “physical” represen-
tation of the density matrix (described in detail in Sec. 2.4.2), then varied the free
parameters in ρ to find the most probable representation of the state of the system
[63]. In this analysis, I used the 17 tomographic pulse combinations (see Table 7.3)
for each cavity measurement power Pβ to get an overcomplete set of measurements
Mβ ,τ [64].
145
7.4. QUANTUM STATE TOMOGRAPHY
The likelihood function L is given by
 ∑   
 	
17 7 (β) † 2
∑∑ Mβ ,τ − C G ρ̂G
L = ν ν
τ τ νν , (7.14)
” —
∑
E (β)
 
 2
τ=1 †β=1
ν ν Gτρ̂Gτ νν
where ρ̂ is the parameterized, explicitly-physical density matrix [63]. Here τ is the in-
dex for applied tomographic gates, β is the index for cavity mapping and measurement
power Pβ , ν is the index for system eigenstates {|gg〉 , |ge〉 , |eg〉 , |ee〉}, and Gτ are the
tomographic gates used for QST (see Table 7.3). The numerator of L quantifies the
closeness of measured signalsMβ ,τ to those expected from ρ̂. In general the different
measurementsMβ ,τ will have different uncertainties associated with them. Thus, the
closeness is weighed by using the estimated square of the uncertainty inMβ ,τ in the
denominator of L . By minimizing L with respect to the 16 free parameters in ρ̂ (see
Appendix D), the most probable physical density matrix ρ was extracted.
Qubit 
Cavity Measurement
Operations (𝐿) (𝐻)
𝐺𝜏 𝐺𝜏 𝑡
Fig. 7.6 Pulse train used during QST. Following qubit manipulation (red), tomographic pulses
G(L)τ on qubit L (green) and G
(H)
τ on qubit H (blue) were applied. As in Fig. 7.4, up to two
mapping pulses (yellow and gold) were then applied and finally the cavity measurement pulse
(black) was applied.
146
Map 1
Map 2
7.4. QUANTUM STATE TOMOGRAPHY
7.4.2 QST During SWIPHT Evolution
To verify that the system was evolving properly during the SWIPHT gate, the
SWIPHT pulse was started at t = 0, but stopped at times nδt < τg for n = 0 to
n = 181, and δt = 5ns; QST was performed for each timestep. For the measurements,
I first initialized the system in the desired state (|gg〉, |ge〉, |eg〉, or |ee〉), applied the
part of the SWIPHT pulse from time t = 0 to nδt, applied tomographic pulses G(L)
τ
and G(H), and finally performed a measurement (see Fig. 7.6). At each 5 ns timestep,
τ
this process was repeated for each of the seven cavity powers Pβ for 1000 single-shot
averages, to obtainMβ ,τ.
The points in Fig. 7.7 show the measured probabilities (Pgg, Pge, Peg, and Pee)
versus t, with the system initialized in the |gg〉 state (Fig. 7.7(a)) and in the |ge〉 state
(Fig. 7.7(b)). The curves in Fig. 7.7 show reasonably good agreement between the
data and master equation simulations (as discussed below), provided I include a drive
phase φd = 245◦. Note that when qubit L is not excited, e.g. |gg〉, a bit-flip operation
takes place and when qubit L is excited, e.g. |eg〉, the bit-flip operation is suppressed. I
note that the maximum eigenstate population at the end of the operations is not 100%,
and the master equation simulations suggest that most of this loss can be attributed to
qubit relaxation.
The master equation simulations only included the relaxation rates of the two qubits,
since the devices were mostly T1 limited. The example code for QST in Appendix D
show the procedure for the full SWIPHT pulse length. For the case of simulating the
evolutions in Fig. 7.7, I executed the program and extracted the density matrix for all
147
7.4. QUANTUM STATE TOMOGRAPHY
times t < 908ns.
(a) 1.0
0.8 gg ge
0.6
0.4
0.2
0.0
0 200 400 600 800
t (ns)
(b) 1.0
0.8 eg
0.6
0.4
0.2 ee
0.0
0 200 400 600 800
t (ns)
Fig. 7.7 (a) Evolution of populations during partial SWIPHT gate lasting time t, after initializing
the system in state |gg〉 and then performing QST. Discrete points are experimental data and the
solid curves are obtained using master equation simulations of the two-qubit system. This data
was obtained with a SWIPHT drive phase φd = 245◦. (b) Corresponding population evolution
during partial SWIPHT pulse lasting time t, when system was prepared in state |eg〉.
148
𝜌𝑖,𝑖 𝜌𝑖,𝑖
7.4. QUANTUM STATE TOMOGRAPHY
7.4.3 Verification of Coherence and Phase Control
To verify that the phase ϕd of the SWIPHT gate (see Eq. (7.13)) is set by the phase of
the drive, I variedφd of the drive from−2π rad to +2π rad while generating a Bell state
between the two qubits. To do this, I initialized qubit L in one of four superposition
€ Š
states R±π/2x ,y ⊗ I |gg〉, where I is the identity operator, and R is the rotation operator
on qubit L with a given axis (x or y) and phase (±π/2). I then applied, the SWIPHT gate
and performed QST. Since the initial state was an equal superposition of the control
qubit’s |g〉 and |e〉 states, applying the generalized CNOT gate generated a Bell state
between the two qubits. Figure 7.8 shows the resulting dependence of the off-diagonal
term ρge,eg in the density matrix versus φd. The plots showed a clear oscillating pattern
with a period 2π in φd, an average amplitude of 0.41. The master equation simulations
0.50
𝜋/2
0.25 𝑅𝑥
−𝜋/2
𝑅𝑦
0.00 −𝜋/2𝑅𝑥
𝜋/2
𝑅
-0.25 𝑦
-0.50
2𝜋 𝜋 0 𝜋 2𝜋
𝜙d (rad)
 

Fig. 7.8 Plot of Im ρge,eg versus the phase φd of the SWIPHT pulse, for four different initial
€ Š
superposition states of qubit L R±π/2x ,y ⊗ I |gg〉. Points are from QST measurements and the
solid curves are from master equation simulations of ρ with a fixed added phase offset of 245°.
149
7.5. QUANTUM PROCESS TOMOGRAPHY
Cavity Measurement
(𝐿) (𝐻)
𝑅 𝐿 𝑅 𝐻 SWIPHT 𝐺𝜏 𝐺𝜏 𝑡
Fig. 7.9 Pulse train for QPT. The two qubits were initialized in one of the states given by
{I , Rπ ±π/2x , Rx ,y }
⊗2 |gg〉 (green and blue striped pulses for qubit L and qubit H, respectively). The
SWIPHT gate was then applied (red), followed by the tomographic pulses (green and blue solid
pulses), mappings to higher levels (yellow and gold), and the measurement pulse (black).
of ρge,eg (curves) show good agreement with data, provided I include a phase offset of
245°. This offset is likely due to the phase shift along the heavily-attenuated > 2 m
long input line.
7.5 Quantum Process Tomography
To provide a more complete characterization of the SWIPHT gate, I also performed
quantum process tomography (QPT) [67, 177]. The essential idea of QPT is to prepare
a complete set of initial states and then to perform QST on all the resulting states
to quantify the performance of the gate. I adopted the method discussed in O’Brien
et al. [67], in which a χ-matrix is used to characterize a quantum gate. An equivalent
method for characterizing quantum processes using the Pauli transfer matrix has been
discussed by Chow et al. [177].
Similar to QST, I needed to ensure that the χ-matrix I extracted from my measure-
ments was not un-physical. I used the 16 two-qubit Pauli operators {Âm} = { Î Î , Î X̂ , Î Ŷ ,
Î Ẑ , X̂ Î , X̂ X̂ , X̂ Ŷ , X̂ Ẑ , Ŷ Î , Ŷ X̂ , Ŷ Ŷ , Ŷ Ẑ , Ẑ Î , Ẑ X̂ , Ẑ Ŷ , Ẑ Ẑ} as a basis, where the index
150
Map 1
Map 2
7.5. QUANTUM PROCESS TOMOGRAPHY
i = 1, . . . , 16 denotes the operator number. χ was then a 16× 16 matrix. Now, I can
write an ideal CNOT gate as
1 1 1 1
CNOT = Î Î + Î X̂ − Ẑ Î + Ẑ X̂ . (7.15)
2 2 2 2
Similarly the SWIPHT operation can be written as
1 1
SWIPHT = i0.4577 Î Î + Î X̂ − i0.4577 Ẑ Î + Ẑ X̂ + 0.2012 Î Ẑ − 0.2012 Ẑ Ẑ ,
2 2
(7.16)
where contributions to the norm smaller than 10−4 have been neglected in the real
components of the coefficients for Î Î and Ẑ Î , and the imaginary components of the
coefficients for Î Ẑ and Ẑ Ẑ . The resulting χ-matrix has 36 non-zero elements, 20 of
which are real and 16 which are imaginary. The visual representation of the χ-matrix
can serve as a gate diagnostic tool, with discrepancies between ideal and experimental
χ-matrices providing insight into sources of systematic errors and possible remedies
[3].
∑
To ensure thatχ represents a trace-preserving process, the constraint mnχmnÂ
†
nÂm =
I must be met [67]. To enforce this constraint, I introduced a Lagrange multiplier λ
and a cost function:
16
∑¦
 2 © 2
Ξ= Re(Zi j)− Ii j + Im(Zi j) , (7.17)
i, j=1
151
7.5. QUANTUM PROCESS TOMOGRAPHY
∑
where Z = χ †mn mn nÂm. I then defined a likelihood function
¦ ©
∑   
  2
36 17 7 M (σ) (β) (σ) †∑∑∑
β , − C G ρ Gτ ν ν τ out τ
L̃ νν= 2 +λΞ, (7.18)” —∑ ( )  
β (σ)
σ=1 =1 †τ β=1
ν Eν Gτρout Gτ νν
whereσ is an index for the overcomplete set of 36 initial states given by {I , Rπ ±π/2 ⊗2x , Rx ,y } |gg〉
(see Fig. 7.9). In this expression, ρ(σ)out denotes an explicitly physical output density
matrix for a given initial state obtained using the χ-matrix as,
16
∑
(σ) = Â (σ)ρout χmn mρin Â
†
n, (7.19)
m,n=1
where (σ)ρin is the density matrix corresponding to the initial state index σ.
Minimizing L̃ with respect to each complex quantity χmn and λ yields χ̃, the
maximum-likelihood estimate for the χ-matrix that describes the underlying process.
Figure 7.10(a) shows χ̃ I extracted from the measurements of the SWIPHT gate. Fig-
ure 7.10(b) and Fig. 7.10(c) show, respectively, the χ-matrices I obtained from master
equation simulations of the SWIPHT gate incorporating decoherence (χsim), and the
ideal decoherence-free case (χSW). The effects of decoherence can be seen by compar-
ing χsim to the ideal χSW; for example all the non-zero components in χSW are slightly
smaller in magnitude in χsim. The presence of some additional non-zero components
in χ̃ and χsim, that do not occur in χSW, indicate the presence of coherent errors [16].
For example, the real part of χ̃ and χsim have non-zero contributions from (I I , IX ) and
(IX , I I), while the contributions vanish in the ideal χSW. In addition, by comparing
χ̃ with χsim, I infer that additional coherent z-rotations are present in χ̃. I note that I
152
7.5. QUANTUM PROCESS TOMOGRAPHY
(a) Re[𝜒ǁ] (b) Re 𝜒 (c)sim Re 𝜒. SW.
II
IX +0.25
IY
IZ
XI
XX
XY
XZ
YI
YX
YY
YZ
ZI
ZX
ZY -0.25
ZZ
Im[𝜒ǁ] Im 𝜒sim Im 𝜒SW.
II
IX +0.25
IY
IZ
XI
XX
XY
XZ
YI
YX
YY
YZ
ZI
ZX
ZY
-0.25
ZZ
Fig. 7.10 Process matrices from QPT. Real and imaginary parts of experimental (χ̃) , simulated
(χsim) and ideal (χSW) process matrices for the SWIPHT operation with qubit L as the control.
The maximum theoretical magnitude for any element is 0.25.
also observed extra z-rotations using QPT on single qubit gates (see Appendix E).
To compare the extracted χ̃ to the expected χSW from the ideal decoherence-free
SWIPHT operation, I computed the process fidelity
Fp = Tr (χ̃χSW) , (7.20)
153
II II
IX IX
IY IY
IZ IZ
XI XI
XX XX
XY XY
XZ XZ
YI YI
YX YX
YY YY
YZ YZ
ZI ZI
ZX ZX
ZY ZY
ZZ ZZ
II II
IX IX
IY IY
IZ IZ
XI XI
XX XX
XY XY
XZ XZ
YI YI
YX YX
YY YY
YZ YZ
ZI ZI
ZX ZX
ZY ZY
ZZ ZZ
II II
IX IX
IY IY
IZ IZ
XI XI
XX XX
XY XY
XZ XZ
YI YI
YX YX
YY YY
YZ YZ
ZI ZI
ZX ZX
ZY ZY
ZZ ZZ
7.5. QUANTUM PROCESS TOMOGRAPHY
Table 7.4 Gate times τgate and measured QPT performance metrics Fp, Fg, and Tr (ρ2) for
several gates. For the four SWIPHT gates at the bottom of the table, the subscript denotes the
control qubit, and the superscript denotes the phase φd of the drive.
Gate τgate (ns) Fp Fg Tr (ρ2)
I ⊗ I 0 0.94 0.95 0.97
I ⊗ Rπx 37 0.93 0.94 0.97
I ⊗ Rπy 37 0.91 0.93 0.95
I ⊗ Rπ/2x 37 0.91 0.93 0.96
I ⊗ Rπ/2y 37 0.90 0.92 0.95
I ⊗ R−π/2x 37 0.90 0.92 0.96
I ⊗ R−π/2y 37 0.90 0.92 0.95
Rπx ⊗ I 72 0.89 0.92 0.95
Rπy ⊗ I 72 0.87 0.90 0.93
Rπ/2x ⊗ I 72 0.86 0.89 0.93
Rπ/2y ⊗ I 72 0.85 0.88 0.93
R−π/2x ⊗ I 72 0.82 0.85 0.94
R−π/2y ⊗ I 72 0.84 0.88 0.95
SWIPHT0L 907 0.76 0.81 0.72
SWIPHT0H 907 0.75 0.80 0.73
SWIPHT49π/36L 907 0.77 0.82 0.72
SWIPHT7π/12H 907 0.77 0.82 0.74
and the mean gate fidelity
F d + 1
F pg = , (7.21)d + 1
where d = 4 is the dimensionality of the two-qubit system [67, 178, 179]. I found that
  
 
Fp = 0.76,Fg = 0.81, and the average purity of the gate Tr (ρ2) = d Tr χ̃2 + 1 /(d+
1) = 0.72 [67]. Average purity is a metric for the degree of mixture introduced by a
gate and indicates how much the process was affected by incoherent errors [178].
To understand if the SWIPHT gate was significantly limited by decoherence in the
154
7.5. QUANTUM PROCESS TOMOGRAPHY
qubits, I performed QPT for master equation simulations; i.e. I performed QPT on
a complete set of simulated results (see Appendix D). Since the simulated results
only possessed relatively small numerical errors from the limited precision of the
numerical calculation, I ignored the denominator of the first term in Eq. (7.18) during
the minimization to calculate the simulated process matrix χsim. I observed that the
SWIPHT performance metrics from the simulations were Fp = 0.84, Fg = 0.87, and
Tr (ρ2) = 0.77. These were significantly better (5%) than the experimental results, but
otherwise showed reasonable agreement with the experiment, and revealed that the
main reason for infidelity as relaxation in the qubits.
