ABSTRACT
Title of Dissertation: QUANTUM COMPUTING WITH FLUXONIUM:
DIGITAL AND ANALOG DIRECTIONS
Aaron Joseph Somoroff, Doctor of Philosophy, 2022
Dissertation Directed by: Professor Vladimir E. Manucharyan
Department of Physics
This dissertation explores quantum computing applications of fluxonium super-
conducting circuits. Fluxonium?s high coherence time T2 and anharmonicity make it
an excellent platform for both digital quantum processors and analog quantum simula-
tors. Focusing on the digital quantum computing applications, we report recent work on
improving the T2 and gate error rates of fluxonium qubits. Through enhancements in fab-
rication methods and engineering of fluxonium?s spectrum, a coherence time in excess
of 1 millisecond is achieved, setting a new standard for the most coherent superconduct-
ing qubit. This highly coherent device is used to demonstrate a single-qubit gate fidelity
greater than 99.99%, a level of control that had not been observed until now in a solid-
state quantum system. Utilizing the high energy relaxation time T1 of the qubit transition,
a novel measurement of the circuit?s parity-protected |0? ? |2? transition relaxation time
is performed to extract additional sources of energy loss.
To demonstrate fluxonium?s utility as a building block for analog quantum simu-
lators, we investigate how to simulate quantum dynamics in the Transverse-Field Ising
Model (TFIM) by inductively coupling 10 fluxonium circuits together. When the fluxo-
nium loops are biased at half integer values of the magnetic flux quantum, the spectrum is
highly anharmonic, and the qubit transition is well-approximated by a spin-1/2. This re-
sults in an effective Hamiltonian that is equivalent to the TFIM. By tuning the inter-qubit
coupling across multiple devices, we can explore different regimes of the TFIM, estab-
lishing fluxonium as a prominent candidate for use in near-term quantum many-body
simulations.
QUANTUM COMPUTING WITH FLUXONIUM:
DIGITAL AND ANALOG DIRECTIONS
by
Aaron Joseph Somoroff
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2022
Dissertation Committee:
Professor Vladimir E. Manucharyan, Chair
Professor Steven M. Anlage
Professor Victor M. Galitski
Professor Alicia J. Kolla?r
Professor Ichiro Takeuchi, Dean?s Representative
?Copyright by
Aaron Joseph Somoroff
2022
ACKNOWLEDGMENTS
To give anything less than your best is to sacrifice the gift. -Steve Prefontaine
Over the past eight years, my pursuit of a PhD in physics was made possible by many great
people. Some of them I met along the way, others were there from the beginning. I begin my
dissertation by thanking my parents, Michael and Samantha, for their endless love and support.
From my early passion for track and field, to aspirations of becoming a titan of Wall Street, finally
to land on physics, they always encouraged me to do whatever made me happy, as long as I did my
best. Mom and Dad, this dissertation is dedicated to you.
This work would not be possible without my advisor, Vladimir Manucharyan. It has been a
privilege to learn under the father of fluxonium. Vlad made sure that every student and postdoc
in the lab had everything they needed for experimental success. He also assembled an amazing
group of people making up the Superconducting Circuits group at UMD, which optimized every-
one?s work output. Thank you, Vlad, for giving me the opportunity to work in such a wonderful
environment, and for your guidance through my PhD years. I also extend a warm thank you to the
other members of my dissertation committee: Professors Anlage, Galitski, Kolla?r, and Takeuchi.
Among these great people that I had the pleasure of working with in my early years of graduate
study were postdoc Yen-Hsiang Lin and graduate student Long Nguyen. Both joined the lab before
even the first dilution refrigerator was set up, and were key to cementing the infrastructure that
allowed me to hit the ground running in research when I started my time at UMD in 2017. For my
first two years of graduate research I worked with Long, who taught me everything he knew about
superconducting quantum computing. I have fond memories of our late night dinners together
after long days in the lab, and intense conversations ranging in topics from experimental setups to
international soccer.
It was a pleasure to work with postdoc Quentin Ficheux, who joined the lab in the spring
of 2019. Quentin taught me how to be a professional scientist, always attacking problems by
deconstructing them to the most fundamental level. His vast knowledge of quantum measurements
was an invaluable resource for my experimental progress. Quentin?s arrival in the lab coincided
with many significant experimental achievements for our group, and I hope that we can one day
work together again. I also thank him for the valuable input he provided for this manuscript.
ii
I thank my colleagues: Nitish Mehta, Ray Mencia, and Haonan Xiong, who joined the lab as
graduate students at around the same time as I did. I worked with all three on various projects,
each one providing their own area of expertise. Nitish?s knowledge of quantum circuit theory was
of great value in the fluxonium-based spin chain project. Ray?s skill in coding and RF engineering
was instrumental in conducting millisecond coherence experiments. Haonan?s expertise in low
temperature filtering and quantum measurements was also a boon to the millisecond coherence
project. Other lab members who I did not work with directly on any projects, but contributed to
the overall positive culture of our group were postdocs Roman Kuzmin and Ivan Pechenezhskiy,
as well as fellow graduate students Natalia Pankratova, Nick Grabon, and Hanho Lee. I thank you
all for your friendship and support during our time together in the lab.
I acknowledge the many influential people who contributed to my PhD journey from The City
College of New York. There are too many to name here, but key figures are Professors Javad Sha-
bani (now at NYU), V. Parameswaran Nair, and Timothy Boyer. I took a somewhat nontraditional
route into science, deciding late as an undergraduate at Boston University that I wanted to study
physics. Upon graduating from BU, I spent two years doing undergraduate physics coursework at
City College so that I could apply to graduate programs. I would not be where I am today without
my first research advisor, Javad. In addition to the time Javad invested in showing me how to be
an experimental physicist, he also introduced me to Vlad and UMD, encouraging me to do my
graduate work there. Looking back, this was no doubt the right decision. Professors Nair and
Boyer taught a number of my courses at City College, playing significant roles in developing my
burgeoning love of physics at the start of this journey. They also gave valuable advice on how to
navigate my path to a PhD in uncertain times. In addition to the faculty members at City College,
my peers made the process of admission into graduate school all the more possible. My entrance
at City College coincided with a number of other motivated second bachelor degree-seekers who,
like me, discovered their passion for physics and wanted to pursue PhD?s in the field. Without
being pushed and inspired by them, my time at City College would not have been as successful.
I thank the rest of my immediate family members: my sisters Sofie, Illiana, and Lara, and
my stepmother Irina. I don?t know where I would be without such a supportive group of people.
Though at this juncture in our lives we are scattered across the globe, I think about you often. It
would be quite remiss of me if I didn?t also thank my more extended family members: my best
friends. I could not have made it this far without you.
Finally, I thank the most important person in my life, my partner Jenny. Jenny?s intelligence,
calmness, and love made the challenging times during my PhD studies tolerable. Jenny, I love
you, and look forward to this next chapter together.
Washington, DC, April 2022
iii
TABLE OF CONTENTS
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
LIST OF APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
CHAPTER
1 Introduction to Quantum Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Bloch Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Quantum Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Density Matrix Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6.2 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6.3 Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6.4 Qubit Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6.5 Qubit Dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Fluxonium Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Superconducting Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1 Circuit Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.2 The Josephson Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Fluxonium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.2 Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.3 Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.4 Decoherence Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Coupled Fluxonium Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3.1 Capacitive Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3.2 Inductive Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3.3 Fluxonium-Based Spin Chain . . . . . . . . . . . . . . . . . . . . . . . 48
iv
3 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1 Cryogenic Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1.1 Single-Qubit Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1.2 Spin Chain Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 Room Temperature Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.1 RF Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.2 Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.3 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3 Device Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3.1 Silicon Fabrication Recipe . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3.2 Sapphire Fabrication Recipe . . . . . . . . . . . . . . . . . . . . . . . . 63
4 Fluxonium Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1 One-Tone Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Two-Tone Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3 Single Shot Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3.1 Readout Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.4 Time-Domain Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4.1 Rabi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4.2 Relaxation Time T1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4.3 Coherence Time T2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.5 Decoherence Mechanism Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.5.1 Dielectric Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.5.2 Quasiparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.5.3 Measurement of T 021 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5.4 1/f Flux Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.5.5 Thermal Cavity Photons . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.6 Gate Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.6.1 Gate Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.6.2 Single-Qubit Randomized Benchmarking . . . . . . . . . . . . . . . . . 92
4.6.3 Purity Benchmarking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.7 Fluxonium-Based Spin Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.7.1 Device Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.7.2 One-Tone Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.7.3 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.7.4 Circuit Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.1 Millisecond Coherence in a Superconducting Qubit . . . . . . . . . . . . . . . . 103
5.2 Fluxonium-Based Spin Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
v
LIST OF FIGURES
FIGURE
1.1 Qubit states as vectors on the Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Example of Bloch vector undergoing rotations . . . . . . . . . . . . . . . . . . . . . 5
1.3 Bloch vector under free evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Bloch vectors and their corresponding density matrices . . . . . . . . . . . . . . . . . 10
1.5 Bloch vector with corresponding density matrix undergoing a rotation . . . . . . . . . 13
2.1 LC oscillator circuit diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Josephson junction diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Fluxonium circuit diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Fluxonium eigenstates at integer and half-integer flux quanta . . . . . . . . . . . . . . 23
2.5 Fluxonium spectrum sweeping Josephson energy EJ . . . . . . . . . . . . . . . . . . 24
2.6 Fluxonium spectrum sweeping inductive energy EL . . . . . . . . . . . . . . . . . . . 25
2.7 Fluxonium spectrum sweeping charging energy EC . . . . . . . . . . . . . . . . . . . 26
2.8 Charge and phase matrix elements versus external flux . . . . . . . . . . . . . . . . . 28
2.9 Dispersive readout diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.10 Thermal factors as a function of qubit frequency . . . . . . . . . . . . . . . . . . . . 32
2.11 Simulated T1 limit due to dielectric loss versus external flux . . . . . . . . . . . . . . 34
2.12 Simulated T1 limit due to dielectric loss as a function of EL, EJ . . . . . . . . . . . . 34
2.13 Simulated T1 limit due to quasiparticle tunneling versus external flux . . . . . . . . . 36
2.14 Simulated T1 limit due to 1/f flux noise versus external flux . . . . . . . . . . . . . . . 37
2.15 Theory curves for 1/f noise induced dephasing time . . . . . . . . . . . . . . . . . . . 38
2.16 Bose-Einstein distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.17 Dephasing time due to thermal photons T th? . . . . . . . . . . . . . . . . . . . . . . . 40
2.18 Circuit diagram of two flunoniums coupled via mutual capacitance CM . . . . . . . . 41
2.19 Spectrum of capacitively coupled fluxoniums with varying coupling strength JC . . . 44
2.20 Circuit diagram of two fluxoniums coupled via mutual inductance LM . . . . . . . . . 45
2.21 Spectrum for inductively coupled fluxoniums with varying coupling strength JL . . . . 47
2.22 Truncated fluxonium qubit transition . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.23 Circuit diagram for a section of the fluxonium-based spin chain . . . . . . . . . . . . 50
2.24 Ground to first excited state transition frequency versus spin-spin coupling strength
and number of spins N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.25 Spin chain spectrum for N = 10 spins, and 0 spin-spin interaction . . . . . . . . . . . 51
2.26 Spin chain spectrum for N = 10 spins, with weak and intermediate spin-spin interaction 52
2.27 Spin chain spectrum for N = 10 spins, for strong spin-spin interaction . . . . . . . . . 53
3.1 Cryogenic setup diagram for single qubit experiments . . . . . . . . . . . . . . . . . 56
vi
3.2 Cryogenic setup at the base plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3 Cryogenic setup for the fluxonium-based spin chain experiments . . . . . . . . . . . . 58
3.4 Cu waveguide with 3 dB cutoff frequency of 6 GHz . . . . . . . . . . . . . . . . . . . 58
3.5 Room temperature electronics diagram . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.6 Fluxonium fabrication recipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1 One-tone spectroscopy of the cavity resonance . . . . . . . . . . . . . . . . . . . . . 66
4.2 One-tone spectroscopy of the cavity resonance versus flux . . . . . . . . . . . . . . . 66
4.3 Two-tone spectroscopy measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4 Two-tone spectroscopy over the full spectrum for Qubit J . . . . . . . . . . . . . . . . 68
4.5 Two-tone spectroscopy for Qubit J near HFQ . . . . . . . . . . . . . . . . . . . . . . 69
4.6 Single shot histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.7 |0? state population versus readout pulse duration . . . . . . . . . . . . . . . . . . . . 71
4.8 Single shot histograms in the IQ plane . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.9 Rabi measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.10 T1 measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.11 T ?2 measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.12 Interleaved T , T ?1 2 loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.13 Bloch sphere representation of Hahn-Echo measurement . . . . . . . . . . . . . . . . 78
4.14 TE2 measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.15 Interleaved T E1, T2 loop with a varying external flux . . . . . . . . . . . . . . . . . . 79
4.16 T2 versus T1 for selected devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.17 Dielectric loss study of previously measured fluxonium qubits . . . . . . . . . . . . . 81
4.18 T1 versus external flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.19 T1 of the |0? ? |2? transition versus external flux . . . . . . . . . . . . . . . . . . . . 83
4.20 T 021 measurement pulse diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.21 T 021 measurement scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.22 TE2 versus external flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.23 Interleaved measurement of T1, TE2 , T
?
2 . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.24 Simulated gate error rate as a function of EJ , EL . . . . . . . . . . . . . . . . . . . . 90
4.25 Pulse train measurement diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.26 Example of a pulse train measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.27 Single-qubit randomized benchmarking . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.28 Purity benchmarking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.29 Fluxonium-based spin chain device . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.30 One-tone spectroscopy of the spin chain device resonator . . . . . . . . . . . . . . . . 98
4.31 Spectrum of the fluxonium-based spin chain . . . . . . . . . . . . . . . . . . . . . . . 100
4.32 Fit for one of the spin chain transitions to the fluxonium Hamiltonian . . . . . . . . . 101
5.1 Spectrum of a fluxonium-based spin chain device with larger coupling . . . . . . . . . 106
A.1 Experimental realization of a fluxonium circuit . . . . . . . . . . . . . . . . . . . . . 107
A.2 ELS-Elionix G-100 Electron Beam Lithography system . . . . . . . . . . . . . . . . 113
A.3 Fluxonium chip development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
A.4 Fluxonium circuit mask . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
A.5 Small Josephson junction area deviation . . . . . . . . . . . . . . . . . . . . . . . . . 116
vii
A.6 Plassys deposition system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
A.7 Chip on the Plassys loading puck . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
A.8 Chip on the Plassys loading puck ready for deposition . . . . . . . . . . . . . . . . . 118
B.1 T1 of the |1? ? |2? transition versus flux . . . . . . . . . . . . . . . . . . . . . . . . . 121
B.2 Optical images of Qubit J along with computed charge matrix elements . . . . . . . . 122
viii
LIST OF APPENDICES
A Fluxonium Fabrication Recipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
B Additional Qubit J Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
ix
LIST OF SYMBOLS
|?? - Wavefunction or state vector
??I (I?) - Identity operator
??x (X?) - Pauli x-operator
??y (Y? ) - Pauli y-operator
??z (Z?) - Pauli z-operator
??+ - Raising operator
??? - Lowering operator
th - Quantum gate accuracy threshold
? - Angular frequency
?jk - Angular transition frequency between eigenstates |j?, |k?
f, ? (?/2?) - Frequency
t, ? - time
R?i(?) - Rotation operator about axis i by angle ?
H - Hamiltonian
L - Lagrangian
K - Kinetic Energy
U - Potential Energy
LK - Kinetic energy term of the Lagrangian
LU - Potential energy term of the Lagrangian
?? - Density matrix
P~ - Qubit polarization
P - Qubit state purity
pi - Population in eigenstate |i?
pthi - Population in eigenstate |i? at thermal equilibrium
L?i - Lindblad operator
T1 - Energy relaxation time of the qubit transition
T jk1 - Energy relaxation time of the transition between eigenstates |j?, |k?
T? - Dephasing time of the qubit transition
T2 - Coherence time of the qubit transition
?1 - Energy relaxation rate of the qubit transition
?? - Dephasing rate of the qubit transition
?2 - Coherence rate of the qubit transition
Q - Charge, Quadrature signal component
? - Magnetic flux
n? - Cooper pair number operator
?? - Phase operator
x
? - Superconducting phase
?ext - External magnetic flux
?ext (2?
?ext ) - Reduced external magnetic flux
?0
C - Capacitance
L - Inductance
Z - Impedance
V - Voltage
Cg - Capacitance to ground
LJ - Josephson inductance
CJ - Josephson capacitance
LarrayJ - Josephson inductance of a Josephson junction in the superinductance array
CarrayJ - Josephson capacitance of a Josephson junction in the superinductance array
IC - Josephson critical current
?r - Resonance frequency
?p - Plasma frequency
Ei - Eigenenergy of eigenstate |i?
EJ - Josesphson energy
EC - Charging energy
EL - Inductive energy
EarrayJ - Josephson energy of a Josephson junction in the superinductance array
EarrayC - Charging energy of a Josephson junction in the superinductance array
a? - Photon annihilation operator
a?? - Photon creation operator
D - Hilbert space dimension
? - Anharmonicity
A - Area, fit constant
T - Temperature
n?jk - Charge matrix element between eigenstates |j?, |k?
??jk - Charge matrix element between eigenstates |j?, |k?
? - Dispersive shift
g - Qubit-resonator coupling rate
? - Resonator linewidth
?? - Excitation rate
?? - Relaxation rate
S(?) - Noise spectral density
 - Dielectric constant
tan ?C (1/Qdiel) - Dielectric loss tangent
Qdiel - Dielectric quality factor
xqp - Quasiparticle density normalized by the Cooper pair density
?qpjk - Energy relaxation rate due to quasiparticle tunneling between eigenstates |j?, |k?
? - Pi, the area of a unit circle
Afl - 1/f Flux noise amplitude
??01 - Energy relaxation rate due to 1/f flux noise
??? - Pure dephasing rate of the qubit transition due to 1/f flux noise
xi
T?? (1/?
?
? ) - Pure dephasing time due to 1/f flux noise
nth - Average thermal photon number
LM - Mutual Inductance
CM - Mutual Capacitance
JC - Capacitive coupling constant
JL - Inductive coupling constant
hz - z-component energy for a spin-1/2
hx - x-component energy for a spin-1/2
Jxx - Inter-spin coupling constant/strength
Qint - Internal quality factor
Qext - External quality factor
S(t) - Measurement signal
I - In-phase signal component
TRO - Time duration of the readout pulse
? - Rabi frequency
d - Qubit driving field amplitude (units of energy)
?d - Phase offset of the qubit drive
T ?2 - Ramsey coherence time
TE2 - Hahn-Echo coherence time
?? - Ramsey frequency
T 021 - Energy relaxation time of the |0? ? |2? transition
Teff - Decay time constant of experimental signal for measurement of T 021
?eff (1/Teff ) - Decay rate of experimental signal for measurement of T 021
tg - Gate time duration
t?g - Gate time duration of a ?-pulse
C? - Clifford gate
C?r - Clifford recovery gate
p - Depolarization parameter
r - Gate error rate
F (1? r) - Gate fidelity
Fg - Gate fidelity of physical gate g
rdec - Gate error rate due to decoherence
u - Unitarity
Physical Constants
e - Elementary charge, 1.6022? 10?19 C
kB - Boltzmann constant, 1.3806? 10?23 J ?K?1
h - Planck constant, 6.6261? 10?34 J ? s
~ (h/2?) - Reduced Planck constant, 1.0546? 10?34 J ? s
?0 (h/2e) - Magnetic flux quantum, 2.0678? 10?15 Wb
?0 (~/2e) - Reduced magnetic flux quantum, 0.3291? 10?15 Wb
xii
RQ (h/(2e)2) - Superconducting resistance quantum, 6, 453.2 ?
? - Superconducting energy gap of aluminum (Al), 1.8? 10?4 eV
Abbreviations
QEC - Quantum Error Correction
ODE - Ordinary Differential Equation
BCS - Bardeen, Cooper, and Schrieffer
cQED - Circuit Quantum Electrodynamics
AlOx - Aluminum Oxide
JJ - Josephson Junction
IFQ - Integer Flux Quanta
HFQ - Half Flux Quanta (also referred to as the sweet spot)
QP - Quasiparticle
TFIM - Transverse-Field Ising Model
DR - Dilution Refrigerator
HEMT - High Electron Mobility Transistor
DUT - Device Under Test
DRC - Directional Coupler
RF - Radio Frequency
IF - Intermediate Frequency
LO - Local Oscillator
AWG - Arbitrary Waveform Generator
ADC - Analog to Digital Converter
EBL - Electron Beam Lithography
BOE - Buffered Oxide Etch
DRAG - Derivative Removal by Adiabatic Gate
RB - Randomized Benchmarking
SPAM - State Preparation And Measurement
PB - Purity Benchmarking
xiii
CHAPTER 1
Introduction to Quantum Computing
Nature isn?t classical, dammit, and if you want to make a simulation of Nature, you?d better make
it quantum mechanical, and by golly it?s a wonderful problem because it doesn?t look so easy.
Richard Feynman
1.1 Introduction
The objective of this dissertation is to demonstrate the powerful applications of fluxonium su-
perconducting qubits in digital and analog quantum computing systems. Digital computers solve
problems by processing information in units of binary digits, or bits. Analog computers use con-
tinuous physical quantities such as pressure, voltage, and length to solve problems. Generally
speaking, the analog approach is designed to solve specific problems, while a digital computer can
solve a much broader array of them. In an analog quantum simulator, quantum systems would be
designed to emulate a Hamiltonian found in Nature, and the artificial system?s dynamics would
be used to simulate the physical system under study. The digital quantum processor would be the
quantum version of the classical ones ubiquitous to life in the 21st century, performing gates and
algorithms on a register of quantum bits. A universal digital quantum computer would be able to
do everything the analog quantum simulator could do and more, with the proper programming.
The drawback of digital is that it is generally more complex than analog, although both imple-
mentations come with unique challenges [3, 5]. In the Noisy Intermediate Scale Quantum (NISQ)
era [86], it is likely that analog quantum simulators will be the first manifestation of a nontrivial
quantum computer. By nontrivial, we mean able to simulate quantum dynamics which cannot be
done on today?s classical computers in a reasonable time scale. Indeed, quantum simulations of
systems of more than 50 atoms have already been demonstrated [13].
This work is organized as follows: in the remaining sections of this chapter, basic quantum
computing concepts will be introduced. In chapter 2, we present the theory of the fluxonium su-
perconducting circuit, starting with a general introduction to superconducting circuits in section
1
2.1. Section 2.2 will cover the key properties of fluxonium, and its main decoherence mechanisms.
Section 2.3 covers coupled fluxonium circuits, and presents our concept for using fluxonium to
simulate a chain of coupled spin-1/2 systems in 1D. Chapter 3 will go over the experimental meth-
ods used in this work. In chapter 4, the experimental data from a case study of a highly coherent
fluxonium superconducting qubit will be presented. The initial sections will go over tune up and
characterization experiments. Section 4.5 will be a characterization of the decoherence mecha-
nisms in this device, as well as the presentation of a novel measurement of the relaxation time of
a parity-forbidden transition between higher energy levels of the circuit. Section 4.6 will present
an analysis of the single qubit gates in this device for digital quantum computing applications.
Finally, section 4.7 will explore fluxonium?s utility as an analog quantum simulator for probing
quantum many-body physics by presenting our experiments on a circuit composed of 10 coupled
fluxoniums. In the final chapter, we will summarize the main experimental results, outlining their
significance, and give an outlook for future studies of fluxonium.
