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Tunable Coupling and Its Applications

in Circuit Quantum Electrodynamics

Gengyan Zhang

A Dissertation

Presented to the Faculty

of Princeton University

in Candidacy for the Degree

of Doctor of Philosophy

Recommended for Acceptance

by the Department of

Electrical Engineering

Adviser: Professor Andrew A. Houck

June 2018

Abstract

Circuit quantum electrodynamics (cQED) uses superconducting circuit elements as

its building blocks for controllable quantum systems and has become a promising ex-

perimental platform for quantum computation and quantum simulation. The ability

to tune the coupling rate between circuit elements extends the controllability and

flexibility of cQED devices and can be utilized to improve device performance.

This thesis presents the study, implementation and application of tunable cou-

pling devices in cQED. The tunability originates from the basic principles of quan-

tum superposition and interference, and unwanted interactions can be suppressed by

destructive interference. Following this principle, we design and conduct two experi-

ments that demonstrate the utility of tunable coupling for better device performances

in quantum information processing.

The first experiment aims to improve the coherence of qubits against noise. We

implement a qubit whose frequency and dispersive coupling to a readout resonator

can be tuned independently. When the coupling rate is tuned to near zero, the

qubit becomes immune to photon number fluctuations in the resonator and exhibits

robust coherence time in the presence of noise. The second experiment extends to

a multi-qubit system where crosstalk between qubits causes error in quantum gates.

We develop a two-qubit device and suppress crosstalk by tuning the ZZ coupling

rate between the qubits. The tunable dispersive coupling can also be parametrically

modulated to implement a two-qubit entangling gate in the low crosstalk regime.

Those devices provide flexible and promising building blocks for cQED systems.

iii

Acknowledgements

I would like to thank, first and foremost, my advisor Professor Andrew Houck for

his guidance and support in my thesis work. Andrew’s enthusiasm about science and

talent in research can only be matched by his amazing personality and charisma. His

insightful suggestions, encouraging attitude and contagious optimism are invaluable

resources when I face difficulties in research, and his willingness to share opinions on

non-academic topics makes him an excellent mentor besides a great advisor. I would

also like to thank Professor Nathalie de Leon and Professor Alejandro Rodriguez for

sparing time to be my thesis reader.

Srikanth Srinivasan was my senior mentor when I joined the group and trained me

on almost everything from doing simulation and fabrication to using lab equipment

and performing measurements. His kindness and responsibility set an excellent ex-

ample of senior student for us. Devin Underwood and James Raftery helped me with

my early experiments. Devin helped me measure my very first qubit, and James and

I worked together during the “𝑇1 crisis”. Will Shanks’s highly organized and systemic

research style has deep influence on me. He is also really resourceful when it comes to

cryogenic experiment techniques and equipment maintenance. Darius Sadri’s knowl-

edge in theoretical physics is encyclopedic and his daily physics lecture at coffee break

is missed by us all. His strong commitment to research ethics and integrity makes him

my role model for a scientist. Yanbing Liu’s persistent pursuit of “hardcore physics”

always reminds me to keep high standards as a physicist. My experiments would not

have been done without him taking care of the fridge. Neereja Sundaresan’s patient

and careful approach to experiments makes her fabrication recipes the most reliable

in the lab. She also thoughtfully takes on many housekeeping duties in the lab and

makes our life much easier. Mattias Fitzpatrick is the official organizer for our annual

APS reunion and other group bonding events. He is always passionate about sharing

his research experiences and trying new ideas, and I enjoy exchanging thoughts withiv

him about teaching, time management, work-life balance and so on. Andrei Vraji-

toarea is a human search engine for literature and always available when there is lab

emergency. It is a great pleasure discussing all kinds of problems with him, although

his jokes are sometimes hard to understand. Zhaoqi Leng took over my title of “fab

machine of the lab” and I am really impressed and inspired by his work ethic. An-

drás Gyenis, Alicia Kollár and Tom Hazard joined our lab more recently and brought

knowledge from other research fields. I learned a lot from the discussions with them.

It was my privilege to provide some guidance for the next generation group members,

Pranav Mundada and Basil Smitham, at the start of their academic career. Pranav

made major contributions to the second experiment in this thesis and I feel fortunate

to have the opportunity to work with him.

I would like to thank Bert Harrop, Yong Sun and other cleanroom staff, Nan

Yao, Jerry Poirier and other staff members at Imaging and Analysis Center, Barbara

Fruhling, Sarah McGovern and other administrative staff, for their help and support

during my graduate program.

Finally I would like to thank my family for their love, encouragement and support.

v

To my parents.

vi

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1 Introduction 1

1.1 Controllable quantum systems . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Circuit quantum electrodynamics . . . . . . . . . . . . . . . . . . . . 3

1.3 Transmon qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Tunable coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Theoretical Tools 12

2.1 Quantization of the circuit . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 Derivation of the Hamiltonian . . . . . . . . . . . . . . . . . . 13

2.1.2 Diagonalization of the Hamiltonian . . . . . . . . . . . . . . . 16

2.1.3 Coupling between TCQ and cavity . . . . . . . . . . . . . . . 20

2.2 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Black-box quantization . . . . . . . . . . . . . . . . . . . . . . . . . . 26

vii

CONTENTS

3 Experimental Techniques 36

3.1 Device design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.1 Dolan bridge technique . . . . . . . . . . . . . . . . . . . . . . 41

3.2.2 Bridge-free technique . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.3 Junction resistance . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3 Packaging and shielding . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.4 Measurement setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4 Suppression of Photon Shot Noise Dephasing 53

4.1 Photon shot noise dephasing . . . . . . . . . . . . . . . . . . . . . . . 54

4.2 Dispersive coupling rate . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3 Device design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4 Device calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.5 Readout method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.6 Coherence measurements . . . . . . . . . . . . . . . . . . . . . . . . . 66

5 Suppression of Qubit Crosstalk 70

5.1 Qubit crosstalk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2 Device design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3 Device calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.4 Parametric coupling and iSWAP gate . . . . . . . . . . . . . . . . . . 79

6 Conclusion and Outlook 85

A Fabrication Recipes 88

A.1 Photolithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

A.2 Niobium etching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

viii

CONTENTS

A.3 Electron beam lithography . . . . . . . . . . . . . . . . . . . . . . . . 89

A.4 Aluminum evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . 90

B Derivation for ZZ Coupling Rate 92

B.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

B.2 Second order terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

B.3 Fourth order terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

B.4 Final result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

C Single-qubit Clifford Gates 101

D Publications and Presentations 104

Bibliography 105

ix

List of Figures

1.1 Circuit quantum electrodynamics . . . . . . . . . . . . . . . . . . . . 4

1.2 Transmon qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Circuit diagram for TCQ . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Theoretical calculation for TCQ . . . . . . . . . . . . . . . . . . . . . 22

2.3 Model and circuit for the black-box quantization method . . . . . . . 28

2.4 Two-port network with parallel impedance . . . . . . . . . . . . . . . 29

2.5 Imaginary part of admittance 𝑌1(𝜔) . . . . . . . . . . . . . . . . . . . 30

2.6 TCQ energy levels calculated by black-box quantization . . . . . . . 33

3.1 Transmission line resonator . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Coplanar waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 A fabricated cQED device . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4 Dolan bridge technique . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.5 Short circuit caused by bridge collapse . . . . . . . . . . . . . . . . . 42

3.6 Bridge-free technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.7 A Josephson junction fabricated using bridge-free technique . . . . . 44

3.8 Device packaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.9 Device shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.10 ECCOSORB® filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.11 Measurement setup for cQED experiments . . . . . . . . . . . . . . . 50

x

LIST OF FIGURES

3.12 Cryogenic setup for cQED experiments . . . . . . . . . . . . . . . . . 51

4.1 Dispersive coupling between qubit and cavity . . . . . . . . . . . . . 55

4.2 Photon shot noise dephasing . . . . . . . . . . . . . . . . . . . . . . . 56

4.3 Straddling regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.4 Black-box quantization calculation for dispersive coupling rate of TCQ 59

4.5 TCQ device for suppression of photon shot noise . . . . . . . . . . . . 60

4.6 TCQ calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.7 TCQ spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.8 Transmission for TCQ readout cavity . . . . . . . . . . . . . . . . . . 64

4.9 TCQ readout scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.10 TCQ readout signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.11 Measurement scheme for TCQ coherence time . . . . . . . . . . . . . 67

4.12 Coherence measurements for TCQ . . . . . . . . . . . . . . . . . . . 68

5.1 Crosstalk in multi-qubit systems . . . . . . . . . . . . . . . . . . . . . 71

5.2 Tunable 𝜁 device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.3 Calibration for tunable 𝜁 device . . . . . . . . . . . . . . . . . . . . . 75

5.4 Qubit crosstalk characterization . . . . . . . . . . . . . . . . . . . . . 78

5.5 Parametric modulation for iSWAP gate . . . . . . . . . . . . . . . . . 81

5.6 iSWAP gate between two qubits . . . . . . . . . . . . . . . . . . . . . 82

5.7 Thermal population of coupler qubit . . . . . . . . . . . . . . . . . . 83

B.1 ZZ coupling rate calculation . . . . . . . . . . . . . . . . . . . . . . . 99

xi

List of Tables

2.1 Black-box quantization parameters . . . . . . . . . . . . . . . . . . . 35

3.1 CPW parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Measured junction resistance and qubit frequency . . . . . . . . . . . 46

4.1 TCQ device parameters . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.1 Tunable 𝜁 device parameters . . . . . . . . . . . . . . . . . . . . . . . 76

B.1 Second order terms for ZZ coupling . . . . . . . . . . . . . . . . . . . 93

B.2 Type (I) fourth order terms for ZZ coupling . . . . . . . . . . . . . . 94

B.3 Type (II) fourth order terms for ZZ coupling . . . . . . . . . . . . . . 96

B.4 Parameter configurations for ZZ coupling rate calculation . . . . . . . 100

C.1 Single-qubit Clifford gates . . . . . . . . . . . . . . . . . . . . . . . . 101

xii

Nomenclature

Abbreviations

CPB Cooper pair box, see section 1.3.

CPW Coplanar waveguide, see section 3.1.

cQED Circuit quantum electrodynamics, see section 1.2.

PCB Printed circuit board, see section 3.3.

RB Randomized benchmarking, see section 5.3.

RWA Rotating wave approximation, see section 2.1.

SQUID Superconducting quantum interference device, see section 1.4.

TCQ Tunable coupling qubit.

Physical constants

𝑐 Speed of light in vacuum (2.99792458 × 108 m⋅s−1).

𝑒 Elementary charge (1.6021766208 × 10−19 C).

ℎ Planck constant (6.626070040 × 10−34 J⋅s).

ℏ Reduced Planck constant (ℎ/2𝜋 = 1.054571800 × 10−34 J⋅s).

𝑘B Boltzmann constant (1.38064852 × 10−23 J⋅K−1).xiii

NOMENCLATURE

Φ0 Flux quantum (ℎ/2𝑒 = 2.067833831 × 10−15 Wb).

𝜙0 Reduced flux quantum (ℏ/2𝑒 = 3.29105976 × 10−16 Wb).

Latin letters

𝑎†, 𝑎𝑏†, 𝑏

Creation and annihilation operations.

𝐶 Capacitance.

𝐸 Energy.

𝐹 Fidelity of primary gates, see section 5.3.

𝑔 Coupling rate, see (2.25).

𝐻 Hamiltonian.

𝐼 Current.

𝐽 Hopping rate, see (2.14) and (5.8).

𝐿 Inductance; Lagrangian.

𝑚 Mode index for black-box quantization, see section 2.3.

𝑛 Number of Cooper pairs; Number of photons.

𝑝 Port index for black-box quantization, see section 2.3.

𝑄 Electric charge; Quality factor.

𝑅 Resistance.

𝑇 Temperature.

𝑇1 Relaxation time.xiv

NOMENCLATURE

𝑇2 Coherence time.

𝑈 Unitary matrix/operator.

𝑉 Voltage.

𝑌 Admittance.

𝑍0 Characteristic impedance.

[𝑍𝑖𝑗] Impedance matrix, see (2.45).

𝒵eff Effective impedance for black-box quantization, see (2.47).

Greek letters

𝛼 Anharmonicity, see (2.14); Attenuation constant for coplanar waveguide,

see section 3.1.

𝛽 Voltage division ratio, see (2.23); Propagation constant for coplanar waveg-

uide, see section 3.1.

𝜒 Dispersive coupling rate, see chapter 4.

Δ Frequency detuning; Gap energy of superconductor.