To understand why the SWIPHT gate gave somewhat lower fidelity than found in
the simulation, I also performed QPT on other gates, including three simple variations
on the SWIPHT gate (see Table 7.4 and Appendix E). All the single qubit gates had
relatively high purity due to fast gate times, while all the SWIPHT gates showed lower
purity, in part due to qubit relaxation during the longer gate times. Of particular note
was the I ⊗ I gate, which I set up to be a zero-length operation undergoing QPT. For
this simple null gate, I found Fg = 0.95, supporting that there were ∼ 5% errors
introduced due to SPAM during QPT. I note that the I ⊗ I operation had the highest
performing metrics, compared to all other gates (see Table 7.4).
155
7.6. SUMMARY
7.6 Summary
In this chapter, I discussed my results on the SWIPHT gate, a generalized CNOT op-
eration between two fixed-frequency transmon qubits [170, 171]. Four different
implementations of SWIPHT yielded gate fidelities Fg ' 80− 82%, similar to initial
demonstrations of other all-microwave gates [165, 167, 180]. Overall, I found good
agreement between the data and master equation simulations that incorporated deco-
herence. In the presence of decoherence, the maximum theoretical gate fidelity was
Fg = 86%, while experimental fidelities were about 5% lower, likely due to SPAM. The
relatively rapid decay of the cavity with a lifetime 1/κ= 50 ns, directly affected the
qubit lifetimes due to the Purcell effect, but otherwise had a minimal impact on the
fidelity of the SWIPHT gate.
As described by Deng et al. [171], the gate time τg could be significantly reduced
by using qubits with |g〉 ↔ |e〉 frequencies that were closer together, similar to the
situation for CR gates [164, 165]. I should also emphasize that all the gate fidelities
I reported were obtained through QPT, and not corrected for state preparation and
measurement (SPAM) associated errors. Detection and measurement of the two-
qubit states was performed using a joint-readout technique, which could be further
optimized, or superseded by a more efficient dispersive readout [64]. I also note
that the single-qubit gate fidelities (see Table 7.4) could have been improved by
derivative pulse-shaping techniques [181], and this would have possibly given overall
higher SWIPHT fidelities, as process tomography is sensitive to SPAM errors in the
156
7.6. SUMMARY
state initialization and tomography pulses. Finally, I found that adjusting the drive
phase allowed excellent control of the phase of the target qubit, and that the phase
accumulation on the control qubit could be corrected through single qubit rotations.
157
CHAPTER 8
Conclusion
8.1 Summary of Main Results
In this dissertation, I described my research on superconducting qubits. This work
included the design, fabrication, characterization, measurement and simulation of
transmons and cavities. All of the major results I presented were taken using the two
transmons SPP-v1-Q6 and SPP-v1-Q7, and the two cavities 6D and 6G. Depending on
the set of parameters I needed, the couplings to the cavities were modified between
different measurement runs. When not using the devices, I stored them in a dry-
box. Nevertheless, there were also some run-to-run variations and variations in the
transmons due to aging, which I had little control over.
8.1.1 Fock State Generation
In Chapter 6, I described my work on generating Fock states in a superconducting
microwave cavity. I used the STIRAP (Stimulated Raman Adiabatic Passage) technique,
158
8.1. SUMMARY OF MAIN RESULTS
to generate the Fock states by transferring an excitation from the transmon to the cavity.
I was able to perform this transfer successively to generate Fock states up to n = 3.
The results showed good agreement with master-equation simulations I performed
using MATLAB (see Appendix C).
To confirm that the correct states were being generated, Wigner tomography [144]
would have been the method of choice. However, given the relatively short lifetime
of our cavity (κ−1 = 2.4µs), and the relatively short anharmonicity in the cavity,
performing Wigner tomography would have been challenging. Instead, I verified the
generation of the Fock states by performing Rabi oscillations on the states generated
by STIRAP. This was sensitive enough to reveal the change in cavity frequency due to
the dispersive shift. I was also able to generate superpositions between cavity Fock
states, as well as Bell states involving the transmon and the cavity.
8.1.2 A Generalized CNOT Gate
In Chapter 7, I presented my results on implementing a generalized CNOT gate
using the SWIPHT (Speeding up Waveforms by Inducing Phases to Harmful Transitions)
technique. I repackaged two qubits, which I had measured previously into a single
cavity that I had modified to hold two chips. The output coupling Q of the cavity was
set low (Qoutput ∼ 3000) to obtain a fast readout time. I initially expected this setup to
only serve as a test-bed for me to learn how to characterize two qubits, and develop
two-qubit control and measurement techniques. To my surprise, the results I obtained
far exceeded my expectations.
159
8.1. SUMMARY OF MAIN RESULTS
One of the major issues I faced was reading out the qubit states. The qubits had
transition frequencies that were relatively close to the cavity frequency, and the coupling
g of each qubit to the cavity was not too different from the single-qubit cases I had
worked with previously. I found that the high-power-readout (see Sec. 5.2.2) worked
well for the readout of individual qubit readout for the individual qubit QH. However,
to read out both qubits efficiently I had to develop a least-squares joint-state-detection-
method, which involved measuring at multiple cavity powers and mapping the qubit
states to higher transmon levels.
The results from the SWIPHT experiment showed good agreement with the master-
equation simulations I performed using Mathematica. To fully characterize the SWIPHT gates,
I used quantum process tomography with maximum-likelihood-estimation. The con-
struction and numerical search for the minimum of the likelihood function, which
contained 257 parameters and 4284 individual terms, proved difficult using Mathe-
matica. I therefore apportioned the likelihood function for the 36 initial states, used
MATLAB to generate those 36 components in parallel, and finally imported the grand
likelihood function function into Mathematica for numerical minimization. With this
approach, I was able to complete QPT characterization of gates in a timely manner.
160
8.2. POSSIBLE IMPROVEMENTS
8.2 Possible Improvements
8.2.1 Improving Fock State Generation
The major limitation in my experiment to generate Fock states was the lifetime of
the cavity. Having a longer lifetime would have resulted in higher fidelities, as well as
improved detection and verification methods. As I discussed in Chapter 6, I set the
cavity quality factor to ensure a reasonably fast readout time (∼ 3µs). Unfortunately,
this relatively short time was detrimental to the STIRAP transfer of the excitation; the
photons suffered significant loss since they were stored in the same cavity that I used
to read out the qubit state.
One way to improve the process would be to use two cavities — a storage cavity
for storing the generated Fock states and a measurement cavity for reading out the
transmon state. Two possible geometries for such an approach have been discussed by
Reagor et al. [147] and Axline et al. [182]. In both cases, they used a high-quality-factor
seamless cylindrical cavity as the storage cavity, while the readout cavity was another
3D cavity or 2D resonator, respectively.
Having a larger qubit-cavity dispersive shift would also facilitate a faster operation
time; the STIRAP pulses were limited in Rabi drive strength by the limited anhar-
monicity. For the parameters of my device, more intense pulses would have resulted in
unwanted excitations into other cavity states. By increasing the coupling of the qubit
to the cavity, or decreasing the qubit-cavity detuning, a larger qubit-cavity dispersive
shift would be generated. This allows the use of stronger drives with shorter durations.
161
8.2. POSSIBLE IMPROVEMENTS
8.2.2 Improving the SWIPHT gate
There are many possible avenues for improving the generalized CNOT gate that I
implemented using the SWIPHT technique. As I discussed in Chapter 7, a major limiting
factor in the gate time was the small qubit-qubit dispersive shift χqq. This quantity can
be increased by reducing the qubit-qubit detuning, and also by having the |g〉↔ |e〉
of one qubit close in frequency to the |e〉↔ |f〉 transition of the second qubit [171].
However, having a higher χqq would significantly affect the operation of single-qubit
gates, since this is a fixed coupling term. It would then be necessary to implement
special techniques to perform single-qubit gates [183], or engineer a SWIPHT-like pulse
to perform identical operations on both transitions.
I should note, that the technique introduced by Economou and Barnes [170] is fairly
general, and applies to any two transitions which are close in frequency. Thus, the exact
mechanism which lifts the degeneracy between the two close transitions is irrelevant.
When the degeneracy is lifted, it is possible to appropriately address the system using
SWIPHT to execute a CNOT gate. The speed and performance in SWIPHT however seems
to come at the cost of the simplicity of single-qubit gates. Examination of the trade-offs
of different parameter choices would be an interesting topic for further investigation.
162
APPENDIX A
Experimental Determination of Quality Factors for a
Bandpass Cavity
A.1 Modeling the Cavity
A bandpass cavity having two ports can be modeled using a simplified lumped
element circuit. This allows for analytical expressions that can be used to extract the
scattering parameters (S-parameters).
1 : p q : 1
V C L R1 V2
Fig. A.1 Simplified effective circuit model for a bandpass resonator.
163
A.2. ABCD MATRIX ANALYSIS FOR THE MODEL
I used inductive coupling in Fig. A.1 to model the input and output coupling from
the transmission lines to the cavity. This simplifies the analysis considerably. L, C ,
R are the effective inductance, capacitance and resistance of the fundamental mode
of the cavity. G0 is the characteristic admittance of the transmission lines. p and q
represent the ratios of the turns of the input and output transformers, respectively.
The internal, input, output, loaded, and external quality factors can be defined
respectively as [184]
√
√C R
Q0 = R = (A.1)L ω0 L
√
p2√C p2
Qin = = (A.2)G0 L ω0 LG0
√
q2√C q2
Qout = = (A.3)G0 L ω0 LG0
1 1 1 1
= + + (A.4)
QL Q0 Qin Qout
1 1 1
= + . (A.5)
QE Qin Qout
A.2 ABCD Matrix Analysis for the Model
ABCD matrices are a convenient method to model circuits with many different
components [103]. The ABCD matrices for the circuit in Fig. A.1 are, from left to right,
         
1
p 0 1 0 1 0 1 0 q 0
=   =   =      K , M , N , P= , Q= .
         
0 p 1 1 1R iωL 1 iωC 1 0
1
q
164
A.3. OBTAINING THE S-PARAMETERS
The total ABCD matrix for the complete circuit is obtained by multiplying the matrices
for individual components, preserving the order, and one finds:
 
q
p 0
U= ·  K M ·N · P ·Q= .
 
 

pq 1 − i + iC pR Lω ω q
A.3 Obtaining the S-parameters
The S-parameters can be found from the ABCD matrix [103] using the following
relations:
€ Š
A+ BG C0 − G − D 2
QL
Q
S = 011 C = −1+
in
A+ BG0 + G + D 1+ i
QL 2
ωω (ω −ω
2
0 0 0
)
2iQ
= −1− 0
Qoutωω0 (A.6)
Q Q Q (ω2 −ω20 in out 0)− i (QinQout +Q0Qout +Q0Qin)ωω0
€ Š
−A+ BG0 −
C + D 2 QLG0 QS = = −1+ out22
A+ BG + C + D 1+ i QL (ω2 20 G −ω )0 ωω0 0
2iQ Q
= −1− 0 in
ωω0 (A.7)
Q0Q 2
2
inQout(ω −ω0)− i (QinQout +Q0Qout +Q0Qin)ωω0
 ‹
2 p QL
2 QinQout
S21 = =
A+ BG + C0 G + D 1+ i
QL 2 2
0 ωω
(ω −ω
0 0
)
p
2Q
= 0
ωω0 QinQout . (A.8)
(QinQout +Q0Qout +Q0Qin)ωω0 + iQ0QinQout(ω2 −ω20)
165
A.4. APPROXIMATIONS FOR CALCULATION OF QUALITY FACTORS
In obtaining the above expressions, I have made the following substitutions and
simplified to eliminate the variables p, q, L, C , R and G0.
R=ω0 LQ0
1
C =
ω20 L
p
p = ω0 LG0Qin
p
q = ω0 LG0Qout
A.4 Approximations for Calculation of Quality Factors
A.4.1 Qout =Qin = 2QE
With the assumption Qout = Qin = 2QE, I obtain the following formula for S21
(Eq. (A.8))
€ Š
QL
Q
S E21 =
1+ i Q
. (A.9)
L 2 2
ωω (ω −ω0 0)
This reduces to the following simplified formula under the approximation ω ≈
ω0 +∆,
€ Š
QL
Q
S = E21
1+ 2iQ ω−ω
. (A.10)
0
L ω0
166
A.4. APPROXIMATIONS FOR CALCULATION OF QUALITY FACTORS
A.4.2 ω=ω0
For the case ω =ω0, the formulas for the S-parameters are considerably simplified
allowing for back of the envelope calculations of Qin and Qout. One finds:
S(ω=ω )
2Q
0 = L11 − 1 (A.11)Qin
(ω=ω ) 2QS 0 L22 = − 1 (A.12)Qout
S(ω=ω0)
2QL
21 = p . (A.13)QinQout
167
APPENDIX B
Rabi Oscillations after FSG
Rabi oscillations were performed immediately after |g1〉 and |g2〉 generation. The
amplitude and frequency of the resulting oscillations allowed me to verify that the
correct state was generated. I compare the resulting oscillations to master equation
simulations, and observe good agreement.
168
B. Rabi Oscillations after FSG
(a) 1.0 (b) 1.0
1.5 MHz 3.1 MHz
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0
0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700
t (ns) t (ns)
(c) 1.0 (d) 1.0
4.6 MHz 6.2 MHz
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0
0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700
t (ns) t (ns)
(e) 1.0 (f) 1.0
7.7 MHz 9.2 MHz
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0
0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700
t (ns) t (ns)
Fig. B.1 Rabi oscillations performed after |g1〉 generation. Immediately following |g1〉 gener-
ation, Rabi oscillations were performed with microwave drives of frequency ωq and ωg1→e1.
Different drive amplitudes were used, resulting in resonant oscillations when driven at ωg1→e1
(black) and off-resonant oscillations when driven at ωq (red) as expected. The oscillations
from driving at ωg1→e1 can be easily distinguished from higher-frequency and lower-amplitude
oscillations produced by driving at ωq. The nominal strengths for the Rabi drives are (a)
1.5 MHz (b) 3.1 MHz (c) 4.6 MHz (d) 6.2 MHz (e) 7.7 MHz and (f) 9.2 MHz.
169
𝒫𝑒 𝒫𝑒 𝒫𝑒
𝒫𝑒 𝒫𝑒 𝒫𝑒
B. Rabi Oscillations after FSG
(a) 1.0 (b) 1.0
1.5 MHz 3.1 MHz
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0
0 200 400 600 0 200 400 600
t (ns) t (ns)
(c) 1.0 (d) 1.0
4.6 MHz 6.2 MHz
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0
0 200 400 600 0 200 400 600
t (ns) t (ns)
(e) 1.0 (f) 1.0
7.7 MHz 9.2 MHz
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0
0 200 400 600 0 200 400 600
t (ns) t (ns)
(g) 1.0 1.0
10.8 MHz (h) 13.8 MHz
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0
0 200 400 600 0 200 400 600
t (ns) t (ns)
Fig. B.2 Rabi oscillations performed after |g1〉 generation under ωg1→e1 drive. Data (black
points) are overlaid with simulated evolution (solid red). Examination of these curves reveals
that the oscillations contain components at two different frequencies: a majority corresponding
to resonant Rabi oscillations and a minority component at the off-resonant Rabi frequency. The
nominal strengths for the Rabi drives are (a) 1.5 MHz (b) 3.1 MHz (c) 4.6 MHz (d) 6.2 MHz
(e) 7.7 MHz (f) 9.2 MHz (g) 10.8 MHz (h) 13.8 MHz.