1.2 The Qubit
A quantum computer harnesses the unique characteristics of superposition and entanglement in
quantum mechanics to perform algorithms that are inaccessible to a classical computer. Modern
day classical computers process information by manipulating bits of information. A bit is simply a
yes or a no, or rather a 0 or a 1, in binary. The value of a bit can be stored as the on or off state of a
transistor, and many transistors can be coupled together to make an integrated circuit where these
bits are manipulated at high speeds to do algorithms and process information.
A quantum bit (qubit for short) is the quantum version of a bit, with basis states denoted by |0?
and |1?, using Paul Dirac?s bra-ket n(ota)tion. By(con)vention, we work in the ?z (see equation 1.6)
1 0
basis, where in vector form: |0? = , |1? = . Like any quantum system, a wavefunction
0 1
|?? describes the general state of a qubit:
|?? = a|0?+ b|1?. (1.1)
The coefficients a, b are complex numbers called amplitudes such that |a|2 and |b|2 are the probabil-
ities of the qubit being measured in state |0? or state |1? (assuming measurement in the ?z basis),
respectively. Therefore, |a|2 + |b|2 = 1, leading us to the normalization condition: ??|?? = 1
which is the requirement that the sum of the probabilities of each possible outcome must equal
1. The qubit wavefunction |?? is really a vector living within a complex vector space, called the
Hilbert space. For a single qubit, the dimension of the Hilbert space is 2, and for N qubits, the
2
dimension of the associated Hilbert space is 2N . As the number of qubits N increases, it quickly
becomes a daunting task to simulate such a system on a classical computer!
Consider the qubit state:
| ? ?1? = (|0?+ |1?) (1.2)
2
?
where the coefficients a, b = 1/ 2. This means that the probability of the qubit being detected in
either |0? or |1? from a measurement is 0.5 for each outcome. This is an example of superposition,
which has no classical analog. The qubit is in both states at the same time, and a measurement in the
computational basis will ?collapse? the state vector into one of the basis states |0?, |1? (analogous to
Schrodinger?s cat [91]). The distribution of measurement outcomes is determined by the complex
coefficients a, b.
1.3 Bloch Sphere
Though the coefficients a, b in equation 1.1 are complex numbers, they do not each have two
degrees of freedom; one is removed due to the normalization condition |a|2+|b|2 = 1. Furthermore,
another degree of freedom can be removed when considering that the global phase of a quantum
state vector (i.e. a complex exponential factor) has no physical significance. We can then write the
coefficients as:
?
a = cos
2 (1.3)
b = ei?
?
sin
2
yielding:
|?? ? ?= cos |0?+ ei? sin |1?. (1.4)
2 2
The choice of polar coordinates to represent the amplitudes a, b allows us to conveniently visualize
the qubit state vector as a point on the surface of a sphere with radius 1. This sphere is known as the
Bloch sphere (see figure 1.1). Note that states with the same value of ? but different relative phase
? will yield identical measurement outcomes when measuring in the qubit basis. All possible qubit
states (there are infinitely many) can by defined with a unique ? and ?. Qubit states that can be
represented with a state vector |?? are known as pure states. In a closed system where the qubit
doesn?t interact with its environment, the state vector is enough to model the system. In reality,
no qubit can ever be fully decoupled from its environment, and therefore in practice state vectors
alone are not sufficient for representing a qubit state.
3
Figure 1.1: (a) A general qubit state vector |?? with polar coordinates ?, ? that define the vector.
(b) Four equal superposition states with ? = ?/2 and relative phase ? = 0 (brown), ?/2 (green), ?
(blue), and 3?/2 (red).
1.4 Quantum Gates
Single qubit gates are operators that rotate the state vector |?? around a given axis within the
Bloch sphere. Since they are rotations, the length of the vector is unchanged. Any operator that
preserves the length of the state vector is a unitary operator. Mathematically, an operator U? is
unitary if and only if:
U? U? ? = I? (1.5)
meaning that an operator is unitary if its hermitian adjoint (complex transpose) is also its inverse:
U? ? = U??1. With the single-qubit rotations around the x, y, and z axes, we can prepare any qubit
state we?d like. To write down these operators we first need to remind ourselves of the Pauli
operators: ( ) ( )
1 0 0 1
??I ? I? = (0 1 ) ??x ? X?(= 1 0) (1.6)
0 ?i 1 0
??y ? Y? = ??z ? Z? = .
i 0 0 ?1
4
The rotation operators are then:
? ? ?
R?X(?) ? e?i X?2 = cos I? ? i sin X?
2 2
? ? ?
R?Y (?) ? e?i Y?2 = cos I? ? i sin Y? (1.7)
2 2
R? ?i
? Z? ? ?
Z(?) ? e 2 = cos I? ? i sin Z?
2 2
where we have used the following algebraic properties of the Pauli operators:
??2i = I? (1.8)
and
??i??j = ?ij I? + iijk??k, (1.9)
as well as the Taylor expansion of the complex exponential. With these rotation operators, any
qubit state vector can be prepared. In fact, only two of these rotation operators in equation 1.7 are
needed to prepare any single qubit state, which follows from equation 1.9.
Figure 1.2: An example of a qubit in initial sate |0? (green), with a ?/2 rotation around x applied,
followed by a 5?/4 rotation around z. The final state is: R?Z(5?/4)R?X(?/2)|0? = ?1 (|0? +2
ei
3?
4 |1?). Note that had the rotations been applied in the opposite order, the final state would have
been different, demonstrating that these rotation operators do no commute.
It is known that any multi-qubit unitary operation can be composed of only single and two qubit
gates [25, 8]. A couple of important examples of two qubit gates are the CNOT and CPhase gates.
The ?C? stands for control, meaning if the first qubit (called the control qubit in this case) is in
5
state |1?, apply the corresponding gate to the second qubit (called the target qubit). For CNOT,
this means flip the bit of the target qubit, and for CPhase this means add a relative phase factor ei?
to the target qubit if it too is in state |1?. Below are the gates written in matrix form, along with
examples of their action on the four two qubit states. Note that we use matrix and bra-ket notations
interchangeably.
?? ? ? ??1 0 0 0? ?1 0 0 0?? ??0 1 0 0?? ?0 1 0 0 ?CNOT = CPhase = ? ?0 0 0 1?? ??0 0 1 0 ??
0 0 1 0 0 0 0 ei?
(1.10)
CNOT|00? = |00? CPhase|00? = |00?
CNOT|01? = |01? CPhase|01? = |01?
CNOT|10? = |11? CPhase|10? = |10?
CNOT|11? = |10? CPhase|11? = ei?|11?
The CNOT gate is also found in classical computing, whereas CPhase has no classical analog.
Along with the single qubit rotations in equation 1.7, these two qubit gates form a universal set.
If we have a physical method for doing the single qubit rotations (gates), along with the two-qubit
gates in equation 1.10, any quantum operation on an arbitrary number of qubits is within reach.
The work of eminent theoretical physicists and computer scientists (Peter Schor, David Deutsch,
and David Divincenzo, to name a few of them) [94, 26, 11, 8, 29] have helped to reduced the
problem of building a quantum computer to the hardware and engineering levels, although more
work is still necessary to implement efficient quantum algorithms with imperfect qubits. Among
these theoretical achievements is quantum error correction (QEC), which relies on encoding a
logical qubit which is more robust to errors out of an assembly of physical qubits [95]. With QEC,
the Threshold Theorem proves that when the error rate of physical qubit gates is below a certain
threshold value th, the error rate of the logical qubit can be made arbitrarily small [2]. Improved
error correcting codes [55] have now been shown to boast a lower bound on the accuracy threshold
as high as th > 1.04? 10?3 [4], bringing the promise of fault-tolerant quantum computers closer
than ever to reality.
1.5 Coherence
The state vector (or wavefunction) |?? gives a complete description of the qubit state. Any state
vector describes a pure quantum state. Coherence is the existence of the relative phase ? between
the qubit basis states in a superposition (see figure 1.1b). A qubit can be defined by two energy
6
levels, or a quantum two-level system. Consider the time dependent Schrodinger Equation for a
qubit Hamiltonian H? = ~?01 ??
2 z
:
H?| d?(t)? = i~ |?(t)? (1.11)
dt
where |?(t)? is an equal superposition of the stationary states (or the eigenbasis) of H?:
| 1?(t)? = ? (|?0(t)?+ |?1(t)?). (1.12)
2
Plugging equation 1.12 into 1.11 yields:
E0|?0(t)?
d
+ E1|?1(t)? = i~ (|?0(t)?+ |?1(t)?). (1.13)
dt
We equate the like-terms, yielding two first-order ODE?s:
| i~ d?0(t)? = |?0(t)? (1.14)
E0 dt
| i~ d?1(t)? = |?1(t)? (1.15)
E1 dt
The solutions to equations 1.14, 1.15 are:
|? (t)? = e?
E
i 0
0 ~
t|0? (1.16)
? E|?1(t)?
1
= e i ~ t|1?. (1.17)
E
After plugging equations 1.16, 1.17 into equation 1.12, we factor out the global phase ?i 0e ~ t and
arrive at an expression for |?(t)?:
1 ? (E1?E| ? ? | ? i 0
)
?(t) = ( 0 + e ~ t|1?). (1.18)
2
7
Figure 1.3: Depiction of the equal superposition qubit state evolving in time under Hamiltonian
H? = ~?01 ??
2 z
The wavefunction 1.18 is visualized on the Bloch sphere as a vector rotating about the z-axis at
the qubit transition frequency (figure 1.3), ?01 = (E1?E0)/~. Noise due to coupling to the external
world causes ?01 to fluctuate unpredictably in time, resulting in dephasing. The state of a quantum
system (prior to measurement) cannot be deduced with a single projective measurement. Rather,
an ensemble of identical systems must be prepared and measured, and then the statistics of these
measurement results allows a deduction of the quantum state prior to measurement. If the system is
coupled to an external noise source normally distributed around the qubit frequency ?01, repeated
measurements of the ensemble will reveal a state vector with an exponentially decaying length. The
time constant of this decay, T?, is the dephasing time. Herein lies a significant obstacle to realizing
a quantum computer; in such a processor the qubits must be strongly coupled to one another in a
controlled way for doing gates and reading out computational results, all while maintaining a level
of quantum coherence that it high enough to run a quantum algorithm.
1.6 Density Matrix Formalism
1.6.1 Introduction
All state vectors represent pure quantum states. To model the decoherence of a qubit state, we
must introduce the density matrix formalism, first introduced by John von Neumann [102]. For a
pure state, the density matrix ?? of a qubit is the projector of the state vector:
8
( )
??00 ??01
?? = |????| = . (1.19)
??10 ??11
A fundamental condition on the density matrix is that Tr(??) = 1, which is the statement that the
sum of probabilities must always equal 1. The diagonal terms ??00, ??11 are called populations, and
correspond to the probabilities of measuring a state in the |0? or |1? state, respectively. The off-
diagonal terms, ??01 = ???10 are called coherences, they are zero when describing a maximally mixed
state, rather than a pure one. A mixed state cannot be written as a state vector, and so the density
matrix is a more general way to model a qubit that captures its coupling to external noise sources.
The purity P is defined by: P =Tr(??2) ? 1, the equality condition is met when the density matrix
corresponds to a pure quantum state which can be written down as a vector |??. Purity, therefore,
is a continuous quantity, ranging from 0.5 (maximally mixed state, with coherences equal to 0) to
1 (pure state).
The density matrix is connected to the Bloch sphere representation by:
1
?? = (I? + P~ ? ~?), (1.20)
2
where I? is the identity operator, P~ = (x, y, z) is the polarization of the qubit, and ~? =
(??x, ??y, ??z), the vector formed by the Pauli matrices. The tip of the polarization vector P~ lies
on the Bloch sphere for pure states and inside of(it for)mixed states, therefore, |P~ | ? 1. For the
1 1 0maximally mixed state corresponding to ?? = , |P~ | = 0. We connect the purity of a
2 0 1
quantum state to the length of the polarization vector via [104]:
|P~ |2 1= 2(P ? ). (1.21)
2
The relations between the components of P~ and the density matrix are given by [40]:
x = ??01 + ??10
y = i(??01 ? ??10) (1.22)
z = ??00 ? ??11.
Notice that the x and y components of the polarization vector |P~ | depend on the coherences of
the density matrix, while the z component only depends on the populations. The z component,
therefore, determines the value of the measurement outcomes:
z = ???z?. (1.23)
9
Figure 1.4: (a) Bloch sphere representation of a qubit in a mixed state with population ??00 = 0.6
and ??11 = 0.4 (b) Visualization of the density matrix of the qubit state in (a). Bloch sphere
representation of the pure state |?? = ?1 (|0? ? i|1?) , with corresponding density matrix in (d).
2
Color denote phase |?? in the exponential phase factor ei? for each matrix element. Only the
coherences, ??01, ??10 can be complex since it is unphysical to have complex populations.
It is the expectation value of the ??z operator which ranges between 1 and -1, inclusive. Another
quantity of importance is the von Neumann entropy:
?1 + |P
~ | 1 + |P~ | ? 1? |P
~ | 1? |P~ |
S = log2( ) log ( ). (1.24)2 2 2 2 2
For a pure state, it is straightforward to calculate that S = 0. From a thermodynamic point of
view, nature moves to higher entropy, and less purity in quantum states, assuming coupling to an
external bath at finite temperature.
1.6.2 Measurement
Let us describe a quantum measurement as a set of measurement operators M?i that satisfy the
completeness relation:
10
?
M? ?i M?i = I? . (1.25)
This is simply the statement that the sum of the probabilities of all measurement outcomes must
be equal to unity. Each measurement operator in the set has an associated measurement outcome,
mi. The probability of measuring a specific outcome mi of a qubit with density matrix ?? is:
pi = Tr(M?i??M?
?
i ). (1.26)
After such a measurement is performed, the qubit that was initially described by ?? is left in a new
state described by the density matrix [40]:
M?i??M?
?
?? = ii . (1.27)
pi
When the measurement operators obey the relation:
M?jM?k = ?jkM?j, (1.28)
they are orthogonal projectors, and measurements associated with them are called projective mea-
surements. We can perform spectral decomposition of such an operator:
?
M? = miM?i (1.29)
where M?i projects onto the eigenspace of M? with associated eigenvalue mi. The expectation value
of the observable associated with the projective measurement operator M? for a qubit described by
?? is:
?M?? = Tr(??M?). (1.30)
By convention, the qubit is described in the Pauli operator ??z basis, and therefore we measure
the z-component of the polarization vector (equation 1.20) with the associated operator Z? ? ??z.
Looking back at equation 1.29, we see that for projective measurements with Z?: mi = 1,?1 and
M?i = |0??0|, |1??1|. Note that we use the bra-ket notation and matrix notation interchangeably for
brevity.
1.6.3 Time Evolution
It follows from the relation |?(t)? = U?(t)|?(0)? that:
11
??(t) = U?(t)??(0)U? ?(t). (1.31)
We can model the action of a quantum gate on a qubit using this equation. In practice, the rotations
in equation 1.7 are performed by applying an electromagnetic drive for time t, and the rotation
angle is linearly dependent on this drive time. Figure 1.5 displays the action of a unitary operation
on the Bloch/state vector and corresponding density matrices. For unitary processes (i.e. the length
of the polarization vector P~ remains constant), the Schrodinger equation in terms of the density
matrix is:
d??
= ? i [H?, ??]. (1.32)
dt ~
To expand our analysis of qubit time dynamics to include decoherence, we expand equation
1.32 to form the Lindblad Master Equation:
d?? ? i
? 1 1
= [H?, ??] + (L? ??L??i i ? L?
? ?
~ i
L?i??? ??L?i L?i). (1.33)dt 2 2
i
The additional term models decoherence processes that reduce the purity Tr(??2) of the density
matrix. The Lindblad operators L?i represent each decoherence process affecting the qubit system.
For a full derivation of equation 1.33, consult reference [40]. This is the power of the density
matrix formalism, for it becomes possible to model the evolution of a pure quantum state with
vector |?? into an incoherent mixed state, which cannot be written down as a state vector.
12
Figure 1.5: (a) Bloch sphere representation of a qubit initialized in the ground state, |0?. (b)
Visualization of the density matrix ??0 of the qubit state in (a). Bloch sphere representation of the
state |0? after a rotation of ?/2 about the y-axis R?Y (?/2) is applied. (d) The corresponding density
matrix ??1 to the state in (c): ?? ?1 = R?Y (?/2)??0R?Y (?/2) .
1.6.4 Qubit Relaxation
For energy relaxation of the qubit (i.e. decay from |1? ? |0?), the associated Linbladian opera-
?
tor is L?1 = ?1???, where ??? = |0??1| is the lowering operator and ?1 is the relaxation, or decay
rate (relaxation time T1 = 1 ), corresponding to the frequency of a relaxation event. Plugging L?? 11
into equation 1.33 yields:
d??
= ?i?01 1 1[??z, ??] + ?1(???????+ ? ??+?????? ????+???). (1.34)
dt 2 2 2
Here we have used H? = ~?01 ??z, the two level Hamiltonian with qubit transition frequency ?2 01, and
the fact that the hermitian conjugate of the lowering operator is the raising operator: ??+ = |1??0|.
Upon evaluating the terms on the right side of equation 1.34, we set the like-terms of the density
matrix elements equal to one another and are left with four differential equations describing the
13
dynamics of the coherences ?? , ?? (?? = ???01 10 01 10) and populations ??00, ??11 :
d??00 d??11 ? d??10 ?1= ?1??11, = ?1??11, = ?i?01??10 ? ??10. (1.35)
dt dt dt 2
As expected, the excited state population ??11 decays exponentially with decay rate ?1. Notice too
that the coherences will decay with decay rate ?1/2 due to a relaxation event, establishing the
upper bound on qubit coherence time set by T1.
1.6.5 Qubit Dephasing
We will now examine the effects of pure depha?sing using the Linblad master equation. The
Linbladian associated with pure dephasing is ?L? = ?2 ??z, which follows from the fact that pure2
dephasing manifests as unpredictable rotations about the z-axis in the Bloch sphere (figure 1.3).
Plugging L?2 into equation 1.33, and using the fact that ???z = ?? and ??
2
z z = 1, we arrive at:
d?? ?i?01 ??= [??z, ??] + (??z????z ? ??). (1.36)
dt 2 2
Equating the matrix elements on either side yields:
d??01/10
= ?i?01??01/10 ? ????01/10 (1.37)
dt
The diagonal terms of the matrix on the right side of equation 1.36 are zero and therefore the
differential equations for the populations are trivial. Dephasing will cause the coherences to decay
with decay rate ??. Combining the effects of energy relaxation and pure dephasing, we find that
the coherences will exponentially decay with rate:
?1
?2 = + ??. (1.38)
2
Which, upon converting the decay rates into times, finally brings us to the well-known relation:
1 1 1
= + . (1.39)
T2 2T1 T?
We see then that in principle, if we can eliminate any noise-induced pure dephasing from our qubit,
the only obstacle to increasing coherence time T2 is improving T1, the energy relaxation time. In
the following sections we will outline how we can do this in fluxonium through fabrication methods
and engineering of the circuit?s spectrum.
14
CHAPTER 2
Fluxonium Theory
Imagination is more important than knowledge.
Albert Einstein
2.1 Superconducting Circuits
In the past two decades, superconducting circuits have emerged as a prominent platform for
quantum computing. Though they are macroscopic objects composed of an Avogadro?s number
of particles, when the superconductor is cooled below the critical temperature, the particles col-
lectively enter the lowest energy state, allowing quantum mechanical effects to be observed. This
manifests itself as a current flowing through the superconductor with zero resistance. Supercon-
ductivity was first discovered by Heinke Onnes in 1911, and decades later the microscopic theory
(BCS Theory in 1957) was developed by Bardeen, Cooper and Schrieffer [7]. Once cooled below
the critical temperature, (around 1 K for the aluminum (Al) films of the circuits in this work), we
simply model these circuits as if they had no resistors, and treat them quantum mechanically. For
quantum computing applications, two energy levels of the circuit define the qubit, typically the
ground and first excited states.
Macroscopic quantum mechanical behavior in superconducting circuits was first observed in
1999 with the Cooper pair box circuit by Nakamura, Pashkin, and Tsai [74]. From there, no-
table leaps took place with the invention of quantronium [100] and 3D-transmon qubits [81],
the latter leading to a widespread use of transmons and related circuits, such as X-mons [9] and
capacitively-shunted (C-shunt) flux qubits [110]. As a result, the study of superconducting circuits
has ballooned into its own research area. Beyond the digital quantum computing applications,
superconducting circuits allow us to study quantum mechanics in a controlled setting, where the
researcher can change system parameters simply by tuning classical circuit elements. This level of
control is not possible in nature, where the systems? (such as atoms) properties are set. This field
is known as circuit quantum electrodynamics (cQED) [105], where the superconducting circuits
15
take the place of atoms.
The sustained growth in quantum computing research based on superconducting circuits has
come from the orders of magnitude improvement in qubit coherence times, which ultimately sets
the lower limit on the gate error rates. Starting from the order of nanoseconds with the Cooper
pair box, over the past two decades a near six-order of magnitude improvement in T2 has been
achieved [28, 52]. Coherence is still limited by material defects in the circuit, and no fundamen-
tal limit on the coherence time seems to be present. With the quantum gate accuracy tolerance
threshold in sight [4], we are closer than ever to achieving a digital quantum processor based on
superconducting circuits.
2.1.1 Circuit Quantization
We begin our analysis of superconducting circuits by quantizing the LC oscillator. Consider the
LC circuit with zero resistance (conveniently taken care of by superconductivity):
Figure 2.1: LC oscillator circuit diagram with magnetic flux ? threading the inductor with induc-
tance L and charge Q on the capacitor with capacitance C.
where the voltage V across the inductor and capacitor is given by V = ??? and V = Q/C,
respectively. These follow from the voltage-charge relation for the capacitor: Q = CV , and the
current-flux relation for the inductor: ? = LI . Since these elements are in parallel, they share
the same voltage V , resulting in the relation: Q/C = ?LI? . The magnetic energy stored in the
2
inductor, the inductive energy, is given by E = ?L . The electric energy stored in the capacitor,2L
2
the charging energy, is given by E = QC .2C
To obtain the Hamiltonian of the LC oscillator, we write down the Lagrangian: L = K ? U
(K, U are the kinetic and potential energies, respectively) in terms of the generalized coordinates
?, ??:
1 ?2L = C??2 ? . (2.1)
2 2L
16
Note that we could have chosen the generalized coordinates to be Q, Q?, where then the energy
stored in the inductor would play the role of mechanical kinetic energy. The Hamiltonian is ob-
tained by applying the Legedre transformation:
H ?L= ?? ? L, (2.2)
???
and writing the result in terms of the canonical momentum: ?L = C?? = Q. Finally, we arrive at
???
the total energy of the LC oscillator, given by the HamiltonianH:
2 2
H Q ?= + . (2.3)
2C 2L
To quantize the Hamiltonian, we map these classical variables H, Q,? to quantum mechanical
operators: H?, Q?, ??, with canonical commutation relation:
[??, Q?] = i~. (2.4)
The classical Hamiltonian for the mechanical oscillator is:
H p
2 1
= + m?2x2. (2.5)
2m 2
In this representation of the LC oscillator, the charge stored on the capacitance is analogous to
the momentum in the mechanical one (spring/mass), and flux is analogous to the position. The
capacitance is analogous to a particle?s mass, and inductance to the inverse of the spring constant
(1/k). For convenience, we rewrite the charge Q? and flux ?? operators as dimensionless operators:
n? = Q?/2e
?? = ??/?0 (2.6)
[??, n?] = i.