𝛿𝑖𝑗 Kronecker delta.

𝜖eff Effective permittivity for coplanar waveguide, see section 3.1.

𝜖r Dielectric constant, see section 3.1.

Γ𝜙 Photon shot noise dephasing rate, see (4.5).

𝜅 Photon decay rate.

xv

NOMENCLATURE

𝜔 Angular frequency.

Φ Magnetic flux.

𝜎𝑥,𝑦,𝑧,± Pauli spin and ladder operators.

𝜑 Phase difference across a Josephson junction.

𝜁 ZZ coupling rate, see chapter 5.

xvi

Chapter 1

Introduction

The ability to build complex systems and operate them in a controllable way has been

the driving force for scientific advances and the central task of engineering. In the

last two decades, controllable quantum systems, where multiple quantum mechanical

objects can be manipulated without destroying their coherence, have evolved from

theoretical proposals to concrete experimental realizations. The potential of exploring

new physical phenomena and exploiting quantum properties for computation in those

systems has attracted increasing interest in both academia and industry, and inspired

rapid research progress.

This thesis presents the study and application of tunable coupling devices in

one of the experimental platforms of controllable quantum systems, circuit quantum

electrodynamics. The ability to tune the coupling rate between different components

adds to the controllability and flexibility of those devices, provides tools to suppress

undesirable interactions and improves device performance. This chapter serves as an

introduction to the background and motivation of this thesis work. In section 1.1 we

discuss controllable quantum systems and the prospects of quantum computation and

quantum simulation enabled by those systems. Section 1.2 introduces circuit quantum

electrodynamics and briefly reviews the historical development and recent progress

1

CHAPTER 1. INTRODUCTION

in this field. Section 1.3 discusses the transmon qubit that is used for experiments in

this thesis. The importance of tunable coupling in quantum systems and devices is

explained in section 1.4 and a preview of the thesis is given in section 1.5.

1.1 Controllable quantum systems

Physical systems which show explicit quantum mechanical effects, such as superpo-

sition, interference, entanglement and many-body effects, etc., have been inspiring

research and technological progress in various fields in the last few decades. On the

one hand, it enables scientists to perform experiments in a systematic way to reveal

physical mechanisms behind observed phenomena. On the other hand, it allows quan-

tum effects to be utilized as resources to tackle interesting and challenging problems.

Quantum computers are machines that exploit the potential power of many-body

quantum systems to solve computational problems [1]. In 1994, Shor developed a

quantum algorithm which shows exponential speedup over known classical algorithms

for factorizing large numbers [2, 3]; In 1996, Grover designed a quantum algorithm for

database search that provides quadratic speedup [4, 5]. These discoveries, together

with the theory of quantum error correction [6, 7] and fault tolerance [8, 9], have led to

the rapid development of the field of quantum computation and information [10, 11].

The criteria for physical implementation of quantum computation is addressed by

DiVincenzo [12]. The basic element of a quantum computer is a quantum bit (qubit)

that can store arbitrary superposition of two quantum states. To perform computa-

tion, quantum gates are applied to qubits to transform one quantum state to another.

A fully functional quantum computer requires large numbers of interconnected qubits

and the ability to apply single- and two-qubit gates with high fidelity. The fragile

nature of quantum information and the difficulty in scaling up individual qubits to

2


large complex systems make it a challenging task to build and operate a quantum

computer.

Another research field that utilizes controllable quantum systems is quantum sim-

ulation [13, 14]. Due to the exponential growth of parameter space, simulating quan-

tum many-body systems is generally intractable on a classical computer. Originally

proposed by Feynman [15], quantum simulation maps physical systems or models onto

experimentally controllable and measurable devices. It would inspire the exploration

of new physical phenomena and new insights into complex quantum systems.

The ultimate goal of quantum computation and quantum simulation is to solve

problems that are prohibitively hard for classical computers. This task is both promis-

ing and challenging, and has attracted interests in a wide range of research fields, in-

cluding condensed matter and atomic physics, quantum chemistry, computer science,

electrical engineering and so forth.

Different physical implementations for controllable quantum systems have been

developed as candidate testbeds for quantum computation and simulation, includ-

ing trapped ions [16, 17], ultracold atoms [18, 19], photonics [20, 21] and quantum

dots [22, 23], etc. In the next section, we discuss the experimental platform for this

thesis, circuit quantum electrodynamics.

1.2 Circuit quantum electrodynamics

Circuit quantum electrodynamics (cQED) [24, 25] uses superconducting circuit el-

ements as its building blocks for controllable quantum systems and has undergone

rapid progress in the last two decades. It realizes interacting quantum systems analo-

gous to cavity quantum electrodynamics [26, 27] and is one of the promising physical

implementations for quantum computation and simulation.

3


Superconductivity [28] has been known as a macroscopic quantum phenomenon

since the 1950s [29] and in the 1980s condensed matter physicists began to explore

the properties of “macroscopic quantum variables” in superconductors [30]. These

research efforts eventually led to the development of superconducting qubits. Pio-

neering works date back to the study of macroscopic degrees of freedom and their

quantum behavior in Josephson junctions by Clarke [31], Devoret [32] and Marti-

nis [33]. In the late 1990s and early 2000s, coherent superpositions of quantum states

were observed in Josephson junction devices [34–36]. Shortly afterwards, different

types of qubits were developed, including charge [37, 38], flux [39], and phase [40]

qubits. Strong coupling between a qubit and a cavity was demonstrated in 2004 [41]

and two-qubit interaction mediated by a bus cavity is was achieved in 2007 [42, 43].

(a) (b)

Figure 1.1: Circuit quantum electrodynamics device. (a) Schematic representation ofa cQED device, where a qubit (Cooper-pair box atom) interacts with the photons in atransmission line cavity. The device is controlled and measured by input and outputsignals at either side of the cavity. Figure reproduced from [44]. (b) Improvementof relaxation time (𝑇1) and coherence time (𝑇2) for cQED devices. Qubit lifetimewas improved by more than six orders of magnitude between 1998 and 2012, duringwhich several generations of better qubit designs were invented to further suppressdecoherence. Figure reproduced from [45].

In a typical cQED device, one or more qubits are coupled to a transmission line

cavity, as is shown in figure 1.1(a). The physics of such a device is described by

4


Jaynes-Cummings Hamiltonian [46],

𝐻/ℏ = 𝜔𝑞𝜎𝑧/2 + 𝜔𝑟(𝑎†𝑎 + 1/2) + 𝑔(𝜎+𝑎 + 𝜎−𝑎†), (1.1)

where the first two terms represent the qubit with transition frequency 𝜔𝑞 and cavity

with resonance frequency 𝜔𝑟, and the last term represents the coupling between the

qubit and the photons in the cavity at rate 𝑔. The device is operated in the microwave

regime and implements the basic model for atom-light interaction in a solid state

system.

Compared to experimental approaches based on microscopic degrees of freedom

such as atoms, ions or quantum dots, in cQED strong coupling (𝑔/𝜔 ∼ 10 %) can

be achieved relatively easily due to the large electric or magnetic dipole moments

of macroscopic circuit elements. The ability to design the parameters, engineer the

performance and integrate multiple components makes cQED a powerful and flexible

platform for quantum computation [47, 48], quantum optics [49, 50] and quantum sim-

ulation [51–56]. Remarkable progresses have been made in achieving quantum-limited

amplification [57, 58], building error-corrected logical qubits [59–62], demonstrating

quantum supremacy [63, 64], and solving quantum chemistry problems [65]. Readers

can find excellent reviews [44, 45, 66–70], tutorials [71, 72], and theses [73–76] in the

literature for more in-depth introductions to cQED.

As cQED devices are artificial systems, they are susceptible to disorder introduced

in the fabrication process. Their macroscopic nature and large electric/magnetic

dipole moment cause the coupling to uncontrolled degrees of freedom in the envi-

ronment. These factors lead to decoherence processes where quantum information is

lost to the environment irreversibly [77]. The time scale for a qubit to stay in the

excited state (relaxation time, 𝑇1) and maintain a coherent superposition (coherence

5


time, 𝑇2) set the limit for the storage and processing of quantum information and

are important figure of merits for cQED devices. Extensive research and engineering

efforts were made in the past two decades to isolate the qubits from various decay

channels and noise sources in the environment, and both 𝑇1 and 𝑇2 have increased

remarkably by more than six orders of magnitude, as is shown in figure 1.1(b). State-

of-the-art devices have achieved qubit lifetime in the millisecond regime [78–81] and

pushed gate fidelity close to the threshold for fault-tolerant quantum computing [82].

The improvement in coherence time is the result of both physical insight into the

decoherence mechanisms and engineering innovations to mitigate them. The advent

of the transmon qubit is one excellent example of such scientific progress.

1.3 Transmon qubit

The transmon qubit [83, 84] derives its design from the Cooper pair box (CPB) [34],

where two superconducting islands are connected by a Josephson junction [85, 28

(chapters 6 and 7)]. The Hamiltonian for CPB is

𝐻 = 4𝐸𝐶(𝑛 − 𝑛𝑔)2 − 𝐸𝐽 cos 𝜑, (1.2)

where 𝐸𝐶 is the electrostatic charging energy for a single electron, 𝐸𝐽 is the tunneling

energy of the Josephson junction, 𝑛 is the number of Cooper pairs and 𝜑 is the phase

difference across the Josephson junction. The offset charge 𝑛𝑔 is typically induced by

an external voltage bias.

The Josephson junction enables coherent tunneling of Cooper pairs between the

two islands and causes the hybridization of charge states in the CPB, giving rise to

energy levels that can be used as a qubit. From a circuit point of view, a Josephson

junction is effectively a nonlinear inductor, which can be seen from the Josephson6


relations,𝐼 = 𝐸𝐽

𝜙0sin 𝜑,

𝑉 = 1𝜙0

𝑑𝜑𝑑𝑡 ,

⎫⎬⎭

⇒ 𝐿𝐽 = 𝑉𝑑𝐼/𝑑𝑡 = 1

𝐸𝐽 cos 𝜑, (1.3)

where 𝜙0 = ℏ/2𝑒 is the reduced flux quantum. This nonlinearity causes the energy

levels of the device to be anharmonic and when the energy difference 𝐸01 between

the lowest two levels |0⟩ and |1⟩ is sufficiently different from other transition energies,

a qubit can be formed by selectively addressing only those two levels.

The CPB is in the parameter regime of 𝐸𝐽 ≈ 𝐸𝐶 and the frequency of the

qubit is sensitive to the offset charge 𝑛𝑔, as is shown in the case of 𝐸𝐽/𝐸𝐶 = 1 in

figure 1.2(a), and suffers from fluctuations in bias voltage (charge noise). To achieve

best performance, the external voltage bias needs to be tuned such that 𝑛𝑔 = 1/2 [38],

where the device is first order insensitive to charge noise.

−2 −1 0 1 202468

10

E m/E

01

∼ EJ

EJ/EC = 1.0

−2 −1 0 1 20

1

2

3EJ/EC = 5.0

−2 −1 0 1 2ng

0

1

2

E m/E

01

EJ/EC = 10.0

−2 −1 0 1 2ng

0

1

2

∼p

8EJ EC

EJ/EC = 50.0

100 µm

5 µm

(a) (b)

Figure 1.2: Transmon qubit. (a) Charge dispersion of CPB and transmon for different𝐸𝐽/𝐸𝐶 ratios. As 𝐸𝐽/𝐸𝐶 increases, charge dispersion decreases exponentially, anddecoherence due to charge noise is suppressed. The eigenenergies 𝐸𝑚 (𝑚 = 0, 1, 2)of Hamiltonian (1.2) are normalized such that 𝐸1 − 𝐸0 = 1 when 𝑛𝑔 = 1/2. Figurereproduced from [83]. (b) Micrograph for a transmon qubit. The two big super-conducting islands form the shunting capacitor and the inset shows the Josephsonjunctions.

7


The main innovation of transmon is changing the parameter regime to 𝐸𝐽 ≫ 𝐸𝐶.

Theoretical analysis [83] shows that charge dispersion 𝜖𝑚 = max(𝐸𝑚) − min(𝐸𝑚)decreases exponentially with respect to √𝐸𝐽/𝐸𝐶 [see figure 1.2(a)], while the anhar-

monicity remains reasonably large (𝐸12 − 𝐸10 ≈ 𝐸𝐶) as 𝐸𝐽/𝐸𝐶 increases.