170
𝒫𝑒 𝒫𝑒 𝒫𝑒 𝒫𝑒
𝒫𝑒 𝒫𝑒 𝒫𝑒 𝒫𝑒
B. Rabi Oscillations after FSG
(a) 1.0 (b) 1.0
1.5 MHz 3.1 MHz
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0
0 200 400 600 0 200 400 600
t (ns) t (ns)
(c) 1.0 (d) 1.0
4.6 MHz 6.2 MHz
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0.0
0 200 400 600 0.00 200 400 600
t (ns) t (ns)
(e) 1.0 (f) 1.0
7.7 MHz 9.2 MHz
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0
0 200 400 600 0 200 400 600
t (ns) t (ns)
(g) 1.0 (h) 1.0
10.8 MHz 13.8 MHz
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0
0 200 400 600 0 200 400 600
t (ns) t (ns)
Fig. B.3 Rabi oscillations performed after |g1〉 generation under ωq drive. Data black points)
are overlaid with simulated evolution (solid red). Examination of these curves reveals that
the oscillations contain components at two different frequencies: a majority corresponding to
resonant Rabi oscillations and a minority component at the off-resonant Rabi frequency. The
nominal strengths for the Rabi drives are (a) 1.5 MHz (b) 3.1 MHz (c) 4.6 MHz (d) 6.2 MHz
(e) 7.7 MHz (f) 9.2 MHz (g) 10.8 MHz (h) 13.8 MHz.
171
𝒫𝑒 𝒫𝑒 𝒫𝑒
𝒫𝑒
𝒫𝑒 𝒫𝑒 𝒫𝑒 𝒫𝑒
B. Rabi Oscillations after FSG
(a) 1.0 (b) 1.0
1.5 MHz 3.1 MHz
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0
0 200 400 600 800 1000 0 200 400 600 800 1000
t (ns) t (ns)
(c) 1.0 (d) 1.0
7.7 MHz 15.4 MHz
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0.0
0 200 400 600 800 1000 0.0 0 200 400 600 800 1000
t (ns) t (ns)
Fig. B.4 Rabi oscillations performed after |g2〉 generation under ωg2→e2 drive. Following |g2〉
generation, a limited number of Rabi oscillations were performed at various drive strengths to
verify the generated state. Data from experiments (black points) are overlaid with simulated
evolution (solid red). The nominal strengths for the Rabi drives are (a) 1.5 MHz (b) 3.1 MHz
(c) 7.7 MHz and (d) 15.4 MHz.
172
𝒫𝑒 𝒫𝑒
𝒫𝑒 𝒫𝑒
APPENDIX C
MATLAB Code for STIRAP Master Equation Simulations
C.1 The Main Program for Simulating FSG upto n= 3
1 clear all;
2
3 %% System Parameters
4
5 ncav = 25; % number of cavity levels
6 ntot = 2*ncav + 1; % total number of levels
7 pi2= 2*pi;
8 wC = pi2*7.934971E9;
9 wQ = pi2*5.5317455E9;
10 g = pi2*69.8E6;
11
12 piAmp = pi2/(59E-9); %59 seemed without any interference
13 piAmp2 = pi2/(65E-9); %65 seemed without any interference
14 piAmp3 = pi2/(65E-9); %65 seemed without any interference
15 rabiAmp = 0*pi2/(60E-9);
16
17 qubAmp = pi2*9.6E6;
18 cavAmp= pi2*26.2E6;
19 driveAmp=cavAmp;
20 a=qubAmp/cavAmp;
21
22 pulseLength = 340.01E-9;
23 stDevFraction = 4.8;
24 pulseSeparation = 14E-9;
25
26 delta1 = pi2*8.1E6; % detuning
27 delta2 = pi2*9.7E6;
28 delta3 = pi2*11.3E6;
29
30 piLength= 32.01E-9;
31 piLength2= 34.01E-9;
173
C.1. THE MAIN PROGRAM FOR SIMULATING FSG UPTO N = 3
32 piLength3= 32.01E-9;
33 rabiLength= 0*500.01E-9;
34
35 tRed = pulseLength/2 + piLength; % center of first pulse
36 tBlue = tRed + pulseSeparation; % center of second pulse
37
38 tmax = pulseLength + pulseSeparation + piLength;
39 tmax2 = pulseLength + pulseSeparation + piLength2;
40 tmax3 = pulseLength + pulseSeparation + piLength3;
41
42 stDev = pulseLength/stDevFraction; % st.dev of pulses
43 timeStep = 40E-12;
44 finalTime = tmax+tmax2+tmax3 + rabiLength;
45
46 Qdec = 1/(24E-6); % Qubit Decay rate
47 Cdec = 1/(2.5E-6); % Cavity Decay rate
48 Qdph = 1/(41E-6); % Qubit Dephasing Rate
49
50 %% The Bare Hamiltonian
51
52 H0 = zeros(ntot);
53
54 for i = 1:ncav
55 H0(2*i+1,2*i+1)=i*wC;
56 H0(2*i,2*i)=(i-1)*wC+wQ;
57
58 H0(2*i,2*i+1)=sqrt(i)*g;
59 H0(2*i+1,2*i)=sqrt(i)*g;
60 end
61
62 %Obtaining the drive frequencies
63
64 [e_vecs, e_vals] = eig(H0);
65
66 wg0xe0 = e_vals(2,2);
67 wg1xe1 = (e_vals(4,4)-e_vals(3,3));
68 wg2xe2 = (e_vals(6,6)-e_vals(5,5));
69 wg3xe3 = (e_vals(8,8)-e_vals(7,7));
70
71 wRed = (e_vals(4,4)-e_vals(3,3)-delta1);
72 wBlue = (e_vals(4,4)-e_vals(2,2)-delta1);
73 wRed2 = (e_vals(6,6)-e_vals(5,5)-delta2);
74 wBlue2 = (e_vals(6,6)-e_vals(4,4)-delta2);
75 wRed3 = (e_vals(8,8)-e_vals(7,7)-delta3);
76 wBlue3 = (e_vals(8,8)-e_vals(6,6)-delta3);
77
78 H0=e_vals;
79
80 %% Dissipation (Defining Jump Operators)
81
82 Aqub = zeros(ntot);
83 Acav = zeros(ntot);
84 Adph = zeros(ntot);
85
174
C.1. THE MAIN PROGRAM FOR SIMULATING FSG UPTO N = 3
86 for i = 1:ncav
87 Aqub(2*i-1,2*i) = 1;
88 Acav(2*i-1,2*i+1) = sqrt(i);
89 % Acav(2*i+1,2*i-1) = sqrt(0.04*i);
90 Adph(2*i-1, 2*i-1) = 1;
91 Adph(2*i, 2*i) = -1;
92 end
93
94 for i = 1:ncav-1
95 Acav(2*i,2*i+2) = sqrt(i);
96 % Acav(2*i+2,2*i) = sqrt(0.04*i);
97 end
98
99 AcAc = 0.5*(Acav)’*Acav;
100 AqAq = 0.5*(Aqub)’*Aqub;
101 AdAd = 0.5*(Adph)’*Adph;
102
103 %% Initial state for solving the differential equations
104
105 initConditions = zeros(ntot^2,1);
106 initConditions(1)=0.995;
107 initConditions(ntot*2+3)=0.005;
108
109 %% Bare Driving Matrix
110
111 V = zeros(ntot,ntot);
112 for i = 1:ncav
113 V(2*i-1,2*i) = 1;
114 V(2*i,2*i-1) = 1;
115 V(2*i-1,2*i+1) = sqrt(i);
116 V(2*i+1,2*i-1) = sqrt(i);
117 end
118
119 for i = 1:ncav-1
120 V(2*i,2*i+2) = sqrt(i);
121 V(2*i+2,2*i) = sqrt(i);
122 end
123
124 %% Complete Hamiltonian and creating the set of differential equations
125
126 tspan = 0:0.04E-9:finalTime;
127 [tspanM, tspanN] = size(tspan);
128
129 [T, R] = ode45(@(t,rho)myode_Dph_FullPulses_FS3_rabis(t, rho, H0, V, piAmp, piAmp2,
piAmp3, rabiAmp, qubAmp, cavAmp, Acav, AcAc, Aqub, AqAq, Adph, AdAd, Cdec, Qdec,
Qdph, piLength, piLength2, piLength3, rabiLength, tRed, tBlue, tmax, tmax2,
tmax3, stDev, wRed, wBlue, wRed2, wBlue2, wRed3, wBlue3, wg0xe0, wg1xe1, wg2xe2,
wg3xe3), tspan, transpose(initConditions));
130
131 %% Plotting the obtained data
132
133 %converting output data to reals
134 RR = real(R);
135 II=imag(R);
175
C.1. THE MAIN PROGRAM FOR SIMULATING FSG UPTO N = 3
136
137 g0Data = RR(:,1);
138 e0Data = RR(:,ntot+2);
139 g1Data = RR(:,ntot*2+3);
140 e1Data = RR(:,ntot*3+4);
141 g2Data = RR(:,ntot*4+5);
142 e2Data = RR(:,ntot*5+6);
143 g3Data = RR(:,ntot*6+7);
144 e3Data = RR(:,ntot*7+8);
145 eData = RR(:,ntot+2)+RR(:,ntot*3+4)+RR(:,ntot*5+6)+RR(:,ntot*7+8)+RR(:,ntot*9+10)+RR
(:,ntot*11+12);
146
147 eData = RR(:,ntot+2)+RR(:,ntot*3+4)+RR(:,ntot*5+6)+RR(:,ntot*7+8)+RR(:,ntot*9+10)+RR
(:,ntot*11+12)+RR(:,ntot*13+14)+RR(:,ntot*15+16);
148 diagonalelements = RR(:,[1,ntot+2,2*ntot+3,3*ntot+4,4*ntot+5,5*ntot+6,6*ntot+7,7*
ntot+8,8*ntot+9,9*ntot+10,10*ntot+11,11*ntot+12,12*ntot+13,13*ntot+14,14*ntot
+15,15*ntot+16,16*ntot+17,17*ntot+18,18*ntot+19]);
149 diagonalFullelements = RR(:,[1,ntot+2,2*ntot+3,3*ntot+4,4*ntot+5,5*ntot+6,6*ntot
+7,7*ntot+8,8*ntot+9,9*ntot+10,10*ntot+11,11*ntot+12,12*ntot+13,13*ntot+14,14*
ntot+15,15*ntot+16,16*ntot+17,17*ntot+18,18*ntot+19,...
150 19*ntot+20,20*ntot+21,21*ntot+22,22*ntot+23,23*ntot+24,24*ntot+25,25*ntot+26,26*
ntot+27,27*ntot+28,28*ntot+29,29*ntot+30,30*ntot+31,31*ntot+32,32*ntot
+33,33*ntot+34]);
151
152 figure();
153 plot(T,g0Data, T,e0Data, T,g1Data, T,e1Data,T,g2Data, T,e2Data, ’LineWidth’,2);
154 legend(’g0’,’e0’,’g1’,’e1’,’g2’,’e2’);
155 xlabel(’Evolution time (s)’);
156 ylabel(’Population’);
157
158 figure();
159 plot(T,eData, ’LineWidth’,2);
160 legend(’e’);
161 xlabel(’Evolution time (s)’);
162 ylabel(’Population’);
163
164 % dlmwrite(’rr.csv’,RR)
165 % dlmwrite(’ii.csv’,II)
166
167 % dlmwrite(’g0.csv’,g0Data)
168 % dlmwrite(’e0.csv’,e0Data)
169 % dlmwrite(’e1.csv’,e1Data)
170 % dlmwrite(’g1.csv’,g1Data)
171 % dlmwrite(’e2.csv’,e2Data)
172 % dlmwrite(’g2.csv’,g2Data)
173 % dlmwrite(’e3.csv’,e3Data)
174 % dlmwrite(’g3.csv’,g3Data)
175 % dlmwrite(’e.csv’,eData)
176 % dlmwrite(’t.csv’,T)
177 % dlmwrite(’diagonals.csv’,diagonalelements)
178 % dlmwrite(’diagonalsFull.csv’,diagonalFullelements)
176
C.2. SUBPROGRAM FOR THE LINDBLAD MASTER EQUATION
C.2 Subprogram for the Lindblad Master Equation
1 %% Creating the Gaussian Pulses
2
3 function drhodt = myode_Dph_FullPulses_FS3_rabis(t, rho, H0, V, piAmp, piAmp2,
piAmp3, rabiAmp, QdriveAmp, CdriveAmp, Acav, AcAc, Aqub, AqAq, Adph, AdAd, Cdec,
Qdec, Qdph, piLength, piLength2, piLength3, rabiLength, tRed, tBlue, tmax,
tmax2, tmax3, stDev, wRed, wBlue, wRed2, wBlue2, wRed3, wBlue3, wg0xe0, wg1xe1,
wg2xe2, wg3xe3)
4
5 %% Dissipation Operators
6
7 rho = reshape (rho,size(H0));
8
9 % forcing a first operation to set the size of rho
10 Acavrho = Acav*rho;
11 Aqubrho = Aqub*rho;
12 Adphrho = Adph*rho;
13
14 Lcav = Acavrho*Acav’ - AcAc*rho - rho*AcAc;
15 Lqub = Aqubrho*Aqub’ - AqAq*rho - rho*AqAq;
16 Ldph = Adphrho*Adph’ - AdAd*rho - rho*AdAd;
17
18 %% Drive Matrix
19
20 pSep = tBlue - tRed;
21
22 tnow=floor(t/40E-12)*40E-12;
23
24 piPulse1 = heaviside(piLength-tnow).*sin(wg0xe0*tnow);
25 qubPulse1 = heaviside((tnow-piLength).*(tmax-pSep-tnow)).*round(exp(-(tnow-tRed)
.^2/(2*stDev.^2))*sin(wRed*tnow-piLength)*0.58/0.0009765625)*0.0009765625/0.58;
26 cavPulse1 = heaviside((tnow-piLength-pSep).*(tmax-tnow)).*round(exp(-(tnow-tBlue)
.^2/(2*stDev.^2))*sin(wBlue*tnow-piLength-pSep)*0.0455/0.0009765625)
*0.0009765625/0.0455;
27
28 piPulse2 = heaviside((tnow-tmax).*(tmax+piLength2-tnow)).*sin(wg1xe1*tnow-tmax);
29 qubPulse2 = heaviside((tnow-piLength2-tmax).*(tmax+tmax2-pSep -tnow)).*round(exp(-(
tnow-tRed-tmax).^2/(2*stDev.^2))*sin(wRed2*tnow-tmax-piLength2)
*0.58/0.0009765625)*0.0009765625/0.58;
30 cavPulse2 = heaviside((tnow-piLength2-tmax-pSep).*(tmax+tmax2 -tnow)).*round(exp(-(
tnow-tBlue-tmax).^2/(2*stDev.^2))*sin(wBlue2*tnow-tmax-piLength2-pSep)
*0.0455/0.0009765625)*0.0009765625/0.0455;
31
32 piPulse3 = heaviside((tnow-tmax-tmax2).*(tmax+tmax2+piLength3-tnow)).*sin(wg2xe2*
tnow-tmax-tmax2);
33 qubPulse3 = heaviside((tnow-piLength3-tmax-tmax2).*(tmax+tmax2+tmax3-pSep -tnow)).*
round(exp(-(tnow-tRed-tmax-tmax2).^2/(2*stDev.^2))*sin(wRed3*tnow-tmax-tmax2-
piLength3)*0.58/0.0009765625)*0.0009765625/0.58;
34 cavPulse3 = heaviside((tnow-piLength3-tmax-tmax2-pSep).*(tmax+tmax2+tmax3 -tnow)).*
round(exp(-(tnow-tBlue-tmax-tmax2).^2/(2*stDev.^2))*sin(wBlue3*tnow-tmax-tmax2-
piLength3-pSep)*0.0455/0.0009765625)*0.0009765625/0.0455;
35
177
C.2. SUBPROGRAM FOR THE LINDBLAD MASTER EQUATION
36 rabiPulse = heaviside((tnow-tmax-tmax2-tmax3).*(tmax+tmax2+tmax3 + rabiLength -tnow)
).*sin(wg3xe3*tnow);
37
38 pulses = piAmp*piPulse1 + QdriveAmp*qubPulse1 + CdriveAmp*cavPulse1 + piAmp2*
piPulse2 + QdriveAmp*qubPulse2 + CdriveAmp*cavPulse2 + piAmp3*piPulse3 +
QdriveAmp*qubPulse3 + CdriveAmp*cavPulse3 + rabiAmp*rabiPulse;
39
40 V = pulses*V;
41 Htot = H0+V;