The new dimensionless operators are the reduced charge and reduced flux operators. ?0 = ~/2e is
the reduced magnetic flux quantum. The Hamiltonian with the dimensionless operators now takes
the form:
H? 1= 4E n?2C + EL??2, (2.7)
2
where we have redefined the charging and inductive energies:
e2
EC = (2.8)
2C
17
and
?2
E 0L = . (2.9)
L
The dimension of energy is now stored in these constants.
As is the case with the mechanical oscillator, we can rewrite the Hamiltonian of the LC-
oscillator in terms of photon creation and annihilation operators, a?? and a?, respectively: H? =
~?r(a??a?+ 1/2). In terms of a?? and a?, the reduced charge and flux operators read:( ) 1
i E 4? Ln? = ?( ) (a? ? a?)2 8EC 1 (2.10)
1 8E 4C
?? = ? (a?? + a?).
2 EL
The resonant frequency ?r of the oscillator is given by:
1? 1
?r = 8ECEL = ? . (2.11)~ LC
The characteristic impedance of the oscillator i?s: ?
~ 8EC L
Z0 = = . (2.12)
(2e)2 EL C
When the impedance is greater than the resistance quantum RQ = ~/(2e)2, charge fluctuations are
suppressed below the cooper pair level 2e, the regime where this thesis will focus on [66].
2.1.2 The Josephson Effect
The quantum LC oscillator yields a spectrum of equally spaced eigenenergies, the level spacing
being ~?r. If we are to uniquely define a qubit as a pair of energy levels with transition frequency
?q, then a nonlinear circuit element must be introduced to the oscillator, without sacrificing the
superconductivity. In 1962, Brian D. Josephson developed the theory of the Josephson effect: the
phenomenon where a supercurrent can flow through a thin, non-superconducting region (weak
link) that is sandwiched by two superconducting electrodes. This weak link is known as a Joseph-
son junction (JJ) [48, 49].
The supercurrent IJ flowing across the weak link is given by:
IJ = IC sin? (2.13)
where IC is the critical current (i.e. the largest possible supercurrent that can cross the junction,
set by the geometry and materials of the junction) and ? = ?a ? ?b is the superconducting phase
18
Figure 2.2: (a) Diagram of a Josephson junction. The superconducting electrodes have their respec-
tive superconducting phases (?a, ?b) and are separated by a relatively thin non-superconducting
layer (grey). A voltage V results in a supercurrent IJ flowing across the non-superconducting
region. (b) The circuit representation of a Josephson junction. In practice, the JJ carries a self-
capacitance, and should therefore be thought of as a capacitor and an ideal JJ in parallel.
difference across the junction (see figure 2.2). The time-evolution of ? is given by:
?? 2e V (t)
= V (t) = , (2.14)
?t ~ ?0
these equations are known as the first and second Josephson relations, respectively. Using these
relations, we take the time-derivative of the supercurrent IJ :
?IJ ?IJ ?? V (t)
= = IC cos? . (2.15)
?t ?? ?t ?0
In terms of V (t), equation 2.15 reads:
?0 ?IJ ?IJ
V (t) = = L(?) . (2.16)
IC cos? ?t ?t
A circuit element that has a voltage across it proportional to the time-derivative of the current
flowing through it is an inductor. Therefore, the JJ can be thought of as a non-linear inductor with
an inductance L(?) that depends on the superconducting phase difference across it.
19
We define the Josephson inductance as:
?0
LJ = , (2.17)
IC
which sets the scale that the inductance L(?) = LJ/ cos? varies by. Finally, the energy U stored
in this non-linear inductor is?found in the same wa?y we treat a classical, linear one:
U = IJV (t)dt = IC?0 sin?d? = ?EJ cos?, (2.18)
where we have defined the Josephson energy EJ as:
?2
EJ = I
0
C?0 = . (2.19)
LJ
Notice that the Josephson energy EJ takes the same form as the inductive energy EL defined in
equation 2.9, with the inductance L replaced by the Josephson inductance LJ .
The plasma frequency ?p of the JJ is given by:
1? 1
?p = 8EJEC = ? , (2.20)~ LJCJ
where CJ is the self-capacitance of the Josephson junction. The plasma frequency is the upper
limit on the frequency of an electromagnetic wave propagating through the junction. We always
work in frequency regimes where ? < ?p. Since LJ ? 1/A and CJ ? A, where A is the cross
sectional area of the JJ, ?p is independent of the JJ area. It is set by the characteristics of the
weak link material, typically an insulator. For the JJ?s in the devices studied in this work, the
weak link material is aluminum oxide (AlOx), and the plasma frequency falls in a range of ?p ?
2??(20?40) GHz [66]. A major obstacle in scaling quantum hardware based on superconducting
circuits is the the ability to control the parameters of the individual physical qubits, and is a major
area of research today [60, 42].
2.2 Fluxonium
With the nonlinear circuit element (Josephson junction) in hand, we now posses the building
blocks required to make a superconducting qubit. The general Hamiltonian of such a supercon-
ducting circuit is:
H? 1= 4E 2 2C n? + EL?? ? EJ cos(??? ?ext) (2.21)
2
20
where ?? is now defined as the superconducting phase difference across the linear inductance, this
difference being offset by some external magnetic flux ?ext = ?ext/?0 through the loop formed by
the linear inductor and Josephson junction, as seen by the Josephson junction.
Figure 2.3: Generic circuit diagram for a superconducting circuit with inductance, Josephson junc-
tion, and capacitance in parallel. If the inductance and JJ form a closed loop, then there is an
external magnetic flux ?ext which can act as another tuning knob of the circuit?s spectrum. The
experimental realization of this circuit is shown in figure A.1 in Appendix A.
With these three elements, we can construct any superconducting circuit with an anharmonic
spectrum, such that individual transitions can be addressed as qubits. Today, the most prominent
superconducting circuits being researched for quantum computing applications are the transmon
(EL = 0, EJ/EC & 40) [56, 81], the flux qubit (EJ & EL > EC) [110], and fluxonium, the
subject of this work.
2.2.1 Spectrum
Fluxonium?s Hamiltonian is given by equation 2.21, with the parameter constraints ofEJ/EC ?
1 ? 10 and EJ > EL [67, 66]. Fluxonium can be viewed as a flux qubit with a much larger
inductance, or as a transmon with an inductive shunt across the JJ. Importantly, this shunting
inductance which connects the two superconducting islands across the JJ removes the offset-charge
sensitivity. The parameter constraints defining the fluxonium regime require a superinductance,
defined as an inductive element that has 0 resistance in DC and an impedance at the microwave
frequencies of study (1-10 GHz) greater than the resistance quantum RQ = ~/(2e)2 ' 6.5 k?
[66, 71]. The superinductance is achieved by using an array of N ? 100 identical Josephson
junctions. The Hamiltonian of such an array reads:
H?array = ?NEarrayJ cos(??/N). (2.22)
21
Equation 2.22 m?akes two assumptions: the first, that phase slips across the junctions are sup-
pressed: exp{? 8EarrayJ /E
array
C } << 1, E
array
C = e
2/2Carray, where CarrayJ J is the self capaci-
tance of each array JJ. And the second, that the capacitance-to-ground C satisfies Carrayg J /Cg >>
N , which makes the array?s resonance frequency greater than each junction?s plasma frequency ?p
[103, 71], and therefore we can ignore its effect. Taking N ??:
Earray 4H?array = ? J ????2 +O( ) (2.23)
2N N3
which is the energy of a linear inductance withL = NLarray, LarrayJ J being the Josephson inductance
of each identical JJ in the array, which is typically on the order of 1 nH. Therefore, we can treat
this array as a linear inductance, with inductive energy E = EarrayL J /N = ?
2 array
0/NLJ
The potential energy term in fluxonium?s Hamiltonian reads: 1E ??2 ?E cos(??? ?
2 L J ext
). Note
that ?ext = 2?(?ext/?0), where ?ext is the magnitude of magnetic flux threading the loop formed
by the JJ and the inductance and ?0 = h/2e = 2??0 is the magnetic flux quantum. In order to
study the spectrum and dynamics of fluxonium, we numerically diagonalize the Hamiltonian 2.21
in the basis of the harmonic oscillator states, i.e. the case where EJ = 0. To do this, we insert the
reduced charge and phase operators in terms of the harmonic oscillator creation and annihilation
operators into the fluxonium Hamiltonian:
( ) 1
4
n? = ?i EL (a??( ) ? a?)2 8EC 1
1 8E 4C
?? = ?? (a?? + a?)2 EL (2.24)
H? = 2E E ((a??C L a?+)a?a??)?1 ( ) 1
1 4 4
EJ(exp{?
i 8EC ?i 8EC
(a?? + a?)? ? ?ext}+ exp{? (a? + a?)? ?ext}).
2 2 EL 2 EL
Note that we rewrote the cosine function in terms of complex exponentials (Euler?s formula).
Using the QuTip python package, we set the dimension D of the Hilbert space by defining
the annihilation operator a? as a D ? D matrix, and then the matrix formed by equation 2.24 is
numerically diagonalized to find its eigenvalues. This process is done for each external flux point
?ext. With a Hilbert space as small as D = 20, we get an excellent agreement between theory
and experiment for the first few transition energies in a very short computational time. Figure
2.4 shows the potential energy of fluxonium as a function of ?? at integer and half-integer external
magnetic flux quanta. At integer flux quanta (IFQ), the two lowest energy levels (what we will use
to define a qubit) form a plasmon transition, where the wavefunctions are localized in the same
22
potential well. At half flux quanta (HFQ) the two energy levels form a fluxon transition, where
the wavefunctions are localized in the two potential wells. These two eigenstates correspond to
superpositions of clockwise and anticlockwise current flowing in the superconducting loop. At
HFQ, the qubit transition is about an order of magnitude smaller in frequency than at IFQ, and
is separated by a plasmon gap to the third energy level. For reasons that will become more clear
later in the thesis, it is at HFQ where we will use fluxonum as a qubit. For now, the most apparent
reason is the maximal anharmonicity ?, defined as ? = (?12 ? ?01)/?01, where ?12 and ?01 are
the |1? ? |2? and |0? ? |1? (qubit) transition frequencies. Higher anharmonicity means that we are
less likely to address an unwanted transition when performing a qubit operation, which leads to
computational errors.
Figure 2.4: Fluxonium potential with first 10 eigenenergies and corresponding wavefunctions
(EJ/h = 5 GHz, EC/h = 1 GHz, EL/h = 0.5 GHz) for external flux ?ext at (a) integer and (b)
half-integer values of the magnetic flux quantum ?0. In (a), index n ? Z, and for (b) n ? Z : n 6= 0.
To gain further intuition on how the three energy parameters affect fluxonium?s spectrum, we
look at the transition and eigenenergies across one flux period (i.e. ?0) for varying EJ , EC , EL.
We will also look at the potential energy function with the first few eigenenergies at HFQ for
varying parameters. Unless otherwise stated, energies will be written in units of Joules/h (i.e.
frequency). As we increase EJ , the anharmonicity in the spectrum increases (see figure 2.5). This
follows from the fact that EJ is part of the nonlinear term in the Hamiltonian. Looking at the
potential at HFQ, the height of the double well increases with EJ , therefore the qubit frequency
decreases with higher EJ . Of more significance than the absolute value of EJ is the ratio of EJ
to EC . The tunneling rate across the potential barrier at HFQ is exponentially suppressed with
increasing EJ/EC , resulting in a smaller transition frequency.
23
Figure 2.5: Fluxonium?s spectrum with varying Josephson energies EJ . First column: first six
transition energies starting from the ground state. Second column: first 7 energy levels (eigenen-
ergies). Third column: potential energy at HFQ with first 7 energy levels and wavefunctions.
24
Figure 2.6: Fluxonium?s spectrum with varying inductive energies EL. First column: first six
transition energies starting from the ground state. Second column: first 7 energy levels (eigenen-
ergies). Third column: potential energy at HFQ with first 7 energy levels and wavefunctions.
25
Varying the inductive energyEL sets the steepness of the parabolic envelope of the potential (see
figure 2.6). As EL increases, the number of potential wells accessible to the lowest energy levels
decreases, resulting in an increasing qubit frequency. the ratio EJ/EL sets the number of potential
wells seen by the qubit transition at HFQ, and in the fluxonium regime we aim for two wells. There
is active research in what is nicknamed the blochnium regime [82], where EL << EJ , EC , and
the qubit transition is localized in several wells, effectively leading to a qubit transition frequency
that is independent of external flux.
Figure 2.7: Fluxonium?s spectrum with varying charging energies EC . First column: first six
transition energies starting from the ground state. Second column: first 7 energy levels (eigenen-
ergies). Third column: potential energy at HFQ with first 7 energy levels and wavefunctions.
Varying the charging energyEC obviously does not affect the potential energy, since it is part of
the Hamiltonian term containing n?, analogous to kinetic energy. In the the particle-in-a-potential
picture, EC is the inverse to the particle?s mass. Therefore, as EC increases, the energy levels
26
increase, since the effective particle is less massive and therefore harder to confine in a potential
well. The anharmonicity of the spectrum decreases with increasing EC , since the transitions will
be less likely to be localized in different potential wells. More transition become plasmons, which
resemble harmonic oscillator transitions, and have energy level spacing approximately given by
?
8EJEC . This is apparent in figure 2.7.
2.2.2 Matrix Elements
Fluxonium?s matrix elements between various energy levels of reduced charge n? and phase ??
determine its dynamics. Our most important application of them is modeling loss-mechanisms of
the qubit transition |0?? |1?. Though the operators are of course canonically conjugate, (canonical
commutation relation in equation 2.6), their matrix elements obey a simple relation. Taking the
time-derivative of the ?? matrix element between states |j? ? |k?, we get:
d ?k|??| d 2ej? = ?k| ??|j?
dt dt ~
2e
= ?k|V? |j? (2.25)
~
(2e)2
= ?k|n?|j?.
~C
Here, we used Faraday?s Law: d ?? = V? , and V? = Q? = 2e n? is the operator for voltage across the
dt C C
single Josephson junction of the circuit. We then apply Ehrenfast?s Theorem, which gives the time
derivative of the expectation value of a canonical operato?r O?: ?
d ? iO?? = ?[H? ?O?, O?]?+ . (2.26)
dt ~ ?t
This leads to:
d ?k|??| ? ij = ?k|H???? ??H?|j?
dt ~
i
= (Ek ? Ej)?k|??|j? (2.27)~
= i?jk?k|??|j?,
where we used the fact that ?? has no explicit time dependence, and that states |j?, |k? are eigen-
states of H? with eigenenergies Ej, Ek, respectively. Finally, by equating the results of equations
27
2.25 and 2.27, we arrive at the relation between the reduced charge and phase operators:
? | | i~?jkCk n? j? = ?k|??|j?
(2e)2
~ (2.28)i ?jk
= ?k|??|j?.
8EC
To gain more intuition on how the circuit parameters affect the matrix elements, we numerically
compute them, using the same parameter choices as in figure 2.5.
Figure 2.8: Matrix elements of fluxonium?s first three transitions with EC/h = EL/h = 1 GHz
and EJ/h = 1, 4, 10 GHz (blue, orange, green curves, respectively). It is instructive to see how
the changes in parameters affect the eigenstate distribution with respect to the potential at HFQ in
figure 2.5.
Notice that there is a clear selection rule at IFQ and HFQ, since at these external magnetic flux
points, the potential become symmetric, and therefore the eigenstates have a definite parity. For
transitions between eigenstates of like parity, the matrix elements are zero, and these transitions
are therefore forbidden at IFQ and HFQ (see rightmost column in figure 2.8, the matrix elements
for the |0? ? |2? transition).
28
2.2.3 Readout
Readout of the fluxonium qubit is performed by coupling the circuit to a resonator with
linewidth ?, and detecting the qubit-state dependent shift in resonator frequency. This resonator
frequency shift is called the dispersive shift, ?jk, where jk represent the corresponding circuit
transition causing the shift. Typically the qubit is capacitively coupled to the resonator, though
inductive coupling is possible as well. We operate our fluxonium qubits in the dispersive regime
of cQED [15, 105], where the resonator-qubit frequency detuning |?| >> g, where g is the qubit-
resonator coupling strength.
The Hamiltonian of the combined qubit-resonator system is the bare fluxonium Hamiltonian
(equation 2.21) with the addition of the bare resonator term H?r = ~?r(a??a? + 1/2), and the qubit-
resonator coupling term for capacitive coupling:
H? = ?gn?(a?+ a??c ). (2.29)
In the case of inductive coupling to the resonator, the coupling term reads: H? ?c = ?g??(a? + a? ).
The coupling constant g depends on either the mutual capacitance or inductance between resonator
and qubit.
Using second order perturbation theory, we calculate the shifts in the coupled system energies
for the capacitive case. We denote the uncoupled (unperturbed) fluxonium-resonator eigenstates as
|j, l?, with corresponding eigenenergy ?0j,l = ?j+l?r. Note that we express the energies in terms of
angular frequency ?, factoring out ~. The coupled (perturbed) eigenenergies are: ? 0j,l = ?j,l+??j,l,
the energy shift given by:
? |?k,m|gn?(a?+ a??)|j, l?|2
??j,l =
?0 0
j 6=k j(,l ? ?k,m?l=6 m )
= g2| l l + 1n? 2jk| + (2.30)
? ?6 jk ? ?r ?jk + ?rj=k
2| |2 2l?jk + ?jk ? ?r= g n?jk
?2 ? ?2
j=6 k jk r
where ?jk = ?j ? ?k is the transition frequency between fluxonium eigenstates |j?, |k?. The shift
in the bare resonator frequency ?r due to coupling to a fluxonium circuit in eigenstate |j?, known
as the dispersive shift, is then given by:
29
?j = (?j?,l+1 ? ?j,l)? ?r
2 2 2?jk (2.31)= g |n?jk| .
?2 2
k jk
? ?r
What is measured in the lab is the resonator frequency in either the |j? or |k? state at ?r + ?j or
?r + ?k, respectively. The difference of these frequencies is the dispersive shift of the |j? ? |k?
transition: ?jk = ?j ? ?k. The(fluxonium qubit dispersive shift ?01 is ther)efore [114, 72, 66]:? ?
2 | |2 2?0k?01 = g n?0k ? | 2
2?1k
n?
2 ? 2 1k
| . (2.32)
? ? ?2 2
k 6=0 0k r k=6 1 1k
? ?r
Figure 2.9: Dispersive readout scheme of the qubit. Top: Resonator transmission signal as a
Lorentzian function centered at the resonator frequency with linewidth ?/2? = 10 MHz. Bottom:
Phase of transmitted signal. Transmitted signal picks up a qubit-state dependent phase shift: ?0 or
?1.
From equation 2.32, we expect larger ?01 by engineering a spectrum where there are transitions
involving |0? and |1? that have frequencies close to the resonator frequency, ?r. Higher transitions
contribute to the fluxonium qubit?s dispersive shift, which is why we can still readout the qubit
30
despite working with small qubit frequencies ?01 and matrix elements |n? 201| as compared to the
typically used transmon [56]. This was demonstrated in reference [62], where ?01 maintains its
value despite a vanishing |n? 201| , due to the higher level transitions.
2.2.4 Decoherence Mechanisms
The relation between T1, T2, and T? (equation 1.39) establishes that the upper limit on the
coherence time is set by energy relaxation from |1? ? |0?. As a result, we will begin our analysis
of decoherence mechanisms of fluxonium?s qubit by looking at the main source of energy loss.
The first consideration to take is that the environment has a finite temperature, which leads to
absorption and emission with the qubit. This effect becomes important when the qubit transition
energy become comparable to the temperature of the environment: ~?01 ? kBT , where kB is the
Boltzmann constant. The environment causes a thermal excitation rate, ??, while there is also the
relaxation rate, ??, caused by dissipative sources. The relation between these two rates is:
?? ?~?01
= e kBT , (2.33)
??
which results in the following relation between qubit state populations p0, p1 at thermal equilib-
rium:
p ?~?1 01
= e kBT . (2.34)
p0
The measured energy relaxation time T1 = 1/?1 is the time the system takes to reach thermal
equilibrium. This leads to the relation:
?~?01
?1 = ?? + ?? = (1 + e kBT )??. (2.35)
A more significant effect due to the temperature of the environment comes from the fluctuation-
dissipation theorem, where the thermal noise spectral density associated with a lossy circuit ele-
ment with impedance Z(?) is given by [27]: [ ( ) ]
~?
S(?) = ~?Re[Z(?)] coth + 1 . (2.36)
2kBT
In the succeeding sections, we will use Fermi?s Golden rule (derived by Paul Dirac, and named
after Enrico Fermi) to determine energy decay rates for various processes. Fermi?s Golden rule
reads:
??
1
jk = |?k|O?|j?|2S?(?jk), (2.37)~2
31
Figure 2.10: Thermal factors discussed in a typical fluxonium qubit frequency range for 10, 25,
50, and 100 mK. Typical operating temperature in experiments is 10-25mK.
where ??jk is the energy relaxation rate between energy eigenstates |j?, |k? due to noise source ?
with associated noise spectral density S?(?jk). |?k|O?|j?| is the matrix element of the operator that
couples the circuit to the noise source.
2.2.4.1 Dielectric Loss
Dielectric loss causes energy relaxation in the qubit due to the presence of lossy dielectric
materials in or on the circuit. These materials absorb the energy in the electric field generated
by the energy states [69, 31, 106]. This effect is apparent in all types of superconducting qubits
and is the leading limitation on qubit T1 today. We model dielectric loss as the capacitance in the
fluxonium circuit being lossy with complex dielectric constant  = 1 + i2, allowing us to define
the dielectric loss tangent tan ?C [85]:
2 1
tan ?C = = , (2.38)
1 Qdiel
where Qdiel is the quality factor of a resonator composed of this lossy capacitance. The impedance
Zdiel of this lossy capacitance is:
1
Zdiel(?) = (2.39)
i?C(1 + i tan ?C)
We then find the noise spectral density of the dielectric loss, Sdiel(?). The real part of Zdiel is:
32
tan ?C tan ?C 1
ReZdiel(?) = ? = , (2.40)
?C(1 + tan2 ?C) ?C ?CQdiel
where we used the fact than tan ?C << 1 << Qdiel. Plugging equation 2.40 into equation 2.36,
we arrive at the noise spectral density due to die[lectric(loss: ) ]
~ tan ?C ~?
Sdiel(?) = coth + 1 . (2.41)
C 2kBT
To find ?diel01 from Fermi?s Golden rule, we identify the associated operator coupling to the noise
source as Q? = 2en?, the charge across the capacitance. Finally, we come to the relaxation rate of
the qubit due to dielectric loss: [ ( ) ]
diel 8EC |? | ~?01?01 = 1 n?|0?|2 tan ?C coth + 1 . (2.42)~ 2kBT
Using the relation between the phase and charge matrix elements (equation 2.28), we can also
write ?diel01 in terms of the phase matrix element: [ ( ) ]
~?2diel 01 |? ~?01?01 = 1|??|0?|2 tan ?C coth + 1 , (2.43)8EC 2kBT
where now there is an explicit frequency dependence in the relaxation rate. We see then that as we
engineer the spectrum such that ?01 or |n?01| is small, we will be decoupled from dielectric loss and
can expect higher T = 1/?diel1 01 , and therefore a higher bound on the coherence time T2. One of
the key advantages of fluxonium is the ability to decouple from dielectric loss in-situ by tuning the
magnetic flux to HFQ where the qubit transition frequency is minimal, as demonstrated in figure
2.11. Upon analyzing the experimental results, we deduce that dielectric loss is the main limiting
factor on fluxonium?s coherence time. Using figure 2.12, we can select desirable circuit parameters
to optimize the limit on T1 due to dielectric loss.