Figure 1.2(b) shows a transmon qubit, where the Josephson junctions are shunted

by a large capacitor. This leads to reduced 𝐸𝐶 and typical devices are designed and

operated in the regime of 𝐸𝐽/𝐸𝐶 ≳ 50. In this regime charge dispersion is greatly

reduced and coherence time of 𝑇2 ≈ 𝑇1 can be achieved without the need of a voltage

bias [86]. The transmon therefore combines simple design, good performance and

easy operation in an elegant way, and has become the most widely used type of qubit

in cQED.

More detailed discussions on transmon can be found in [72, 86].

1.4 Tunable coupling

To utilize the potential power of quantum systems we need to prepare, manipulate

and measure various degrees of freedom in the system. This is often realized by

adjusting some parameters (energy levels, tunneling/interaction strength, external

fields/potentials, etc.) and observing the response. Tunability therefore plays an

important role in the realization of quantum computation and quantum simulation.

The increasing number of tuning knobs for quantum systems is a signature of research

progress on all physical platforms and allows us to perform experiments in a more

flexible way and larger parameter space.

In particular, the ability to control the coupling or interaction strength between

various degrees of freedom is an important experimental tool for the study and engi-

neering of quantum systems. In the field of quantum simulation using cold atoms [19],

8


the tunable attractive interaction between fermions near Feshbach resonances [87]

leads to the remarkable observation of BEC-BCS crossover [88]; tunable interaction

between Rydberg states [89] is used to study the dynamics of Ising-like systems [90].

In trapped ion systems [91], variable-range effective spin-spin interactions are gen-

erated by applying spin-dependent optical dipole forces [92] and allow scientists to

demonstrate a small-scale quantum computer [93] and explore many-body localiza-

tion in a disordered system [94]. In cavity opto- and electro-mechanics [95], the

linearized coupling rate between mechanical and optical modes are controlled by a

driving laser/microwave field to achieve strong coupling [96, 97] and resolved sideband

cooling [98, 99].

In cQED the tuning mechanism for most devices are introduced by Josephson junc-

tion based elements. For transmon qubit, the qubit frequency can be made tunable by

splitting the Josephson junction into a ring configuration [See inset of figure 1.2(b)]

and forming a superconducting quantum interference device (SQUID) [28 (section

6.5)]. As a consequence of quantum interference and flux quantization, the effective

Josephson energy can be tuned by an external magnetic flux Φ,

𝐸𝐽 = √(𝐸𝐽1 + 𝐸𝐽2)2 cos2(𝜋Φ/Φ0) + (𝐸𝐽1 − 𝐸𝐽2)2 sin2(𝜋Φ/Φ0), (1.4)

where 𝐸𝐽1, 𝐸𝐽2 are the Josephson energies of the two junctions and Φ0 = ℎ/2𝑒 is

the flux quantum.

The ability to tune coupling strength between different components adds to the

flexibility of cQED devices. It enables the exploration of new parameter regimes and

provides a useful tool when frequency tuning is limited by concerns such as spectral

crowding, flux noise induced dephasing, etc. Current biased Josephson junctions [100,

101], rf [102–104] and dc [105–107] SQUIDs are used to statically and dynamically

9


control qubit-qubit and qubit-cavity coupling rates and lead to the implementation

of two-qubit gates, parametric interactions, novel readout methods and qubit state

stabilization.

In our lab we utilize the quantum interference effect between two frequency-

tunable transmons to achieve tunable qubit-cavity coupling [108, 109]. Previous

works include the demonstration of both static [110] and dynamic [111] tunability

and the application in pulse shaping of single photon emission [112]. This thesis ex-

tends those works and discusses the engineering and application of tunable dispersive

coupling in cQED devices.

1.5 Thesis overview

This thesis studies and implements tunable qubit-cavity and qubit-qubit interactions

in the dispersive regime and realizes two applications of tunable coupling devices

in cQED. These efforts extend the controllability and flexibility of cQED devices

and provide promising building blocks for cQED systems. The rest of the thesis is

organized as follows.

Chapter 2 presents the theoretical tools that are used in this thesis, including

circuit quantization, perturbation theory, and black-box quantization. Those tools

provide physical intuition and quantitative calculation that help the understanding,

analysis and design of the devices used in later chapters. In particular, we explain how

tunable coupling mechanism can be achieved using quantum interference in detail.

Chapter 3 turns to the experiment side and introduces techniques in micro/nano

fabrication, device packaging/shielding and microwave/cryogenic engineering. The

development and improvement of those techniques help to produce devices reliably

and ensure their good performance in the experiments.

10


The next two chapters discuss two applications of tunable coupling devices. A

common theme in both projects is the utilization of quantum interference and tun-

able coupling as resources to suppress unwanted interactions in cQED systems. In

chapter 4 a tunable coupling qubit is used to suppress photon shot noise dephasing.

We explain this dephasing mechanism and its connection to dispersive coupling, and

demonstrate 𝑇1 limited coherence time that is robust against injected noise when the

device is operated in zero dispersive coupling regime.

In chapter 5, we explore the application of tunable coupling in multi-qubit sys-

tems and develop a two-coupler device that suppresses static ZZ crosstalk between

two qubits. The origin of the crosstalk is explained and its suppression is observed

and characterized experimentally. An efficient two-qubit entangling gate is also im-

plemented by parametrically modulating the dispersive coupling rate.

A brief summary and outlook of the thesis is presented in chapter 6 and detailed

discussions on fabrication recipes, ZZ coupling rate calculation and Clifford gates can

be found in the appendices.

11

Chapter 2

Theoretical Tools

This chapter introduces the theoretical methods used in this thesis for the design and

analysis of cQED devices. These methods provide physical understanding and quan-

titative calculation for the tunable coupling devices in later chapters. In section 2.1

we present the circuit quantization procedure, where we start from a circuit descrip-

tion of a device and derive its Hamiltonian. Section 2.2 discusses perturbation theory,

a useful tool for approximating qubit-qubit and qubit-cavity interactions when the

devices are operated in the dispersive regime. In section 2.3 we introduce the more

recently developed black-box quantization method in cQED community, which is es-

pecially suitable for weakly nonlinear circuit elements such as the transmon qubit.

The derivation and calculation follow closely the treatments in [71, 72, 108, 113], and

the main purpose of this chapter is to apply those methods to the tunable coupling

qubit (TCQ) as a concrete example and explain how tunable coupling arises as a

consequence of quantum superposition and interference.

2.1 Quantization of the circuit

In cQED experiments the temperature (∼ 20 mK) is low enough so that the spacing

between energy levels of the electromagnetic modes of the circuit (∼ 5 GHz) is much12

CHAPTER 2. THEORETICAL TOOLS

larger than thermal fluctuations. Therefore, a quantum mechanical treatment is

appropriate for the description and analysis of the circuit. In this section we discuss

the circuit quantization procedure for TCQ, following the treatments in [71, 72].

At the end of this section we show that when a TCQ is coupled to a cavity, both

the qubit frequency and qubit-cavity coupling can be controlled independently by

adjusting quantum superposition and interference between two coupled transmons.

2.1.1 Derivation of the Hamiltonian

The starting point of circuit quantization is the observation that in an LC oscillator

consisting of superconductors, there is only one low energy degree of freedom, the

flux variable Φ (or equivalently the charge variable 𝑄). The current-flux relations for

the basic circuit elements are

Capacitor ∶ 𝐼 = 𝐶Φ,

Inductor ∶ 𝐼 = Φ/𝐿,

Josephson junction ∶ 𝐼 = 𝐼𝑐 sin 𝜑 = 𝐼𝑐 sin(Φ/𝜙0),

(2.1)

where Φ is the flux across the element, 𝜙0 = ℏ/2𝑒 is the reduced flux quantum, 𝐼𝑐

and 𝜑 are the critical current of and phase difference across a Josephson junction, re-

spectively. The currents in different elements of the circuit are related by Kirchhoff’s

law, from which we can obtain the Lagrangian of the system. Legendre transforma-

tion and canonical quantization are then performed to derive the Hamiltonian of the

circuit.

We now apply the above procedure to the TCQ, whose image is shown in fig-

ure 2.1(a). It is a three-island device that contains two coupled transmon qubits.

Each transmon consists of a shunting capacitor 𝐶𝐵1(2) and two Josephson junctions

13


VV

(a) (b) a

1 2

3

3

2a

1

50 µm

−Vg

0

EJ1 CB1 CB2 EJ2

CI

Cg1 Cg2

Figure 2.1: Device image and circuit diagram for TCQ. (a) False-colored opticalmicrograph of a TCQ, which consists of two transmon qubits (left and right halvesof the device) coupled via a capacitor in the middle. The frequencies of the twotransmons can be varied by applying external magnetic flux to the two SQUID loopsthrough the two bias lines. Figure adapted from [110]. (b) Simplified circuit modelfor the device, where only capacitors between neighboring islands are considered. Thetwo SQUID loops are represented by single junctions with tunable 𝐸𝐽 for simplicity.

forming a SQUID loop, which makes the Josephson energy 𝐸𝐽1(2) tunable by apply-

ing magnetic flux. The two transmons are coupled to each other via capacitor 𝐶𝐼

and to the transmission line cavity via capacitors 𝐶𝑔1 and 𝐶𝑔2, respectively. The

voltage source represents the voltage drop between the center conductor and ground

of the transmission line, at the position where the TCQ is coupled to the cavity. The

effective circuit containing these elements is shown in figure 2.1(b) and we denote the

four nodes in the circuit by 1, 2, 3 and 𝑎.

We define the flux variables at nodes 1, 2, 𝑎 as Φ1, Φ2, Φ𝑎, respectively, and

set Φ3 = 0. Applying Kirchhoff’s current law to nodes 1 and 2, we get

𝐶𝑔1(Φ𝑎 − Φ1) + 𝐶𝐼(Φ2 − Φ1) − 𝐶𝐵1Φ1 − 𝐼𝑐1 sin 𝜑1 = 0,

𝐶𝑔2(Φ𝑎 − Φ2) + 𝐶𝐼(Φ1 − Φ2) − 𝐶𝐵2Φ2 − 𝐼𝑐2 sin 𝜑2 = 0,(2.2)

where the phase differences 𝜑𝑖 are related to the flux variables through 𝜑𝑖 = Φ𝑖/𝜙0.

The Lagrangian that produces (2.2) as its equation of motion is

𝐿 =2

∑𝑖=1

12𝐶𝑔𝑖(Φ𝑖 + 𝑉𝑔)2 + 1

2𝐶𝐵𝑖Φ𝑖 + 𝐸𝐽𝑖 cos 𝜑𝑖 + 12𝐶𝐼(Φ1 − Φ2)2, (2.3)

14


where 𝐸𝐽𝑖 = 𝐼𝑐𝑖𝜙0 is the Josephson energy and we use the relation Φ𝑎 = −𝑉𝑔. The

canonical momenta conjugate to Φ𝑖’s are charge variables,

𝑄1 = 𝜕𝐿𝜕Φ1

= (𝐶𝑔1 + 𝐶𝐵1 + 𝐶𝐼)Φ1 − 𝐶𝐼Φ2 + 𝐶𝑔1𝑉𝑔,

𝑄2 = 𝜕𝐿𝜕Φ2

= −𝐶𝐼Φ1 + (𝐶𝑔2 + 𝐶𝐵2 + 𝐶𝐼)Φ2 + 𝐶𝑔2𝑉𝑔.(2.4)

We can express Φ𝑖 in terms of 𝑄𝑖,

Φ1 = (𝐶Σ2 + 𝐶𝐼)(𝑄1 − 𝐶𝑔1𝑉𝑔) + 𝐶𝐼(𝑄2 − 𝐶𝑔2𝑉𝑔)𝐶2

𝐸,

Φ2 = 𝐶𝐼(𝑄1 − 𝐶𝑔1𝑉𝑔) + (𝐶Σ1 + 𝐶𝐼)(𝑄2 − 𝐶𝑔2𝑉𝑔)𝐶2

𝐸,

(2.5)

where

𝐶Σ𝑖 = 𝐶𝑔𝑖 + 𝐶𝐵𝑖, 𝐶𝐸 = √𝐶Σ1𝐶Σ2 + 𝐶𝐼(𝐶Σ1 + 𝐶Σ2). (2.6)

Using (2.3), (2.4) and (2.5), we obtain the Hamiltonian (omitting constant terms)

𝐻 =2

∑𝑖=1

𝑄𝑖Φ𝑖 − 𝐿

= (𝐶Σ2 + 𝐶𝐼)2𝐶2

𝐸[𝑄1 − (𝐶𝑔1 + 𝐶𝑔2𝐶𝐼

𝐶Σ2 + 𝐶𝐼) 𝑉𝑔]

2− 𝐸𝐽1 cos 𝜑1

+ (𝐶Σ1 + 𝐶𝐼)2𝐶2

𝐸[𝑄2 − (𝐶𝑔2 + 𝐶𝑔1𝐶𝐼

𝐶Σ1 + 𝐶𝐼) 𝑉𝑔]

2− 𝐸𝐽2 cos 𝜑2

+ 𝐶𝐼𝐶2

𝐸𝑄1𝑄2.