42
43 %% Final output Density matrix
44
45 drhodt = (Qdph/2)*Ldph + Cdec*Lcav + Qdec*Lqub + 1j*rho*Htot - 1j*Htot*rho;
46 drhodt=drhodt(:);
178
APPENDIX D
Code for Quantum Tomography
D.1 Mathematica Program for Performing Quantum State
Tomography
Preparation---------------------------------------------------------------------------------------------------
ClearAll["`*"]; Clear["Global`*"
SetDire[cto]ry=[NotebookDirectory[]
]]; ClearSystemCache[]; $HistoryLength = 0;;
ket2dm M_ : M.M ;
$Assum∈ptions = {t1∈∈ Reals, t2∈∈ Reals, t3 ∈∈Reals, t4 ∈ Reals, t5 ∈ Reals, t6 ∈ Reals, t7 ∈ Reals,t8 ∈Reals,}t9 Reals, t10 Reals, t11 Reals, t12 ∈ Reals, t13 ∈ Reals, t14 ∈ Reals, t15 ∈ Reals,t16 Reals ;
Importing Data-----------------------------------------------------------------------------------------------
cavBiases=={{14.6, 16.1, 17.3, 17.6, 18.7, 19.5, 20.5};dataSets = allData146, allData161, allData173, allData176, allData187, allData195, allData205};dataSets
Im(port /@"SWIPHT__LAmp=0.495_HAmp=0.000_cPwr=" <> ToString[NumberForm[#, {4, 2}]] <>[ = "dBm<=LAngle=0deg_HAngle=0deg.dat" & /@ cavBiases);F{or i 1, i Length[dataSets], i++, dataSets[[i]] = dataSets[[i, 1 ;; 17]]];allData146, allData161, allData173, allData176, allData187, allData195, allData205} = dataSets;
S146 == allData146[[[[All, 2]]]]; S161 = allData161[[All, 2]];S173 = allData173 All, 2 ; S176 = allData176[[All, 2]];S187 = allData187[[[[All, 2]]]]; S195 = allData195[[All, 2]];S205 allData205 All, 2 ;
179
D.1. MATHEMATICA PROGRAM FOR PERFORMING QUANTUM STATE
TOMOGRAPHY
Calibration values-------------------------------------------------------------------------------------------
{(*The followi}ng= set of equations contains the weights from S-curves used during MLE*)b0MLE, a0ML*E +Coefficie*ntA+rrays[{0.00554* gg + 0.00724* eg 0.0369* ge + 0.116* ee ⩵ S146,0.00657* gg + 0.00943* eg + 0.1635* ge + 0.1986* ee ⩵ S161,0.00836* gg+ 0.0962* eg ++ 0.0588** ge ++ 0.2135* ee ⩵ S173,0.0105* gg + 0.02129 eg 0.2140 ge 0.2246* ee ⩵ S176,0.0240* gg + 0.2123** eg ++ 0.1806* ge + 0.23405* ee ⩵ S187,0.0402* gg + 0.1824* eg + 0.2342* ge + 0.24003* ee ⩵ S195,{0.1506 gg 0.2344 eg 0.23959* ge + 0.24367* ee ⩵ S205},(* gg, ge, eg, ee}];{ The following set}o=f equations will co[n{tain the errors associated with each of the measurements*)b0Errors, a*0Err+ors Coe*ffi+cientArr*ays0.00033* gg + 0.00065 eg 0.0060 ge + 0.026* ee ⩵ 0,0.00052* gg + 0.00099* eg + 0.0092* ge + 0.0025* ee ⩵ 0,0.00082* gg+ 0.018** eg ++ 0.0041** ge ++ 0.0020** ee ⩵ 0,0.0011* gg + 0.0018* eg 0.0022 ge 0.0016 ee ⩵ 0,0.0020* gg + 0.0024* eg + 0.0051* ge + 0.00086* ee ⩵0.0027* gg + 0.0045* eg ++ 0.0013**ge ++0.00065**ee ⩵
0,
0,
{0.0042 gg 0}.0]014 eg 0.00081 ge 0.00025 ee ⩵ 0},gg, ge, eg, ee ;
System Parameters------------------------------------------------------------------------------------------
Eδe== 2ππ 6.75427;((**targe-t qubit frequency*)ϵ = 2 π 0.00103; δqubit qubit shift*)2 6.07135 - ;
Γωp == Ee/; (*
2
DriΓve=frequency*)1= 1 350Δ0;= δ2 1/ 9030; (*target and control qubit decay rates*)τA =138.9; ;5.p [8Δ7] ; (*Length of SWIPHT pulse*)Abs
Gate Definitions----------------------------------------------------------------------------------------------
θ = π/ 2; =θ1 = -π/ 2;identit=y{{Iden}tit{yMatr}i}x[2];sig=max 0, 1 ,[ 1, 0 ; s⊗igmay = {{]0, -I}, {I, 0}}; sigmaz = {{1, 0}, {0, -1}};id = ArrayFlatten identity identity ;sx = ArrayFlatten[sigmax⊗identity]sz = ArrayFlatten[[sigmaz⊗i⊗dentity]
; sy = ArrayFlatten[sigmay⊗identity];
;
tx = ArrayFlatten[identity⊗sigmax]]; ty = ArrayFlatten[identity⊗sigmay];tz =Arra[yFθl/at]ten -identit[yθ/si]gmaz ;sx2 = Cos[θ/ 2] id - I Sin[θ/ 2] sx; sy2 = Cos[θ/ 2] id - I Sin[θ/ 2] sy;sz2 = Cos[θ/ 2] id - I Sintx2 = Cos[θ/ 2] id - I Sin[[θθ/
2] sz;/ 2] tx; ty2 = Cos[θ/ 2] id - I Sin[θ/ 2] ty;tz2 =Cos [θ 2/ i]d I-Sin [θ2 /tz;snx2 = Cos 1 2 id I Sin 1 2] sx;sny2 = Cos[θ1/ 2] id - I Sin[θ1/ 2] sy;tnx2 = Cos[θ1/ 2] id - I Sin[θ1/ 2] tx;tny2 Cos[θ1/ 2] id - I Sin[θ1/ 2] ty;
180
D.1. MATHEMATICA PROGRAM FOR PERFORMING QUANTUM STATE
TOMOGRAPHY
Initial Conditions---------------------------------------------------------------------------------------------
ξ = (245 - 0) Degree; (*Adjustment phase for the SWIPHT pulse*)
1
initialState = N@Flatten@ket2dmsx2. 00 ;
0
Initial Definitions---------------------------------------------------------------------------------------------
0 0 0 0
= 0 EeH0 0 0 δ 0+ ϵ 00
2
0 0 0 Ee - δ + ϵ ;
2
0 0 0 0
H0RWA = 0 ωp0 0 ω0 0p 0 ;
0 0 0 2 ωp
Ω [ ] ⅇ-0ⅈ (ω +ξ) Ω [t] ⅇⅈ (ωp t+ξ)= 0 0t p tV 0 0
0 0 Ω [ ] ⅇ-0ⅈ (ω +ξ) Ω [ ] ⅇ
0
t ⅈ (ω ;p t+ξ)
( = 0 0 t
p t 0
(Ht=ot H0 + V)[// Mat]r)ix/F/orm;(U MatrixExp I H0 t MatrixForm;Hrotρ= Ch[op]@FρullS[im]pliρfy[U[.H]toρt.In[ve]rse[U] + I D[U, t].Inverse[U]]) // MatrixForm;ρ = ρ0x0[t] ρ0x1 t 0x2 t 0x3 tρ1x0[t] ρ1x1[[t] ρ1x2[t]ρ2x0[t] ρ2x1[t]] ρρ2x2[[t] ρ
ρ1x3[[t]2x3 t] ;
3x0 t 3x1 t 3x2 t] ρ3x3[t]
ass[ume = Flatten[ρ /. t → 0];For i = [1[, i]≤] L=ength[assume], i++,a[ssume ]i = ToEx-pression[T-oString[as(su*me[[i]]] <> "==" <> ToString@initialState[[i]]];]1 1A_, ρ_ : A.ρ.A A .A.ρ ρ.A .A; Lindblad operator definition*)
(* 2 2Creating the jump operators*)
= 0 1 0 0 0 0 1 0Aq1 0 0 0 0 0 0 0 10 0 0 1 ; Aq2 = 0 0 0 0 ;
0 0 0 0 0 0 0 0
Creating the SWIPHT Pulses-------------------------------------------------------------------------------
χχ[ ] = τt 4 - τt 4 πt_ A 1 + ;p p 4
Ω[ ] = D[χχ[t], {t, 2}] Δ2_ 2t - [χχ[ ] ] - - D[χχ[t], t] Cot[2 χχ[t]] ;Δ 422 D t , t 2
4
Ω[τp/ 2]
maxRabiDrive = π * 106;2
181
D.1. MATHEMATICA PROGRAM FOR PERFORMING QUANTUM STATE
TOMOGRAPHY
The complete Hamiltonian--------------------------------------------------------------------------------
equationsR == ([Iρ*(ρ].Hrot - Hrot.ρ)) + Γ1*[Aq1, ρ] + Γ2*[Aq2, ρ];equationsL D , t ;
Solving the system of equations--------------------------------------------------------------------------
solutions =]NDSolve[Jo[iρn][F{latten[τTa}ble[equatio→nsL[[i, j]]]⩵ equationsR[[i, j]], {i, 1, 4}, {j, 1, 4}]],ρa=ssume[(,{Fρl/atten , t},/0, →pτ,)M[a[xSteps I]n]f]inity ;thy Chop = . solutio[ns .[t p[ 1, 1, [All ρ];tℱh=yP(lot[Matrix Part[ition Join R[ifflρe Flat]ten tρhy , 0, 5], [{0, 0ρ, 0, 0, 0, 0}], 5];Tr MatrixPower MatrixPower thy , 0.5 .exp .MatrixPower thy , 0.5], 0.5]])2;
Maximum Likelihood Estimation-------------------------------------------------------------------------
= t1 0 0 0T t5 ++ I t6 +t2 0 0t11 + I t12 t7 I t8 t3 0 ;t15 I t16 t13 + I t14 t9 + I t10 t4ρ = T .Tp FullSimplify@ ;
(* Tr[T.T]List of
tomoMeρas[u[res[
gat]es== II, IY, YI, IX, XI, XX, YY, YX, XY *)i]_] : Transpose@{{(p i, i ,(ty2.ρρp.ty2))[[[[i, i]]]], ((sy2.ρp.sy2)[[i, i]],(tx2. p.tρx2 i, i , sx2.ρp.sx2)[[i, i]],(sx2.tx2.ρp.tx2.sx2))[[[[i, i]]]], (sy2.ty2.ρp.ty2.sy2)[[i, i]],(sy2.txρ2. p.tx)2[.[sy2 ]] i(, i , (sx2.ty2.ρp.ty2.sx2)[[i, i]],(tny2.ρp.tny2)[[i, i]], (sny2.ρρp.sny2)[[i, i]],(tnx2. p.tnρx2 i, i , snx2. p.snx2)[[i, i]],(snx2.tnx2.ρp.tnx2.snx2))[[[[i, i]]]], ((sny2.tny2.ρρp.tny2.sny2))[[[[i, i]],sny2.tnx2. p.tnx2 .sny2 i, i , snx2.tny2. p.tny2 .snx2 i, i]]}};
lik
Forelik =hood = 01, k <=;Lengt=h[cavBiases], k++,sumForAGivenBias Sum[a0MLE[[k, j]]* tomoMeasures[j], {j, 1, 4}];
differenceForAGivenBias = sumForAGivenBias - dataSets[[k, All, 2]];
measurementError = Sum(a0Errors[[k, j]]* a0MLE[[k, j]]* tomoMeasures[j])2, {j, 1, 4};
differenceForAGivenBias2
likelihood += ;;
measurementError
likelihood = To=tal@Flatten@likelihood;noiseTriangle
RandomReal[{-
RandomComplex[[{
1-, 1}] 0 0 01, 1}] RandomReal[{-1, 1}] 0 0
RandomComplex[{{--1, 1}}]] RandomComplex[{-1, 1}] RandomReal[{-1, 1}] 0 ;RandomComplex 1, 1 RandomComplex[{-1, 1}] RandomComplex[{-1, 1}] RandomReal[{-1, 1}]= 1 noiseTriangle .noiseTrianglenoise FullSimplify@ [ // MatrixForm;( 1000 Tr noiseTriangle.noiseTriangle]guess = thyρ) // MatrixForm;
cleanSample = guess + noise;
182
D.1. MATHEMATICA PROGRAM FOR PERFORMING QUANTUM STATE
TOMOGRAPHY
Constructing the Initialization for the T - Matrix------------------------------------------------------
level1Minors == Map[[Reverse, Minors[cleanSample], {0, 1}];level2Minors Map Reverse, Minors[cleanSample, 2], {0
Det[, 1}];cleanSample]
TInitializingData = IfDet[cleanSample] ≠ 0, , 0, 0, 0, 0,
level1Minors〚1, 1〛
If 〚 〛 ≠ level1Minors〚1, 2〛level1Minors 1, 2 0, , 0,
level1Minors〚1, 1〛 level2Minors〚1, 1〛
level1Minors〚1, 1〛
Iflevel1Minors〚1, 1〛 ≠ 0, , 0, 0, 0,
level2Minors〚1, 1〛
  〚 〛 ≠ leve〚l2Mi〛nors〚1, 4〛If level2Minors 1, 4 0, , 0,cleanSample 4, 4 level2Mi
 〚 〛 ≠ leve〚l2Mi〛nors〚1, 2〛
nors〚1, 1〛
If level2Minors 1, 2 0, 〚 〛 , 0,cleanSample 4, 4 level2Minors 1, 1
level2Minors〚1, 1〛
Iflevel2Minors〚1, 1〛 ≠ 0, , 0, 0,
cleanSample〚4, 4〛
  〚 〛 ≠ cleanSample〚〚4, 1〛〛   〚 〛 ≠ cleanSample〚4, 2〛If cleanSample 4, 1 0, , 0 , If cleanSample 4, 2 0, , 0,cleanSample〚 4, 4〛 cleanSample〚4, 4〛 cleanSample 4, 3If cleanSample〚4, 3〛 ≠ 0, 〚 〛 , 0, cleanSample〚4, 4〛 ;cleanSample 4, 4
TInitializingData = Chop[TInitializingData - I* DiagonalMatrix[Im@Diagonal[TInitializingData]]];[TInitializingData .TInitializingData // MatrixForm;Tr TInitializingData .TInitializingData]
Results of MLE------------------------------------------------------------------------------------------------
initializeParameters =
Union[Parti[tion[Rif[fle[FullSimplify@Re@Flatten[T], FullSimplify@Re@Flatten[TInitializingData]], 2Par-tit]i]on Riffle FullSimplify@Im@Flatten[T], FullSimplify@Im@Flatten[TInitializingData]], 2]][[
],
2 ;; 1 ;
(sol ==Fiρnd/Minimum[likelihood, initializeParameters, MaxIterations → Infinity];ℱs=ss( [p . sol[[2][]) // MatrixForm; expρ = sss;Tr MatrixPower MatrixPower[thyρ, 0.5].expρ.MatrixPower[thyρ, 0.5], 0.5]])2;
Experimental matrix and Result from Master - equation Simulations-------------------------
thyPlotMatrix == Partition[[Join[[Riffle[[Flatten[[thyρρ]], 0, 5]], {{0, 0, 0, 0, 0, 0}}], 5];expPlotMatr=ix Partition Join Riffle Flatten exp , 0, 5 , 0, 0, 0, 0, 0, 0 ], 5];colorThe=me "Rainbow["; plotHeightRange = {-0.6, 0.6};thyPlot List→Plot3D Re@thyPl→otMatrix, InterpolationOrder → 0, Mesh → NoneImageSize 500, PlotR→ange p]lotHeightRange, ColorFunction → ColorData[
,{Filling → Axis,colorTheme, plotHeightRange}],
Color=FunctionScaling False ;expPlot List→Plot3D[Re@expPlotMatrix, InterpolationOrder → 0, Mesh → None, Filling → Axis, ImageSize → 500,PlotRange plotHeight→Range,]ColorFunction → ColorData[{colorTheme, plotHeightRange}],ColorFunctionScaling False ;