33
Figure 2.11: Simulated T1 limit due to dielectric loss across one full flux period for varying loss
tangents. Limit on T1 increases by more than an order of magnitude from IFQ to HFQ. Fluxonium
parameters in the simulation are: EJ/h = 4.0 GHz, EL/h = EC/h = 1.0 GHz. Temperature is
set to T = 25 mK. Black curve corresponds to right y-axis and is the qubit frequency as a function
of external magnetic flux.
Figure 2.12: (a) Simulated T1 as a function of EL, EJ with EC/h = 1.08 GHz and tan ?C =
5? 10?7. Temperature is set to 25 mK, a typical value for our cryogenic setup. (b) Corresponding
qubit frequency as a function of the same parameters.
34
2.2.4.2 Quasiparticles
Quasiparticles (QP?s) are excitations out of the BCS ground state, and can be thought of as the
occurrence of broken Cooper-pairs. The observed density of QP?s in present-day superconducting
circuits cannot be explained by thermal effects, and their source remains an open question in the
field [98, 92, 68, 93]. We therefore refer to these types of QP?s as out-of-equilibrium quasiparti-
cles. Out-of-equilibrium QP?s remain a major limiting factor on the coherence of superconducting
circuits [20, 84]. When a quasiparticle tunnels across a Josephson junction (either the single JJ
which sets EJ , or a JJ in the superinductance array setting EL), unwanted transitions occur, which
manifest as energy relaxation of the qubit. Following the procedure in reference [34], and ap-
plying Fermi?s Golden rule, the energy relaxation rate between two fluxonium eigenstates due to
QP-tunneling across a single JJ is: ?
qp 8EJ 2? |? | ??? ?ext?jk = k sin( )|j?|
2xqp. (2.44)
?~ ~?jk 2
Here, ? ' 180 ?eV is the superconducting gap of Aluminum and xqp is the quasiparticle density
normalized by the Cooper pair density in the circuit. It can be thought of as the fraction of Cooper
pairs that have been broken in the circuit. Using this equation, we can extrapolate to the case where
QP?s tunnel across the JJ?s in the superinductance array. In this case, the rate depends on the sum
of all possible tunneling events across each of the N identical junctions with Earray
? J
/h ? 100 GHz
in the JJ array:
8Earrayqp,array J 2? |? | ??? ?ext?jk = k |j?|
2x
~ ~ qp
. (2.45)
?N ?jk 2
To arrive at this rate, we used the fact that the junctions are identical and have large enoughEJ such
that each ?n across each junction is small and identical. We can therefore use the approximation
sin? ? ?, and set E = EarrayL J /N . Finally, since ?k|
?ext |j? = 0 for j =6 k, we arrive at the decay
2
rate induced by QP?s tunnelling across the arra?y JJ?s:
?qp,array
8EL 2? |? | ??jk = k |j?|
2xqp. (2.46)
?~ ~?jk 2
The even parity of sin( ????ext ) at ? = ? (HFQ) results in a rate of 0 for ?qpext 01 , as well as for2
any other opposite-parity transition. Therefore, at HFQ, the qubit is protected from QP-induced
decay due to tunneling across the single JJ. Tunneling events across the array junctions, however,
can limit T1 of the qubit. Recall that since |?2|??|0?| = 0, the energy relaxation of the |2? ? |0?
transition is protected from dielectric loss at HFQ. This transition?s decay rate is limited by QP?s
tunneling across the single JJ, since it is a transition between states of like-parity. Measurements
35
of this decay time, T 021 , can therefore provide stringent estimates on the normalized quasiparticle
density in the circuit.
Figure 2.13: T1 limit on the qubit transition for circuit parameters EJ/h = 4.0 GHz, EL/h =
EC/h = 1.0 GHz across a flux period for varying xqp. Solid line: limit on T1 due to QP?s
tunneling across the single JJ (equation 2.44). At HFQ, the curve diverges as ?qp01 ? 0. Dashed
lines are T1 limit due to tunneling in the JJ array (equation 2.46).
.
2.2.4.3 1/f Flux Noise
Being a flux-sensitive device, fluxonium is susceptible to 1/f flux noise ubiquitous to solid state
systems [70, 61]. This noise can limit qubit coherence by inducing a direct relaxation rate between
qubit states. It also causes pure dephasing (outlined in section 1.5), due to the external noise
changing the flux bias ?ext through the loop, thereby changing the qubit frequency. The noise
spectral density function for 1/f flux noise reads:
A2
S?(?) = 2?
fl , (2.47)
?
where Afl is defined as the flux noise amplitude at 1 Hz. This parameter is dependent on the
experimental setup and the environment of the circuit. By applying Fermi?s Golden rule, we find
36
the decay rate of the qubit transition due to 1/f flux noise [87]:
? 2? 1
2
?01 = |?1|??| ?|2
A
0 fl , (2.48)
(2e)2 L2 ?01
where L is the inductance (coming from the JJ array forming the superinductance) of the loop.
L appears because the coupling operator to the noise source is the loop current I? = ??/L =
?0??/L that arises from the varying ?ext through the loop. Typical reported values of the flux noise
amplitude are on the order ofAfl ? 1??0 [110, 79]. Due to fluxonium?s extremely high inductance
L (? 100 nH), 1/f noise has no appreciable affect on the qubit?s T1, even though we operate the
qubit where its frequency is at a minimum and therefore maximally coupled to the 1/f noise. This
is extremely important, for if fluxonium?s loop inductance was that of a flux qubit (? 1 nH), we
could not decouple from dielectric loss by tuning the external flux to HFQ without introducing a
new source of energy relaxation.
Figure 2.14: T1 limit due to 1/f flux noise on the qubit transition for circuit parameters EJ/h =
5.0 GHz, EC/h = 1.0 GHz, and varying EL/h = 0.33, 1.09, 4.09 GHz across one flux period.
These EL?s correspond to L = 500, 150, 50 nH, respectively. Black curve is the same quantity, but
for flux qubit with typical parameters (EJ/h = 200, EL/h = 160, EC/h = 1 GHz). The value of
EL corresponds to L = 1 nH. The limit on T1 for the flux qubit is orders of magnitude lower than
for fluxonium. Flux noise amplitude Afl = 1??0 for all curves.
37
Though the fluxonium qubit?s T1 is practically unaffected by 1/f flux noise, dephasing time T?
is. To first order, 1/f flux noise will introduce a dep?hasing? rate given by [89]:? ? ??01 ?? 1?? = Afl ln(2) ?? ?? = . (2.49)?? ?ext T?
For a measurement of T2, this 1/f noise induced dephasing will result as a signal with a Gaussian
decay envelope:
? t
2
?2
p1(t) = Ae
T
? +B, (2.50)
where p1(t) is the population of state |1? as function of time and A,B are fitting constants that
account for initial state of the qubit and its temperature.
Figure 2.15: Theory curves for 1/f noise induced dephasing time T?? (dashed lines), for two differ-
ent values of EJ , and EL/h = EC/h = 1 GHz. Solid lines correspond to the qubit frequency for
given EJ . Singularity at the sweet spot is where first order flux noise dephasing rate becomes 0,
and coherence time T2 is mainly limited by energy relaxation with upper limit 2T1.
At HFQ (?ext = 0.5?0), the qubit frequency becomes insensitive to 1/f flux noise to first order,
38
since:
??01
= 0, (2.51)
??ext
which leads to ??? = 0. At this flux point, which we nickname ?the sweet spot?, T2 should become
mainly limited by the energy relaxation, setting the upper limit at 2T1. Note that there is also
a sweet spot at IFQ, but we do not operate fluxonium as a qubit there since the qubit transition
frequency will be many times higher than at HFQ, and therefore maximally coupled to dielectric
loss. Figure 2.15 shows how varying circuit parameters affects the qubit?s sensitivity to 1/f noise.
LoweringE ?J will results in a ?sweeter? sweet spot, where T? drops off less rapidly when detuning
the external flux from 0.5?0. Remarkably, even flux points off sweet spot, T?? ? 1? 10 ?s, which
is important for certain flux-gate schemes [113]. This is again attributed to the high inductance of
fluxonium.
2.2.4.4 Thermal Photons
Thermal photons in the readout resonator are another source of pure qubit dephasing [109]. In
the low photon number nth limit, the dephasing rate due to residual thermal photons reads [107]:
n ??2th th? = 01? , (2.52)?2 + ?201
where ? and ?01 are the resonator linewidth and the dispersive shift of the qubit transition, re-
spectively. The average number of thermal photons nth in the resonator at frequency ?r obeys the
Bose-Einstein distribution:
1
nth = ~? . (2.53)r
e kBT ? 1
The resonator frequency we readout with in our experiments is ?r = 2? ? 7.54 GHz. The base
temperature of our experiments is set at 10 mK, although the sample itself is likely at a higher
temperature due to imperfect thermalization. The Bose-Einstein distribution is plotted in figure
2.16 as a function of temperature for the resonator frequency ?r.
39
Figure 2.16: Bose-Einstein distribution for the average thermal photon number in the resonator at
?r = 2? ? 7.54 GHz.
Figure 2.17: (a) Dephasing time due to thermal photons T th = 1/?th? ? as a function of ?01 for fixed
?/2? = 18 MHz, the value for the readout resonator used in the main experiments. (b) Same as
(a) but as a function of ? with fixed ?01/2? = 1 MHz. In both plots, we vary nth over the same
three values with their corresponding temperature shown.
Examining the curves in figure 2.17, we can deduce favorable readout and resonator param-
eters to limit the pure dephasing due to residual resonator photons. Though 20 mK is readily
achievable in modern day dilution refrigerators, nth?s corresponding to temperatures in excess of
50 mK have been reported in the superconducting circuits community [111, 107]. Increasing T th?
40
remains an ongoing effort, either by lowering nth via improved thermalization and line-filtering, or
by optimizing resonator and qubit parameters.
2.3 Coupled Fluxonium Circuits
To scale up to a fluxonium-based quantum processor, we must of course couple individual
circuits. In this section we outline the theory behind two coupled fluxonium circuits. We will also
present how to couple multiple fluxoniums for analog quantum simulations of a 1D spin-chain.
2.3.1 Capacitive Coupling
Consider two fluxonium circuits that are coupled via a mutual capacitance CM , depicted in
figure 2.18. We write down the Lagrangian L(?A,?B, ??A, ??B) = LK(??A, ??B) ? LU(?A,?B),
Figure 2.18: Circuit diagram of two flunoniums coupled via mutual capacitance CM , labeled flux-
oniums A and B. Charge operators Q?A,B = 2en?A,B and flux operators ??A,B = ?0??A,B are depicted
as well.
where the ??s are the magnetic fluxes through each inductor in the two circuits, analogous to
position for mechanical oscillator. We also write the Lagrangian in terms of the kinetic energy
term LK , and potential energy term LU . We will restrict our attention to the kinetic energy term
in the Lagrangian, LK(??A, ??B) since the coupling is arising from the shared capacitance. The
resulting Lagrangian will have a potential energy term LU(?A,?B) equivalent to the sum of the
two uncoupled fluxonium circuits.
L 1 1 1K(??A, ??B) = C ??2A A + C ??2 2B B + CM(??A ? ??B) (2.54)2 2 2
41
To find the Hamiltonian, we apply the Legendre Transformation:
?
H ?L= ??A,B ? L. (2.55)
???
A,B A,B
We then write the Hamiltonian in terms of the generalized momenta (in this case, charge):
?L
QA = = (CA + CM)??A ? CM ??B
???A (2.56)
?L
QB = = ?CM ??A + (CB + CM)??B
???B
We can solve for the ???s directly by writing this linear system in vector form: Q~ = C?~? , and then
taking the inverse of the capacitance matrix [103]:
?~? = C?1Q~ . (2.57)
Finally, arrive at the Hamiltonian where we have used equation 2.57 to write equation 2.55 in terms
of the generalized momentaQA, QB, and then switching back to the dimensionless charge number
n? and reduced phase ?? operators:
H? = 4EAn?2 1 A 2 A 1C A+ EL ??A?EJ cos(?? ??A B 2 B 2 B BA2 ext
)+4EC n?B+ EL ?? ?E2 B J
cos(??B??ext)+JC n?An?B.
(2.58)
The capacitances ?CA,B associated with
A,B 2 ?EC = e /2CA,B are now:
? CACB + CACM + CBCM
CA = CB + CM (2.59)
? CACB + CACM + CBCM
CB = CA + CM
And the coupling constant JC is:
2 CMJC = (2e) . (2.60)
CACB + CACM + CBCM
42
In the weak coupling limit, where CM << CA, CB, we get:
?
CA ? CA
?
CB ? CB (2.61)
CM
JC ? (2e)2 .
CACB
Following the same method in section 2.2.1, we numerically diagonalize the two-qubit Hamilto-
nian 2.58 at each external flux point. In the succeeding figure, we plot the spectra of identical fluxo-
nium circuits (EA/h = EBJ J /h = 4 GHz,E
A/h = EBC C /h = E
A
L /h = E
B
L /h = 1 GHz, ?
A B
ext = ?ext)
for varying values of coupling strength JC to gain intuition on the characteristics of a capacitively-
coupled fluxonium spectrum. More information on capacitively coupled fluxonium qubits and
potential two-qubit gating schemes can be found in references [76, 75].
43
Figure 2.19: Spectrum across one flux period for two capacitively coupled fluxoniums with varying
coupling strength JC . First five transitions starting from the ground state are shown. Since the
qubits couple via the operator n?An?B, the level repulsion is much greater when the single qubit
charge matrix elements |n?01| is large (see figure 2.8). This is why at IFQ, the interaction is much
greater than at HFQ.
2.3.2 Inductive Coupling
Now consider two fluxonium circuits coupled via a mutual inductance LM , depicted in figure
2.20.
44
Figure 2.20: Circuit diagram of two flunoniums coupled via mutual inductance LM , labeled fluxo-
niums A and B. Charge operators Q?A,B = 2en?A,B and flux operators ??A,B = ?0??A,B are depicted
as well.
In the same approach we took for the capacitive case, we will now restrict our attention to the linear
terms of the potential energy in the Lagrangian, LlinU (?A,?B), since the coupling is arising from
the shared linear inductance. The resulting Hamiltonian will have a kinetic energy term equivalent
to the sum of the two uncoupled fluxonium circuits, and the same goes for the nonlinear terms in
the potential arising from the Josephson junctions. Note that:
?2 ? ? ?A ?2 ? ? ?BLnonlin A BU (?A,?B) = 0 cos( ext ) + 0 cos( ext ), (2.62)LA ? LBJ 0 J ?0
where LU(? ,? ) = LnonlinA B U (?A,?B) + LlinU (?A,?B).
2
Llin (?A ? ?M) (?B ? ? )
2
M ?
2
U (?A,?B) = ? ? ? M (2.63)2LA 2LB 2LM
Which is equivalent to:
2 2 2
Llin ? ?A ?A?M ? ?B ?B?MU (?A,?B) = + + ?
?M? (2.64)2LA LA 2LB LB 2L
where L? is the equivalent inductance of the three inductances in parallel:
1? 1 1 1= + +L LA LB LM
? (2.65)LMLALBL = .
LMLA + LMLB + LALB
It follows from Kirchhoff?s circuit laws that the current flowing through LM must equal the sum
45
of the currents flowing through LA and LB, which leads to the relation:
?M ?A ? ?M ?B ? ?M
= +
LM ?LA LB (2.66)L
?M = (LB?A + LA?B).
LALB
Upon plugging equation 2.66 into equation 2.64, we arrive at the simplified linear terms of the
potential energy in the Lagrangian:
?
lin (LA ? L ) 2 (LB ? L?) L?LU (? 2A,?B) = ? ?A ? ?2L2 2L2 B + ?A?B. (2.67)A B LALB
The full Lagrangian is:
L 1 1(? ,? , ?? , ?? ) = C ??2 + C ??2 + Llin(? ,? ) + LnonlinA B A B A A B B U A B U (?A,?B). (2.68)2 2
Upon applying the Legendre transformation (equation 2.55), we write the Hamiltonian in terms
of the generalized momenta: QA = CA??A, QB = CB??B. We also convert charge and flux back
to their dimensionless quantum operators: QA,B = 2en?A,B and ?A,B = ?0??A,B. Finally, the
Hamiltonian is:
H? = 4EA 1 1C n?2A+ EA??2L A?EAJ cos(?? ??A BA ext)+4EC n?2 + EB??2 ?EBB L B J cos(??B??Bext)?JL??A??B.2 2
(2.69)
The inductances ?LA,B associated with E
A,B ?
L = ?
2
0/2LA,B are now:
L2?
L = AA L ? L?A (2.70)
? L2
L BB = LB ? L?
And the coupling constant JL is:
L?
JL = ?
2
0 . (2.71)LALB
In the weak coupling limit, where LM << LA, LB, we get:
?
LA ? LA
?
LB ? LB (2.72)
? 2 LMJL ?0 .LALB
46
In the succeeding figure, we plot the spectra of identical fluxonium circuits (EAJ /h = E
B
J /h =
4 GHz, EAC/h = E
B
C /h = E
A
L /h = E
B
L /h = 1 GHz, ?
A
ext = ?
B
ext) for varying values of coupling
strength JL to gain intuition on the characteristics of an inductively-coupled fluxonium spectrum.
Figure 2.21: Spectrum across one flux period for inductively coupled fluxoniums with varying
coupling strength JL. First five transitions starting from the ground state are shown. Since the
qubits couple via the operator ??A??B, the level repulsion is much greater when the single qubit
phase matrix elements |??01| are large (see figure 2.8). This is opposite the capacitive case, where
now the level repulsion of the qubit transition is much greater at HFQ than at IFQ.
47
2.3.3 Fluxonium-Based Spin Chain
In this section, we present a model for simulating many body spin-1/2 systems with fluxonium;
the Tranverse-Field Ising Model (TFIM) [46], in particular. At HFQ, the qubit transition of fluxo-
nium is well-approximated by a spin-1/2 system, due to its extraordinarily high anharmonicity. A
typical transmon anharmonicity |?| ? 0.1, while fluxonium is over an order of magnitude higher.
Flux qubits also have high anharmonicity, but their coherence time greatly degrades when the
external magnetic flux bias is detuned from HFQ.
Near HFQ, the qubit transition of fluxonium can be approximated as a spin-1/2 by truncating
the fluxonium Hamiltonian 2.21 to:
H?spinfl = hz??z + hx(?ext)??x
1
hz = ~?01 (2.73)
2
hx(?ext) = hx|?ext ? ?|.
Recall that ?ext = 2?(?ext/?0) is the dimensionless external magnetic flux threading the fluxo-
nium loop. The truncated fluxonium Hamiltonian described a spin-1/2 in a magnetic field with
z and x components. We now work in the basis of the first two fluxonium eigenstates at HFQ.
At HFQ, hx = 0, and the Hamiltonian is simply that of a spin-1/2 with transition frequency ?01,
which is obtained by diagonalizing the exact fluxonium Hamiltonian. As a rule of thumb, this ap-
proximation is very good for anharmonicity |?| > 1 and at external flux in the proximity of HFQ
? (0.3? 0.7)?0.
Figure 2.22 shows the fluxonium qubit transition as a function of external flux for circuit param-
eters EJ/h = EC/h = 5 GHz, EL/h = 0.5 GHz. The corresponding hz, hx are then calculated
and the qubit transition using the truncated Hamiltonian is also plotted as a function of flux. An-
harmonicity is also plotted, showing that it reaches a maximum at HFQ. In the vicinity of HFQ, the
truncated and true fluxonium Hamiltonians yield qubit frequencies that are in excellent agreement.
48
Figure 2.22: Qubit transition of fluxonium from diagonalizing the exact fluxonium Hamiltonian
with parameters EJ/h = EC/h = 5 GHz, EL/h = 0.5 GHz (blue curve) and the corresponding
truncated spin-1/2 Hamiltonian (dashed red curve) for ?01/2? = 2.48 GHz and hx/h = 1.20 GHz.
Anharmonicity ? is plotted in green, corresponding to the right y-axis.
Now consider a chain of N identical, coupled fluxonium qubits. Truncating each individual
fluxonium?s Hamiltonian into the spin-1/2 Hamiltonian, we arrive at the Hamiltonian for a 1D spin
chain:
N??1 N??2
H?chainfl = h ??iz z + h (? ix ext)??x + Jxx ??ix??i+1x
i=0 i=0
1 (2.74)
hz = ~?01
2
hx(?ext) = hx|?ext ? ?|,
where Jxx is the inter-spin coupling strength. At HFQ, where ?ext = ?, the Hamiltonian resembles
the Tranverse-Field Ising Model [80]:
N??1 N??2
H?chainfl = hz ??iz + Jxx ??i i+1x??x
i=0 i=0 (2.75)
1
hz = ~?01.
2
To realize this Hamiltonian with fluxoniums, we inductively couple 10 of them (N = 10), one
of which is coupled to a resonator for readout. We choose inductive coupling since it is easily
49
controlled by selecting the fraction of shared superinductance array junctions each fluxonium loop.
Figure 2.23: Circuit diagram for a section of the fluxonium-based spin chain. We assume the exter-
nal flux threading each loop is the same, and that each fluxonium?s circuit elements are identical.
The inter-qubit coupling is achieved by shared superinductance array junctions, giving rise to a
mutual inductance LM .
Since we assume identical fluxonium circuits across the chain of 10, the coupling constant Jxx
reads (from equation 2.71 with all inductances equal):
Jxx = |?0?
LM
1|??|0?|2 , (2.76)
L2 + 2LML
where the qubit phase matrix element at HFQ |?1|??|0?|2 comes from the fact that we are using
??i ??i+1x x for the inter qubit coupling term rather than ??i??i+1 in the full fluxonium Hamiltonian. The
truncated fluxonium Hamiltonian is in the basis of the |0?, |1? fluxonium eigenstates at HFQ. As a
reminder, L in equation 2.76 is the total unshared inductance of each fluxonium.
When Jxx = hz for N ? ? (the thermodynamic limit), the ground state becomes two-fold
degenerate, and the system undergoes a quantum phase transition [97, 43, 14]. For finite number
of spins N , there is technically no phase transition, although the transition energy between ground
and first excited state becomes extremely small, and we therefore recover signatures of the phase
transition. These signatures have been predicted to occur in long chains of inductively coupled
fluxoniums [73], and similar experiments have been performed on two inductively coupled flux-
oniums [57]. For a quantitative analysis on the critical spin-spin coupling strength for a phase
transition, we plot the transition energy between the ground and first excited states of the Ising
Hamiltonian 2.75 versus number of spins N (figure 2.24a), and relative coupling strength Jxx/hz
(figure 2.24b).
50
Figure 2.24: (a) The ground to first excited state transition frequency versus of number of spins N in
the chain for varying Jxx/hz. For larger Jxx/hz, the transition frequency exponentially converges
to 0 faster. (b) Same quantity now verses Jxx/hz for number of spinsN = 10. As Jxx/hz increases,
the transition frequency converges to 0, although never reaching it due to the finite value ofN . The
convergence to 0 transition frequency is the signature of a quantum phase transition.
We now numerically compute the spectrum of Hamiltonian 2.74 for various relative coupling
strengths and N = 10 spins. For simplicity, we choose the individual qubit frequencies at HFQ to
be ?01/2? = 2 GHz so that the value of Jxx/hz will be equivalent to Jxx/h in GHz.
Figure 2.25: Spectrum of the Hamiltonian 2.74 for N=10 spins, ?01/2? = 2 GHz, hx/h =
1.59 GHz, and Jxx = 0. Red, blue, and green curves correspond to single, double, and triple
spin flips, respectively. Black dashed cure is the ground state energy.