(2.7)

Defining the number of Cooper pairs 𝑛𝑖, gate charge 𝑛𝑔𝑖, charging energy 𝐸𝐶𝑖, and

coupling energy 𝐸𝐼 as

𝑛𝑖 = 𝑄𝑖2𝑒 , 𝑛𝑔1 = (𝐶𝑔1 + 𝐶𝑔2𝐶𝐼

𝐶Σ2 + 𝐶𝐼) 𝑉𝑔

2𝑒 , 𝑛𝑔2 = (𝐶𝑔2 + 𝐶𝑔1𝐶𝐼𝐶Σ1 + 𝐶𝐼

) 𝑉𝑔2𝑒 ,

𝐸𝐶1 = 𝑒2 𝐶Σ2 + 𝐶𝐼2𝐶2

𝐸, 𝐸𝐶2 = 𝑒2 𝐶Σ1 + 𝐶𝐼

2𝐶2𝐸

, 𝐸𝐼 = 𝑒2 𝐶𝐼𝐶2

𝐸,

(2.8)

15


we can rewrite the Hamiltonian as

𝐻𝑇 =2

∑𝑖=1

4𝐸𝐶𝑖(𝑛𝑖 − 𝑛𝑔𝑖)2 − 𝐸𝐽𝑖 cos 𝜑𝑖 + 4𝐸𝐼𝑛1𝑛2. (2.9)

Recalling the Hamiltonian for a CPB (1.2), we can see that 𝐻𝑇 describes two qubits

that are capacitively coupled to each other. This coupling leads to the hybridization

of qubit states and is the origin of the tunable coupling mechanism.

Canonical quantization is performed by promoting 𝑄𝑖 (𝑛𝑖) and Φ𝑖 (𝜑𝑖) to operators

which obey the commutation relations

[Φ𝑖, 𝑄𝑗] = 𝑖ℏ𝛿𝑖𝑗 ⇔ [𝜑𝑖, 𝑛𝑗] = 𝑖𝛿𝑖𝑗. (2.10)

Due to the coupling between the two qubits the Hamiltonian (2.9) is not diagonal.

To find the eigenmodes of the system we proceed to diagonalize the Hamiltonian.

2.1.2 Diagonalization of the Hamiltonian

We follow the approach in appendix C of [83] to approximate the Hamiltonian in the

transmon regime where 𝐸𝐽𝑖/𝐸𝐶𝑖 ≫ 1. In this limit charge dispersion (the effect of

𝑛𝑔𝑖) can be neglected and the two qubits can be approximated as weakly anharmonic

oscillators. Following the usual treatment for quantum harmonic oscillators, we ex-

press the charge and phase variables in terms of creation and annihilation operators

𝑏†𝑖 and 𝑏𝑖,

𝑛𝑖 = −𝑖 ( 𝐸𝐽𝑖8𝐸𝐶𝑖

)1/4 1√

2(𝑏𝑖 − 𝑏†

𝑖 ),

𝜑𝑖 = (8𝐸𝐶𝑖𝐸𝐽𝑖

)1/4 1√

2(𝑏𝑖 + 𝑏†

𝑖 ),

[𝜑𝑖, 𝑛𝑗] = 𝑖𝛿𝑖𝑗 ⇔ [𝑏𝑖, 𝑏†𝑗 ] = 𝛿𝑖𝑗.

(2.11)

16


Substituting (2.11) into (2.9) and making proper approximations, we find

𝐻𝑇 /ℏ ≈2

∑𝑖=1

4𝐸𝐶𝑖𝑛2𝑖 − 𝐸𝐽𝑖 cos 𝜑𝑖 + 4𝐸𝐼𝑛1𝑛2 (Neglecting 𝑛𝑔𝑖)

≈2

∑𝑖=1

4𝐸𝐶𝑖𝑛2𝑖 − 𝐸𝐽𝑖(1 − 𝜑2

𝑖 /2 + 𝜑4𝑖 /24) + 4𝐸𝐼𝑛1𝑛2

⎛⎜⎜⎝

Fourth order

expansion in 𝜑𝑖

⎞⎟⎟⎠

=2

∑𝑖=1

√8𝐸𝐶𝑖𝐸𝐽𝑖(𝑏†𝑖 𝑏𝑖 + 1/2) − 𝐸𝐽𝑖 − 𝐸𝐶𝑖

12 (𝑏𝑖 + 𝑏†𝑖 )4

− 𝐸𝐼 ( 𝐸𝐽1𝐸𝐽24𝐸𝐶1𝐸𝐶2

)1/4

(𝑏1 − 𝑏†1)(𝑏2 − 𝑏†

2) [Substituting (2.11)]

≈2

∑𝑖=1

(√8𝐸𝐶𝑖𝐸𝐽𝑖 − 𝐸𝐶𝑖)(𝑏†𝑖 𝑏𝑖 + 1/2) − 𝐸𝐶𝑖

2 𝑏†𝑖 𝑏†

𝑖 𝑏𝑖𝑏𝑖

+ 𝐸𝐼 ( 𝐸𝐽1𝐸𝐽24𝐸𝐶1𝐸𝐶2

)1/4

(𝑏1𝑏†2 + 𝑏†

1𝑏2) − 𝐸𝐽𝑖 + 𝐸𝐶𝑖4

(2.12)

In the last step we keep the diagonal terms for (𝑏𝑖 + 𝑏†𝑖 )4,

⟨𝑗|(𝑏𝑖 + 𝑏†𝑖 )4|𝑗⟩ = ⟨𝑗|(6𝑏†

𝑖 𝑏†𝑖 𝑏𝑖𝑏𝑖 + 12𝑏†

𝑖 𝑏𝑖 + 3)|𝑗⟩, (2.13)

and use rotating wave approximation (RWA) for (𝑏1 − 𝑏†1)(𝑏2 − 𝑏†

2).Defining frequency 𝜔𝑖, anharmonicity 𝛼𝑖 and hopping rate 𝐽 as

𝜔𝑖 = (√8𝐸𝐽𝑖𝐸𝐶𝑖 − 𝐸𝐶𝑖)/ℏ, 𝛼𝑖 = 𝐸𝐶𝑖/ℏ,

𝐽 = 𝐸𝐼(𝐸𝐽1𝐸𝐽2/𝐸𝐶1𝐸𝐶2)1/4/√

2ℏ.(2.14)

and dropping constant terms in (2.12), we obtain the approximate Hamiltonian for

TCQ in the transmon regime,

𝐻𝑇 /ℏ =2

∑𝑖=1

𝜔𝑖𝑏†𝑖 𝑏𝑖 − 𝛼𝑖

2 𝑏†𝑖 𝑏†

𝑖 𝑏𝑖𝑏𝑖 + 𝐽(𝑏1𝑏†2 + 𝑏†

1𝑏2). (2.15)

17


The Hamiltonian 𝐻𝑇 describes two anharmonic oscillators with frequencies 𝜔1(2),

anharmonicities 𝛼1(2) and hopping rate 𝐽 between each other. The three-island design

and large coupling capacitor 𝐶𝐼 in TCQ lead to extremely large hopping rate 𝐽/2𝜋 >1 GHz (In comparison, 𝐽/2𝜋 = 26 MHz in [42] when two qubits are coupled via a bus

cavity), resulting in strong hybridization of the two qubits even when they are detuned

from each by a few GHz. When we adjust the amount of detuning using external

magnetic fluxes, the hybridization coefficients are tuned accordingly and causes the

frequency and coupling rate of the normal modes to change. This is the basic tuning

mechanism for TCQ. To show this more concretely, we now follow the analysis in [108]

and approximately diagonalize the Hamiltonian using the transformation

𝑏+ = 𝑏1 cos 𝜃 + 𝑏2 sin 𝜃, 𝑏− = −𝑏1 sin 𝜃 + 𝑏2 cos 𝜃. (2.16)

where 𝜃 determines the superposition coefficients for the normal modes. We note that

the normal (uncoupled) modes are labeled by ±, and the original (coupled) transmon

modes are labeled by 1, 2. Diagonalization of 𝐻𝑇 requires quadratic cross terms like

𝑏†+𝑏− to vanish when we substitute (2.16) into (2.15) and leads to

tan 2𝜃 = 2𝐽/𝜁, (2.17)

where 𝜁 = 𝜔1 − 𝜔2 + (𝛼1 − 𝛼2)/2. The transformation (2.16) exactly diagonalizes

(2.15) when 𝛼1,2 = 0, and is a valid approximation when 𝛼1,2/𝐽 is small.

Combining (2.15), (2.16), and (2.17) and keeping the diagonal fourth order terms,

we get the diagonalized Hamiltonian

𝐻𝑇 /ℏ = 𝜔+𝑏†+𝑏+ − 𝛼+

2 𝑏†+𝑏†

+𝑏+𝑏+ + 𝜔−𝑏†−𝑏− − 𝛼−

2 𝑏†−𝑏†

−𝑏−𝑏− − 𝛼𝑐𝑏†+𝑏+𝑏†

−𝑏−, (2.18)

18


with parameters

𝜔+ = (𝜔1 + 𝛼12 ) cos2 𝜃 + (𝜔2 + 𝛼2

2 ) sin2 𝜃 + 2𝐽 sin 𝜃 cos 𝜃

− 𝛼1 + 𝛼22 sin2 𝜃 cos2 𝜃 − 𝛼+

2= 𝜔1 + 𝛼1 − 𝛼+

2 + √4𝐽2 + 𝜁2

2 − 𝜁2 − 𝛼1 + 𝛼2

2𝐽2

4𝐽2 + 𝜁2 ,

𝜔− = (𝜔1 + 𝛼12 ) sin2 𝜃 + (𝜔2 + 𝛼2

2 ) cos2 𝜃 − 2𝐽 sin 𝜃 cos 𝜃

− 𝛼1 + 𝛼22 sin2 𝜃 cos2 𝜃 − 𝛼−

2= 𝜔2 + 𝛼2 − 𝛼−

2 − √4𝐽2 + 𝜁2

2 + 𝜁2 − 𝛼1 + 𝛼2

2𝐽2

4𝐽2 + 𝜁2 ,

𝛼+ = 𝛼1 cos4 𝜃 + 𝛼2 sin4 𝜃

= 𝛼1 + 𝛼24 (1 + 𝜁2

4𝐽2 + 𝜁2 ) + 𝛼1 − 𝛼22

𝜁√4𝐽2 + 𝜁2 ,

𝛼− = 𝛼1 sin4 𝜃 + 𝛼2 cos4 𝜃

= 𝛼1 + 𝛼24 (1 + 𝜁2

4𝐽2 + 𝜁2 ) − 𝛼1 − 𝛼22

𝜁√4𝐽2 + 𝜁2 ,

𝛼𝑐 = 2(𝛼1 + 𝛼2) sin2 𝜃 cos2 𝜃

= 2(𝛼1 + 𝛼2) 𝐽2

4𝐽2 + 𝜁2 ,

(2.19)

In the above derivation, we choose 0 < 𝜃 < 𝜋/4 when 𝜁 > 0, and 𝜋/4 < 𝜃 < 𝜋/2when 𝜁 < 0 (Assuming 𝐽 > 0), to ensure 𝜔+ > 𝜔−.

We find from (2.18) that in the normal basis the TCQ consists of two qubits with

self and cross anharmonicities, and the energy levels form a V-shaped structure, as is

shown in figure 2.2(a). It is worth noting that the cross anharmonicity 𝛼𝑐 enables the

indirect readout scheme in section 4.5. In a TCQ both 𝜔1,2 can be tuned by external

19


magnetic flux Φ1,2 respectively,

𝜔𝑖 = 𝜔max𝑖 √| cos(𝜋Φ𝑖/Φ0)|, 𝑖 = 1, 2, (2.20)

and from the first two equations in (2.19) we see that the frequencies 𝜔± of the normal

modes can be tuned accordingly. We now proceed to show that their coupling rates

𝑔± are also tunable.