thyPlotIm = ListPlot3D[Im@thyPlotMatrix, InterpolationOrder → 0, Mesh → None, Filling → Axis,
ImageSize → 500, PlotR→ange → p]lotHeightRange, ColorFunction → ColorData[{colorTheme, plotHeightRange}],ColorFu=nctionScaling False ;expPlotIm Li→stPlot3D[Im@exp→PlotMatrix, InterpolationOrder →→0, Mesh → None, Filling → Axis,ImageSize 500, PlotR→ange p]lotHeightRange, ColorFunction ColorData[{colorTheme, plotHeightRange}],C[olorFunctionS[ca{l{ing False ;Show Graphic→sGrid ] thyPlot, expPlot, thyPlotIm, expPlotIm, BarLegend[{colorTheme, plotHeightRange}]}}],ImageSize 2500 ;
183
D.1. MATHEMATICA PROGRAM FOR PERFORMING QUANTUM STATE
TOMOGRAPHY
Ideal SWIPHT Results---------------------------------------------------------------------------------------
equationsRPerfect = (I*(ρ.Hrot - Hrot.ρ));
solutionsPerfect =
NDSolve[J]oin[Flatt[eρn][Ta{ble[eqτua}tionsL[[i, →j]] ⩵ equationsRPerfect[[i, j]], {i, 1, 4}, {j, 1, 4}]],assρum=e , Fl[a(t{tρen/ , t, 0, p , Max}S/teps→ τInfinity];perfectℱ = C(hop[ . solu[tionsPerfect[ . t ρp)[[1], 1, Aρll]]];p(erfect Tr Matri=xPower Mat[rixPower perfect , 0.5 .exp .MatrixPower[perfectρ, 0.5], 0.5]])2;perfectPlotMa=trix Parti[tio@n Join[Riffle[Flatten[perfectρ], 0, 5], {0, 0, 0, 0, 0, 0}], 5]) // MatrixForm;perfectPlot→Re ListPlot3D Re→ perfectPlotMat→rix, InterpolationOrder → 0, Mesh → None,Filling Axis, ImageSize 500, PlotRange plotHeightRange,
ColorFuncti=on → ColorDa[ta[@{colorTheme, plotHeightRange}], ColorFu→nctionSca→ling → False];perfectPlot→Im ListPlot3D Im→ perfectPlotMat→rix, InterpolationOrder 0, Mesh None,Filling Axis,→ImageSize [{500, PlotRange plotHeightRange,C[olorFunction [{C{olorData colorTheme, plotHeightRange}], ColorFunctionScaling → False];Show GraphicsGrid perfectPlotRe, perfectPlotIm}}], ImageSize → 1000];
Generalized CNOT Results---------------------------------------------------------------------------------
ζ = (245) Degree; ι1 =ζ (113.67) Degree; ι2 = (66.21) Degree;-0 ζ EI 0 0 1I
cnotphase = E 0 0 0 ; cnotρ = ket2dmcnotphase.sx2. 0 ;
0 0 EI ι2 0ι 00 0 0 EI 1 0
cnotℱ = (Tr[ℱMa=t(rix[Power[MatrixPower[cnotρ, 0.5].expρ.MatrixPower[cnotρ, 0.5], 0.5]])2;cnotperfect= *Tr M[atrixPower[M[a(trixPρo-wer[cnotρρ,)0(.5].pρer-fectρ.Maρt)rixPow]e]r[cnotρ, 0.5], 0.5]])2;Tr@D(istance 0.5 Tr= MatrixPow[er c[not perfect . cnot perfect , 0.5 ;N cnotPlot=Matrix Par[tit@ion Join Riffle[Flatten[cnotρ], 0, 5], {0, 0, 0, 0, 0, 0}], 5]) // MatrixForm;cnotPlotRe L→istPlot3D Re cn→otPlotMatrix, InterpolationOrder→→ 0, Mesh →[N{one, Filling → Axis,ImageSize 500, PlotR→ange p]lotHeightRange, ColorFunction ColorData colorTheme, plotHeightRange}],ColorFunctionScaling False ;
cnotPlotIm = L→istPlot3D[Im@cn→otPlotMatrix, InterpolationOrder→→ 0, Mesh → None, Filling → Axis,ImageSize 500, PlotR→ange p]lotHeightRange, ColorFunction ColorData[{colorTheme, plotHeightRange}],ColorFunctionScaling False ;
Show[GraphicsGrid[{{cnotPlotRe, cnotPlotIm}}], ImageSize → 1000];
184
D.2. MATLAB PROGRAMS FOR CONSTRUCTING THE LIKELIHOOD FUNCTION
D.2 MATLAB Programs for Constructing the Likelihood Func-
tion
D.2.1 MATLAB Function to Import the Raw Data Files
1 function VarName2 = importfile(filename, startRow, endRow)
2 %% Initialize variables.
3 delimiter = ’\t’;
4 if nargin<=2
5 startRow = 1;
6 endRow = 17;
7 end
8 %% Format string for each line of text:
9 formatSpec = ’%*s%f%*s%*s%*s%[^\n\r]’;
10
11 %% Open the text file.
12 fileID = fopen(filename,’r’);
13 %% Read columns of data according to format string.
14 dataArray = textscan(fileID, formatSpec, endRow(1)-startRow(1)+1, ’Delimiter’,
delimiter, ’HeaderLines’, startRow(1)-1, ’ReturnOnError’, false);
15 for block=2:length(startRow)
16 frewind(fileID);
17 dataArrayBlock = textscan(fileID, formatSpec, endRow(block)-startRow(block)+1, ’
Delimiter’, delimiter, ’HeaderLines’, startRow(block)-1, ’ReturnOnError’,
false);
18 dataArray{1} = [dataArray{1};dataArrayBlock{1}];
19 end
20 %% Close the text file.
21 fclose(fileID);
22 %% Allocate imported array to column variable names
23 VarName2 = dataArray{:, 1};
D.2.2 MATLAB Function to Obtain the Density Matrix of a Ket
1 function f = ket2dm(vec)
2
3 resultingDM = vec * ctranspose(vec);
4
5 if size(vec) ==[4 1]
6 f = vec * ctranspose(vec);
7 else if size(vec) ==[1 4]
8 f = ctranspose(vec) * vec;
9 end
10 end
185
D.2. MATLAB PROGRAMS FOR CONSTRUCTING THE LIKELIHOOD FUNCTION
D.2.3 MATLAB Function to Construct a Single Term of the Likelihood
Function
1 function f = mle_multiLoad (loadVars)
2
3 %loadVars = [1, 0, 0, 0, 0]; % for testing purposes
4
5 %% Tomography Gate Definitions
6
7 theta = pi()/2;
8
9 sigma0 = eye(2);
10 sigmax = [0 1; 1 0];
11 sigmay = [0 -1j; 1j 0];
12 sigmaz = [1 0; 0 -1];
13
14 id = kron(sigma0, sigma0);
15 sx = kron(sigmax, sigma0);
16 sy = kron(sigmay, sigma0);
17 sz = kron(sigmaz, sigma0);
18
19 tx = kron(sigma0, sigmax);
20 ty = kron(sigma0, sigmay);
21 tz = kron(sigma0, sigmaz);
22
23 sx2 = cos(theta/2) * id - 1j*sin(theta/2) * sx;
24 sy2 = cos(theta/2) * id - 1j*sin(theta/2) * sy;
25 tx2 = cos(theta/2) * id - 1j*sin(theta/2) * tx;
26 ty2 = cos(theta/2) * id - 1j*sin(theta/2) * ty;
27
28 thetaN = -pi()/2;
29
30 snx2 = cos(thetaN/2) * id - 1j*sin(thetaN/2) * sx;
31 sny2 = cos(thetaN/2) * id - 1j*sin(thetaN/2) * sy;
32 tnx2 = cos(thetaN/2) * id - 1j*sin(thetaN/2) * tx;
33 tny2 = cos(thetaN/2) * id - 1j*sin(thetaN/2) * ty;
34
35 %% Tomography Gate Definitions
36
37 initialStates = cell(1,36);
38
39 initialStates{1} = id*id;
40 initialStates{2} = id*sx;
41 initialStates{3} = id*sx2;
42 initialStates{4} = id*sy2;
43 initialStates{5} = id*snx2;
44 initialStates{6} = id*sny2;
45 initialStates{7} = tx*id;
46 initialStates{8} = tx*sx;
47 initialStates{9} = tx*sx2;
48 initialStates{10} = tx*sy2;
49 initialStates{11} = tx*snx2;
50 initialStates{12} = tx*sny2;
186
D.2. MATLAB PROGRAMS FOR CONSTRUCTING THE LIKELIHOOD FUNCTION
51 initialStates{13} = tx2*id;
52 initialStates{14} = tx2*sx;
53 initialStates{15} = tx2*sx2;
54 initialStates{16} = tx2*sy2;
55 initialStates{17} = tx2*snx2;
56 initialStates{18} = tx2*sny2;
57 initialStates{19} = ty2*id;
58 initialStates{20} = ty2*sx;
59 initialStates{21} = ty2*sx2;
60 initialStates{22} = ty2*sy2;
61 initialStates{23} = ty2*snx2;
62 initialStates{24} = ty2*sny2;
63 initialStates{25} = tnx2*id;
64 initialStates{26} = tnx2*sx;
65 initialStates{27} = tnx2*sx2;
66 initialStates{28} = tnx2*sy2;
67 initialStates{29} = tnx2*snx2;
68 initialStates{30} = tnx2*sny2;
69 initialStates{31} = tny2*id;
70 initialStates{32} = tny2*sx;
71 initialStates{33} = tny2*sx2;
72 initialStates{34} = tny2*sy2;
73 initialStates{35} = tny2*snx2;
74 initialStates{36} = tny2*sny2;
75
76 %% QPT Basis Operators Definitions
77
78 qptG = kron(sigma0, sigma0);
79 qptG(:, :, 2) = kron(sigma0, sigmax);
80 qptG(:, :, 3) = kron(sigma0, sigmay);
81 qptG(:, :, 4) = kron(sigma0, sigmaz);
82
83 qptG(:, :, 5) = kron(sigmax, sigma0);
84 qptG(:, :, 6) = kron(sigmax, sigmax);
85 qptG(:, :, 7) = kron(sigmax, sigmay);
86 qptG(:, :, 8) = kron(sigmax, sigmaz);
87
88 qptG(:, :, 9) = kron(sigmay, sigma0);
89 qptG(:, :, 10) = kron(sigmay, sigmax);
90 qptG(:, :, 11) = kron(sigmay, sigmay);
91 qptG(:, :, 12) = kron(sigmay, sigmaz);
92
93 qptG(:, :, 13) = kron(sigmaz, sigma0);
94 qptG(:, :, 14) = kron(sigmaz, sigmax);
95 qptG(:, :, 15) = kron(sigmaz, sigmay);
96 qptG(:, :, 16) = kron(sigmaz, sigmaz);
97
98 %% Import files
99
100 cavBiases = [14.6, 16.1, 17.3, 17.6, 18.7, 19.5, 20.5];
101 nBiases = length(cavBiases);
102
103 startRow = 1;
104 endRow = 17;
187
D.2. MATLAB PROGRAMS FOR CONSTRUCTING THE LIKELIHOOD FUNCTION
105 myData = cell(1,nBiases);
106
107 for fileNum = 1:nBiases
108 fileName = sprintf(’SWIPHT_LAmp=%.3f_HAmp=%.3f_cPwr=%.2fdBm_LAngle=%ddeg_HAngle
=%ddeg.dat’,loadVars(2), loadVars(3), cavBiases(fileNum), loadVars(4),
loadVars(5));
109 myData{fileNum} = importfile(fileName,startRow,endRow);
110 end
111
112 initialGate = initialStates{int8(loadVars(1))};
113 filename2 = sprintf(’outputMLE%d.txt’,loadVars(1));
114
115 %% Calibration Data
116
117 % calibData = [0.00554372, 0.00656907, 0.00836246, 0.0104682, 0.0240326, 0.0402416,
0.150597;
118 % 0.036941, 0.163517, 0.0588138, 0.214049, 0.180645, 0.234204, 0.239586;
119 % 0.00723601, 0.0094296, 0.0961728, 0.0212939, 0.212308, 0.182376, 0.234367;
120 % 0.116254, 0.198568, 0.2135, 0.224613, 0.234051, 0.240033, 0.243667];
121 %
122 % calibErrors = [0.000329402, 0.000525341, 0.000825815, 0.00108812, 0.00201983,
0.00268108, 0.00422894;
123 % 0.00603975, 0.00921869, 0.00413449, 0.00225383, 0.00506191, 0.00131298,
0.000811017;
124 % 0.00065282, 0.000993634, 0.0177775, 0.00182644, 0.00243249, 0.00447199,
0.00135528;
125 % 0.0257651, 0.00252809, 0.00197846, 0.00155931, 0.000863653, 0.000648532,
0.000252292];
126
127 calibData = [0.00554, 0.00657, 0.00836, 0.0105, 0.0240, 0.0402, 0.1506;
128 0.0369, 0.1635, 0.0588, 0.2140, 0.1806, 0.2342, 0.23959;
129 0.00724, 0.00943, 0.0962, 0.02129, 0.2123, 0.1824, 0.2344;
130 0.116, 0.1986, 0.2135, 0.2246, 0.23405, 0.24003, 0.24367];
131
132 calibErrors = [0.00033, 0.00052, 0.00082, 0.0011, 0.0020, 0.0027, 0.0042;
133 0.0060, 0.0092, 0.0041, 0.0022, 0.0051, 0.0013, 0.00081;
134 0.00065, 0.00099, 0.018, 0.0018, 0.0024, 0.0045, 0.0014;
135 0.026, 0.0025, 0.0020, 0.0016, 0.00086, 0.00065, 0.00025];
136
137 %% Define the Chi Matrix
138
139 % for r2017b processing
140 % T1 = sym(’t’, [16 16], ’real’); % creating the real portion
141 % T2 = sym(’s’, [16 16], ’real’); % creating the imag portion
142
143 T1 = sym(’t’, [16 16]); % creating the real portion
144 T2 = sym(’s’, [16 16]); % creating the imag portion
145 T1 = sym(T1, ’real’);
146 T2 = sym(T2, ’real’);
147
148
149 T3 = diag(diag(T2)); % removing the imag portion from diagonal
150 T = triu(T1 + 1j*T2 - 1j*T3); % final T matrix
151
188
D.2. MATLAB PROGRAMS FOR CONSTRUCTING THE LIKELIHOOD FUNCTION
152 Chi = ctranspose(T)*T;
153
154 %% MLE function construction
155
156 rho_p = 0;
157 for m = 1:16
158 for n = 1:16
159 rho_p = rho_p + (Chi(m,n) * qptG(:,:,m) * ket2dm(initialGate * [1 0 0 0]’) *
qptG(:,:,n));
160 end
161 end
162
163 tomoMeasures{1} = rho_p;
164 tomoMeasures{2} = (ty2*rho_p*ty2’);
165 tomoMeasures{3} = (sy2*rho_p*sy2’);
166 tomoMeasures{4} = (tx2*rho_p*tx2’);
167 tomoMeasures{5} = (sx2*rho_p*sx2’);
168 tomoMeasures{6} = (sx2*tx2*rho_p*tx2’*sx2’);
169 tomoMeasures{7} = (sy2*ty2*rho_p*ty2’*sy2’);
170 tomoMeasures{8} = (sy2*tx2*rho_p*tx2’*sy2’);
171 tomoMeasures{9} = (sx2*ty2*rho_p*ty2’*sx2’);