51
Figure 2.25 shows the transition energies and eigenenergies in units of GHz for the trivial case of
zero spin-spin interaction. Only the first three bands of excitations are plotted, the first band being
single spin flips (red curves), the second being double flips (blue curves), and the third band being
triple spin slips (green curves). Any combination of flipped spins has the s(am)e energy, therefore
all transitions in a given band have the same energy. For N spins, there are N transitions within
M
the M th band. For N = 10, there are 10 transitions in the first band, 45 in the second, and 120 in
the third.
Figure 2.26: Spin chain spectrum now with the spin-spin interaction turned on, for relative coupling
strengths Jxx/hz = 0.2, 0.5.
Figure 2.26 now displays the spin chain spectrum for weak and intermediate relative coupling
strengths: (Jxx/hz = 0.2, 0.5, respectively). The degeneracy of intra-band transitions is lifted. In
52
the case of Jxx/hz = 0.5, the bands begin to hybridize, and it is no longer appropriate to class
these transitions within a separate band. There are more transitions and energies above the third
band (green curves), but they are not plotted for brevity.
Figure 2.27: Spin chain spectrum in the strong relative coupling regime, for Jxx/hz = 1.0, 2.0.
The bands have now hybridized with one another, and the ground to first excited state transition
energy is approaching 0, the signature of a quantum phase transition. Notice that for Jxx/hz = 2.0,
the ground state and first excited state level energies nearly overlap one another.
Finally, figure 2.27 shows the spectra for strong relative coupling strengths Jxx/hz = 1.0, 2.0.
For Jxx/hz = 2.0, the bands are fully hybridized, and at HFQ, the ground to first excited state
transition energy is nearly zero. Due to the finite number of spins, this transition energy will never
equal 0, but its convergence to 0 as the coupling strength increases is the signature of a quantum
53
phase transition that we seek to replicate in physical devices.
54
CHAPTER 3
Experimental Methods
I do not feel obliged to believe that the same God who has endowed us with sense, reason, and
intellect has intended us to forgo their use.
Galileo Galilei
3.1 Cryogenic Setup
All the experiments reported in this thesis were conducted in a BlueFors LD400 cryogen-free
dilution refrigerator (DR). The samples were bolted to the base plate of the DR at approximately
10 mK.
3.1.1 Single-Qubit Experiment
Figure 3.1 shows the wiring diagram inside the DR used to conduct the single qubit fluxonium
experiments. The setup above the 10 mK stage is essentially identical to the spin chain experimen-
tal setup. The actual DR is also depicted. 20 dB attenuators are bolted to the 4K, 100mK, and 10
mK stages to thermalize the measurement line. There is also a home-made low pass filter (labelled
as cryogenic filter) bolted to the 10 mK stage, which has an attenuation of approximately 20 dB at
the resonator frequency of ? 7.5 GHz. Homemade eccosorb CR110 low pass filters [41, 39] are
also thermalized to the 4 K, 800 mK, and 10 mK stages, which cut high frequency radiation from
entering the measurement line, potentially causing quasiparticle excitations in the sample. With
that in mind, a 12 GHz K&L low pass filter is also added at the 10 mK stage.
On the readout line, two 4-12 GHz circulators are placed immediately after the sample to pre-
vent any backaction to the device. There is another K&L low pass filter to protect against high
frequency radiation, and finally a high-electron-mobility transistor (HEMT) at the 4 K stage is
used to amplify the readout signal. To control the external flux bias, a superconducting coil made
of niobium is placed beneath the sample and is thermalized to the 40 K and 4 K stages. The coil,
55
along with the sample, is housed in a cryoperm magnetic shield, to protect against any spurious
fields that can affect the flux-sensitive qubit.
Figure 3.1: Cryogenic setup diagram for single qubit experiments, with the actual DR and stages
highlighted to the right. The wiring diagram is not drawn to scale.
56
Figure 3.2: (a) 3D copper (Cu) cavity used as the readout resonator, with qubit chip mounted. The
cavity is sealed shut with indium. (b) Mounted cavity with external magnetic flux bias coil. (c)
Cryoperm shield housing the mounted cavity and coil depicted in (b).
The input (driving line) mainly consists of a stainless steel coaxial cable. Stainless steel is a
poor thermal conductor, which is important since we want to minimize the thermal link between
the temperature stages of the DR. Below the base plate at 10 mK, only copper (Cu) coaxial cables
are used to maximize the thermal link to the base plate. The output (readout) line consists of a
Copper-Nickel (Cu-Ni) alloy coaxial cable, which again limits thermal conductivity between the
stages, but also introduces minimal losses to the readout signal.
3.1.2 Spin Chain Experiment
Above the 10 mK stage, the experimental setups for the single qubit and spin chain experiments
in the DR are the same, except for the removal of the low pass filters. In the spin chain experiments,
we only focused on spectroscopy and therefore did not require high coherence. Instead of a 3D
cavity, the chip is housed in a 3D Cu waveguide, and the readout resonator is fabricated on the chip
with the device. Unlike the single qubit experiments, the spin chain device is inductively coupled
to the readout resonator, rather than capacitively coupled. This is necessary to ensure that each
fluxonium circuit has the same circuit parameters. A directional coupler (DRC) is used to send the
measurement signal into the waveguide, which subsequently reflects the readout signal through the
DRC and out through the readout line.
57
Figure 3.3: Diagram of the setup inside the DR for the fluxonium-based spin chain experiments.
Right figure shows the configuration below the 10 mK stage.
Figure 3.4: Left: Cu waveguide with a 3dB cutoff frequency of 6 GHz, with sample mounted.
Right: Transmission signal of the waveguide versus frequency.
58
3.2 Room Temperature Electronics
Figure 3.5: The room temperature electronics diagram for both experiments. We use a heterodyne
detection scheme to readout the qubit state, where a local oscillator (LO) is mixed with the readout
frequency down to an intermediate frequency (IF) which is then digitized by an analog-to-digital
converter. The readout tone is split in two: one part goes into the fridge for qubit readout (chan-
nel A), while the other goes directly to the digitizer which serves as a reference signal (channel
B). The pulses for qubit control and readout are generated by RF sources that are equipped with
IQ-modulation, and the shape of the pulses themselves are controlled by an arbitrary waveform
generator (AWG). All instruments are synchronized to a 10 MHz clock.
3.2.1 RF Controls
The radio-frequency (RF) tones used to drive the qubit and other fluxonium transitions are gen-
erated by Rohde & Schwarz SGS100A SGMA IQ vector sources (labelled as Qubit RF, Qubit 2
RF, and Readout RF in figure 3.5). We typically use a second RF tone to drive either the qubit or
higher transitions for more complex experiments. A Tektronix 5014C arbitrary waveform gener-
ator (AWG) shapes the output tones from the RF sources into the pulses used to control the qubit
and do readout. It operates at a sampling rate of 1 GSa/s, giving us a pulse resolution of 1 ns.
59
Typical time duration of qubit drive pulses is 10-1000 ns. The Readout and Qubit 2 RF tones are
combined via a Minicircuits room temperature RF combiner, and then combined with the qubit RF
tone via the -20 dB decoupled port of a directional coupler before entering the fridge. We choose
a directional coupler to minimize the insertion loss of the Qubit RF tone, to allow for maximal
driving power and therefore minimal qubit gate time.
3.2.2 Readout
Placement of qubits in 3D cavities has been shown to greatly improve coherence [81, 88] due
to the increased isolation of the electromagnetic environment. For the single-qubit experiments,
we use the fundamental mode of a 3D Cu cavity for readout. For the spin chain experiment, we
use an on-chip resonator for readout, the details of which will be discussed in the experimental
section. The room temperature electronics for both experiments are the same. We use transmission
measurements to determine the state of the cavity, with an input port and an output port which
connect to the coaxial cables of the driving and readout lines. Control and readout of the qubit
is done wirelessly, where each port has a pin that capacitively couples to the circuit antenna. To
determine the characteristics of the input and output cavity pins prior to a cooldown, we take
reflection measurements of the cavity using each port, and fit the resulting reflected signal S11(?)
to:
2i(? ? ?r)/?r ? 1/Qext + 1/Qint
S11(?) = , (3.1)
2i(? ? ?r)/?r + 1/Qext + 1/Qint
where ?r is the resonance frequency and Qext, Qint are the external and internal quality factors of
the resonator. Qext is, in general, a complex number. The length of the pin determines the real part
of Qext. During cavity preparation, we systematically cut the length of the pins until we reach the
desired Qext. Qext for the input pin is typically 2000, and for the output pin it is around 500. We
want a lower Qext for the output port so that the readout signal predominantly exits through the
readout line.
Once the resonator-fluxonium system is cooled down, the readout RF tone is generated and
split in two. One part is mixed with the LO (generated by a HP 8763B signal generator) to an
IF frequency of ?IF = 50 MHz. This signal is then sent to channel B of the analog-to-digital
converter (ADC). For the ADC, we use an AlazarTech ATS9870 Card, which is integrated with
the measurement computer. The other part of the readout signal is sent into the fridge, and once it
emerges from the fridge output, it is mixed with the LO to form another IF signal, which is sent
to channel A of the the Alazar card. The signals SA.B(t) emerging from the fridge, and reference,
respectively are demodulated into I (in-phase) and Q (quadrature) components:
60
?
1 t+TRO ? ? ?
IA,B(t) = ? cos(?IFt )SA,B(t )dtTRO t (3.2)
1 t+TRO ? ? ?
QA,B(t) = sin(?IFt )SA,B(t )dt
TRO t
which are the I and Q signals averaged over the integration time which we set to equal the duration
of the readout pulse, TRO.
3.2.3 Software
We use Labber to control all instruments and perform data acquisition. For data processing and
measurement automation, the Labber python package is used. All data analysis and experimental
simulations are performed using the NumPy, SciPy, and QuTip python packages.
3.3 Device Fabrication
All devices reported in this work were fabricated on high resistivity 330 ?m-thick silicon (Si)
and 430 ?m-thick sapphire, diced into 9 ? 4 mm chips. We will briefly outline the fabrication
steps in this section. For the comprehensive fabrication recipe, see Appendix A. All recipes for
the quantum circuits fabricated in our lab employ the Dolan bridge method, also known as the
shadow evaporation technique. This fabrication method allows small, overlapping structures to be
fabricated with very high accuracy and reproducibility [30].
At the heart of the Dolan bridge technique is electron beam lithography (EBL). In EBL, a
polymer called a resist is put on the surface of the substrate that the device will be fabricated
on. This polymer is then exposed to high energy electrons in the desired pattern of the device.
The regions of resist that were exposed to the electrons become soluble in a chemical known as a
developer. Upon placing the substrate into the developer, the electron-exposed regions of the resist
are washed away, yielding what is essentially a stencil of the device, which we call the mask. Once
a mask is obtained, the material that one wants their device to be made of is deposited onto the
substrate. The remaining resist is then removed (we do this with acetone) and one is left with only
the device on the substrate.
In the Dolan bridge method, two layers of resist are used. The bottom layer is known as the
overlay layer, and the top, the main layer. The two layers need different doses, i.e. critical amount
of electrons required to make the resist soluble in the developer. Generally, the overlay dose is
much lower than the main dose (about 10% of the main dose). By exposing the bilayer resist to
the main dose, both layers will be removed, but by exposing it to the overlay dose, only the bottom
layer will be removed. By selectively applying main and overlay doses to different regions of the
61
resist bilayer, bridges can be made with the top layer of resist. Once the we have a mask with these
bridges, the material for the device is deposited at two angles, allowing tightly packed structures
to be fabricated with ease at the sub-micron scale.
3.3.1 Silicon Fabrication Recipe
The recipes for fabricating devices on Si and sapphire differ slightly. We will first present the
recipe for Si since it is simpler. We follow the steps shown in figure 3.6:
1. Clean 9 ? 4 mm Si chip by sonicating in acetone for 3 minutes, then isopropanol (IPA)
for 3 more minutes. Optional: 1 minute dip in water-ammonium flouride-hydroflouric acid
solution prior to step 1.
2. Spin overlay resist later (MMA EL 13) at 5000 rpm for 1 minute.
3. Bake the overlay resist on hotplate at 180?C for 1 minute.
4. Spin main resist layer (950 PMMA A3) at 4000 rpm for 1 minute.
5. Bake the main resist on hotplate at 180?C for 30 minutes.
6. Main dose electron beam exposure using 100 kV electron beam at a current of 1 nA,
7. Overlay dose electron beam exposure using 100 kV electron beam at a current of 1 nA,
8. Mask is developed for 2 minutes in a 3:1 IPA:DI solution at 6?C. Chip is lightly shaken back
and forth by hand at around 1-2 Hz while in the developer.
9. First Al deposition: 20 nm is deposited at 1 nm/s at an angle of 23.83?
10. Chip with first Al layer.
11. In-situ oxidation at 100 mBar for 10 minutes to grow aluminum oxide.
12. Second Al deposition: 40 nm is deposited at 1 nm/s at an angle of ?23.83?
13. Chip is bathed in acetone for 3 hours at 60?C for resist liftoff. Then it is sonicated in the
acetone for 5 seconds, followed by 10 seconds of sonication in IPA. Finally, chip is blown
dry with N2.
62
Figure 3.6: Fluxonium fabrication recipe
3.3.2 Sapphire Fabrication Recipe
Sapphire is an insulator, while Si is a semiconductor. This means that during an electron beam
exposure, charge will accumulate on the surface of the sapphire chip. Si provides sufficient electri-
cal conductivity to provide access to ground for the electrons, while sapphire does not. To remedy
this, an anti-charging layer of Al must be deposited on the surface of the chip with resist prior to
63
the electron beam exposure. This layer is grounded and serves as an equipotential surface for the
electrons for the duration of the electron beam writing. In our recipe, we deposit 11 nm of Al at
1 nm/s after step 5 and before step 6 in figure 3.6. After step 7 in figure 3.6, this anti-charging
layer must be removed prior to step 8. We do this by etching the Al in a 0.1 M aqueous solution of
potassium hydroxide (KOH) for about 30 seconds. Aside from these extra steps, the sapphire and
Si fabrication recipes are the same.
64
CHAPTER 4
Fluxonium Experiments
To myself I am only a child playing on the beach, while vast oceans of truth lie undiscovered
before me.
Isaac Newton
In this chapter, we present a case study of a single-qubit fluxonium device (labelled Qubit J)
with coherence and relaxation times in excess of 1 ms. We start by outlining tune up experiments
to characterize the system, such as spectroscopy and time-domain measurements. We then con-
duct experiments aimed at characterizing the predominant loss and decoherence mechanisms using
these time domain measurements. This includes, for the first time, a measurement of the energy
relaxation time of the parity-protected |0? ? |2? transition to establish bounds on the quasiparticle
density in this device. We conclude our case study by measuring the error rates of quantum gates on
the qubit, and study their main limiting factors. In the last section of this chapter, we present novel
work on a network of inductively coupled fluxonium circuits used to simulate the Transverse-Field
Ising Model. We characterize one of these devices and fit its spectrum to the TFIM Hamiltonian.
4.1 One-Tone Spectroscopy
At the start of all device cooldowns, we characterize the readout resonator using one-tone spec-
troscopy. We simply send a single RF signal to the cavity anchored to the base plate of the fridge,
sweeping its frequency, and measure the magnitude of the transmitted signal. This signal is fit to a
Lorentzian to extract the resonance frequency ?r and linewidth ?:
(?/2)2
f(?) = A + y , (4.1)
(? ? ? )2 0r + (?/2)2
where A, y0 are fitting constants.
65
Figure 4.1: One-tone spectroscopy measurement of the cavity resonance used for qubit readout.
Red line is the fit to a Lorentzian, with ?r/2? = 7.5417 GHz and linewidth ?/2? = 18.1 MHz.
Figure 4.2: Left: One-tone spectroscopy measurement of the cavity resonance across one flux
period. Circuit energy transition crossings are observed and fitted to the fluxonium Hamiltonian
with the additional circuit-resonator coupling term. Right: Close up of the anti-crossing between
the |0? ? |2? transition and the resonator. The fit yields a circuit-resonator coupling rate g/2? =
50 MHz.
66
These one-tone measurements are performed across a full flux period (depicted in figure 4.2),
which displays the interaction of the resonator with the circuit. The data is fit to the fluxonium
Hamiltonan 2.21 with the additional resonator term ~?r(a??a?+ 1/2) and the circuit-resonator cou-
pling term ?gn?(a? + a??). Because we work in the dispersive regime of cQED, the additional res-
onator terms are not necessary for fitting fluxoniums?s spectrum. The bare fluxonium Hamiltonian
is sufficient, as we will show in the next section.
4.2 Two-Tone Spectroscopy
As the name suggests, two-tone spectroscopy involves two RF tones: one to excite a fluxonium
transition, and the other for readout. Sweeping the frequency of the first tone allows us to perform
spectroscopy of the circuit?s transitions.
Figure 4.3: (a) Two-tone spectroscopy measurement of the qubit transition at ?ext = 0.501?0. (b)
Diagram of pulses involved in two-tone spectroscopy. Each RF source is controlled by a different
AWG channel, which sets the shapes and timing of pulses. An RF tone of varying frequency aimed
at exciting a fluxonium transition is followed by the readout tone. Typically the first tone uses a
Gaussian edge for improved spectral purity, while the readout tone is a square pulse.
We can also sweep the external flux, and do two-tone spectroscopy at varying flux points, ob-
serving the changing transition lines. By doing this, we can fit the transition lines to the bare
fluxonium Hamiltonian to extract the circuit parameters. For improved precision, we used four of
the transitions in the spectroscopy data to fit the data to theory (figures 4.4 and 4.5): |0? ? |1?,
|0? ? |2?, |0? ? |3?, and |1? ? |2?. Note that the transition lines in the spectroscopy data must be
accurately identified for the fit to be successful.
67
Figure 4.4: Two-tone spectroscopy over the full spectrum. Using the |0? ? |1?, |0? ? |2?, |0? ?
|3?, and|1? ? |2? transitions, we fit to the bare fluxonium Hamiltonian 2.21. Dotted lines are the
spectrum for the given transitions from the fit: EJ/h = 5.571 GHz, EC/h = 1.083 GHz, EL/h =
0.638 GHz. The fit error for all three values is below 0.001 GHz.
68
Figure 4.5: Close up of the four fitting transitions near the sweet spot. The fit is the same as in
figure 4.4.
4.3 Single Shot Readout
Single shot readout is the process of driving the resonator and making histograms out of the
resulting I and Q signals [65, 59]. This is in contrast to typical experiments involving thousands
of ?shots?, which are then averaged to yield averaged demodulated I and Q signals. Single shot
measurements yield a Gaussian distribution of I and Q measurement statistics, which is the Husimi
Q function [44, 19] of the coherent cavity field. Each Gaussian distribution observed corresponds
to a state of the circuit. Once this data is obtained and the corresponding states are identified, it is
possible to determine the qubit state in a single ?shot?, since it is now known which I, Q signals
correspond to given qubit states.
A qubit at finite temperature T will have thermal state populations pth0 , p
th
1 determined by clas-
69
sical Boltzmann statistics:
1
pth0 = ? ~?01
1 + e kBT
? ~?01 (4.2)
kBT
pth
e
1 = ? ~?
.
01
1 + e kBT
?(x?x )2i
By fitting the Gaussian distributions (f(x) = Aie ?2 ) corresponding to the qubit states, we
determine the qubit populations:
A0
p0 =
A0 + A1 (4.3)
A1
p1 = .
A0 + A1
Figure 4.6: Single shot histogram (bin size of 40) for 15,000 realizations of the experiment (i.e.
15,000 shots). The two distributions are labelled by their corresponding qubit states. Black dashed
?(x?x0)
2 ?(x?x1)
2
line is the fit to the sum of the two Gaussian distributions: f(x) = A 20e ? + A1e ?2 .
Solid lines are the individual Gaussian distributions. The fit yields a measured qubit ground state
population pm0 = 0.558? 0.007.
70
4.3.1 Readout Characterization
We observe that the readout pulse causes transitions between qubit states, an effect also reported
in transmon qubits [90]. To take this spurious effect into account, we calibrate the transition
rate induced by applying a tone at the cavity frequency. In the succeeding figure, we apply two
successive cavity pulses. The first pulse is used to induce the transition between qubit states and
reproduce what is happening during the readout with an increasing duration. The second pulse is
used to readout qubit populations.
Figure 4.7: Measured qubit |0? state population (pm0 ) after the application of a readout pulse of
varying duration. The population is extracted from the single-shot measurement. The dashed line
is an exponential fit to p0(t) with a decay time of 0.204? 0.008 ms.
For simplicity, we will restrict our analysis to the ground state probability p0. In the presence
of cavity photons, p0 reads
? t
? ? TROp0(t) = [p0(0) p0( )]e 1 + p0(?) (4.4)
where p0(0), p0(?) are p0 at time equalling 0 and infinity, respectively, and TRO1 is the qubit
relaxation time in the presence of cavity photons. The signal that we actually measure, pm0 , is the
average population during the duration of the re?adout pulse TRO = 20 ?s, given by:
1 TRO
pm0 = p0(t)dt. (4.5)TRO 0
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Integrating and solving for p0(0), we arrive at:
? p
m
0(? p0(?)p0(0) = p0( ) + )T . (4.6)
TRO ? RO1 1? TROe 1
TRO
The parameters p0(?) = 0.166 ? 0.002, TRO1 = 204 ? 8 ?s, and pm0 = 0.558 ? 0.007 are
determined from the fit in figure 4.7. With these values, we find that the thermal population in
state |0?, before any readout pulse is applied, is p0(0) = 0.58? 0.03, corresponding to an effective
qubit temperature of T = 24? 6 mK (for qubit frequency ?01/2? = 163.06 MHz). The operating
temperature of the fridge was 8 mK. All errors are taken by rounding the resulting fit errors to one
significant figure, and then applying the standard propagation of uncertainties.
To further characterize the readout, we observe the distributions in the IQ plane.
Figure 4.8: Measured (a) and fitted (b) single shot histograms of the readout signal in the IQ plane.
The histograms are calculated from 5,000 experimental ?realizations when the qubit starts in the(x?x )2+(y?y )2i i
thermal state. This distribution is fitted to 2D Gaussians: ?2i=0,1Aie .
In the limit of ?01 << ?, the angle between the center of the two Gaussians (magenta lines
in Figure 4.8) is given in terms of ?01/? = 0.064 ? 0.001, and hence we extract ?01/2? =
1.16? 0.02 MHz (we used ? found from the fit in figure 4.1).
72
4.4 Time-Domain Measurements
4.4.1 Rabi
Measurement of Rabi oscillations between the qubit states is the backbone of quantum gate
characterization. A driving field resonant with the qubit frequency induces transitions between the
qubit states, which can be visualized as the Bloch vector rotating about the x- or y- axis. Consider
a driving signal on resonance with the qubit with I (in-phase) and Q (quadrature) components, in
the rotating frame of the drive [112]:
I = d cos(?d)
(4.7)
Q = d sin(?d)
where d is the driving field amplitude and ?d is the phase offset of the drive. The qubit drive
Hamiltonian then reads:
H? ~?d = (I??x +Q??y) (4.8)
2d
where ? is the Rabi frequency, the rate at which the qubit oscillates between the |0? and |1? states.
I and Q components of the drive therefore correspond to rotations of the Bloch vector about the x-
and y- axes, respectively.
The Rabi frequency is given by (when the drive is on resonance with the qubit):
1
? = d|?0|n?|1?|. (4.9)~
We normally apply a pulse envelope, given by some function f(t) which is typically a Gaussian
(or a square with Gaussian edges), to increase the spectral purity of the pulse, thereby mitigating
unwanted transitions outside of the computational subspace. The angle of rotation of the Bloch
vector is given by: ? t
?