2.1.3 Coupling between TCQ and cavity

In the previous sections we treat the gate voltage 𝑉𝑔 as a classical quantity. To

include the quantized electromagnetic field of the cavity mode, we treat the cavity

as an effective LC oscillator [24] with inductance 𝐿𝑟 and capacitance 𝐶𝑟. The gate

voltage 𝑉𝑔 then becomes

𝑉𝑔 → 𝑉𝑔 + 𝑉rms(𝑖𝑎 − 𝑖𝑎†), (2.21)

where 𝑎 and 𝑎† are the annihilation and creation operators for the cavity photons,

𝑉rms = √ℏ𝜔𝑟/2𝐶𝑟 is the root-mean-square voltage of the cavity mode, and 𝜔𝑟 =1/√𝐿𝑟𝐶𝑟 is the resonance frequency of the cavity mode.

Substituting 𝑉𝑔 in (2.8) with (2.21), we find

4𝐸𝐶𝑖(𝑛𝑖 − 𝑛𝑔𝑖)2 → 4𝐸𝐶𝑖(𝑛𝑖 − 𝑛𝑔𝑖)2 + 2𝑒𝛽𝑖𝑉rms𝑛𝑖(−𝑖𝑎 + 𝑖𝑎†),

= 4𝐸𝐶𝑖(𝑛𝑖 − 𝑛𝑔𝑖)2 +√

2𝑒𝛽𝑖𝑉rms ( 𝐸𝐽𝑖8𝐸𝐶𝑖

)1/4

(𝑏𝑖 − 𝑏†𝑖 )(𝑎† − 𝑎),

(2.22)

where

𝛽1 =𝐶𝐼(𝐶𝑔1 + 𝐶𝑔2

) + 𝐶𝑔1𝐶Σ2𝐶2

𝐸, 𝛽2 =

𝐶𝐼(𝐶𝑔1 + 𝐶𝑔2) + 𝐶𝑔2𝐶Σ1

𝐶2𝐸

, (2.23)

20


are the voltage division ratios and in the last step in (2.22) we use (2.11). The term

2𝑒𝛽𝑖𝑉rms𝑛𝑖(−𝑖𝑎 + 𝑖𝑎†) in (2.22) represents interaction between the electric dipole of

the transmons and electric field of cavity photons. We rewrite it as the coupling

Hamiltonian,

𝐻𝐶/ℏ =2

∑𝑖=1

𝑔𝑖(𝑎𝑏†𝑖 + 𝑎†𝑏𝑖), (2.24)

where we use RWA and define the coupling rates for the coupled transmon modes as

𝑔𝑖 =√

2𝑒𝛽𝑖𝑉rmsℏ

( 𝐸𝐽𝑖8𝐸𝐶𝑖

)1/4

, 𝑖 = 1, 2. (2.25)

Applying transformation (2.16), we express 𝐻𝐶 in the normal basis for TCQ,

𝐻𝐶/ℏ = 𝑔+(𝑎𝑏†+ + 𝑎†𝑏+) + 𝑔−(𝑎𝑏†

− + 𝑎†𝑏−), (2.26)

where

𝑔+ = 𝑔1 cos 𝜃 + 𝑔2 sin 𝜃, 𝑔− = −𝑔1 sin 𝜃 + 𝑔2 cos 𝜃, (2.27)

are the coupling rates for the normal modes. We see that 𝑔+ and 𝑔− reflect the

superposition (2.16) and can be tuned by adjusting 𝜔1 − 𝜔2 through the relations

(2.17) and (2.27). In particular, 𝑔− can be tuned to zero when the superposition

satisfies tan 𝜃 = 𝑔2/𝑔1.

The full Hamiltonian, including the TCQ, the resonator and their coupling, is

therefore

𝐻 = 𝐻𝑇 + 𝐻𝑅 + 𝐻𝐶,

𝐻𝑇 /ℏ = 𝜔+𝑏†+𝑏+ − 𝛼+

2 𝑏†+𝑏†

+𝑏+𝑏+ + 𝜔−𝑏†−𝑏− − 𝛼−

2 𝑏†−𝑏†


−𝑏−,

𝐻𝑅/ℏ = 𝜔𝑟𝑎†𝑎, 𝐻𝐶/ℏ = 𝑔+(𝑎𝑏†+ + 𝑎†𝑏+) + 𝑔−(𝑎𝑏†

− + 𝑎†𝑏−).

(2.28)

21


To illustrate the tunability of frequency and coupling rate in TCQ, we calculate

𝜔±, 𝑔±, 𝛼±,𝑐 as a function of 𝐸𝐽2 while 𝐸𝐽1 is fixed, as is shown in figure 2.2. The

parameters are symmetric for the two transmons. The analytical expressions (2.19),

(2.27) are compared with numerical results from diagonalizing (2.9) and evaluating

matrix element of (2.24) in the normal basis and show qualitative agreement. The pa-

rameters for the upper qubit show more deviation from the numerical results, because

they are more affected by the presence of higher energy levels, whose contributions

are not fully captured in the analytical expressions.

12 14 16 18 20 22 24 26EJ2/h (GHz)

5

6

7

8

9

!/2ı

(GHz

)

!+

!−

!1

!2

12 14 16 18 20 22 24 26EJ2/h (GHz)

0

40

80

120

160

|g|/

2ı(M

Hz) g+

g−

NumericalAnalytical

12 14 16 18 20 22 24 26EJ2/h (GHz)

150

200

250

300

350

400

¸/2ı

(MHz

)

¸c

¸+

¸−

|0−0+⟩

|1−1+⟩!− + !+ − ¸c

|1−0+⟩!−

|2−0+⟩2!− − ¸−

|0−1+⟩!+

|0−2+⟩2!+ − ¸+

g−

2g−

g+

2g+

g+

g−

(a) (b)

(c) (d)

Figure 2.2: Theoretical calculation for TCQ. The V-shaped energy level diagram andmatrix elements for transitions between levels are shown in (a). Analytical expressions(2.19), (2.27) and numerical diagonalization are used to calculate (b) qubit frequencies(c) qubit-cavity coupling rates and (d) anharmonicities. The parameters are 𝐸𝐶1/ℎ =𝐸𝐶2/ℎ = 350 MHz, 𝐸𝐽1/ℎ = 19 GHz, 𝐸𝐼/ℎ = 180 MHz, 𝛽1 = 𝛽2 = 0.1, 𝑉rms =1.8 μV, and 𝐸𝐽2 is varied linearly. In (b), black dashed lines correspond to qubitfrequencies when 𝐸𝐼 = 𝐽 = 0. In (c), the coupling rates are calculated using (2.22),ℏ𝑔± = ⟨0| ∑2

𝑖=1 2𝑒𝛽𝑖𝑉rms𝑛𝑖|±⟩. In (d), (2.19) gives the same value for 𝛼± (blue dashedlines), while numerical calculation yields different values.

22


We look at two special cases for the symmetric design where 𝛼1 = 𝛼2 = 𝛼 and

𝑔1 = 𝑔2 = 𝑔.

1. When the two transmon frequencies are degenerate, i.e., 𝜔1 = 𝜔2 = 𝜔, we find

𝜁 = 0 and 𝜃 = 𝜋/4, so (2.19) and (2.27) reduce to

𝜔+ = 𝜔 + 𝐽,

𝛼+ = 𝛼2 ,

𝑔+ =√

2𝑔,

𝜔− = 𝜔 − 𝐽,

𝛼− = 𝛼2 , 𝛼𝑐 = 𝛼,

𝑔− = 0.

(2.29)

In this case the upper (lower) normal mode corresponds to the symmetric (an-

tisymmetric) superposition of the bare qubit modes,

|±⟩ = 1√2

(|10⟩ ± |01⟩), (2.30)

and constructive (destructive) interference between the dipole transitions leads

to enhanced (suppressed) coupling rate 𝑔±. This can be seen in figure 2.2(c)

when 𝐸𝐽1/ℎ = 𝐸𝐽2/ℎ = 19 GHz.

2. When the two transmon frequencies are far detuned, i.e., 𝜔1 − 𝜔2 ≫ 𝐽 , we find

𝜁 ≫ 𝐽 and 𝜃 = 0, so (2.19) and (2.27) reduce to

𝜔+ = 𝜔1, 𝜔− = 𝜔2

𝛼+ = 𝛼− = 𝛼, 𝛼𝑐 = 0

𝑔+ = 𝑔− = 𝑔.

(2.31)

In this limit the normal modes correspond to two uncoupled qubits with the

bare parameters,

|+⟩ ≈ |10⟩, |−⟩ ≈ |01⟩. (2.32)23


Between those two cases, 𝑔− can be tuned smoothly from typical coupling rate in

cQED devices (on the order of 10 to 100 MHz) down to less than 100 kHz as a

function of 𝐸𝐽2, as is shown in figure 2.2(c). When 𝐸𝐽1 is tuned as well, we can

achieve independent control over 𝜔− and 𝑔−, which is demonstrated experimentally

in [110]. We note that the anharmonicities 𝛼±,𝑐 cannot be tuned independently, as

can be seen from (2.19), but remain large enough (> 𝛼/2 for symmetric design) for

the device to be used as a qubit [See figure 2.2(d)].

2.2 Perturbation theory

In cQED experiments, the devices are often operated in the dispersive regime [114],

where the frequency detuning between two states |𝑖⟩, |𝑗⟩ are much large than their

interaction strength,

|𝐸𝑖 − 𝐸𝑗| ≫ ⟨𝑖|𝐻|𝑗⟩. (2.33)

This regime is of particular interest in cQED because it minimizes the disturbance

to the qubits while maintaining their accessibility. For example, dispersive coupling

between a qubit and a cavity allows quantum non-demolition (QND) readout of the

qubit by measuring the cavity response [24]. In the “cat code” scheme, dispersive

coupling is utilized to prepare Schrödinger-cat states and perform parity measure-

ment [48]. The criterion (2.33) coincides with the condition for perturbation theory

and makes perturbative treatment a natural tool for calculations in the dispersive

regime.

As a concrete example, we discuss a TCQ dispersively coupled to a cavity, i.e.,

|Δ±| = |𝜔± − 𝜔𝑟| ≫ |𝑔±|, and use second order perturbation theory to calculate the

approximate dispersive Hamiltonian for (2.28). We treat 𝐻𝑇 +𝐻𝑅 as the unperturbed

24


Hamiltonian and 𝐻𝐶 as the perturbation, i.e.,

𝐻0/ℏ = 𝜔+𝑏†+𝑏+ + 𝜔−𝑏†

−𝑏− + 𝜔𝑟𝑎†𝑎

− 𝛼+2 𝑏†

+𝑏†+𝑏+𝑏+ − 𝛼−

2 𝑏†−𝑏†


−𝑏−

𝐻′/ℏ = 𝑔+(𝑎𝑏†+ + 𝑎†𝑏+) + 𝑔−(𝑎𝑏†

− + 𝑎†𝑏−).

(2.34)

From non-degenerate perturbation theory, the eigenenergy up to second order is given

by𝐸𝑛 = 𝐸(0)

𝑛 + 𝐸(1)𝑛 + 𝐸(2)

𝑛

= ⟨𝑛0|𝐻0|𝑛0⟩ + ⟨𝑛0|𝐻′|𝑛0⟩ + ∑𝑚≠𝑛

|⟨𝑛0|𝐻′|𝑚0⟩|2𝐸(0)

𝑛 − 𝐸(0)𝑚

.(2.35)

We label the states of the TCQ and cavity as |𝑖+𝑗−𝑛⟩, where 𝑖+, 𝑗−, and 𝑛 denote

the excitation number of the two qubits and the cavity photons respectively. Using

(2.35) we find

𝐸(0)|0+0−𝑛⟩ = 𝑛ℏ𝜔𝑟, 𝐸(2)

|0+0−𝑛⟩ = −𝑛ℏ𝑔2+

Δ+− 𝑛ℏ𝑔2

−Δ−

,

𝐸(0)|1+0−𝑛⟩ = ℏ𝜔+ + 𝑛ℏ𝜔𝑟, 𝐸(2)

|1+0−𝑛⟩ = (𝑛 + 1)ℏ𝑔2+

Δ+− 2𝑛ℏ𝑔2

+Δ+ − 𝛼+

− 𝑛ℏ𝑔2−

Δ− − 𝛼𝑐,

𝐸(0)|0+1−𝑛⟩ = ℏ𝜔− + 𝑛ℏ𝜔𝑟, 𝐸(2)

|0+1−𝑛⟩ = (𝑛 + 1)ℏ𝑔2−

Δ−− 2𝑛ℏ𝑔2

−Δ− − 𝛼−

− 𝑛ℏ𝑔2+

Δ+ − 𝛼𝑐,

(2.36)

First order correction 𝐸(1) is zero for all states and the eigenenergies to second order

are given by

𝐸|0+0−𝑛⟩ = 𝑛ℏ(𝜔𝑟 − 𝑔2+

Δ+− 𝑔2

−Δ−

) ,

𝐸|1+0−𝑛⟩ = ℏ𝜔+ + ℏ𝑔2+

Δ++ 𝑛ℏ(𝜔𝑟 + 𝑔2

+Δ+

− 2𝑔2+

Δ+ − 𝛼+− 𝑔2

−Δ− − 𝛼𝑐

) ,

𝐸|0+1−𝑛⟩ = ℏ𝜔− + ℏ𝑔2−

Δ−+ 𝑛ℏ(𝜔𝑟 + 𝑔2

−Δ−

− 2𝑔2−

Δ− − 𝛼−− 𝑔2

+Δ+ − 𝛼𝑐

) .