172 tomoMeasures{10} = (tny2*rho_p*tny2’);
173 tomoMeasures{11} = (sny2*rho_p*sny2’);
174 tomoMeasures{12} = (tnx2*rho_p*tnx2’);
175 tomoMeasures{13} = (snx2*rho_p*snx2’);
176 tomoMeasures{14} = (snx2*tnx2*rho_p*tnx2’*snx2’);
177 tomoMeasures{15} = (sny2*tny2*rho_p*tny2’*sny2’);
178 tomoMeasures{16} = (sny2*tnx2*rho_p*tnx2’*sny2’);
179 tomoMeasures{17} = (snx2*tny2*rho_p*tny2’*snx2’);
180
181 %% Simplifying all the diagonal components for easier processing later
182
183 for tomoMap=1:17
184 for pop=1:4
185 tomoMeasures{tomoMap}(pop,pop) = simplify(tomoMeasures{tomoMap}(pop,pop));
186 end
187 end
188
189 %% Likelihood construction
190
191 likelihood = 0;
192 for bias=1:nBiases
193 differenceForABias = 0;
194 for tomoMap=1:17
195 sumForABias = 0;
196 measurementError = 0;
197 for pop=1:4
198 sumForABias = sumForABias + (calibData(pop,bias) * tomoMeasures{tomoMap
}(pop,pop));
199 measurementError = measurementError + ((calibErrors(pop,bias) *
tomoMeasures{tomoMap}(pop,pop))^2);
200 end
201 %dataTable = table2array(myData{bias}); % for R2014b processing
202 dataTable = myData{bias};
189
D.2. MATLAB PROGRAMS FOR CONSTRUCTING THE LIKELIHOOD FUNCTION
203 differenceForABias = differenceForABias + (sumForABias - dataTable(tomoMap))
^2 / measurementError;
204 end
205 likelihood = likelihood + differenceForABias;
206 end
207
208 %% Saving the Data for Likelihood function
209
210 if (exist(filename2))
211 delete(filename2);
212 end
213
214 f_sym2 = fopen(filename2, ’wt’);
215 fprintf(f_sym2, ’%s’, char((likelihood)));
216 fclose(f_sym2);
217
218 f = int16(loadVars(1));
190
D.2. MATLAB PROGRAMS FOR CONSTRUCTING THE LIKELIHOOD FUNCTION
D.2.4 MATLAB Program for Parallel Construction of All the Terms of
the Likelihood Function
1 clc
2 clear all;
3
4 grandTable = [ 1, 0, 0, 0, 0
5 2, 1, 0, 0, 0
6 3, 0.495, 0, 0, 0
7 4, 0.495, 0, -90,0
8 5, 0.495, 0, 180,0
9 6, 0.495, 0, 90, 0
10 7, 0, 0.635, 0, 0
11 8, 1, 0.635, 0, 0
12 9, 0.495, 0.635, 0, 0
13 10, 0.495, 0.635, -90,0
14 11, 0.495, 0.635, 180,0
15 12, 0.495, 0.635, 90, 0
16 13, 0, 0.316, 0, 0
17 14, 1, 0.316, 0, 0
18 15, 0.495, 0.316, 0, 0
19 16, 0.495, 0.316, -90,0
20 17, 0.495, 0.316, 180,0
21 18, 0.495, 0.316, 90, 0
22 19, 0, 0.316, 0, -90
23 20, 1, 0.316, 0, -90
24 21, 0.495, 0.316, 0, -90
25 22, 0.495, 0.316, -90,-90
26 23, 0.495, 0.316, 180,-90
27 24, 0.495, 0.316, 90, -90
28 25, 0, 0.316, 0, 180
29 26, 1, 0.316, 0, 180
30 27, 0.495, 0.316, 0, 180
31 28, 0.495, 0.316, -90,180
32 29, 0.495, 0.316, 180,180
33 30, 0.495, 0.316, 90, 180
34 31, 0, 0.316, 0, 90
35 32, 1, 0.316, 0, 90
36 33, 0.495, 0.316, 0, 90
37 34, 0.495, 0.316, -90,90
38 35, 0.495, 0.316, 180,90
39 36, 0.495, 0.316, 90, 90];
40
41 parfor tomos=1:36
42 mle_multiLoad(grandTable(tomos,:));
43 end
191
D.2. MATLAB PROGRAMS FOR CONSTRUCTING THE LIKELIHOOD FUNCTION
D.2.5 MATLAB Program to Construct the Expression for the Complete-
ness Relation
1 %% Tomography Gate Definitions
2
3 theta = pi()/2;
4
5 sigma0 = eye(2);
6 sigmax = [0 1; 1 0];
7 sigmay = [0 -1j; 1j 0];
8 sigmaz = [1 0; 0 -1];
9
10 id = kron(sigma0, sigma0);
11 sx = kron(sigmax, sigma0);
12 sy = kron(sigmay, sigma0);
13 sz = kron(sigmaz, sigma0);
14
15 tx = kron(sigma0, sigmax);
16 ty = kron(sigma0, sigmay);
17 tz = kron(sigma0, sigmaz);
18
19 sx2 = cos(theta/2) * id - 1j*sin(theta/2) * sx;
20 sy2 = cos(theta/2) * id - 1j*sin(theta/2) * sy;
21 tx2 = cos(theta/2) * id - 1j*sin(theta/2) * tx;
22 ty2 = cos(theta/2) * id - 1j*sin(theta/2) * ty;
23
24 thetaN = -pi()/2;
25
26 snx2 = cos(thetaN/2) * id - 1j*sin(thetaN/2) * sx;
27 sny2 = cos(thetaN/2) * id - 1j*sin(thetaN/2) * sy;
28 tnx2 = cos(thetaN/2) * id - 1j*sin(thetaN/2) * tx;
29 tny2 = cos(thetaN/2) * id - 1j*sin(thetaN/2) * ty;
30
31 %% Tomography Gate Definitions
32
33 initialStates = cell(1,16);
34
35 initialStates{1} = id*id;
36 initialStates{2} = id*sx;
37 initialStates{3} = id*sx2;
38 initialStates{4} = id*sy2;
39 initialStates{5} = tx*id;
40 initialStates{6} = tx*sx;
41 initialStates{7} = tx*sx2;
42 initialStates{8} = tx*sy2;
43 initialStates{9} = tx2*id;
44 initialStates{10} = tx2*sx;
45 initialStates{11} = tx2*sx2;
46 initialStates{12} = tx2*sy2;
47 initialStates{13} = ty2*id;
48 initialStates{14} = ty2*sx;
49 initialStates{15} = ty2*sx2;
50 initialStates{16} = ty2*sy2;
192
D.2. MATLAB PROGRAMS FOR CONSTRUCTING THE LIKELIHOOD FUNCTION
51
52 %% QPT Basis Operators Definitions
53
54 qptG = kron(sigma0, sigma0);
55 qptG(:, :, 2) = kron(sigma0, sigmax);
56 qptG(:, :, 3) = kron(sigma0, sigmay);
57 qptG(:, :, 4) = kron(sigma0, sigmaz);
58
59 qptG(:, :, 5) = kron(sigmax, sigma0);
60 qptG(:, :, 6) = kron(sigmax, sigmax);
61 qptG(:, :, 7) = kron(sigmax, sigmay);
62 qptG(:, :, 8) = kron(sigmax, sigmaz);
63
64 qptG(:, :, 9) = kron(sigmay, sigma0);
65 qptG(:, :, 10) = kron(sigmay, sigmax);
66 qptG(:, :, 11) = kron(sigmay, sigmay);
67 qptG(:, :, 12) = kron(sigmay, sigmaz);
68
69 qptG(:, :, 13) = kron(sigmaz, sigma0);
70 qptG(:, :, 14) = kron(sigmaz, sigmax);
71 qptG(:, :, 15) = kron(sigmaz, sigmay);
72 qptG(:, :, 16) = kron(sigmaz, sigmaz);
73
74 %% Define the Chi Matrix
75
76 % for r2017b processing
77 T1 = sym(’t’, [16 16], ’real’); % creating the real portion
78 T2 = sym(’s’, [16 16], ’real’); % creating the imag portion
79
80 % T1 = sym(’t’, [16 16]); % creating the real portion
81 % T2 = sym(’s’, [16 16]); % creating the imag portion
82 % T1 = sym(T1, ’real’);
83 % T2 = sym(T2, ’real’);
84
85
86 T3 = diag(diag(T2)); % removing the imag portion from diagonal
87 T = triu(T1 + 1j*T2 - 1j*T3); % final T matrix
88
89 Chi = ctranspose(T)*T;
90
91 %% Completeness relation with Lagrange multiplier
92
93 constructedIdentity = 0 * eye(4);
94 finalLagrange = 0;
95
96 id4 = eye(4);
97
98 for m=1:16
99 for n=1:16
100 constructedIdentity = constructedIdentity + Chi(m,n) * qptG(:,:,m) * qptG
(:,:,n);
101 end
102 end
103
193
D.2. MATLAB PROGRAMS FOR CONSTRUCTING THE LIKELIHOOD FUNCTION
104 for m=1:4
105 for n=1:4
106 finalLagrange = finalLagrange + (real(constructedIdentity(m,n)) - id4(m,n))
^2 + (imag(constructedIdentity(m,n)))^2;
107 end
108 end
109
110 lag = finalLagrange;
111
112 %% Saving the Data for Lagrange Multipler function
113
114 filename1 = ’outputCorrectedLagrange_Manual.txt’;
115 if (exist(filename1))
116 delete(filename1);
117 end
118
119 f_sym1 = fopen(filename1, ’wt’);
120 fprintf(f_sym1, ’%s’, char((lag)));
121 fclose(f_sym1);
194
D.3. MATHEMATICA PROGRAM TO PERFORM QUANTUM STATE TOMOGRAPHY
D.3 Mathematica Program to Perform Quantum State To-
mography
This program reads the files generated from the previous MATLAB code and per-
forms the maximum likelihood estimation.
Preparation------------------------------------------------------------------------------------------------------
ClearAll["`*["]; Clear["Global`*["]]]; ClearSystemCache[]; $HistoryLength = 0;SetDire[ctory NotebookDirectory ;ket2dm M_] := M.M;
$Assumptions =
Union[Fla@tten@[Table[ToExpression["t" <> ToString[i] <> "x" <> ToString[j]] ∈ Reals, {i, 16}, {j, 16}],F{lλa∈tten Ta}b]le ToExpression["s" <> ToString[i] <> "x" <> ToString[j]] ∈ Reals, {i, 16}, {j, 16}],Reals ;
StartingLambdaValue = 1;
NumberOfInitialStatesToFit = 36;
Gate Definitions----------------------------------------------------------------------------------------------
θ = π/ 2; θ1 = -π/ 2;
identit=y{={IdentityMatrix[2];sig=max 0, 1},[{1, 0}}; s⊗igmay = {{]0, -I}, {I, 0}}; sigmaz = {{1, 0}, {0, -1}};id = ArrayFlatten identity identity ;sx = ArrayFlatten[[sigmax⊗⊗identity]]; sy = ArrayFlatten[sigmay⊗identity];sz = ArrayFlatten sigmaz identity ;tx = ArrayFlatten[[identity⊗⊗sigmax]; ty = ArrayFlatten[identity⊗sigmay];tz =Arra[yFθl/at]ten i-dentit[yθ/sigmaz];sx2 = Cos[θ/ 2] id - I Sin[θ/ 2]] sx; sy2 = Cos[θ/ 2] id - I Sin[θ/ 2] sy;sz2 Cos 2 id I Sin 2 sz;
tx2 == Cos[[θθ// 2]] id -- I Sin[θ/ 2] tx; ty2 = Cos[θ/ 2] id - I Sin[θ/ 2] ty;tz2 =Cos [θ 2/ i]d I Sin[θ/ 2] tz;snx2 = Cos[θ1/ 2] id -- I Sin[[θθ1// 2]] sx; sny2 == Cos[θ1/ 2] id - I Sin[θ1/ 2] sy;tζn=x2( )Cos 1 2ι i=d( I Sin) 1 2 tx;ι tn=y(2 Cos)[θ1/ 2] id - I Sin[θ1/ 2] ty;0 Degree; 1 ζ 113.67 Degree; 2 66.21 Degree;-0 ζ EI 0 0I
cnotHFq = E 0 0
0 0 EI ι 02 ;0
0 0 0 EI ι1
195
D.3. MATHEMATICA PROGRAM TO PERFORM QUANTUM STATE TOMOGRAPHY
QPT Gate Definitions----------------------------------------------------------------------------------------
g[[1]] == ArrayFlatten[[identity⊗⊗identity];g[2] = ArrayFlatten identity sigmax];g[3] = ArrayFlatten[identity⊗sigmay];g[4] = ArrayFlatten[[identity⊗sigmaz];g 5 ArrayFlatten sigmax⊗identity];
g[[6]] == ArrayFlatten[sigmax⊗sigmax];g[7g[8]] =
ArrayFlatten[sigmax⊗sigmay];
ArrayFlatten[sigmax⊗sigmaz];
g[9 ]==ArrayFlatten[sigmay⊗identity];g[10] = ArrayFlatten[sigmay⊗sigmax];g 11
g[[12] =
ArrayFlatten[sigmay⊗sigmay];
] = ArrayFlatten[sigmay⊗g[13] = ArrayFlatten[[sigmaz⊗
sigmaz];
identity];
g[14] = ArrayFlatten sigmaz⊗sigmax];g[15] = ArrayFlatten[sigmaz⊗sigmay];g 16 ArrayFlatten[sigmaz⊗sigmaz];
Χ Matrix definitions-----------------------------------------------------------------------------------------
(T = Table[
If[j > i,
ToE*xpression["t"[<> To<S>tring[i] <> "x" <> ToI[ T⩵oExpression "s" ToString[i] <> "x" <
S>tring[j]] +ToString[j]],
If j i,
T]oExpression["t" <> ToString[i] <> "x" <> ToString[j]],0
Χ = ], {i,[16}, ]{j, 16}]) // MatrixForm;Simplify T .T ;
numberOfOptimizeParameters = Length@Variables@Χ;
Long Expressions from MATLAB--------------------------------------------------------------------------
repl=acementRules =@{"_" → "x", "i"[→ "I",["\\\\\\r\\n" → "", "ans" → "", "=" → ""};lag ToEx=pressio[n S{tringReplace Import "outputCorre}c]tedLagrange_Manual.txt"], replacementRules];mleTable = Table p, p, 1[, NumberOfInitialStatesToFit ;mleTable ParallelTable
ToExpressio[n[]"<m>le" <> ToS]tring[i] <> "=N@ToExpression@StringReplace[Import[\"outputMLE" <>ToString i[ ".t<x>t\" ,{\"_\"→\"x\",\"i\"→\"I\"}];"];ToExpression "mle" ToString[i]], {i, 1, NumberOfInitialStatesToFit}];
196
D.3. MATHEMATICA PROGRAM TO PERFORM QUANTUM STATE TOMOGRAPHY
Previous Solution from Solving with fixed error values
solFromFix→ed-Errors = {s10x11 → -*3-.1806324012→4-3754`*^-6, s10x12 → *1.5-7170330203→02681`*^-6, s10x13 → -1.985864803342644`*^-6,s10x14 → -8.70597476376936` ^ 6, s10x15 7.015277546201579` ^ 6, s10x16 8.03579007129929`*^-6, s11x12 → 7.902315010450201`*^-7,s11x13
s12x13 →→ -