?(t) = ? f(t )dt?. (4.10)
0
The rotation angle can therefore be tuned by changing the Rabi frequency directly (i.e. changing
the driving amplitude) or by changing the duration of the pulse. Typically, we leave the amplitude
fixed and sweep the pulse duration to determine the duration that corresponds to ? = ? and ? =
?/2. These are known as ?? and ?/2? pulses and correspond to a qubit state flip (i.e. a bit flip),
or half a bit flip, respectively.
73
Figure 4.9: (a) Rabi measurement of the qubit at HFQ. Signal is fit to a cosine function (red
curve), yielding a Rabi frequency ? = 2? ? 5.722 ? 0.007 MHz and a ??pulse duration of
t? = 87.4? 0.1 ns. The pulse in this experiment is a 1 ns-width Gaussian-edge square pulse, and
the duration of the plateau is swept. Truncation range of the Gaussian edges is 3-sigmas, so that
the total duration of the ?-pulse is 90.4 ns. (b) Diagram of pulses involved in the Rabi experiment.
It is similar to the two-tone diagram in figure 4.3, but now the qubit drive frequency is fixed and
the duration of the pulse (plateau) is swept. Gaussian edges are not shown to scale.
4.4.2 Relaxation Time T1
The energy relaxation time, T1, is the time scale at which the qubit state reaches equilibrium.
Typically, T1 is measured by exciting the qubit via application of a ?-pulse, and sweeping the delay
time before readout. Averaging the measurement outcomes of an ensemble of measurements yields
an exponentially decaying signal: ? e?t/T1 , where the time constant is T1. In practice, any qubit
driving pulse that forces the qubit from thermal equilibrium is sufficient to measure T1. Recall
that the energy decay rate ?1 = ?? + ??, the sum of excitation and relaxation rates, due to the
temperature of the environment. This results in:
1
T1 = . (4.11)
?? + ??
We can write the thermal excitation and relaxation rates in terms of the thermal populations in
equation 4.2:
?? = p
th
1 ?1 (4.12)
? th? = p0 ?1,
Note that as T ? 0, ?? ? 0. This means that the zero temperature limit, the environment has no
energy to excite the qubit.
74
Figure 4.10: (a) T1 measurement of the qubit at HFQ. Signal is fit to a decaying exponential
function (black curve), yielding a decay time constant T1 = 1.20? 0.06 ms, (b) Diagram of pulses
involved in the T1 measurement. We excite the qubit from the thermal state via application of a
?-pulse, and sweep the delay time before readout.
4.4.3 Coherence Time T2
4.4.3.1 Ramsey Coherence Time: T ?2
The most direct way to measure the coherence time of a qubit is by the Ramsey method. In the
Ramsey method, two ?/2- pulses are applied to the qubit, sweeping the delay time between them.
Readout follows immediately after the second ?/2-pulse. The idea is to put the Bloch vector onto
the equator of the Bloch sphere, and vary the time of free evolution of the Bloch vector, before
sending another ?/2-pulse so that we can measure the z-component of the Bloch vector (recall that
all measurements are performed in the ?z basis). In the coordinate frame rotating at the frequency
of the driving pulse ?d, the measured signal versus the pulse delay time oscillates at the qubit-
drive detuning frequency known as the Ramsey frequency ?? = |?d ? ?01|/2?, yielding Ramsey
fringes. This experiment allows us to perform an extremely precise measurement of the qubit
transition frequency, since we know the drive frequency, ?d. This method was first developed for
characterization of atomic systems, and is ubiquitous today in characterization of superconducting
qubits [17, 101].
Analogous to case of T1, we average these measurements over a statistical ensemble, and if
there is random dephasing, the envelope of the Ramsey fringes will decay exponentially. The time
constant of this decay is the Ramsey coherence time, T ?2 . In practice, slow drifts of the qubit
frequency taking place over the measurement time (on the order of minutes) results in a shorter
T ?2 , and therefore other methods which filter out the low frequency noise are typically used to
75
establish the relevant coherence time from a quantum computing perspective. Filtering out the low
frequency drifts yields a more relevant measure of the coherence time since a quantum algorithm
would generally be run on a much shorter timescale, making these drifts irrelevant.
Figure 4.11: (a) T ?2 measurement of the qubit at HFQ. Signal is fit to a cosine with an exponentially
decaying amplitude (black curve), yielding a decay time constant T ?2 = 1.53 ? 0.15 ms. This is
the highest value for T ?2 ever reported in a superconducting qubit. The fitted Ramsey frequency
is ?? = 2.67 ? 0.15 kHz. (b) Diagram of pulses involved in the T ?2 measurement. We rotate the
Bloch vector by ?/2 so that it is on the equator of the Bloch sphere, and vary the delay time before
applying another ?/2-pulse and reading out.
We can learn a great deal about the stability of the qubit by probing the system with Ramsey
measurements over time. In figure 4.12, we perform an interleaved T , T ?1 2 loop over about 12
hours. Interleaved means that at each time step we perform both measurements before going to
the next one. To speed up the measurement time, we initialize the qubit such that p1 = 0.64, using
a cavity pulse of duration 160 ?s and amplitude 80% of the normal readout pulse. This type of
initialization has been demonstrated in other fluxonium experiments [108, 33, 77], and we verified
that using it does not change the resulting T , T ?1 2 . Other, more efficient, methods for initializing
low frequency fluxonium qubits have also been demonstrated in reference [113]. Using single shot
histograms, we calibrate the readout signal such that it corresponds to the qubit populations.
We also use this measurement to track the fluctuations in qubit frequency over time, and find
that it is confined to within 100 Hz, much narrower than what is commonly reported [53]. By
averaging the data across each of the measurement traces (figure 4.12(b,c)) we demonstrate the
stability in both times, as the average curves have an excellent fit to functions decaying with a
single exponential. The reported value of T ?2 exceeds the state of the art value for both transmons
and fluxoniums by an order of magnitude [79, 83]. This coherence time is attainable thanks to the
76
long average energy relaxation time. We will explore the reasons for the improved relaxation time
in the succeeding sections.
Figure 4.12: (a) Ramsey coherence time T ?2 and energy relaxation time T1, measured simultane-
ously and repeatedly over a period of 12 hours. Lower panel shows the Ramsey fringe frequency
??, the fluctuations of which are contained within 100 Hz. (b) Ramsey fringe data averaged over
the entire 12-hour period. The solid line is the fit to a decaying cosine with characteristic time
T? ?2 = 1.16 ? 0.05 ms. (c) Energy relaxation data averaged across the same 12-hour period. The
solid line is an exponential fit with a time constant T?1 = 1.20? 0.03 ms.
4.4.3.2 Hahn-Echo Coherence Time: TE2
We can filter out the effect of slow drifts in the qubit frequency that take place over the time
scales of the measurement. Consider the Bloch vector after the initial ?/2-pulse in the Ramsey
experiment. In the ensemble of measurements, the Bloch vectors will point in various directions,
rather than the same one, due to phasing. These Bloch vectors will fan out even more over increas-
ing time step. We can ?refocus? the Bloch vectors by applying a single ?-pulse in the center of the
77
two ?/2-pulses. When the measurement is performed, this process will usually result in a signal
decaying exponentially with a larger time constant than compared to the Ramsey method. This
method was developed by the nuclear magnetic resonance community (NMR) and is known as a
Hahn-Echo measurement [38].
Figure 4.13: (a) Initial qubit state with Bloch vector pointing along the z-axis. (b) Various Bloch
vectors over the measurement ensemble after the initial ?/2-pulse. They pick up additional random
phases ??(t) due to dephasing. We assume the additional phases to have a normal distribution, with
???(t)? = 0. Black arrows represent the noisy contributions of ??(t) to the Bloch vector. (c) The
Bloch vectors after the middle ?refocusing? ??pulse. The effect of the random dephasing gets
reversed. We of course assume that t? << TE2 . (d) Bloch vector after the final ?/2-pulse. Average
over the ensemble leads to exponentially decaying vector length.
The coherence time measured by the Hahn-Echo protocol is called T2 echo, which we denote
as TE2 . It is this coherence time that is typically considered the relevant T2, since the drifts that
are being picked up over the course the the measurement time in the Ramsey experiment will not
occur over the much shorter timescale of a quantum algorithm. Due to the extremely high stability
of our system, we found that the TE2 of our qubit was only marginally improved (less than a factor
78
of two) compared to the T ?2 .
Figure 4.14: (a) TE2 measurement of the qubit at HFQ. Signal is fit to a decaying exponential (black
curve), yielding a decay time constant TE2 = 1.75 ? 0.25 ms, (b) Diagram of pulses involved in
the TE2 measurement. The total delay time is ? , while the free evolution times between the pulses
is ?/2.
Figure 4.15: An interleaved loop of T1, TE2 , and T
?
2 measurements over ? 60 hours. Each loop
index took approximately 15 minutes. T ?2 measurement is performed to track the qubit frequency.
The lab directly below ours was sweeping a magnetic field (sweep rate ? 0.5 mT/s). The times of
active sweeping coincide with the changing Ramsey frequency ??. The field is swept, and during
the sweep this extra field reaching our qubit sweeps ?ext through HFQ, where TE2 ? 2T1. The
peaks in TE2 correspond to minima in ?? (28 kHz). The qubit drive frequency was 163.04 MHz.
Off the sweet spot, the coherence time is still typically ? 0.3 ms in this experiment.
79
4.5 Decoherence Mechanism Analysis
In this section we will analyze the main sources of decoherence in this millisecond-coherence
device, Qubit J. Our approach is to perform time domain measurements, and sweep the external
magnetic flux. By observing how T1, T2 change versus ?ext, we extract the loss mechanisms. We
can make further constraints on the loss parameters by measuring the energy decay time of higher
transitions as well. Using the study of previous fluxonium devices in our lab [79], we determine
the main reasons why we were able to reach the millisecond coherence threshold. Our overarching
method for coherence improvement is pushing T1 higher. Over the past 5 years in our lab, T2 has
risen with T1.
Figure 4.16: Summary of the steady improvement in coherence time T2 versus T1 in our fluxonium
qubits over the past 5 years.
4.5.1 Dielectric Loss
At HFQ, we find that the main limit on T1 arises from dielectric loss. Figure 4.17 is taken
from reference [79] with the addition of Qubit J. We also denote the substrate each device was
fabricated on. We plot the distribution of T1 measured at HFQ for each qubit, and normalize
them by multiplying by the matrix element squared. By doing this, we compare the inverse noise
spectral densities associated with dielectric loss across all devices. Most devices were fabricated
80
on silicon without any pre-fabrication treatments to remove the native oxide. A range of dielectric
loss tangent tan ?C = (2.0 ? 3.6) ? 10?6 at 6 GHz explains the measured T1?s of these devices
(yellow dashed lines in figure 4.17a, b). A small phenomenological frequency dependence is added
to the loss tangents for better agreement with the data, and the values are defined at 6 GHz, a typical
transmon frequency: ( )a
?
tan ?C(?) = tan ?C(?/2? = 6 GHz) (4.13)
2? ? 6 GHz
where the power dependence a = 0.15.
Figure 4.17: T1?s of all single-junction fluxonium devices of past study multiplied by their squared
matrix elements at HFQ. Device labels are the same as in reference [79]. According to Fermi?s
Golden rule, this quantity is proportional to the inverse of the noise spectral density S(?01). (a):
Qubit T1?s multiplied by the phase matrix element squared versus qubit frequency ?01. According
to equation 2.43, this is ? 1/?201 tan ?C(1 + coth( ~?01 )). Yellow dashed curves are from simu-2kBT
lation with tan ? ?6C = (2.0 ? 3.6) ? 10 at 6 GHz. Purple dashed curve is the same quantity for
tan ?C = 0.8 ? 10?6 at 6 GHz. (b): Qubit T1?s multiplied by the charge matrix element squared
versus qubit frequency. According to equation 2.42, this is ? 1/ tan ?C(1 + coth( ~?01 )). The two2kBT
representations are equivalent, and they fit to the same loss tangent values. The temperature T is
set to 20 mK. Note that the theory curves include the temperature factor given by equation 2.35,
since we compared them to the measured T1 values. We do not explicitly write it here for brevity.
The fabrication process was upgraded by including a buffered oxide etch (BOE) of the silicon
81
chips during preparation. Qubit J was fabricated on sapphire without any prefabrication surface
treatments. The details of both fabrication procedures are found in Appendix A. Both methods
improved the loss tangent to 0.8? 10?6 at 6 GHz (purple dashed in line figure 4.17a,b). Knowing
from past studies that T1 was mainly limited by dielectric loss, we lowered the loss tangent by
fabrication methods. Further decoupling from dielectric loss was realized by reducing the qubit
frequency and charge matrix elements by engineering the spectrum of Qubit J. Once T1 > 1 ms
was achieved, millisecond coherence naturally followed.
Focusing on Qubit J, we measure T1 and sweep the external magnetic field. Figure 4.18 shows
the measurement of T1 vs. qubit frequency ?01 (top x-axis) and magnetic flux (bottom x-axis).
Each data point has been taken by applying a proper ?-pulse on resonance with the new qubit
frequency, and we have manually checked that each relaxation trace was single-exponential. The
data reveals reproducible fluctuations of T1 in frequency. These fluctuations qualitatively eliminate
out-of-equilibrium quasiparticles [84] as the dominant relaxation mechanism, and rather suggest
absorption by material defects [53]. The local peaks in T1 do follow the theory curve for dielectric
loss with tan ?C = 0.8? 10?6 at 6 GHz.
Figure 4.18: T1 versus magnetic flux (bottom x-axis) and frequency (top x-axis). Color indicates
two separate scans taken 24 hours apart. A reproducible frequency dependence points to losses by
material defects rather than quasiparticles. Purple dashed curve is the simulated T1 limit due to
dielectric loss with tan ? ?6C = 0.8? 10 at 6 GHz.
4.5.2 Quasiparticles
Effects due to out-of-equilibrium quasiparticles do not manifest in the study of qubit T1. QP?s
tunneling across the single JJ do not limit T1 at HFQ, since the relevant matrix element goes to
82
0. QP tunneling events in the JJ array do affect T1 at HFQ (see figure 2.13). The measured value
of T1 ? 1 ms corresponds to xqp ? 5 ? 10?10, on the low end for values typically reported in
superconducting circuits [84, 37]. To better estimate xqp, we study the flux and frequency depen-
dence of the |0??|2? transition relaxation time, T 021 [96]. At HFQ, this transition is party-protected
from dielectric loss, since |?0|??|2?| = |?0|n?|2?| = 0. These relations also lead to protection from
quasiparticle tunneling in the JJ array, leaving QP tunneling across the single JJ as the main cause
of a direct |2? ? |0? transition. Off sweet spot, this transition again becomes mainly limited by
dielectric loss.
Figure 4.19: T 021 versus magnetic flux (bottom x-axis) and frequency (top x-axis). Dashed curves
are the theoretical limit due to dielectric loss with tan ? ?6C = (1.5 ? 4.5) ? 10 at 6 GHz. Solid
magenta curve is the theoretical limit due to quasiparticles tunneling across the single JJ with
x ?9qp = 5 ? 10 . We determine that this value is the upper bound on xqp, as it is likely that T 021
becomes limited by a thermal excitation of |2? ? |3? followed by direct relaxation |3? ? |0?
(see section 4.5.3.2) as we approach HFQ. Details of this experiment are found in the succeeding
section.
4.5.3 Measurement of T 021
In order to measure the relaxation rate of the direct |2? ? |0? transition, we saturate the |1??|2?
transition to even up the populations of states |1?, |2? (p1, p2) for duration ? , while monitoring the
time evolution of the ground state population p0(?) (see Figures 4.20 and 4.21a). The increase
in population of the |0? state is associated with direct relaxation events from states |2? and |1?.
Owing to the large value of T 011 > 1 ms, our measurement is extremely sensitive to low values of
T 02  T 011 1 . We record p0 as a function of the duration of the |1? ? |2? drive (time ? ) and fit it to
83
a decaying exponential. Figure 4.21b shows examples of traces obtained with our protocol. The
measured decay constant of p0 (denoted as Teff ), as well as knowledge of the qubit temperature
and T 011 enable us to determine T
02
1 unambiguously. Note that at each external magnetic flux, we
measure the qubit?s thermal populations pth0 , p
th
1 and T1.
We now outline the 3-level system (qutrit) model used to determine T 021 via relaxation time
of the experimental signal, Teff . Consider a qutrit in the presence of a continuous drive of the
|1? ? |2? transition at Rabi frequency ?12. This system is described by the 3? 3 density matrix ??.
The dynamics of the system is determined by the Lindblad master equation 1.33, with associated
Linbladian operators: ?
L1 = ??? |1??0|
L2 = ??? |0??1|? (4.14)L3 = ?02 |0??2|
L4 = ?12 |1??2|
and Hamiltonian representing the presence of the |1? ? |2? drive:
H?
= ?12(|2??1|+ |1??2|), (4.15)~
where ??, ?? are given by equation 4.12, T 021 = 1/?02, and T
12
1 = 1/?12. Note that any time
constant or rate without the transition denoted corresponds to the qubit transition: |0? ? |1?. Only
the excitation rate of the qubit transition ?? needs to be taken into account, as the frequencies ?ij
of the other transitions involved satisfy ~?ij >> kBT . The dynamics of the three populations
p0, p1, p2 (= ??00, ??11, ??22, respectively) are then given by the following linear system:
dp0
= ??01p + ?01? 0 ? p1 + ?02p2dt
dp1
= ?01? p
01
0 ? ?? p1 + ?12p2 + ?12?12dt (4.16)
dp2
= ??02p2 ? ?12p2 ? ?12?12
dt
d??12 ?12
= ? ??12 + ?12(p2 ? p1).
dt 2
Here, ?12 is the off-diagonal element of the density matrix, linking states |1? and |2?. We assume
that p1 ? p2, due the the drive ?12 saturating the transition, and p0 + p1 + p2 = 1, which is
reasonable considering the temperature of the system and the transition frequencies when external
flux is detuned from HFQ. Note that ?12  all ??s in the system.
We define the variables p+ = p1 + p2 and p? = p1 ? p2. Recall that the drive ?12 is applied
for a duration long enough to even the populations in |1? and |2? so that p? ' 0. Using 4.16, the
84
linear system representing the dynamics of p+, p0 is:
dp ?010 + ?
= ??01 ? 02? p0 + p+dt 2 (4.17)
dp+ ?
01
? + ?02
= ?01? p0 ? p+.dt 2
Inspecting this simplified linear system, we see that p0 and p+ exponentially saturate to their equi-
librium values with the rate
?01
01 ? + ?02?eff = ?? + . (4.18)2
The effective decay constant that we extract from fitting p0(?) as a function of |1? ? |2? drive
duration ? in figure 4.21b is therefore Teff = 1/?eff . Solving for ?02, and converting the ??s to
times, we arrive at the formula for T 021 :
01
02 T1 TeffT1 = . (4.19)2T 011 ? (2? pth0 )Teff
Figure 4.20: Pulse diagram of the T 021 measurement. The |1? ? |2? transition is driven for a
duration long enough to even the |1?, |2? state populations. This duration is varied over the range
of ? 10 ? 10, 000 ?s. After the |1?, |2? drive, we wait ? 5 ? T 121 to apply the readout tone so
that remaining population in the |2? state will decay back to the computational subspace for the
single shot measurement to determine p . Due to T >> T 120 1 1 , this readout delay doesn?t have an
appreciable effect on the measured p0.
85
Figure 4.21: (a) Diagram of the three lowest fluxonium eigenenergies making up the qutrit system
under investigation. Wavy arrows represent incoherent relaxation and the straight arrow depicts a
coherent Rabi drive. We drive transition |1? ? |2? at a Rabi frequency ?12  ?i, faster than any
relaxation rate in the system, and monitor the |0? state population p0(?) as a function of the drive
duration ? . (b) Experimental traces used to extract the lifetime T 021 in figure 4.19 at two separate
flux points. For a lower T 021 , p0 saturates to a larger value. Blue and red curves are fitted to decaying
exponentials with time constants Teff equal to (338 ? 26) and (651 ? 48) ?s, respectively. This
time constant is used to extract T 021 according to equation (4.19). (c) Energy relaxation time T
02
1
with corresponding p0 saturation value for each measurement of T 021 (figure 4.19). The data agree
well with the simulated curve using the theory derived in equation 4.16. When T 021 is in the range
where the pth0 error bound intersects the black theory line, we do not measure an exponential decay
in p0 since the initial and final values are nearly the same. We observe this when measuring at
?ext/?0 = 0.5 and therefore estimate T 021 at the sweet spot implicitly. For T
02
1 greater than the
value of the intersection point, p0 will saturate to a value below pth0 , thereby heating the qubit.
4.5.3.1 Constraining T 021 at ?ext/?0 = 0.5
For relaxation times T 021  T 011 , the protocol depicted above results in an increase of p0. In
this case, one can use it as a cooling protocol. We first saturate the |1? ? |2? transition and wait for
population to accumulate in |0?. After turning off the drive and subsequently waiting a few T 121 , the
population is back to the computational subspace with a lower effective temperature. The cooling
fidelity (i.e. the p0 saturation value) depends on the ratio of T 01 to T 021 1 . The larger this ratio, the
higher the cooling fidelity. For all of our measurements, T 011 ? 1 ms. We have established through
single-shot measurements that at HFQ, pth0 = 0.58 ? 0.03. When the ratio T 01/T 021 1 ? 0.63, the
p0 saturation value falls very close to pth0 , such that the value will be within the error bound of p
th
0 .
86
In this case, we lack the precision to reliably measure T 021 . This phenomenon is outlined in figure
4.21c. Since T 021 given by equation (4.19) seems to saturate to around 1.5 ms from the data points
close to ?e/?0 = 0.5 in figure 4.19, we estimate that T 021 & 1.5 ms at the sweet spot.
4.5.3.2 Relaxation Through State |3?
As the value of T 021 grows, it becomes necessary to extend the qutrit model to higher states of
fluxonium. For example, given the qubit temperature of T = 25 mK and the transition frequency
?23/2? = 1.66 GHz, one expects a thermal excitation process |2? ? |3?, the rate of which can
be estimated as follows. We assume that the rate ?231 of decay |3? ? |2? is due to dielectric loss
(tan ? = 1 ? ?6 ?~? /k T10 ) and hence the upward rate is given by ?23 = e 23 B ?23 ? (1 ms)?1C ? .1+e?~?23/kBT 1
This estimate is remarkably close to the maximal measured value of T 021 . Since we use relatively
low temperature and loss tangent values, it is likely that the implied value of T 021 at ?e/?0 = 0.5 is
indeed limited by the thermal excitation to state |3?, followed by a rapid decay to state |0?, rather
than by a direct quasiparticle-induced relaxation to state |0?.
4.5.4 1/f Flux Noise
We now sweep ?ext and measure T2 to quantify the pure dephasing mechanisms. Near HFQ,
we measure the coherence time of the qubit using the Hahn-Echo protocol, in order to obtain TE2 .
According the the theory for 1/f noise-induced dephasing, the signal should decay with a Gaussian
envelope. If coherence is limited by energy relaxation rather than pure dephasing due to 1/f noise,
the decay will be exponential. This is shown in figure 4.22a, where on the sweet spot, the signal
is fit to a decaying exponential, and off-sweet spot, a decaying Gaussian. The 1/f-noise induced
dephasing rate ??? is given by equation 2.49. The dashed curve in figure 4.22b is the simulated
value of T? ?? = 1/?? for Qubit J?s circuit parameters and a flux noise amplitude Afl = 2.0 ??0,
nearly the same value as reported in reference [79].