(2.37)

25


Equations (2.37) indicate that the interaction between the qubits and cavity results

in dressed frequencies

+ = 𝜔+ + 𝑔2+

Δ+, − = 𝜔− + 𝑔2

−Δ−

, 𝑟 = 𝜔𝑟 − 𝑔2+

Δ+− 𝑔2

−Δ−

, (2.38)

and dispersive couplings

𝜒+ = 2𝑔2+𝛼+

Δ+(𝛼+ − Δ+) + 𝑔2−𝛼𝑐

Δ−(𝛼𝑐 − Δ−),

𝜒− = 2𝑔2−𝛼−

Δ−(𝛼− − Δ−) + 𝑔2+𝛼𝑐

Δ+(𝛼𝑐 − Δ+).(2.39)

The Hamiltonian in the dispersive regime can be approximated by

𝐻/ℏ = +𝑏†+𝑏+ + −𝑏†

−𝑏− + 𝑟𝑎†𝑎 + 𝜒+𝑎†𝑎𝑏†+𝑏+ + 𝜒−𝑎†𝑎𝑏†

−𝑏−+

− 𝛼+2 𝑏†

+𝑏†+𝑏+𝑏+ − 𝛼−

2 𝑏†−𝑏†


−𝑏−.(2.40)

We will use (2.39) and (2.40) in chapter 4 to study the tunability and suppression

of dispersive interaction. Perturbation theory is also used in the calculation of ZZ

coupling rate and details can be found in appendix B.

2.3 Black-box quantization

The black-box quantization method is developed in [113] for the analysis of cQED de-

vices. In this method, the system is divided into linear and nonlinear parts [115], and

the latter is treated as perturbation. The advantage of this method is the coupling

between linear parts can be treated classically using analytical or numerical methods.

This is particularly suitable for weakly nonlinear circuits (e.g., transmon qubits dis-

persively coupled to cavities), because the normal modes of the linear part provide

26


a good basis for quantizing the circuit. The addition of the nonlinear part gives rise

to self and cross interactions between the normal modes, which are directly related

to experimentally measurable quantities such as anharmonicity and dispersive shift.

Good agreement between simulations using this method and measured data has been

reported [79, 116].

In this section, we discuss the application of black-box quantization to a TCQ

coupled to a 3D cavity. The procedure is summarized in the following steps.

1. Divide the full system Hamiltonian 𝐻 into linear part 𝐻0 and nonlinear part

𝐻nl.

2. Calculate the impedance matrix [𝑍𝑖𝑗(𝜔)] for linear part of the circuit.

3. Find the frequency 𝜔𝑚 and impedance 𝒵eff𝑚 for eigenmodes 𝑚 = 1, 2, ⋯ , 𝑀 of

𝐻0.

4. Quantize the linear part as 𝑀 independent harmonic oscillators with annihila-

tion and creation operators 𝑎𝑚, 𝑎†𝑚.

5. Express the flux across each junction in terms of 𝑎𝑚, 𝑎†𝑚.

6. Treat 𝐻nl as perturbation and find the eigenmodes of 𝐻.

We now explain each step in detail.

1. The model for a TCQ inside a 3D cavity is shown in figure 2.3. We associate

a port (labeled 𝑝 = 1 and 2) with each Josephson junction, and the Hamiltonian for

either junction is

𝐻𝐽𝑝 = 4𝐸𝐶𝑝𝑛2𝑝 − 𝐸𝐽𝑝 cos 𝜑𝑝

= 4𝐸𝐶𝑝𝑛2𝑝 − 𝐸𝐽𝑝 (1 − 1

2𝜑2𝑝 + 1

24𝜑4𝑝 + 𝒪(𝜑6

𝑝))

= 𝑄2𝑝

2𝐶𝐽𝑝+ Φ2

𝑝2𝐿𝐽𝑝

− Φ4𝑝

24𝐿𝐽𝑝𝜙20

− 𝐸𝐽𝑝 + 𝒪((Φ𝑝/𝜙0)6).

(2.41)

27


Here 𝑄𝑝 and Φ𝑝 are the charge and flux variables, 𝐶𝐽𝑝 is the junction capacitance,

and 𝐿𝐽𝑝 = 𝜙20/𝐸𝐽𝑝 is the linearized inductance of the junction. The Φ4

𝑝 and higher

order terms are the nonlinear part of the junctions, represented by the spider symbols

in figure 2.3(c).

CavitySubstrateIslands

(a) (b)

(c)

(d)

5 mm

500 µm

LJ1 LJ2

CJ1 CJ2

’1 ’2

LJ1 CJ1 LJ2CJ2[Zij] [Zij ]

L1 L2 LM

C1 C2 CM

Figure 2.3: Model and circuit for the black-box quantization method. (a) The linearpart of the system consists of a 3D cavity, the sapphire substrate and qubit islands.(b) Zoomed in picture for the dashed box in (a) shows the qubit islands and Josephsonjunctions on the chip. (c) Circuit representation for the model includes linear partenclosed by the dashed box and nonlinear part represented by the spider symbols.The linear part is a lossless two-port network and can be described by impedancematrix [𝑍𝑖𝑗(𝜔)]. (d) The impedance 𝑍11(𝜔) can be synthesized by the equivalentcircuit of a series of uncoupled LC oscillators.

Assuming the cavity and chip are linear and lossless, we can combine them with

𝐶𝐽𝑝 and 𝐿𝐽𝑝 and describe the linear part of the system, represented by the dashed

box in figure 2.3(c), as a two-port network. The Hamiltonian for the system can then

be divided into two parts,

𝐻 = 𝐻0 + 𝐻nl, (2.42)

28


where 𝐻0 corresponds to the linear part described by a purely imaginary impedance

matrix [𝑍𝑖𝑗(𝜔)], and the nonlinear part is

𝐻nl = −2

∑𝑝=1

Φ4𝑝


+ 𝒪((Φ𝑝/𝜙0)6). (2.43)

2. For the linear part, we first numerically calculate the impedance matrix

[ 𝑍𝑖𝑗(𝜔)] for “cavity + substrate + islands” using commercial software ANSYS HFSS.

To incorporate 𝐿𝐽𝑝, 𝐶𝐽𝑝 into [𝑍𝑖𝑗(𝜔)], we consider the circuit shown in figure 2.4. Us-

ing circuit analysis and the definition of impedance matrix, we have

𝑉1 = (𝐼1 − 𝑉1𝑍1

) 𝑍11 + (𝐼2 − 𝑉2𝑍2

) 𝑍12,

𝑉2 = (𝐼1 − 𝑉1𝑍1

) 𝑍21 + (𝐼2 − 𝑉2𝑍2

) 𝑍22.(2.44)

+

−V1

I1

Z1

I2

Z2

+

−V2[Zij]

[Zij ]

Figure 2.4: Two-port network with parallel impedance. The impedance matrix [ 𝑍𝑖𝑗]is calculated by ANSYS HFSS and 𝑍1,2 can be incorporated into (2.45) to obtain[𝑍𝑖𝑗] for the full linear network.

For reciprocal network we have 𝑍12 = 𝑍21. Solving for 𝑉1, 𝑉2 in terms of 𝐼1, 𝐼2,

we obtain the impedance matrix for the full network,

𝑍11 = 𝑉1𝐼1

∣𝐼2=0

=𝑍11(𝑍2 + 𝑍22) − 𝑍2

12(𝑍1 + 𝑍11)(𝑍2 + 𝑍22) − 𝑍2

12𝑍1,

𝑍12 = 𝑍21 = 𝑉1𝐼2

∣𝐼1=0

=𝑍12𝑍1𝑍2

(𝑍1 + 𝑍11)(𝑍2 + 𝑍22) − 𝑍212

,

𝑍22 = 𝑉2𝐼2

∣𝐼1=0

=𝑍22(𝑍1 + 𝑍11) − 𝑍2

12(𝑍1 + 𝑍11)(𝑍2 + 𝑍22) − 𝑍2

12𝑍2.

(2.45)

29


−4

−2

0

2

4

6(a) LJ1 = 5.45 nH

LJ2 = 5.45 nH(b) LJ1 = 3.55 nH

LJ2 = 5.40 nH

6 7 8 9 10 11−4

−2

0

2

4

6(c) LJ1 = 4.78 nH

LJ2 = 4.78 nH

6 7 8 9 10 11

(d) LJ1 = 6.81 nHLJ2 = 6.81 nH

Frequency (GHz)

ImY 1

(kΩ−

1 )

Figure 2.5: Calculated imaginary part of admittance 𝑌1(𝜔) (blue curves) and normalmode frequencies (black circles) for different combinations of (𝐿𝐽1, 𝐿𝐽2). (a) Bothqubits are far detuned from the cavity. (b) The lower qubit is on resonance withthe cavity and has moderate coupling rate of 𝑔−/2𝜋 = 51.5 MHz. (c) The lowerqubit is on resonance with the cavity and has low coupling rate of 𝑔−/2𝜋 = 175 kHz.(d) The upper qubit is on resonance with the cavity and has high coupling rate of𝑔+/2𝜋 = 112.1 MHz.

Figure 2.5 shows the calculated imaginary part of admittance 𝑌1(𝜔) = 𝑍11(𝜔)−1

for the model in figure 2.3 with different junction inductances (𝐿𝐽1, 𝐿𝐽2). The zeros

of 𝑌1(𝜔) (black circles) correspond to the frequencies of the normal modes. The three

modes considered here are the fundamental mode (TE101) of the 3D cavity around 8

GHz and the two qubit modes whose frequencies are determined by (𝐿𝐽1, 𝐿𝐽2). Im-

portantly, the two qubit modes correspond to the normal modes |±⟩ of the TCQ, and

the linear part of the system already captures the tunable dipole coupling 𝑔± between

the TCQ and cavity, because it includes the substrate of the chip, islands of the TCQ

and linearized inductance of the Josephson junctions. The coupling can be seen when

30


the lower or upper qubit is on resonance with the cavity [See figure 2.5(b)∼(d)], and

the frequency difference between the two normal modes correspond to an avoided

crossing of 2𝑔± due to the coupling. When 𝐿𝐽1 = 𝐿𝐽2, the lower/upper normal mode

becomes antisymmetric/symmetric superpositions of the two transmon qubits and

results in small 𝑔−/large 𝑔+, as is discussed in section 2.2. When 𝐿𝐽1 and 𝐿𝐽2 are

sufficiently different, the qubit modes have moderate coupling rates 𝑔±. The mod-

erate, small and large avoided crossings in figure 2.5(b)∼(d) agree with the above

analysis.

3. We arbitrarily choose port 1 as the reference port (Choosing port 2 gives the

same result). The impedance 𝑍11(𝜔) can be synthesized by a series of parallel LC

circuits, shown in figure 2.3(d),

𝑍11(𝜔) =𝑀

∑𝑚=1

(𝑗𝜔𝐶𝑚 + 1𝑗𝜔𝐿𝑚

) . (2.46)

The resonance frequencies 𝜔𝑚 are determined by the zeros of 𝑌1(𝜔) (black circles in

figure 2.5), and the capacitances 𝐶𝑚, inductances 𝐿𝑚 and effective impedances 𝒵eff𝑚

are determined by

𝐶𝑚 = 12Im𝑌 ′

1 (𝜔𝑚), 𝐿𝑚 = 1𝜔2𝑚𝐶𝑚

= 2𝜔2𝑚Im𝑌 ′

1 (𝜔𝑚),

𝒵eff𝑚 = √𝐿𝑚

𝐶𝑚= 2

𝜔𝑚Im𝑌 ′1 (𝜔𝑚).

(2.47)

The Hamiltonian for the linear part is therefore

𝐻0 =𝑀

∑𝑚=1

𝑄2𝑚

2𝐶𝑚+ Φ2

𝑚2𝐿𝑚

. (2.48)

31


This describes 𝑀 uncoupled harmonic oscillators, corresponding to the normal modes

of the linear system.