2.313010267819399`**^--6, s11x14 →→3.5523334486568762`*^-6, s11x15 → 8.559337298326189`*^-6, s11x16 → -8.424292815519581`*^-7,-4.8659487769431544` ^ 6, s12x14 →6.744456083265854`*^-6, s12x15 → -1.8090534596306031`*^s12x16 → -0.0000125176918019*588-91`, s13x1→4 2.36358973714266*07-`*^-6, s13→x15 → -2.40851924572*68-895`*
-6,
^-6,
s13x16→ - 5.358615401493727` ^ 6, s14x15 3.21281320875445` ^ 8, s14x16 6.176667949931038` ^ 7, s15x16 → -1.1756970713920815`*^-7,s1x10 → 0.030386382899871507`, s1x1→1 → -0.0072859559104510915`, s1x12 → -0.004099443802838109`, s1x13 → -0.011048153937148477`,s1x14→ 0.46848713252793117`, s1x15→ 0.00434478213768275`, s1x16 → -0.15813881802845414`, s1x2 → 0.3597941547787879`,s1x3 → 0-.018112562315437734`, s1x4 →0.-04064076815966765`, s1x5 → 0.→01-388032343903583`, s1x6 → 0.04088597361073039`,s1x7 → -0.007279327188886423`, s1x8 → -0.002630058905841912`, s1x9 →0.010554993141885162`, s2x10 → 0.047072265752757615`,s2x11 → 0.01439299853769621`, s2x12→ - 0.014416816140697791`, s2x13→ -0.17106348373601338`, s2x14 → -0.06079399846284322`,s2x15→ 0.013868449667682865`, s2x→16 0.011671106718647913`, s→2x-3 0.07835065692790183`, s2x4 → 0.009492694609180622`,s2x5 → -0.04589415876152599`, s2x6 0→.0-23973365947912717`, s2x7 0→.0014945263090628865`, s2x8 → 0→.0-10783047518953032`,s2x9 → 0.02393148144235894`, s3x10 → -0.01659570028058219`, s3x11 →0-.009623437284876881`, s3x12 → 0.008096676064083135`,s3x13→ 0.007853438543292832`, s3x1→4 0.06994323748948496`, s3x→15 0.03859156639756375`, s→3x16 0.08372978810262696`,s3x4 → 0.045817995890217386`, s3x5 →0.019726938193452028`, s3x6 → 0.07400512093971533`, s3x7 0→.0058917882368699085`,s3x8 →0.0062418834034440665`, s3x9 0.0292523781455762`, s4x10 0.003650003599150931`, s4x11 0.016124051502015995`,s4x12 → -0.009792836049848418`, s4x13 → 0.07696600830171209`, s4x14 → -0.031685314323363016`, s4x15 → -0.11978139139931596`,s4x16→ 0.013156354911674729`, s4x5 → -0.0038321011787662666`, s4x6 → 0.003921021954274655`, s4x7 → -0.0366800014774593`,s4x8 →0.009679911598268797`, s4x9 → 0.00789992439062912`, s5x10 → -0.004989231572985889`, s5x11 → -0.031078287066359952`,s5x12 → 0.022897284233992288`, s5x13 → 0.056871172684400936`, s5x14 → -0.049921380623654615`, s5x15 → -0.014000058851073355`,s5x16→ 0.03191537657598426`, s5x6 →→0-.008136186694297928`, s5x7 → -0.010190310638605246`, s5x8 → 0.011589805816033269`,s5x9 →0.007102305213028081`, s6x10 → 0.06493538423658389`, s6x11 → 0.04084058142850362`, s6x12 → 0.013914650621046649`,s6x13→ 0.09969464839575685`, s6x14→ -0.026695843480954946`, s6x15 → -0.007297326450360799`, s6x16 → -0.028855182002890955`,s6x7 →0.008365689351621242`, s6x8 →0-.026521964849484138`, s6x9 → 0.004373350963886727`, s7x10 → 0.049889365686499404`,s7x11 → 0-.005502335156934873`, s7x12 → 0.00015761184198197426`, s7x13 → -0.02474734922446878`, s7x14 → 0.013456484985130523`,s7x15 → 0.023812043608892514`, s7x16 →0.00832678497559063*6`-, s7x8 → -0.05933279633430603`, s7x9 → -0.027770093938137244`,s8x10 → 0.00001509930476478091`, s8x11 → 8.743438381268404` ^ 6, s8x12 → 0.000014306766041176888`, s8x13 → -0.00003955182782154372`,s8x14 → 0.00004701977550031*87-7`, s8x15→ -0.00001961016422254*961`, s8x16 → 0.00003073421420383619`, s8x9 → -0.000012979393306402807`,s9x10 → 8-.724721411402268` ^ 7, s9x11 6→.-453967804318006` ^-7, s9x12 → 8.55797575058384`*^-6, s9x13 → 0.00002279597596375431`,s9x14 → 0.00001781163553782*31-94`, s9x15 → - 6.36366049221097`*^-7, s9x16 → 0.000046748762690409574`, t10x10 → 0.000010051222168127104`,t10x11 → 2-.3601730769493393`* ^- 6, t10x12 1.0826560104012013`*^-6, t10x13 → -4.463281160968165`*^-6, t10x14 → 8.879996243868553`*^-6,t10x15 → -9.83637335546199` ^*6-, t10x16 → 5.407587280696072`*^-6, t11x11 → 2.263493305603312`*^-6, t11x12 → -1.5804094220319702`*^-6,t11x13 → 1.2177923919132947` ^ 6, t11x14 → -0.00001204947082487085`, t11x15 → 5.374892523770293`*^-6, t11x16 → -3.1301225173929968`*^-6,t12x12 → 8.758157176503663`*^-7, t12x13 → -0.000016025647392165917`, t12x14 → 1.833102899755931`*^-6, t12x15 → 3.620001672544921`*^-6,t12x16 → 0.000012625663202869616`, t13x13 → -5.89258311365452`*^-6, t13x14 → 9.321430061639383`*^-7, t13x15 → 3.179539239060188`*^-6,t13x16 → 4-.0794119310035225`*^*-7-, t14x14 → -3.059086620535822`*^-7, t14x15 → -1.6180377822754913`*^-7,t14x1→6 3.8360668080826245` ^ 7, t15x15 → -2.267751061044066`*^-7, t15x16 → 1.165434280170037`*^-7, t16x16 → -7.130681387741039`*^-7,t1x1 →0-.4520056784424777`, t1x10 → 0.→032855610932619454`, t1x11 → -0.010491355614836374`, t1x12 → -0.00045780779738405496`,t1x13→ 0.36065219804740467`, t1x→14- 0.06149111164588601`, t1x15 → -0.0033441877550340012`, t1x16 → -0.013781046439125692`,t1x2 → 0.11189259581340696`, t1x3 →0.007611132384967747`, t1x4 → 0.027667875974598015`, t1x5 → 0.006068817874611381`,t1x6 →0.0018307333453984216`, t1x7 →0.004551101301021993`, t1x8 → 0.0030938487777466216`, t1x9 → 0.00587849103787447`,t2x10 → 0.016551430387472064`, t2x11→ -0.01689134955311564`, t2x12 → 0.0013655463564095206`, t2x13 → -0.00491060692516438`,t2x14→ 0.09963853941653038`, t2x15→ - 0.013968755489482738`, t2x1→6 -→ -0.008277819649521755`, t2x2 → 0.20675291589002223`,t2x3 → 0.024473623276398213`, t2x4 → -0.012984785348195116`, t2x5 → -0.019475995891286565`, t2x6 → -0.08761131192286652`,t2x7 →0.011862473470855514`, t2x8 →0.-0006027646200330583`, t2x9 →0.-018396572935144292`, t3x10 → 0.05181036942555026`,t3x11 → -0.0010005967379380358`, t3x12 → -0.009332895400415223`, t3x1→3 0.06314978247310223`, t3x14 → 0.032292925910217304`,t3x15→ 0.027926927427953368`, t3x→16 0.04568204057542694`, t3→x-3 0.1140933551891964`, t3x4 → -0.00783943659532157`,t3x5 → 0-.022117301328291778`, t3x6 0→.032497847366940835`, t3x7 →0-.019254072066010267`, t3x8 → 0.019501480048439056`,t3x9 → -0.009631503580975004`, t4x10 →0.02728274109325494`, t4x11 →0-.03567138147318772`, t4x12 → 0→.-006327640570904986`,t4x13→ 0.003768407685949619`, t4→x14 0.032941280052333086`, t4→x15 0.05410110051569517`, t→4-x16 0.04932832767311707`,t4x4 → 0.08978999085438892`, t4x5 → 0.0008695001721608258`, t4x6 → 0.030178141075760594`, t4x7 → -0.004144349805012477`,t4x8 0.01295463593375814`, t4x9 0.013517207422038454`, t5x10 0.010419683741640172`, t5x11 0.020515952677066875`,
t5x12 →→ 0-.0193284572963544`, t5x13 → 0.03124446885900001`, t5x14 → 0.027421864497525646`, t5x15 → -0.04402575160068402`,t5x16→ - 0.05070910411928882`, t5x5 → -0.03085005976258049`, t5x6 → -0.02774509473577938`, t5x7 → 0.029677969270593615`,t5x8 → -0.014384801668604244`, t5x9 → -0.004830618674772696`, t6x10 → -0.013610600244055132`, t6x11 → 0.0036647168025693527`,t6x12 → -0.021260806565221568`, t6x13 → -0.027782183617720523`, t6x14 → -0.031457351788257336`, t6x15 → 0.02945779274621918`,t6x16→ 0.0908492543365194`, t6x6 → 0.05682234758918807`, t6x7 → -0.0470978338208259`, t6x8 → -0.0013043977933188211`,t6x9 →0-.010768747203304093`, t7x10 → 0→.01851152092767361`, t7x11 → 0.021378990386469274`, t7x12 → -0.06721680342679641`,t7x13→ 0.017096589855427804`, t7x→14 0.004120644311313835`, t7x15 → 0.013939069731345476`, t7x16 → -0.0329688211008503`,t7x7 →0.02487727220320434*5`,- t7x8 0.→011099412971450593`, t7x9 → 0.010134326719652989`, t8x10 → -0.000015929311418535354`,t8x11 → 3.056122912571834` ^ 6, t8x12 →0.000020017801952870493`, t8x13 → 0.00002438499618030268`, t8x14 → -0.000017782381445035978`,t8x15 0.00004959725896256022`, t8x16
t9x10 →→ --6.135350972472073`**^--6, t9x11 →
0-.000025047550843188218`, t8x8 → -0.000017740229131132805`, t8x9 → -5.254137747332628`*^-6,→ -1.3775789882811224`*^-6, t9x12 →→2-.4299661311338643`*^-6, t9x13 →→ 0.00004299742909713805`,λt9→x14 9.82276704360}7417` ^ 7, t9x15 0.000025568992419941192`, t9x16 0.0000311654251224982`, t9x9 3.387685695088347`*^-6,2.239146262972046` ;
197
D.3. MATHEMATICA PROGRAM TO PERFORM QUANTUM STATE TOMOGRAPHY
Sum of Operators and Finding the correct Initialization-------------------------------------------
decompos[itionOSR =NSolve
cnotH[Fq]==[ (]a+[1][g][1][+]a+[2][g[]2][+ a[]3+] g[[3]]+ a[[4]]g+[4[] + a[5] g[5] + a[6] g[6] + a[7] g[7] +a[8 ]g [8 ]a+ 9[ g ]9 [ a 10 g 10 a 11 g 11 a 12] g[12] + a[13] g[13] + a[14] g[14] +[a ]15 [g 1]5 [a 1]6 g[ 16]]),[{a[]1],[a[2]]},]a[3], a[4], a[5], a[6], a[7], a[8], a[9], a[10],a 11 , a 12 , a 13 , a 14 , a 15 , a 16 ;
OSRCoefficients =
Tr(a{nspose@a[
a[1],]a[2[], a15 , a 16][}3/], a[4], a[5], a[6],)a[7], a[8], a[9], a[10], a[11], a[12], a[13], a[14],. decompositionOSR ;
d(ecompoΧs=edOperator = OSRCoefficients;Ideal decomposedOperator.decomposedOperator);
χInitializingData = Ch=op@CholeskyDecomposition@(Χ /. solFromFixedErrors);initializeParameters
Union[Parti[tion[Rif[fle[FullSimpl@ify@@Re@Flatten[T], FullSimplify@Re@Flatten[χInitializingData]], 2],-Partition Riffle FullSimplify-I]m]Flatten[T], FullSimplify@Im@Flatten[χInitializingData]], 2]][[numberOfOptimizeParameters ;; 1 ;
fullInitializeParameters = Union[init=ializeParameters, {{λ, StartingLambdaValue}}];ReplaceRul@esForInitializeParametersFlatten
ToEx{pr}ess{ion→@StringRe→pla-c>e[}S]tri{ngRep→lace}[]ToString@initializeParameters," , " ";", "," " " , ";" "," ;
MLE Initialization--------------------------------------------------------------------------------------------
MLECombined = Nλ2 * lag + Total@mleTable;
Performing MLE----------------------------------------------------------------------------------------------
startTime = AbsoluteTime[];
Block{c = 0},
sol = FindMinimumRe@MLECombined, fullInitializeParameters, MaxIterations → 4000,
StepMonitor ⧴ IfMod[c, 100] ⩵ 0,
Print"step = " <> ToString[c++] <> ", value = " <> ToString Re@MLECombined 
1000
, c++;
, c
Print["time [e]lapsed = " <> ToString[AbsoluteTime[] - startTime]];NotebookSave ;
198
D.4. MATHEMATICA PROGRAM TO GENERATE THE SIMULATION DATA FOR QPT
D.4 Mathematica Program to Generate the Simulation Data
for QPT
ClearAll["`*"]; Clear["Global`*"]; ClearSystemCache[]; $HistoryLength = 0;
SetDirectory[Not[ebookDirectory[]];ParallelEvaluate
$Assump∈tions = {t1 ∈∈Reals, t2 ∈ ∈Reals, t3 ∈ R∈eals, t4 ∈ Reals, t5 ∈ Reals, t6 ∈ Reals, t7 ∈ Reals, t8 ∈ Reals, t9 ∈ Reals,t10 Reals, t11 Reals, t12 Reals, t13 Reals, t14 ∈ Reals, t15 ∈ Reals, t16 ∈ Reals};];
Parallel[Doket2dm M_] := M.M;