At HFQ, fluxonium is protected from 1/f noise to first order, since ??01/?? = 0. This results in
the diverging T?? , and a T2 approaching 2T1. Our approach to reaching millisecond coherence was
therefore to improve T1 by decoupling from dielectric loss, knowing that on sweet spot, the most
significant source of pure dephasing would be suppressed. A remarkable property of fluxonium is
that due to its extremely high inductance, 1/f flux noise does not ruin T2 even off the sweet spot,
where figure 4.22b shows that TE2 ? 100 ?s.
87
Figure 4.22: (a) Top: TE2 measurement trace at ?ext/?0 = 0.5, yielding a exponentially decaying
signal. This implies dephasing due to energy relaxation. Bottom: Measurement trace at ?ext/?0 ?
0.5001, showing a Gaussian decay envelope. The fitted decay time constants are shown above the
plots. (b) TE2 versus external magnetic flux. Dashed line is the limit on pure 1/f noise-induced
dephasing time T?? for flux noise amplitude A
?
fl = 2.0 ??0. At the sweet spot, T? ? ?, and T2
becomes T1- limited.
4.5.5 Thermal Cavity Photons
Though TE2 approaches 2T1 at HFQ, there is still a finite dephasing time. Dephasing time T? is
given by:
2T1T2
T? = (4.20)
2T1 ? T2
and we define T2 as TE2 . To measure T?, an interleaved T , T
E, T ?1 2 2 measurement is performed.
The addition of T ?2 in the measurement is simply for completeness, it is not used in the calculation
of T?. This interleaved measurement is shown in figure 4.23, and from the value of TE2 and T1,
we calculate T? = 6.0 ? 0.2 ms. Given the calculated values of ?01, ? in section 4.3.1, and
assuming the dephasing time to be primarily due to thermal cavity photons, equation 2.52 yields
nth = (4 ? 1) ? 10?4. This corresponds to a cavity temperature Tcav = 46 ? 6 mK, which is
consistent with other reported values in the field of superconducting qubits [111, 107].
88
Figure 4.23: An interleaved measurement of the qubit energy relaxation time T1, Hahn-Echo co-
herence time TE2 , and the Ramsey coherence time T
?
2 . The qubit is initialized to |1? state population
0.64, and then each measurement is made at a particular time step. Note that in the T1 measurement,
the population saturates to the thermal state. Black lines are the fits for each measurement. T1 and
TE2 data are fit to a decaying exponential with time constants 1.04? 0.08 ms and 1.55? 0.10 ms,
respectively. T ?2 is fit to an exponentially decaying cosine function with time constant 1.02? 0.05
ms.
4.6 Gate Characterization
With millisecond coherence in hand, we turn to gate characterization in Qubit J. In the case of
T1-limited T2, the gate error rate scales as r ? tg/T1 [24], where tg is the gate duration. We see
then that high coherence is only part of the story: we also need a sufficiently short game time in
order to harness the power of the high coherence. The real quantity of interest then is the ratio
tg/T1 (again assuming that T ?2 ? 2T1). For a ?-pulse, tg is given by:
? ?tg = (4.21)?
where ? is the Rabi frequency (equation 4.9). Typically a ?/2-pulse gate duration is approximately
t?g/2. In general, tg ? 1/?. Using the equation for dielectric loss in terms of the n? matrix elements
(equation 2.42), it follows that:
89
( ( ))
t ~?01
r ? g ? |?0|n?|1?| ~?01 ?1 + coth (1 + e kBT ) ? |?0|n?|1?|?(?01, T ), (4.22)
T1 2kBT ( ( )) ~?01
where we have defined the function ?(? , T ) = 1 + coth ~?01 ?01 (1 + e kBT ), which sets the2kBT
temperature dependence. Note that ?(?01, T = 0) = 2. We now have a relation between the circuit
parameters and the gate error rate, up to a proportionality constant, giving us an idea of how to
optimize gate error rate in fluxonium. The assumptions made are reasonable, considering in the
previous sections we presented data that shows T2 is T1-limited at HFQ, and that dielectric loss
limits T1. The temperature of the environment plays a significant role as ~?01/kBT ? 1. Note
that by ignoring temperature, one may naively think that reducing |?0|n?|1?| will simply lead to
low error rates. In reality, given a finite temperature, there is an optimal set of circuit parameters
(see figure 4.24). By engineering systems that can reach lower temperatures and/or improving
thermalization of the qubit-resonator system, gate error rates can be further improved. Of course,
there are other ways to improve the error rate, such as increasing the strength of the drive field, or
by lowering tan ?C .
Figure 4.24: Color map of |?0|n?|1?|?(?01, T ) ? r as a function of EL, EJ for T = 25 mK (Left)
and T = 0 mK (Right). EC/h = 1.08 GHz is fixed. Magenta X marker corresponds to Qubit J?s
parameters. As EJ increases and/or EL decreases, ?01 decreases. As ~?01/kBT ? 1, error rate r
increases. We assume that the error rate is only limited by decoherence in this analysis.
90
4.6.1 Gate Tuning
The ?/2-rotations about x and y (denoted as X?/2 and Y? /2, respectively) were created using the
conventional broadband vector modulation technique. We used Gaussian-edged square pulses with
60 ns plateau and 5 ns width edges with a truncation range of 4 sigmas, totaling to ?/2tg = 80 ns.
This pulse duration is chosen as it corresponds to a ?/2-pulse with nearly maximal power. Due to
fluxonium?s high anharmonicity, available driving power and coupling to the drive line are the only
limiting factors on the gate time. No Derivative Removal by Adiabatic Gate (DRAG) correction
or other state leakage mitigation technique is necessary [23, 22]. The relatively long plateau is
an artifact of weak input port coupling in our wireless setup in the 100 MHz frequency range.
Faster gates have been demonstrated in fluxonium qubits with lower frequencies [33, 108], where
the port coupling was stronger. The X? , Y? gates (?-rotations about x, y, respectively) consist of
pairs of concatenated X?/2, Y? /2 pulses with 5 ns separation, and they predictably have about twice
higher error rates (figure 4.27, inset).
Fixing the time-domain parameters of the pulses, we tuned their amplitude to find the optimal
gate parameters. To do this, we performed a pulse train, where the qubit is driven with a sequence
of ?/2 pulses ranging from 0 to 120 pulses, with a step of 12. Each set of 12 ?/2 pulses is
equivalent to a 6? rotation of the Bloch vector, which is tantamount to applying the identity gate.
For the correct pulse amplitude, the expected signal is flat, while it oscillates when the gate is
imperfectly calibrated due to the accumulation of the erroneous rotation angle. We ran this pulse
train, sweeping the pulse amplitude, and used the amplitude corresponding to the signal with the
smallest standard deviation. This procedure was repeated four times, for ??/2 rotations around x
and y to take into account IQ mixer imbalance and offset. The pulse diagram for this experiment
is shown in figure 4.25, and an example of the experimental data is found in figure 4.26.
Figure 4.25: Diagram of the pulse sequence of a pulse train experiment using ?/2-pulses
91
Figure 4.26: (a) Pulse train measurement signal standard deviation ? versus pulse amplitude. The
? is minimized when the pulse amplitude corresponds to a proper ?/2-pulse. (b) Individual mea-
surement traces from the data in (a) for the X?/2 pulse train. At an amplitude of 228.75 mV (light
blue data) the signal is flat as the number of pulses is swept, since in even multiples n, n?/2 pulses
form an identity operation up to a phase factor. Dark blue data correspond to an incorrect ?/2
pulse. In this case, the 12 ?/2 pulses add an erroneous rotation angle ?? to the Bloch vector, which
is amplified as the number of pulses applied increases. The pulse train in this case effectively
amounts to a Rabi measurement.
4.6.2 Single-Qubit Randomized Benchmarking
The randomized benchmarking (RB) technique has emerged as a leading method for character-
izing quantum gate error rates [54, 63, 64]. In a RB sequence,m randomly chosen Clifford gates C?
are performed on the qubit before applying a single recovery gate C?r, aimed at bringing the Bloch
vector back to the initial state (see figure 4.27b). This is performed over some number of random
experimental realizations, and the results are averaged together. We will restrict our attention to
the single qubit case, though RB can be extended to the N qubit level. A single qubit Clifford gate
C? is a gate that normalizes the Pauli group [35] (i.e. the Pauli matrices given in equation 1.6):
C???nC?
? = ??n, (4.23)
where ??n is one of the four Pauli matrices.
A set of single-qubit gates S that generates all 24 elements of the single-qubit Clifford group is
given by:
S = {I? ,?X?,?Y? ,?X?/2,?Y? /2}. (4.24)
92
The Gottesman-Knill theorem states that the Clifford gates can be efficiently simulated on a classi-
cal computer, which is necessary to determine the recovery gate C?r in a reasonable time [1]. In the
absence of gate errors, the entire random sequence of C??s + C?r amounts to an identity operation.
In reality, upon averaging over many random realizations of RB sequences, the excited state prob-
ability (or population) p1 (assuming we initialize the qubit in state |1?) decays with the sequence
length m as:
p m1 = A+Bp , (4.25)
where p is the depolarization parameter, and A,B are constants that absorb state preparation and
measurement (SPAM) errors (see the red curve in figure 4.27a). The average error rate of a single-
qubit Clifford gate is given by:
1? p
rcliff = . (4.26)
2
Because each single-qubit Clifford operation is composed on average of 1.833 physical gates (typ-
ically this number is 1.875, but here we do not count the identity gate), the average physical gate
fidelity is given by:
F = 1? rcliff (4.27)
1.833
After performing the RB protocol presented above (formally known as standard RB). the fidelity
of each physical gate in the list {?X?,?Y? ,?X?/2,?Y? /2} (our single-qubit Clifford generating set
S sans the identity gate) can be extracted using an interleaved RB sequence. For interleaved RB, a
given gate is interleaved between each Clifford operation in the standard RB sequence (see figure
4.27c). The resulting curve follows the same decay profile as the standard RB, but with a new
depolarization parameter pg. The physical gate error is then given by:
1? pg/p
rg = = 1? Fg, (4.28)
2
where p is the depolarization parameter obtained from reference RB and Fg is the fidelity of the
interleaved gate. The error rates of each physical gate, rg = 1 ? Fg, are quoted in the inset
of figure 4.27a. Note that the average of the values Fg listed in figure 4.27a differs from the
average physical gate fidelity F composing a Clifford operation, because there are more ?/2- than
?-rotations, on average, in a Clifford operation. The average physical gate fidelity is given by
F = 1 ? rcliff/1.833 = 0.99991(1). To our knowledge, a significantly higher fidelity number
has been possible only in refined trapped ion demonstrations [12]. Improvement in fluxonium?s
coherence through the years has translated to a level of control never before seen in a solid-state
qubit.
93
Figure 4.27: (a) Results of single-qubit standard and interleaved randomized benchmarking (RB)
for Qubit J. The red dashed curve (solid markers) is the fit to equation 4.25 for the standard RB se-
quence averaged over 50 random experimental realizations (each experimental realization is aver-
aged over 1000 individual runs) with an average Clifford gate error rate of rcliff = (1.7?0.2)?10?4,
which converts to the average fidelity of the physical gates used to generate the Clifford group of
0.99991(1). The eight other curves are the results of the interleaved RB measurements, where each
color marks a given interleaved gate. The relative uncertainty on the gate errors given in the caption
is about 10%. (b) Pulse sequence diagram for standard RB. m randomly chosen Clifford gates are
applied to the qubit before their effect is reversed by the recovery gate C?r. This is performed over
many runs, each time choosing new random Clifford gates, which contain 1.833 physical gates in
set S (equation 4.24) on average (ignoring the identity gate). The results of each measurement run
are then averaged together to yield the data in (a). (c) Pulse sequence for interleaved RB. Now
each physical gate g in the set S is interleaved with the sequence in (b) to determine the physical
gate?s error rate.
94
4.6.3 Purity Benchmarking
Quantum gate errors can be separated into two classes: coherent and incoherent errors. Co-
herent errors preserve the state?s purity (or Bloch vector length), but do not execute the correct
operation. For example, an X? gate that over-rotates the Bloch vector is a coherent error. Incoher-
ent errors arise from the decoherence of the qubit state, and set the lower limit for gate error rates
(or upper limit on the gate fidelity). The decoherence contribution to the gate error can be deter-
mined using the purity benchmarking (PB) procedure [32, 104, 21]. Purity benchmarking consists
of performing state tomography of the qubit at the end of the RB sequence instead of the recovery
gate, where the x, y, and z components of the Bloch vector are measured. The length of the Bloch
vector squared is then:
|P~ |2 = ??? ?2 + ??? ?2 + ??? ?2x y z (4.29)
which is related to the purity by equation 1.21. The protocol is then insensitive to coherence
errors, since they will not change the purity of the state, and we can use this protocol to quantify
the contribution of decoherence to the gate errors rates in figure 4.27.
For the PB measurement, the purity P = tr(??2) of the qubit state decays as [32]:
P = A? +B?um?1, (4.30)
where u is called the unitarity and A?, B? are constants that again absorb SPAM errors. The error
rate due to decoherence per Clifford gate is:
?
1? u
rdec, Cliff = (4.31)
2
and the error rate due to decoherence per physical gate is:
rdec, gate = rdec, Cliff/1.833. (4.32)
Our PB measurement (see figure 4.28) yields an error rate due to decoherence per Clifford
gate of r ?4dec, Cliff = 1.1 ? 10 , and an error rate due to decoherence per gate of rdec, gate =
r /1.833 = 0.6 ? 10?4dec, Cliff . The value of rdec, gate establishes an upper bound on the achiev-
able average gate fidelity of 0.99994, which is consistent with the rough estimate from (tg =
80ns)/(T2 = 1.5 ms) ? 0.5? 10?4. The ratio rdec, Cliff/rCliff ? 65% gives the fraction of the error
rate arising from decoherence. We conclude that most of the gate error is caused by incoherent
processes and hence can be reduced even further by shortening the pulses.
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Figure 4.28: (a) Results of the purity benchmarking experiment. The exact same protocol as
standard RB is performed, but instead of the recovery gate C?r, we perform state tomography to
measure ???x?, ???y?, ???z?. Error bars are larger for lower number of applied Cliffords because there
is a greater spread in the Bloch vector components. As the number applied Cliffords increases,
the effect of decoherence becomes more apparent, and all experimental runs will yield a purity
tending towards 0.5, resulting in a smaller spread. Solid line is the fit to equation 4.30, yielding
an error rate due to decoherence per Clifford gate of rdec, Cliff = 1.1 ? 10?4. The initial value for
purity is relatively low because the qubit is initialized to p1 = 0.64. (b) Pulse sequence of the PB
measurement, showing the gates applied before readout for state tomography. Each measurement
run is performed three times: 1. applying the X?/2 tomography pulse to measure ???y?, 2. applying
Y? /2 to measure ????x?, and 3. applying I? (i.e. no pulse) to measure ???z?. Note that the sign of
each expectation value doesn?t matter since they all get squared in equation 4.29.
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4.7 Fluxonium-Based Spin Chain
We now explore fluxonium?s utility as a building block of an analog quantum processor. We
demonstrate that a chain of inductively coupled fluxoniums has a spectrum that mimics that of
the Transverse-Field Ising Model (Hamiltonian given by equation 2.74). Such a spectrum is not
achievable for weakly anharmonic qubits (transmons), distinguishing fluxonium as a leading solid-
state device for quantum simulations. Simulation of the TFIM has also been explored in trapped
ion systems [50, 51].
4.7.1 Device Design
Our approach to designing this device follows the theoretical model presented in section 2.3.3.
We inductively couple 10 fluxonium circuits via the mutual inductance formed from shared JJ?s
in the superinductance array. One or both of the fluxonium circuits at the end of the chain is also
inductively coupled to an on-chip readout resonator, which couples to propagating electromagnetic
modes in a 3D Copper waveguide [58]. We found that one resonator was sufficient for reading out
all 10 fluxoniums, and therefore the spectroscopy data presented in the succeeding sections only
uses one resonator for readout.
Figure 4.29: Experiemntal realization of the 10 inductively coupled fluxonium circuits used to
simulate the TFIM. Left: Full device with ?bow-tie? antenna, which sets the capacitance of the
readout resonator. Middle: Close-up optical image of the device. Sections of shared junctions
making up the mutual inductance LM for the inter-fluxonium coupling are highlighted in red.
Right: SEM image of a section of the superinductance array.
Each fluxonium circuit in the chain has a superinductance array with a total of 126 junctions, 7
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of which are shared with the neighboring fluxonium. All junctions in the the array superinductance
and the resonator has an area of 0.4? 1.5 ?m2. The resonator contains a total of 29 junctions, 5 of
which are shared with one of the fluxonium circuits for readout.
4.7.2 One-Tone Spectroscopy
We perform one-tone spectroscopy of the readout resonator using a Rohde & Schwarz Vector
Network Analyzer (VNA). The transmitted signal is fit to equation 3.1, since the microwaves are
reflected off of the sample in the waveguide (see section 3.1.2).
Figure 4.30: One-tone spectroscopy near HFQ (all fluxonium loops have approximately the same
external flux) of spin-chain device readout resonator. Red curves are the fit to equation 3.1, yielding
?r/2? = 5.1904 GHz, Qint = 6, 600 ? 300, Re(Qext) = 5, 700 ? 100. The fit error of the
resonator frequency was below the fourth decimal place and is therefore ignored. (a) Real and
imaginary parts of the reflected signal. The data (black circles) are fit to equation 3.1 (red curve).
(b) Magnitude of the reflected signal. (c) Phase of the reflected signal.
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4.7.3 Spectrum
Two-tone spectroscopy is used to obtain the spectrum of this 10-fluxonium chain. The method
is the same as in section 4.2 on the single fluxonium device. Due to the disorder in EJ , EC for
each fluxonium which arises from imperfect fabrication, we modify equation 2.74 for the fit to:
N??1 N??2
H?chainfl = hi ??iz z + hx(?ext)??i + J i i+1x xx ??x??x
i=0 i=0
1 (4.33)
hiz = ~?i2 01
hx(?ext) = hx|?ext ? ?|,
where we have now allowed each fluxonium qubit transition to have a unique frequency denoted
by index i: ?i01. The parameters hx, Jxx remain fixed across all 10 fluxoniums in the chain,
since they are dependent on EL and LM : the inductive energy of each fluxonium and the mutual
inductance from the shared junctions between each fluxonium, respectively. These parameters are
far more robust to fabrication imperfections, since they are set by the number of large area JJ?s
in the superinductance. EJ , EC both depend on the single JJ area which, being on the order of
0.1? 0.1 ?m2, is harder to control.
The spectroscopy data is in very good agreement with this simplified theory (see figure 2.26).
This is remarkable when considering the fact that in reality, these are not perfect spin-1/2 systems
but rather complex artificial atoms with an infinite number of eigenenergies. It is possible because
of the high anharmonicity of fluxonium. The 12-parameter fit to equation 4.33 yields:
f 001 = 3.46? 0.02, f 101 = 3.45? 0.02, f 201 = 3.24? 0.02, f 301 = 3.51? 0.03, f 401 = 3.67? 0.02,
f 501 = 3.58? 0.03, f 601 = 3.27? 0.02, f 701 = 3.19? 0.02, f 801 = 3.33? 0.02, f 901 = 3.31? 0.02,
hx/h = 1.49? 0.01, Jxx/h = 0.251? 0.003,
(4.34)
where all values are in units of GHz, and f i01 = ?
i
01/2?. The fit was performed by using all 10
first band transitions. The resulting second band transitions (blue curves in figure 4.31) from the
fit parameters do coincide with the spectral lines in the data, an important confirmation that the
fit is legitimate. Note that the faint, unfitted spectral lines that are partially cut off are likely to be
inter-band transitions (between first, second, and third bands), which has been confirmed through
simulation.
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Figure 4.31: Left: Results of two-tone spectroscopy while sweeping external flux ?ext =
2?(?ext/?0) for a circuit of 10 inductively coupled fluxoniums. Right: Two-tone spectroscopy
with the fit to equation 4.33. Red curves correspond to the first band transitions, where one there
is one excitation in the chain. Blue curves are the second band transitions, where there are two
excitations in the chain.
4.7.4 Circuit Characterization
To estimate the physical parameters of the circuit, we fit one of the central transitions in the
first band of transitions (see figure 4.32) to the exact fluxonium Hamiltonian 2.21. The central
transitions within the first band are less perturbed by the spin-spin interaction, Jxx, and should be
100
able to fit reasonably well considering this device is still in the low coupling regime (Jxx << hz).
We perform the fit on the |0? ? |4? transition of the spin-chain circuit, shown in figure 4.32. The
results are: EJ = 12.1? 0.7 GHz, EL = 0.425? 0.005 GHz, EC = 10.1? 0.3 GHz.
Figure 4.32: Spectroscopy of the first band transitions of the device. We fit the |0? ? |4? transition
spectral line versus external magnetic flux to the fluxonium Hamiltonian 2.21. Magenta curve is
the fit result, with parameters: EJ/h = 12.1 ? 0.7 GHz, EL/h = 0.425 ? 0.005 GHz, EC/h =
10.1? 0.3 GHz.
Information on the transition frequency around ?ext = 0 is important for determining EJ , EC ,
which is why their fit errors are higher than for EL. The slope of the transition spectral line is
mostly set by EL, which is included in the sample points for the fit.
The value ofEL allows us to determine the inductance L of fluxonium circuit, given by equation
2.9, which is L = 384 ? 5 nH. Each fluxonium in the chain was designed to be identical, so
we will assume this is the inductance of all 10 fluxoniums in the chain. Since the coupling in
this device is relatively weak (LM/L ? 0.06), we can ignore the correction to the individual
fluxonium circuit?s inductance (equation 2.70) due to the mutual inductance. Each fluxonium?s
superinductance array contains 126 JJ?s, which means the Josephson inductance of each junction
in the array is LJ = 3.05 ? 0.04 nH. The junctions in the fluxonium array and the resonator are
identical, and therefore we determine that the resonator which contains 29 JJ?s has an inductance
of: Lr = 88.5 ? 1.2 nH. Recall that the total inductance of a superinductance array is simply
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the product of number of junctions and LJ . Knowing the resonator inductance, Cr can now be
determined since the resonator frequency ?r was directly measured (figure 4.30). We estimate
the resonator capacitance to be Cr = 10.6 ? 0.1 fF. Since this value is set by the geometry of
the resonator antenna, we can assume it will remain fixed from device to device. Importantly,
this value is protected from uncontrollable variations in the junction oxide, which can affect LJ .
Knowing Cr corresponding to the resonator geometry provides a simple way to estimate LJ of the
junctions in future devices, since we can directly measure ?r via one-tone spectroscopy.
102
CHAPTER 5
Conclusions and Outlook
We have begun to contemplate our origins: starstuff pondering the stars...
Carl Sagan
5.1 Millisecond Coherence in a Superconducting Qubit
Building on our group?s previous work [79], we developed a fluxonium qubit with a coherence
time in excess of 1 millisecond. This is the highest T2 reported in a solid-state qubit by over
a factor of three, and an order of magnitude higher than the previously reported state-of-the-art
Ramsey coherence times. Past experiments on fluxonium determined that dielectric loss via the
capacitance formed by the circuit?s antenna was the main limiting factor on T1. Knowing that T2
is T1-limited at the sweet spot (see figure 4.22), the problem of improving coherence further was
reduced to mitigating the effects of dielectric loss.
The millisecond threshold was reached through both fabrication optimization and quantum en-
gineering. By improving our fabrication procedures, we reduced the theoretical limit on energy
decay rate by more than a factor of two (see figure 4.17). This was done by switching the chip
material from silicon to sapphire, and by treating silicon with a buffered oxide etch prior to fab-
ricating the circuit. Both methods reduced the dielectric loss tangent to similar levels. Additional
optimization of our fabrication methods resulted in better precision on the target circuit parameters,
outlined in Appendix A. With a reduced tan ?C in hand, we then turned to quantum engineering
to determine the optimal circuit parameters for decoupling from dielectric loss. This approach
culminated in the production of Qubit J.