4. To quantize the system, we express the flux and charge variables in terms of

creation and annihilation operators,

Φ𝑚 = √ℏ2𝒵eff𝑚(𝑎𝑚 + 𝑎†

𝑚), 𝑄𝑚 = −𝑗√ ℏ2𝒵eff𝑚

(𝑎𝑚 − 𝑎†𝑚), (2.49)

and the Hamiltonian becomes (omitting the zero point energies)

𝐻0 =𝑀

∑𝑚=1

ℏ𝜔𝑚𝑎†𝑚𝑎𝑚, (2.50)

which again describes 𝑀 independent harmonic oscillators.

5. The nonlinear part of the system, represented by Φ4𝑝 and higher order terms

in (2.41), is treated as perturbation. From Kirchhoff’s voltage law, we find the flux

variable Φ𝑝 across the Josephson junctions as

Φ1 =𝑀

∑𝑚=1

Φ𝑚 =𝑀

∑𝑚=1

√ℏ2𝒵eff𝑚(𝑎𝑚 + 𝑎†

𝑚),

Φ2 =𝑀

∑𝑚=1

𝑍21(𝜔𝑚)𝑍11(𝜔𝑚)Φ𝑚 =

𝑀∑𝑚=1

𝑍21(𝜔𝑚)𝑍11(𝜔𝑚)

√ℏ2𝒵eff𝑚(𝑎𝑚 + 𝑎†

𝑚).(2.51)

Since 𝜔𝑚 are the poles of 𝑍21(𝜔) and 𝑍11(𝜔), the ratio 𝑍21(𝜔𝑚)/𝑍11(𝜔𝑚) is obtained

by

lim𝜔→𝜔𝑚

𝑍21(𝜔)𝑍11(𝜔) = lim

𝜔→𝜔𝑚

𝑌11(𝜔)𝑌21(𝜔) = 𝑌 ′

11(𝜔)𝑌 ′

21(𝜔)∣𝜔=𝜔𝑚

(2.52)

32


6. The full Hamiltonian up to fourth order in Φ𝑝 is obtained by substituting (2.50)

and (2.51) into (2.42) and (2.43),

𝐻 = 𝐻0 + 𝐻4 =𝑀

∑𝑚=1

ℏ𝜔𝑚𝑎†𝑚𝑎𝑚 −

2∑𝑝=1

Φ4𝑝


=𝑀

∑𝑚=1

ℏ𝜔𝑚𝑎†𝑚𝑎𝑚 −

2∑𝑝=1

𝑒2

24𝐿𝐽𝑝(

𝑀∑𝑚=1

𝑍𝑝1(𝜔𝑚)𝑍11(𝜔𝑚)√𝒵eff𝑚(𝑎𝑚 + 𝑎†

𝑚))4

,(2.53)

and the eigenmodes can be obtained by diagonalizing the Hamiltonian. In figure 2.6

we show the frequencies |𝑖−𝑗+𝑛⟩ of the eigenmodes for the model in figure 2.3 when

the lower qubit is tuned into and out of resonance with the cavity.

31 32 33 34 357.5

8.0

8.5

|1−0+0⟩|0−0+1⟩

10.0

10.5

11.0 (a)

|0−1+0⟩

31 32 33 34 3515.0

15.5

16.0

16.5

17.0 (b)

|2−0+0⟩

|1−0+1⟩

|0−0+2⟩

H0 + H4

H0

EJ2/h (GHz)

Freq

uenc

y(G

Hz)

Figure 2.6: TCQ energy levels calculated by black-box quantization for the model infigure 2.3. 𝐸𝐽2 is varied linearly and 𝐸𝐽1/ℎ is fixed at 46 GHz. Avoided crossingsappear in both single- and double-excitation energy levels as the lower qubit is tunedacross the cavity. Solid lines correspond to eigenenergies 𝑚 of the fourth orderHamiltonian (2.53) and dashed lines correspond to eigenenergies 𝜔𝑚 of the linearpart 𝐻0. The anharmonicities can be seen in the overall lowering of 𝑚 with respectto 𝜔𝑚.

The nonlinear part 𝐻4 introduces self- and cross-Kerr interactions to normal

modes in the system. Inspecting the last term in (2.53), we find that those interactions

are proportional to both the Josephson energy of the junctions (the 𝑒2/24𝐿𝐽𝑝 factor)

and the effective impedance 𝒵eff𝑚 of each mode. From the perspective of black-box

33


quantization, all modes derive some nonlinearity from the Josephson junctions and

there is no strict separation of qubit and cavity modes. In the dispersive regime, the

qubit/cavity modes can usually be identified as strongly/weakly anharmonic. The

cavity modes have much smaller effective impedance [corresponding to very steep

slope in 𝑌1(𝜔), as is shown in the second mode in figure 2.5(a)], and therefore much

lower anharmonicity (several MHz or lower) compared to that of qubit modes (several

hundred MHz). From the perspective of perturbation theory, the self-Kerr interac-

tion for the cavity is a fourth order effect and therefore much weaker than the second

order cross-Kerr interaction.

In the dispersive regime, the connection between black-box quantization and the

circuit analysis in section 2.1 can be shown explicitly from the observation that the

impedance seen by a qubit [e.g., mode 1 in figure 2.5(a)] is mostly determined by

its shunting capacitance 𝐶Σ = 𝑒2/2𝐸𝐶 and Josephson inductance 𝐿𝐽 , i.e., 𝒵eff =√𝐿𝐽/𝐶Σ. The linear and self-Kerr terms in the Hamiltonian for a qubit mode are

therefore𝐻0 = ℏ𝜔𝑎†𝑎 = ℏ𝑎†𝑎/√𝐿𝐽𝐶Σ = √8𝐸𝐽𝐸𝐶𝑎†𝑎,

𝐻self-Kerr = − 𝑒2

24𝐿𝐽(𝒵eff)2 (𝑎 + 𝑎†)4 = − 𝑒2

24𝐶Σ(𝑎 + 𝑎†)4

≈ − 𝑒2

12𝐶Σ(12𝑎†𝑎 + 6𝑎†𝑎†𝑎𝑎 + 3)

= −𝐸𝐶 (𝑎†𝑎 + 12𝑎†𝑎†𝑎𝑎 + 1

4) ,

(2.54)

where we drop the mode index 𝑚 for simplicity and use (2.13). From (2.54) we

retrieve the result that the frequency and anharmonicity for a transmon are ℏ𝜔 =√8𝐸𝐽𝐸𝐶 − 𝐸𝐶 and ℏ𝛼 = 𝐸𝐶 [See (2.14)].

The anharmonicity (self-Kerr) for each mode and dispersive interactions (cross-

Kerr) between modes can be calculated quantitatively from the eigenenergies and

their values are summarized in table 2.1 for the parameters in figure 2.5(a). For

34


Table 2.1: Calculated black-box quantization parameters for figure 2.5(a). Modes1, 2, 3 correspond to lower qubit, cavity, and upper qubit, respectively. The self- andcross-Kerr rates are related to the anharmonicities and dispersive shifts in (2.40) by𝛼− = −𝜒11, 𝛼+ = −𝜒33, 𝛼𝑐 = −𝜒13, 𝜒− = 𝜒12, 𝜒+ = 𝜒23.

Mode Frequency Impedance Ratio Self-Kerr Cross-Kerr𝑚 𝜔𝑚/2𝜋 𝒵eff

𝑚 𝑍21/𝑍11 𝜒𝑚𝑚/2𝜋 𝜒𝑚𝑛/2𝜋

1 7.224 GHz 128.4 Ω −1.0 −146.3 MHz𝜒12/2𝜋 = −18.0 MHz𝜒13/2𝜋 = −316.8 MHz

2 8.001 GHz 1.97 Ω 1.0 −172.9 kHz𝜒21/2𝜋 = −18.0 MHz𝜒23/2𝜋 = −13.8 MHz

3 8.674 GHz 151.9 Ω 1.0 −171.1 MHz𝜒31/2𝜋 = −316.8 MHz𝜒32/2𝜋 = −13.8 MHz

example, the anharmonicity and dispersive shift for the lower qubit can be calculated

by

𝛼− = 2|1−0+0⟩ − |2−0+0⟩, 𝜒− = |1−0+1⟩ − |1−0+0⟩ − |0−0+1⟩. (2.55)

We note the small effective impedance and anharmonicity of mode 2, as is expected

for a cavity.

In section 4.2 we use black-box quantization to calculate the dispersive coupling

rate between a TCQ and a cavity.

35

Chapter 3

Experimental Techniques

Experimental studies on cQED devices require knowledge and skills in microwave

engineering, micro/nano fabrication, packaging/shielding and cryogenic engineering,

etc. This chapter discusses the experimental techniques developed in our lab, with an

emphasis on the author’s contribution. Section 3.1 introduces the design of coplanar

waveguide resonator, which is an essential building block in cQED. The improvement

in fabrication and testing of Josephson junctions is covered in section 3.2. In sec-

tion 3.3 we discuss the packaging and shielding schemes that are designed and tested

for good device performance. Finally section 3.4 describes the room temperature and

cryogenic setup for performing measurements on cQED devices.

3.1 Device design

Building circuit QED devices involves the design of resonators and qubits. The trans-

mon qubit used in this thesis is introduced in section 1.3. In this section we discuss

the design of microwave resonators.

In the microwave domain, resonators can be realized by inserting open- or short-

circuited terminations into a section of transmission line. The boundary conditions

introduced by the terminations result in the formation of standing waves that corre-36

CHAPTER 3. EXPERIMENTAL TECHNIQUES

spond to the modes of a resonator and the resonance frequencies are determined by

the length of the section. Figure 3.1 summarizes some basic properties of transmission

line resonators.

(b)

(a)

R L

CIV+

−

RLC

I+

−V

BW

BW

0 1

0 1

R

R

0.707R

Zin Z0,˛,¸

‘

Zin

Zin Z0,˛,¸

‘

Zin

|Zin(!)|

R0.707

!=!0

|Zin(!)|

!=!0

Figure 3.1: Transmission line resonators with (a) short-circuited and (b) open cir-cuited termination, their equivalent lumped-element circuits and frequency response.Transmission lines are of length ℓ, characteristic impedance 𝑍0, propagation con-stant 𝛽, and attenuation constant 𝛼. Lumped element circuits consist of resistor 𝑅,capacitor 𝐶, and inductor 𝐿. Figure adapted from chapter 6 in [117].

One commonly used configuration in cQED is half-wavelength resonator with open

circuited (gap capacitor) termination at both ends, as it allows convenient coupling

of input and output signals. The resonance frequency is given by

ℓ = 𝜆/2 ⇒ 𝜔𝑟 = 2𝜋 𝑣𝜆 = 𝑐𝜋

ℓ√𝜖eff, (3.1)

where ℓ is the length of the transmission line, 𝜆 is the wavelength, 𝑣 is the phase

velocity of the microwave signal and 𝜖eff is the effective permittivity. The parameters

37


for lumped-element equivalent circuit in figure 3.1(b) are (see chapter 6 in [117])

𝐿 = 2𝑍0𝜔𝑟𝜋, 𝐶 = 𝜋

2𝜔𝑟𝑍0, 𝑅 = 𝑍0

𝛼ℓ . (3.2)

One particular type of transmission line that is widely used in cQED is the copla-

nar waveguide (CPW) [118, 119]. It consists of a center conductor and ground planes

at both sides on a dielectric substrate, as is shown in figure 3.2. The microwave signal

is carried in the current flowing along the center conductor and the voltage between

center and ground conductors.

a

b

s

s

t

h

›0›r

Figure 3.2: Coplanar waveguide section. The center strip is of width 𝑎 and thickness 𝑡and separates from both ground planes by a distance 𝑠. The substrate is of thicknessℎ and dielectric constant 𝜖r. Figure adapted from chapter 1 in [118].

Coplanar waveguides have the following advantages compared to other transmis-

sion line systems.

1. The planar layout and absence of backside grounding makes it easy to fabricate

CPW resonators in a single step of lithography.

2. CPWs are compatible with coaxial components and systems that are widely

used and commercially available.

38


3. The characteristic impedance of a CPW is determined by the ratio of center

strip and gap width and remains the same as long as the dimensions scale

proportionally. This makes it convenient to design transitions between different

parts in a device and choose proper length scales for different applications.

For example, smaller feature size (a few micrometers) can be used for compact

device and concentrated field distribution, while increasing the length scale

reduces the effect of random disorder introduced in the fabrication process [120].

It is important to connect the two ground planes of a CPW (e.g., via wire bonds)

to ensure a common ground potential. Special care about common ground plane

also needs to be taken in designing structures that involve discontinuities, such as

T-junctions, stubs and crosses, etc.