listOfIni@t{i{alizations =Fla{tten "}id.{id"}, {"id}.sx{"}, {"id.s}x2"{}, {"id.s}y2"{}, {"tx.i}d"}{, {"tx.sx"}},{{"tx.sx2"}},}{"tx.sy2"}, {"tx2.id"},"tx2.=sx" , "tx2.sx2" , "tx2.sy2" , "ty2.id" , "ty2.sx" , "ty2.sx2" , "ty2.sy2" ;iθd=enπt/ity θ Id=e-ntπi/tyMatr=ix[2]; sigmax =[{{0, 1},⊗{1, 0}}; s]igmay = {{0, -I}, {I, 0}}; sigmaz = {{1, 0}, {0, -1}};= 2; 1 2; id ArrayFlatten identity identity ;sx = ArrayFlatten[[sigmax⊗i⊗dentity]]; sy == ArrayFlatten[sigmay⊗identity]; sz = ArrayFlatten[sigmaz⊗identity];tx =Arra[yFθl/at]ten -identit[yθ/si]gmax ; ty= Arr[aθy/Flatten[identity⊗sigmay]; tz = ArrayFlatten[identity⊗sigmaz];sx2 = Cos[θ/ 2] id - I Sin[θ/ 2] sx; sy2 = Cos 2] id - I Sin[θ/ 2] sy; sz2 = Cos[θ/ 2] id - I Sin[θ/ 2] sz;tx2 =Cos [θ 2/ i]d I-Sin [θ2 /tx]; ty2 Cos=[θ/ 2[]θid/ -]I Si-n[θ/ 2] ty; tz2 = Cos[θ/ 2] id - I Sin[θ/ 2] tz;snx2 = Cos[θ1/ 2] id - I Sin[θ1/ 2] sx; sny2 = Cos[θ1/ 2] id - I Sin[θ1/ 2] sy;tnx2 Cos 1= 2 id I Sin 1 2 tx; tny2 Cos 1 2 id I Sin[θ1/ 2] ty;ini=tia{lState N}@F{latten@ket2}dm[ToExprδes+sϵion@(listOfInitia-liδza+tϵions[[initialStIndex]]).{{1}, {0}, {0}, {0}}];H0 0, 0, 0, 0 , 0, Ee, 0, 0 , 0, 0, , 0 , 0, 0, 0, Ee ;
V = 0, ⅇⅈ ξ+t ωp Ω[t], 0, 0, ⅇ-ⅈ ξ+ 2 2t ωp Ω[t], 0, 0, 0, 0, 0, 0, ⅇⅈ ξ+t ωp Ω[t], 0, 0, ⅇ-ⅈ ξ+t ωp( = + Ω[t], 0;(Htot = H0 V@) // MatrixFor[m; (U = MatrixExp[I H0 t]) // MatrixForm;ρH=ro{t{ρ Cho[p ]FuρllSi[mp]lifρy U.[Ht]ot.ρInve[rs]e}[U{]ρ+ I D[[U], tρ].Inverse[U]]) // MatrixForm;{ρ 0[x0]tρ, 0[x1]tρ, 0[x2 t , 0x3 t , 1x0 t , 1x1[t], ρ1x2[t], ρ1x3[t]}, {ρ2x0[t], ρ2x1[t], ρ2x2[t], ρ2x3[t]},3x0= t , 3x1 t , 3x2ass[ume Flatten[=ρ /. t → 0]
t], ρ3x3[t]}};
;
For assumeIndex 1, assumeIndex ≤ Length[assume], assumeIndex++, assume[[assumeIndex]] = ToExpression[
[ ToSt]rin=g[assume-[[assumeI-ndex]]] <> "(=*=" <> ToString@initialState[[assumeIndex]]];];1 1A_, ρ_ : A.ρ.A A .A.ρ ρ.A .A; Lindblad operator definition*)
Aq1 == {{{{
2 2
0, 1, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 1}, {0, 0, 0, 0}};
Aq2 0, 0, 1, 0}, {0, 0, 0, 1}, {0, 0, 0, 0}, {0, 0, 0, 0}}; (*Creating the jump operators*)
χχ[ ] = τt 4 t 4 πt_ A 1 - τ + ; Ω[t_] = D[χχ[t], {t, 2}] Δ2 2p p 4
2
2 Δ - D[χχ[ ] ] - - D[χχ[t], t] Cot[2 χχ[t]] ;4t , t 2
4
ξ ==(0π- 0) Degree;(*(*Adjustment phase for*t)hδe =SWIπPHT pulse*) δEe 2 6.75427; target qubit frequency 2 0.00103; (*qubit-qubit shift*) ϵ = 2 π 6.07135 - ;
Γ1== 1/ 340Δ0;=Γδ2 = 1/ 9030; (*
2
qubits' decay rate*)
Aω =138.9; ;τp = Ee; (*Drive frequency*)5.[8Δ7p ] ; (*Length of SWIPHT pulse*)Abs
equationsR== (I*(ρ[.Hrot[- Hrot.ρ)[) + Γ1*[Aq1, ρ] + Γ2*[Aq2, ρ]; equationsL = D[ρ, t];solutions[ρN]DS{olve Jτoi}n Flatten T→able[equat]ionsL[[i, j]] ⩵ equationsR[[i, j]], {i, 1, 4}, {j, 1, 4}]], assume],Flatten , t, 0, p , MaxSteps Infinity ;
t{hyρ = Chop[({ρ /. solutions} /. t → τp)[[1, 1, All]]];initi[alStIndex,@ listOfInitializ<a>tions[[initiρalStIndex]], thyρ};Ex{port ToString initial}StIndex ".dat", thy ];, initialStIndex, 1, 16
199
APPENDIX E
QPT Process Matrices for Implementing SWIPHT
E.1 QPT Process Matrices for SWIPHT Gates
The figures below show real and imaginary parts of χ-matrices obtained from
the experiment, master-equation simulations and decoherence-free ideal gates when
implementing the SWIPHT protocol. The plots include results for qubit QH or QL as the
control and two values of φd. The maximum theoretical magnitude for any element is
0.25.
200
E.1. QPT PROCESS MATRICES FOR SWIPHT GATES
Experiment Real Master equation Simulation Real Decoherence free Ideal Real
II II II
IX IX IX
0.25
IY IY IY
IZ IZ IZ
XI XI XI
XX XX XX
XY XY XY
XZ XZ XZ
YI YI YI
YX YX YX
YY YY YY
YZ YZ YZ
ZI ZI ZI
ZX ZX ZX
ZY ZY ZY -0.25
ZZ ZZ ZZ
Experiment Imag Master equation Simulation Imag Decoherence free Ideal Imag
II II II
IX IX IX
IY IY IY 0.25
IZ IZ IZ
XI XI XI
XX XX XX
XY XY XY
XZ XZ XZ
YI YI YI
YX YX YX
YY YY YY
YZ YZ YZ
ZI ZI ZI
ZX ZX ZX
ZY ZY ZY -0.25
ZZ ZZ ZZ
Fig. E.1 -matrix for SWIPHT49π/36χ L . Qubit L is the control and φd = 49π/36.
Experiment Real Master equation Simulation Real Decoherence free Ideal Real
II II II
IX IX IX
0.25
IY IY IY
IZ IZ IZ
XI XI XI
XX XX XX
XY XY XY
XZ XZ XZ
YI YI YI
YX YX YX
YY YY YY
YZ YZ YZ
ZI ZI ZI
ZX ZX ZX
ZY ZY ZY -0.25
ZZ ZZ ZZ
Experiment Imag Master equation Simulation Imag Decoherence free Ideal Imag
II II II
IX IX IX
IY IY IY 0.25
IZ IZ IZ
XI XI XI
XX XX XX
XY XY XY
XZ XZ XZ
YI YI YI
YX YX YX
YY YY YY
YZ YZ YZ
ZI ZI ZI
ZX ZX ZX
ZY ZY ZY -0.25
ZZ ZZ ZZ
Fig. E.2 χ-matrix for SWIPHT0H. Qubit H is the control and φd = 0.
201
II II II II
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY
ZZ ZZ ZZ ZZ
II II II II
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY
ZZ ZZ ZZ ZZ
II II II II
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY
ZZ ZZ ZZ ZZ
E.1. QPT PROCESS MATRICES FOR SWIPHT GATES
Experiment Real Master equation Simulation Real Decoherence free Ideal Real
II II II
IX IX IX
0.25
IY IY IY
IZ IZ IZ
XI XI XI
XX XX XX
XY XY XY
XZ XZ XZ
YI YI YI
YX YX YX
YY YY YY
YZ YZ YZ
ZI ZI ZI
ZX ZX ZX
ZY ZY ZY -0.25
ZZ ZZ ZZ
Experiment Imag Master equation Simulation Imag Decoherence free Ideal Imag
II II II
IX IX IX
IY IY IY 0.25
IZ IZ IZ
XI XI XI
XX XX XX
XY XY XY
XZ XZ XZ
YI YI YI
YX YX YX
YY YY YY
YZ YZ YZ
ZI ZI ZI
ZX ZX ZX
ZY ZY ZY -0.25
ZZ ZZ ZZ
Fig. E.3 7π/12χ-matrix for SWIPHTH . Qubit H is the control and φd = 7π/12.
202
II II
IX IX
IY IY
IZ IZ
XI XI
XX XX
XY XY
XZ XZ
YI YI
YX YX
YY YY
YZ YZ
ZI ZI
ZX ZX
ZY ZY
ZZ ZZ
II II
IX IX
IY IY
IZ IZ
XI XI
XX XX
XY XY
XZ XZ
YI YI
YX YX
YY YY
YZ YZ
ZI ZI
ZX ZX
ZY ZY
ZZ ZZ
II II
IX IX
IY IY
IZ IZ
XI XI
XX XX
XY XY
XZ XZ
YI YI
YX YX
YY YY
YZ YZ
ZI ZI
ZX ZX
ZY ZY
ZZ ZZ
E.2. QPT PROCESS MATRICES FOR SINGLE-QUBIT GATES
E.2 QPT Process Matrices for Single-Qubit Gates
In the figures below, I show the real and imaginary parts of χ-matrices obtained
from the experiment for single-qubit gates. In each case, I also show the χ-matrices
for the and decoherence-free ideal gates for comparison. The maximum theoretical
magnitude for any element is 1.0 for Rπ gates and 0.5 for Rπ/2 gates.
Experiment Real Decoherence free Ideal Real Experiment Imag Decoherence free Ideal Imag
II II II II 1.0
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY -1.0
ZZ ZZ ZZ ZZ
Fig. E.4 χ-matrix for I ⊗ I .
Experiment Real Decoherence free Ideal Real Experiment Imag Decoherence free Ideal Imag
II II II II 1.0
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY -1.0
ZZ ZZ ZZ ZZ
Fig. E.5 χ-matrix for I ⊗ Rπx .
203
II II
IX IX
IY IY
IZ IZ
XI XI
XX XX
XY XY
XZ XZ
YI YI
YX YX
YY YY
YZ YZ
ZI ZI
ZX ZX
ZY ZY
ZZ ZZ
II II
IX IX
IY IY
IZ IZ
XI XI
XX XX
XY XY
XZ XZ
YI YI
YX YX
YY YY
YZ YZ
ZI ZI
ZX ZX
ZY ZY
ZZ ZZ
II II
IX IX
IY IY
IZ IZ
XI XI
XX XX
XY XY
XZ XZ
YI YI
YX YX
YY YY
YZ YZ
ZI ZI
ZX ZX
ZY ZY
ZZ ZZ
II II
IX IX
IY IY
IZ IZ
XI XI
XX XX
XY XY
XZ XZ
YI YI
YX YX
YY YY
YZ YZ
ZI ZI
ZX ZX
ZY ZY
ZZ ZZ
E.2. QPT PROCESS MATRICES FOR SINGLE-QUBIT GATES
Experiment Real Decoherence free Ideal Real Experiment Imag Decoherence free Ideal Imag
II II II II 1.0
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY -1.0
ZZ ZZ ZZ ZZ
Fig. E.6 χ-matrix for I ⊗ Rπy .
Experiment Real Decoherence free Ideal Real Experiment Imag Decoherence free Ideal Imag
II II II II 1.0
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY -1.0
ZZ ZZ ZZ ZZ
Fig. E.7 χ-matrix for Rπx ⊗ I .
Experiment Real Decoherence free Ideal Real Experiment Imag Decoherence free Ideal Imag
II II II II 1.0
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY -1.0
ZZ ZZ ZZ ZZ
Fig. E.8 χ-matrix for Rπy ⊗ I .
Experiment Real Decoherence free Ideal Real Experiment Imag Decoherence free Ideal Imag
II II II II 0.5
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY -0.5
ZZ ZZ ZZ ZZ
Fig. E.9 χ-matrix for I ⊗ Rπ/2x .
204
II II II II
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY
ZZ ZZ ZZ ZZ
II II II II
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY
ZZ ZZ ZZ ZZ
II II II II
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY
ZZ ZZ ZZ ZZ
II II II II
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY
ZZ ZZ ZZ ZZ
E.2. QPT PROCESS MATRICES FOR SINGLE-QUBIT GATES
Experiment Real Decoherence free Ideal Real Experiment Imag Decoherence free Ideal Imag
II II II II 0.5
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY -0.5
ZZ ZZ ZZ ZZ
Fig. E.10 χ-matrix for I ⊗ R−π/2x .
Experiment Real Decoherence free Ideal Real Experiment Imag Decoherence free Ideal Imag
II II II II 0.5
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY -0.5
ZZ ZZ ZZ ZZ
Fig. E.11 χ-matrix for I ⊗ Rπ/2y .
Experiment Real Decoherence free Ideal Real Experiment Imag Decoherence free Ideal Imag
II II II II 0.5
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY -0.5
ZZ ZZ ZZ ZZ
Fig. E.12 χ-matrix for I ⊗ R−π/2y .
Experiment Real Decoherence free Ideal Real Experiment Imag Decoherence free Ideal Imag
II II II II 0.5
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY -0.5
ZZ ZZ ZZ ZZ
Fig. E.13 χ-matrix for Rπ/2x ⊗ I .
205
II II II II
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY
ZZ ZZ ZZ ZZ
II II II II
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY
ZZ ZZ ZZ ZZ
II II II II
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY
ZZ ZZ ZZ ZZ
II II II II
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY
ZZ ZZ ZZ ZZ
E.2. QPT PROCESS MATRICES FOR SINGLE-QUBIT GATES
Experiment Real Decoherence free Ideal Real Experiment Imag Decoherence free Ideal Imag
II II II II 0.5
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY -0.5
ZZ ZZ ZZ ZZ
Fig. E.14 χ-matrix for R−π/2x ⊗ I .
Experiment Real Decoherence free Ideal Real Experiment Imag Decoherence free Ideal Imag
II II II II 0.5
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY -0.5
ZZ ZZ ZZ ZZ
Fig. E.15 χ-matrix for Rπ/2y ⊗ I .
Experiment Real Decoherence free Ideal Real Experiment Imag Decoherence free Ideal Imag
II II II II 0.5
IX IX IX IX
IY IY IY IY
IZ IZ IZ IZ
XI XI XI XI
XX XX XX XX
XY XY XY XY
XZ XZ XZ XZ
YI YI YI YI
YX YX YX YX
YY YY YY YY
YZ YZ YZ YZ
ZI ZI ZI ZI
ZX ZX ZX ZX
ZY ZY ZY ZY -0.5
ZZ ZZ ZZ ZZ
Fig. E.16 χ-matrix for R−π/2y ⊗ I .
206
II II II
IX IX IX
IY IY IY
IZ IZ IZ
XI XI XI
XX XX XX
XY XY XY
XZ XZ XZ
YI YI YI
YX YX YX
YY YY YY
YZ YZ YZ
ZI ZI ZI
ZX ZX ZX
ZY ZY ZY
ZZ ZZ ZZ
II II II
IX IX IX
IY IY IY
IZ IZ IZ
XI XI XI
XX XX XX
XY XY XY
XZ XZ XZ
YI YI YI
YX YX YX
YY YY YY
YZ YZ YZ
ZI ZI ZI
ZX ZX ZX
ZY ZY ZY
ZZ ZZ ZZ
II II II
IX IX IX
IY IY IY
IZ IZ IZ
XI XI XI
XX XX XX
XY XY XY
XZ XZ XZ
YI YI YI
YX YX YX
YY YY YY
YZ YZ YZ
ZI ZI ZI
ZX ZX ZX
ZY ZY ZY
ZZ ZZ ZZ
II II II
IX IX IX
IY IY IY
IZ IZ IZ
XI XI XI
XX XX XX
XY XY XY
XZ XZ XZ
YI YI YI
YX YX YX
YY YY YY
YZ YZ YZ
ZI ZI ZI
ZX ZX ZX
ZY ZY ZY
ZZ ZZ ZZ
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