Importantly, we showed that this highly coherent device resulted in probably the lowest single-
qubit gate error rates ever reported in a solid state qubit (see figure 4.27). These error rates were
mainly limited by decoherence (see figure 4.28), and reducing them further should be relatively
straightforward by shortening the gate times. These gates were composed of 80 ns pulses, which
are considerably longer than those used in the more conventional ? 5 GHz transmon qubits. The
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incredibly large anharmonicity of fluxonium?s spectrum at HFQ means that there should be no
issues associated with state leakage errors due to shortened gate times. Our demonstration of single
qubit gates well below the accuracy threshold th for fault tolerance in this work, along with recent
demonstrations of two qubit gate error rates in fluxonium nearing the accuracy threshold [33, 108]
establishes fluxonium as a leading platform for digital quantum processors [78]. The two-qubit
gate error rates in fluxonium qubits are actually limited by the low coherence times of the gate
transitions outside the computation subspace. With two-qubit gate schemes employing transitions
within the computation subspace [75], th for the gate error rate should soon be achieved. From
there, the natural next step is to build a logical qubit out of physical fluxonium qubits for quantum
error correction.
A seemingly disadvantageous feature of low-frequency fluxonium qubits is the increased ther-
mal population in the |1? state, since ~?01 ? kBT . However, initialization protocols are necessary
in all qubit architectures, regardless of the thermal populations, for fast reset when executing al-
gorithms. Though it was not explored in depth in this work, various initialization schemes have
been reported in other studies of low-frequency fluxoniums [113]. Our approach of initialization
via the incoherent readout resonator drive was sufficient for the purposes of this work, namely to
demonstrate a millisecond-coherence qubit with sub-10?4 single qubit error rates. This method
was also employed in two-qubit fluxonium experiments in our lab [33, 108]. More work is neces-
sary to fully understand the cavity-drive initialization, and we verified that using it had no effect
on the quantitative results of this work. Nevertheless, initialization using sideband transitions of
the coupled circuit-resonator system has been shown to work with excellent fidelity and is likely
the best method going forward.
Our novel measurement of fluxonium?s parity-protected |0? ? |2? transition energy relaxation
time T 021 allowed us to verify dielectric loss as the leading source of energy relaxation in the circuit,
as well as establish a stringent upper-bound on the quasiparticle density xqp < 5 ? 10?9. This
measurement scheme should be useful in future fluxonium circuit characterization experiments for
setting bounds on xqp. We stress that the only steps taken to prevent quasiparticle excitations in
the experimental setup were the placement of three homemade eccosorb filters on the driving line
[18, 10] (see figure 3.1). No signatures of quasiparticles were present in our experiments (see
T1 versus flux measurement, figure 4.18), and we conclude that reducing the loss due to material
defects and improving the measurement line thermalization should push coherence even further
into the millisecond range.
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5.2 Fluxonium-Based Spin Chain
For the first time, quantum many-body physics is simulated with a circuit composed of 10 in-
ductively coupled fluxonium superconducting qubits. We demonstrated that in the weak coupling
regime, this circuit?s low energy spectrum can be fitted to the Transverse-Field Ising Model Hamil-
tonian (equation 2.74), despite the fact that each element in the spin chain is actually an artificial
atom and not a spin-1/2 system. This is possible because of the remarkably high anharmonicity of
the fluxonium spectrum. That, along with the high coherence, displays the viability of fluxonium-
based quantum simulators. Quantum many-body simulations have been explored using flux qubits
and transmons with promising results [115, 36], although these aforementioned fluxonium proper-
ties are likely to provide advantages over the other architectures. These experiments on all three
types of superconducting circuits are paving the way for near-term analog quantum simulators in
the NISQ era.
It is possible that for larger relative spin-spin coupling, the approximation of the qubit transition
to a spin-1/2 system breaks down as the higher level energies begin to hybridize with one another.
We explored this regime in subsequent devices, where we raised the number of shared Josephson
junctions in the mutual inductance from 7 (the weakly coupled device studied in this work) to 23.
We were unable to fit the observed spectrum in this device with the level of agreement to theory as
in the weakly coupled one, though signatures of the first and second bands beginning to hybridize
are present. This data is shown in figure 5.1.
Looking ahead, we wish to explore these types of devices in regimes of higher relative coupling
Jxx/hz. The fluxonium-based spin chain has valuable quantum annealing applications, which to
date has mainly been explored using flux qubits [47, 16]. Quantum annealing involves gradually
tuning parameters in the TFIM Hamiltonian, which can be implemented by changing the external
magnetic flux in flux-sensitive devices, such as flux qubits, fluxonium, and flux-tunable transmons
[45]. Fluxonium?s coherence time still remains reasonably high (above 1 ?s) when detuned from
the sweet spot, unlike flux qubits. This is due to its approximately two orders of magnitude larger
loop inductance. This property, along with the demonstration of the weakly coupled fluxonium-
based spin chain, should motivate further studies in fluxonium-based quantum annealing. These
experiments on the first ever fluxonium-based spin chain is an important proof of concept, and we
look forward breadth of analog quantum computing applications such devices will uncover in the
coming years.
105
Figure 5.1: Spectrum of a fluxonium-based spin chain device with larger coupling. The hybridiza-
tion of the first two bands starts to occur at HFQ and at approximately 4.5 GHz. Horizontal lines
at 4.96 GHz and 5.5 GHz are the two readout resonators coupled to the device. All readout was
performed using the higher frequency resonator.
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APPENDIX A
Fluxonium Fabrication Recipe
A.1 Tuning Fluxonium Circuit Parameters
Figure A.1: Experimental realization of a fluxonium circuit. Each of the three circuit elements are
depicted. The loop formed by the single JJ and the Josephson junction array is threaded by external
magnetic flux ?ext.
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A.1.1 Charging Energy: EC
The charging energy is given by the formula:
e2
EC = (A.1)
2C
where e is the elementary charge and C is the effective capacitance of the circuit. This capacitance
is determined by the antenna electrodes of the device. Therefore, the size and geometry of the
antenna will decide EC . Usually, we only change EC if we wish to change the antenna design in
order to modify the qubit-to-resonator coupling rate g. Otherwise, we leave this parameter fixed,
and tune the spectrum with EL and EJ .
A.1.2 Inductive Energy: EL
The inductive energy is given by the formula:
?2
E 0L = , (A.2)
L
where ?0 = ?0/2? = ~/2e is the reduced magnetic flux quantum, and L is the inductance of
the circuit. The L of fluxonium originates not from an electromagnetic inductance formed by a
solenoid, but rather from the kinetic inductance of an array of large area Josephson junctions. If
each junction in the array of N junctions has a Josephson inductance LJ , then the total L of the
circuit is:
L = N ? LJ . (A.3)
Plugging this result into equation A.2 yields:
?2
E 0L = . (A.4)
2NLJ
Therefore, the EL of the fluxonium circuit is inversely proportional to the number of junctions in
the chain array, N . LJ is an experimentally determined parameter. For the 0.4? 2 ?m2 junctions
used in our chain design, LJ is approximately 1 nH. Over short enough periods of time this value
stays constant, but over the range of many months, LJ can fluctuate somewhat unpredictably. This
is due to the changes in oxide growth of the junctions, which depends on atmospheric factors, like
humidity. This usually is not an issue since these changes occur on the scale of a few months.
Therefore, if one device has an EL that is off from the target value, scaling the number of junction
N from there will yield the target value. This effect occurs for EJ as well, as we will discuss in
the next section. This unpredictability would likely be mitigated if we fabricated the junctions in a
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cleanroom.
A.1.3 Josephson Energy: EJ
The Josephson energy is given by the formula:
~
EJ = IC (A.5)
2e
where IC is the critical current of the single Josephson junction. In the approximation of tempera-
ture T ? 0, IC is given by the Ambegoakar-Baratoff formula [6]:
??
IC = (A.6)
2eRN
where ? is the superconducting energy gap of aluminum, RN is room temperature resistance of
the junction, and e is the electron charge. Plugging this back into equation A.5 gives us:
h ? ?Rq
EJ = = (A.7)
(2e)2 2RN 2RN
where we have used the definition of the superconducting resistance quantum: RQ = h/(2e)2. The
formula for normal resistance is: RN = l?/A. So finally, we can write the Josephson energy as:
?RqA
EJ = (A.8)
2l?
where ? is the resistivity of the junction oxide, l is thickness of the oxide tunnel junction forming
the JJ, and A is the JJ?s cross sectional area. Equation A.8 gives us a formula dependent on two
variables, l and A that can be tuned to change EJ . In the fabrication procedures presented in the
succeeding sections, we hold l fixed, and only change the areaA of the single JJ to tuneEJ . Similar
to the case of EL, on the timescale of a few to several months, EJ can change unpredictably. In
these instances, we can simply scale the area with respect to the new value, and usually this will
be sufficient to hitting the target for EJ on the subsequent try.
A.2 AutoCad
Before any fabrication takes place, we must have obtained a .dxf file from AutoCad of our
design. We briefly summarize the steps required to obtain such a file, while assuming the reader
has a working knowledge of AutoCad. Other design tools can be used as well. Usually, a fluxonium
device has 6 different layers in its AutoCad file, each layer corresponds to a different dose applied
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by the electron beam writer. These layers are: Small JJ Main, Small JJ Overlay, Chain Main, Chain
Overlay, Antenna Main, and Antenna Overlay. Most of the time this number can be reduced to 4,
since the main doses for the antenna and chain can be made the same, as well as the overlay doses
for the antenna and small JJ. The sizes of the features for these various layers range from 100?s
of nm to 1000?s of microns, which is why different layer doses is required. Generally speaking,
smaller features need higher doses.
A.3 Chip Preparation
A.3.1 Optional: Buffer Oxide Etch (BOE) of Silicon Chip
Results of the experiments indicate that devices fabricated on silicon chips treated by BOE
yield lower dielectric loss tangents and therefore higher energy relaxation times. BOE has
been found to improve internal losses in other devices as well [99, 83]. The goal of BOE is
to remove the native oxide found on the surface of silicon. We do this by submerging the
chip in a 1:3:6 solution of Hydroflouric Acid:Ammonium Flouride:Water (Buffer HF Improved
from Transene, Inc.). Chip is submerged for 2 minutes, followed by a rinse in DI (deionized) water.
A.3.2 Cleaning
Once we have our cleaved 9 ? 4 mm chips, we must prepare them by cleaning them. In all
remaining steps, it is assumed that chip handling is always done with metal tweezers. Although
we will not be explicitly using the hotplate in this section, at this juncture in the fabrication
process, we must set the hotplate to 180?C and position an aluminum slide on the center of it. The
plate needs time to reach 180?C, and the Al slide needs time to thermalize. The Al slide is used
as a means of keeping the chip off the dirty surface of the hotplate, and allowing the chip to be
heated on a more thermally uniform surface. This second reason is significant because it helps
ensure reproducibility. The hotplate has a significant temperature gradient from the center, so if
the chip is placed on it at different locations for different fabrication processes, this introduces
inconsistencies where the resist may be getting baked at different temperatures. This can lead to
producing unintended features in our devices.
Tools Required
? Two 50 mL Beakers
? Acetone
110
? Isopropanol (IPA)
? Sonicator
Steps
1. Fill one beaker with acetone and the other with IPA. Each beaker should be chemical specific,
this means that one beaker is always designated for acetone, and the other, for IPA.
2. Place the chip into the acetone filled beaker, with the side we will be writing the design on
facing up. From this point on, the chip should never flip onto this side.
3. Place the beaker containing the chip and acetone into the sonicator, and sonicate at low
power for 3 minutes.
4. Remove the chip from the acetone directly into the IPA beaker, then repeat 3.
5. Remove the chip from the IPA and blow dry with nitrogen gas (N2).
6. Inspect the chip by eye. If small defects or dust remain on the chip that are visible to the
naked eye, then this chip is likely not suitable for fabricating the device. If the chip passes the
eye test, then inspect under the optical microscope. If only a few small defects are present,
the chip is acceptable to proceed with.
A.4 Resist Application
Following step 6 in A.3, it is time to move on to the resist application.
Tools Required
? MMA EL 13 Resist
? 950 PMMA A3 Resist
? Spinner
? Hotplate
? Aluminum Slide
? Plastic Pipette
1. Set hotplate to 180?C (with aluminum slide on it so that it is well thermalized).
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2. Spin MMA EL 13. Turn on the vacuum on the chip spinner and carefully place the chip onto
the suction as centered as possible. Make sure that the setting of the spinner is set to 5000
rpm for 1 minute and run the process.
3. Once the spin process is completed, turn off the vacuum and remove the chip from the
spinner. Inspect the resist by eye to check for any defects, if none are present, place on
aluminum slide on the hotplate for 1 minute. Small defects in the resist are permissible if
they are away from the locations where the pattern will be written.
4. Spin PMMA A3. Same as 2, but now spin speed is 4000 rpm, also for 1 minute. Then place
the chip back onto the aluminum slide on the hotplate for 30 minutes (still at 180?C).
5. After 30 minutes, inspect the resist with optical microscope, there should be very little debris
on the surface in the resist. The chip is now ready for electron beam exposure.
A.5 Anti-Charging Layer Depositon (Only for Sapphire Chip)
If the chip being used is sapphire instead of silicon, a thin conducting layer of metal must be
deposited on top of the resist bilayer. Because sapphire is an insulator, charge will accumulate on
the chip, which will create a voltage on the chip?s surface. This voltage will distort the electron
beam, and then the pattern cannot be written accurately. This is not a problem with silicon, a
semiconductor. The anticharging layer is grounded to the sample holder in the electron beam
writer, stopping the accumulation of charge. We deposit 11 nm of Al at a rate of 1nm/s for our
anticharging layer. After the pattern is written, this layer must be removed before development.
A.6 Electron Beam Writing
Electron beam lithography uses electrons rather than UV/optical photons to draw patterns in the
resist. The Superconducting Circuits lab uses the ELS Elionix G-100 electron beam lithography
system, with a 100 kV beam. The advantage of using electrons is that they have a wavelength
much shorter than UV/optical photons, increasing the spacial resolution of device features we are
capable of producing. EBL is, in general, more complicated than its optical counterpart.
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Figure A.2: ELS-Elionix G-100 Electron Beam Lithography system.
There are two key parameters of interest when it comes to fabricating a device with EBL: dose
and beam current. Dose is the total amount of charge applied to the resist (units are Coulombs
per square micron, C/?m2), and beam current is the rate that the dose it delivered at (units are
nanoamperes, nA). Beam current is determined by the size of the features of the device we want
to write. Naturally, a larger beam current corresponds to a wider beam. Dose is determined by the
resist used, baking temperature, and the development procedure (see section A.8 for more details
on development). All devices in this work were written with beam currents of 1nA.
Layer Silicon Dose (C/?m2) Sapphire Dose (C/?m2)
Small JJ Main 3000 3000
Small JJ Overlay 260 260
Chain Main 2000 1000
Chain Overlay 100 100
Antenna Main 2000 1000
Antenna Overlay 260 260
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A.7 Anti-Charging Layer Removal (Sapphire Only)
The anti-charging layer of aluminum is removed once the pattern has been written. To remove
the Al, we perform a wet-etching process. The chip is submerged in a 0.1 M aqueous solution of
potassium hydroxide (KOH) for about 1 minute. After this time, the Al should be visibly gone.
A.8 Development
Development is the process of bathing the electron beam-exposed chip in a liquid known as the
developer. Various resists have their own corresponding developer. For PMMA A3 and MMA EL
13, the developer is simple: a 3:1 IPA:DI by volume solution. Development is a 2 minute dip in
the developer at 6?C. It is very important to have the process streamlined to the point where each
time a new device is fabricated, nothing in the development step has changed. If the temperature
of the developer or length of time in the developer changes, then the small JJ area will change
unpredictably, yielding an EJ that wasn?t targeted.
Tools Required
? 100 mL Beaker
? 500 mL Beaker
? Glass bowl with flat bottom
? 40 mL of 3:1 IPA:DI solution
? 5-7 ice cubes, depending on size
? Timer
? Thermometer
? Regular Tweezers
? Reverse-Action Tweezers
? Nitrogen Gas Gun
Development Steps:
1. Add 6-7 ice cubes into the glass bowl, then pour 200 mL of tap water into it. This will serve
as the bath to hold the temperature of the developer constant during development.
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2. Cover the 100 mL beaker with Al foil to prevent any unwanted debris from entering the
beaker, then insert the beaker into the bath, along with a thermometer to monitor the tem-
perature. Leave for 10 minutes to allow the beaker to thermalize. After 10 minutes, the
temperature should be stable at 5? 6?C.
Figure A.3: The cold bath waiting for 10 minutes before development
3. Prepare chips for development by securing them on the reverse action tweezers. Two chips
can easily be developed at a time.
4. Set timer to 2 minutes.
5. Remove developer bottle from the cold bath that it is stored in (hold the container of de-
veloper in technicloth wipes to insulate the developer from your hand warmth), and fill the
beaker in the cold bath to around the water line of the bath (see figure A.3). Close the
container immediately and set aside to put back into the cold bath after development.
6. Start timer upon dipping the chips into the developer. While submerged, wiggle the chips
back and forth at a frequency of around 1-2 Hz. With 3 seconds remaining on the timer,
remove chips from developer and prepare to blow dry with N2. When the timer goes off, dry
with the N2 gas gun.
115
Figure A.4: Optical image of the completed mask after development at 100x magnification. The
bridges making up the chain are visible.
The cold bath is a key feature of development process, figure A.5 highlights the increased precision
in small JJ area when going from no bath to bath during development.
Figure A.5: Graphs show the deviation in small JJ area between the completed device and the
device design in AutoCad. There should always be a deviation, the goal is to make this deviation
the same every time a new device is fabbed, so it can be properly accounted for, and target circuit
parameters met. (a) The difference between AutoCad design single JJ area and actual device single
JJ area before the cold bath was used to develop. (b) Same value, but the cold bath is now used
in the development step. Standard deviation of the difference between the area in AutoCad design
and true device area decreased by nearly an order of magnitude.
116
A.9 Metal/Oxide Deposition
The Al/AlOx/Al deposition is done using a Plassys deposition system. In this section we will
outline the deposition process. The deposition recipe is entirely automated by the Plassys; the
most involved step in the deposition process is attaching the sample onto the 5 inch wafer puck. A
corner of the chip is held into place by the holding arm as shown in figure A.7.
The edges of the chip tend to not have resist on them, meaning unwanted Al can be deposited
on the edge. Al deposited on the horizontal edges is a problem, because it may introduce a short
along the Cu cavity the chip will be mounted in, resulting in a device that cannot be measured. To
avoid this, aluminum foil must be placed over the horizontal edges of the chip. The final loading
configuration is depicted in figure A.8. Once the device is loaded, we must wait about 20 hours
to begin the deposition, so that the loadlock pressure reaches 1 ? 10?7 mBar. The main chamber
pressure is approximately 3 ? 10?8 mBar, and when the loadlock is opened, the deposition takes
place at around 8? 10?8 mBar.
Figure A.6: Plassys deposition system
117
Figure A.7: A sample clamped in place using the holding harm. Note the specific orientation of of
the chip with respect to the puck. The sample must be loaded onto the puck in this orientation, the
same way it was loaded into the elionix for the e-beam write. The antenna of the device must also
be parallel to the horizontal groove running along the diameter of the puck. This is to ensure that
the first and second depositions will be deposited directly on top of one another.
Figure A.8: The device is ready to be loaded into the Plassys for deposition
118
The Plassys deposition recipe in order is as follows:
1. 20 second 10 mA argon ion beam etch (also known as descum) at 200 V. This is done twice,
for both deposition angles first at 23.83?, then at ?23.83?. This is necessary to improve the
adhesiveness of the Al to the Si surface, and to further clean the chip surface.
2. Titanium (Ti) is deposited in the chamber at 0.1 nm/s for 2 minutes. A shutter covers the
puck holding the chip so that none is deposited on the substrate. The Ti is deposited into
the chamber to lower the chamber pressure, as the Ti coats the chambers, and causes any
particles in it to stick to the Ti along the walls.
3. First Al deposition. 20 nm of Al is deposited at 1 nm/s at an angle of 23.83?.
4. 10 minute oxidation at 100 mBar.
5. Second Al deposition. 40 nm of Al is deposited at 1nm/s at ?23.83?.
6. 20 minutes oxidation (known as capping) at 10 mBar. This is to ensure that no irregular
oxidation takes place on the Al after it is removed from the deposition chamber.
A.10 Liftoff
We have reached the final step in the fluxonium fabrication recipe. Liftoff is the process of
removing the resist mask from the silicon/sapphire substrate so that we are only left with our
fluxonium circuit on the 9? 4 mm chip.
Tools Required
? Two 50 mL beakers
? Acetone
? IPA
? Sonicator
? Hotplate
? Aluminum Foil
Steps
1. Set hotplate to 60?C.
119
2. Fill one beaker with acetone and place the chip face up in the acetone. Fill the other with
IPA.
3. Cover the beaker with aluminum foil so that the acetone will not evaporate
4. Leave the beaker containing the acetone and chip on the hotplate at 60?C for 2-3 hours. This
ensures that all the metal will be removed and that we won?t have residual resist left behind.
5. Sonicate the chip in the same acetone beaker used for the liftoff for 5 seconds. At this point
all the resist and metal that is not part of the device will be removed.
6. Transfer the chip into the other IPA-filled beaker, and sonicate for 10 seconds.
7. Remove and blow dry with N2.
120
APPENDIX B
Additional Qubit J Characterization
B.1 Energy Relaxation Time of the 1 - 2 Transition
To complete our investigation of the qutrit composed of fluxonium?s first three eigenstates:
|0?, |1?, |2?, we also measured the relaxation time of the |1? ? |2? transition: T 121 . Sufficiently
close to HFQ, state |2? decays to |1?much faster than directly to |0? (see figure 4.19). We measured
T 121 by exciting the state |2? and recording the relaxation signal with a characteristic time T 121 .
Similarly to the T 011 data in figure 4.18, we observe reproducible variations in the T
12
1 value as a
function of the transition frequency. The measured decay times are consistent with the dielectric
loss estimates in the main text.
Figure B.1: Energy relaxation time of the |1? ? |2? transition, T 121 as a function of the external
magnetic flux ?ext. Dashed curves are T1 limits imposed by dielectric loss for loss tangents of
1.5?10?6 (red) and 4.5?10?6 (green) at 6 GHz. Blue and light blue data points are the same scan
taken twice, demonstrating the reproducible peaks in T 121 , suggesting loss due to material defects.
121
B.2 Qubit J Device Images and Matrix Elements
We include images of the device used in the single qubit experiments in this work: Qubit J.
Figure B.2c displays the transition frequencies between the first four eigenstates of this device,
along with their corresponding charge matrix elements |?i|n?|j?| at HFQ.
Figure B.2: (a) Optical image of the device under study: Qubit J. The antenna electrodes are
attached directly to the weak junction of fluxonium, contributing to the total shunting capacitance
and coupling the qubit to a copper box readout resonator (see figure 3.2). (b) Close up of the
fluxonium loop formed by the single JJ (top left corner) and the superinductance array, composed
of large area JJ?s. (c) Measured frequencies and calculated charge operator n? matrix elements
for transitions between the lowest three energy levels at the sweet spot. The matrix elements are
computed using the fitted circuit parameters in figures 4.4 and 4.5. Note that the qubit transition
|0??|1? is allowed, albeit suppressed in comparison to the higher frequency transitions. Transitions
|0? ? |2? and |1? ? |3? are dipole-forbidden due to the parity selection rule.
122
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