The characteristic impedance and effective permittivity for a CPW can be calcu-

lated from its material and geometry analytically [121] or numerically using, e.g., NI

AWR Microwave Office. In our lab we use niobium for the center and ground con-

ductors, and sapphire for the substrate. The commonly used parameters are listed in

table 3.1.

Table 3.1: CPW parameters.

Parameter Symbol ValueCenter strip width 𝑎 10 μm

Gap width 𝑠 4.2 μmConductor thickness 𝑡 200 nmSubstrate thickness ℎ 500 μmDielectric constant 𝜖r 11.5

Effective permittivity 𝜖eff 6.25Characteristic impedance 𝑍0 50 Ω

39


3.2 Fabrication

The fabrication techniques used in cQED are determined by the feature sizes and

materials of the devices [122]. Crystalline sapphire is chosen as the material for

substrate because of its low loss at microwave frequency. Thin films of niobium and

aluminum are deposited onto the substrate using sputtering and electron beam (e-

beam) evaporation. Patterns for CPW resonators (gap width ∼ 5 μm) and Josephson

junctions (area ∼ 200 × 200 nm2) are defined by optical and e-beam lithography,

respectively, and transferred to superconducting films using reactive ion etching and

liftoff. Figure 3.3 shows a typical cQED device.

100 µm50 µm

1 mm

(a)

(b) (c)

InputOutput QubitCPW

resonator

Flux biases

Niobium

Aluminum

Figure 3.3: A fabricated cQED device consisting of a meandering CPW resonator withinput and output ports and a TCQ with flux biases. (a) Patterns with feature size≳ 2 μm are defined by photolithography and formed by reactive ion etching of niobiumfilm (bright color) sputtered on sapphire substrate (dark color). (b) The input andoutput capacitors for the CPW are formed by interdigitated fingers separated bygaps. (c) The Josephson junctions and additional coupling capacitor (brighter color)are defined by e-beam lithography and formed by depositing aluminum using e-beamevaporation.

40


Extensive research and engineering efforts have been made in optimizing the fab-

rication process to improve device quality [123–128]. In this section we will discuss

the fabrication for Josephson junctions. Detailed fabrication recipes can be found in

appendix A.

3.2.1 Dolan bridge technique

Dolan bridge technique [129] is commonly used for fabricating sub-micrometer Joseph-

son junctions. The junctions are formed by double-angle shadow evaporation over a

suspended bridge and an oxidation step in between, as is illustrated in figure 3.4.

PMMAMMAAlAlOx

Sapphire

(a) (b) (c)

(d) (e)

Figure 3.4: Dolan bridge technique. (a) Patterns are defined on a double-layer resiststack using e-beam lithography. (b) MMA is more sensitive to e-beam, and a sus-pended bridge can be formed by choosing the proper dose. (c) A layer of aluminumis deposited using e-beam evaporation. Oxygen is then let in to form a thin oxidationlayer on top. (d) A second layer of aluminum is deposited at a different angle. (e) Theresist is removed in a liftoff process and two junctions are formed. A Cross sectionview is shown in steps (b) through (d) for better illustration.

The production of reliable suspended bridges is a crucial and challenging step in

fabricating Dolan-style Josephson junctions. The pattern for the junction is defined

on a PMMA-on-MMA double layer resist stack and the bridge is formed by applying

41


2 µm

(b)

2 µm

(a)

Figure 3.5: Short circuit caused by bridge collapse. Electron micrographs taken after(a) developing resist and (b) evaporating the first layer of aluminum show the fragilenature of the suspended bridge. Inset: The collapse of the bridge leads to a connectionbetween the two arms of the junction and forms a short circuit.

a low e-beam dose that exposes MMA but not PMMA. Although the top PMMA

layer is not sensitized, it becomes weaker due to the exposure and swells after devel-

opment [see figure 3.5(a)]. This leads to cracks or collapse of the bridge due to the

tension and eventually gives rise to shorted junctions [see figure 3.5(b)]. This bridge

collapse problem can be seen in the room temperature resistance measurement (See

subsection 3.2.3) and shorted junctions were found randomly on chips after the same

fabrication process. Switching to ZEP e-beam resist and using cold development in

ice-water mixture, as is reported in section 4.1 of [75], slightly increases the success

rate but still limits our device yield. To improve the robustness of our fabrication

process, we develop a bridge-free technique.

3.2.2 Bridge-free technique

The bridge-free technique [130, 131] is illustrated in figure 3.6. Instead of relying

on suspended bridges, it makes use of the sidewalls of the resist to realize selective

deposition of either arm for a Josephson junction. Aluminum will deposit into the

42


narrow trenches that are in parallel with the evaporation direction and land on the

sidewalls of trenches along the orthogonal direction [see figure 3.6(c) and (d)]. After

liftoff, the aluminum on the sidewalls is removed and junctions are formed in the

overlapping areas of the two orthogonal arms [see figure 3.6(e)].

PMMAMMAAlAlOxSapphire

(a) (b) (c)

(d) (e)

Figure 3.6: Bridge-free technique. (a) Patterns are defined on a double-layer resiststack using e-beam lithography. (b) Undercuts are used in the MMA layer to helpliftoff, but no suspended bridge is formed. (c) A layer of aluminum is deposited usinge-beam evaporation. Oxygen is then let in to form a thin oxidation layer on top. (d)A second layer of aluminum is deposited in the orthogonal direction. (e) The resistis removed in a liftoff process and two junctions are formed. A Cross section view isshown in steps (b) through (d) for better illustration.

The above technique requires steep sidewalls in the mask to prevent the evaporated

material from landing on the substrate. Unintended undercuts in the MMA layer

because of its higher sensitivity to e-beam should therefore be minimized. Using

the 125 kV e-beam system in the cleanroom and mature proximity error correction

schemes, we can produce masks with almost no undercut near the junction area

and the yield of functioning devices is almost 100 %. Figure 3.7 shows the electron

micrograph of a Josephson junction fabricated using bridge-free technique. More

detailed comparison of device yields and statistics between Dolan bridge and bridge-

free junctions can be found in section 4.1.2 of [132].43


2 µm

Figure 3.7: A Josephson junction fabricated using bridge-free technique. The junctionis formed in the overlapped region (∼ 200 × 200 nm2) of two layers of aluminumdeposited in orthogonal directions and the oxidation layer in between.

In addition to higher device yield, the bridge-free technique provides a few more

benefits.

1. The absence of suspended structure makes the e-beam mask mechanically ro-

bust against aggressive fabrication processes such as sonication.

2. The design of e-beam pattern is more intuitive compared to Dolan-style junc-

tions, and the junction dimensions are insensitive to evaporation angle or resist

thickness (to first order).

3. Having proper undercuts in a double-layer mask helps the liftoff process, but in

principle only a single layer of e-beam resist is needed to produce functioning

junctions.

If the arms of a junction are too wide, the sidewalls will not block all the deposition

and aluminum will start to show up in the orthogonal trenches. The ratio of sidewall

44


height to pitch width sets the limit of junction size and for our current recipe (resist

thickness ∼ 700 nm, evaporation angle = 40°), junctions as large as 400 × 400 nm2

have been made reliably. For larger junctions, another bridge-free technique [133] has

also been developed in our lab. ear-

3.2.3 Junction resistance

The relation between the critical current 𝐼𝑐 of a Josephson junction and its room

temperature resistance 𝑅𝑛 is derived in [134],

𝐼𝑐𝑅𝑛 = (𝜋Δ/2𝑒) tanh(Δ/2𝑘B𝑇 ), (3.3)

where Δ is the gap energy of the superconductor. For temperature much lower (∼ 20

mK for a dilution refrigerator) than the critical temperature of the superconductor

(1.2 K for aluminum), we have tanh(Δ/𝑘B𝑇 ) ≈ 1 and (3.3) reduces to 𝐼𝑐𝑅𝑛 = 𝜋Δ/2𝑒.

Using 𝐸𝐽 = ℏ𝐼𝑐/2𝑒 and Δ = 1.7 × 10−4 eV for aluminum, we find

𝐸𝐽𝑅𝑛/ℎ = Δ8𝑒2 = 132.6 GHz ⋅ kΩ. (3.4)

The 𝐸𝐽𝑅𝑛 product depends only on the material properties and not the dimensions

of the junction. This makes room temperature resistance measurement a convenient

way to confirm a functioning junction and predict its 𝐸𝐽 . The experimentally mea-

sured junction resistances and the corresponding qubit parameters are summarized

in table 3.2.

The average measured 𝐸𝐽𝑅𝑛/ℎ is 118.48 GHz ⋅ kΩ and deviates from the theo-

retical value by a factor of 0.89. Similar results are reported in section 6.1 in [135],

section 5.1 in [136], and references therein.

45


Table 3.2: Measured junction resistance and qubit frequency.

Qubit frequency 𝐸𝐶/ℎ 𝑅𝑛 𝐸𝐽/ℎ 𝐸𝐽𝑅𝑛/ℎ(GHz) (MHz) (kΩ) (GHz) (GHz ⋅ kΩ)5.156 340 11.25 11.11 124.936.893 213 3.98 29.63 117.945.563 213 5.94 19.58 116.33.108 340 29.2 4.37 127.632.939 340 29.3 3.95 115.826.469 340 7.92 17.05 135.06.387 340 7.32 16.64 121.784.705 320 11.3 9.86 111.465.15 400 11.8 9.63 113.588.441 619 7.8 16.58 129.295.68 371 9.1 12.34 112.265.42 371 10.0 11.3 112.994.83 371 12.1 9.11 110.285.3 371 10.1 10.84 109.44

3.3 Packaging and shielding

The device fabricated on the chip is mounted to a printed circuit board (PCB) to

couple to the commercial 50 Ω coaxial system. The circuit board is adapted from a

design by IBM and is shown in figure 3.8(a) together with the copper parts that hold

the chip. Microwave signals are sent to the copper traces on the PCB through the

SSMA adapters and to the chip via wire bonding. To ensure a common ground for the

signal, the ground planes at the edges of the chip are wire bonded to the ground of the

PCB [See figure 3.8(b)]. Wire bond or air bridges are also needed to connect ground

planes at either side of a CPW to avoid slotline modes [137, 138]. The dimensions

and shape of the copper sample holder are designed so that the “box mode” [139] is

above the frequency range where we operate the devices. A more detailed description

of the packaging scheme can be found in section 4.2 of [132].

46


ChipSSMA connector

Input Output

GND

(a) (b)

Figure 3.8: Device packaging. (a) The chip for the device is placed in a copperholder and mounted to the PCB. Signals are sent in and out of the device throughthe SSMA connectors. (b) Zoomed-in image for the interface between chip and PCB.The input/output ports and ground planes on the chip are connected to the PCB viaaluminum wire bonds.

Another important issue in performing cQED experiments is the proper shielding

for devices. The frequency of superconducting qubits is usually tuned by an external

magnetic field and susceptible to flux noise. Stray magnetic fields can cause the

formation of vortices in superconductors and lead to loss [140]. Infrared radiation

can create a loss mechanism due to quasiparticle generation [141]. Sufficient and

effective shielding of stray magnetic and radiation fields is therefore a prerequisite for

robust and optimal device performance, and systematic studies have been reported

in the literature [142–144].

We use mu-metal cans and lids as an outer layer to reduce stray magnetic fields

and aluminum as a superconducting shield inside. ECCOSORB® epoxy (CR-124) is

applied to the top, bottom and inner side of the aluminum shield to absorb infrared

radiation. Using the above packaging and shielding, as is shown in figure 3.9, we

are able to measure a highest 𝑇1 of above 30 μs in a single-qubit device (typical 𝑇1

without shielding is below 10 μs). The disadvantage of this shielding scheme is the

superconducting aluminum can is not a good thermal conductor and might cause the

47


effective device temperature to be higher than that of the fridge’s base plate. We are

currently developing a superconductor-plated copper shield to provide both magnetic

shielding and good thermal anchoring, as is reported in [144].

Mu-metal ECCOSORB

Aluminum

SMA adapter

CopperrodSMA

adapter

PCB

Mu-metal Aluminum

Mu-metal

Copperbraid

(a) (b)

SSMA connector

(c)

Figure 3.9: Magnetic and radiation shields. (a) SMA adapters are mounted on mu-metal and aluminum lids, and ECCOSORB® CR-124 is coated on the inner side ofthe aluminum shield. (b) PCB containing the device is anchored to the copper rodand signals are sent through microwave adapters and wires to the device. (c) Theassembly is enclosed by aluminum and mu-metal shields and ready to be mounted tothe base plate of the dilution refrigerator.

In addition to shielding stray magnetic and radiation fields, we use absorptive

filters made from ECCOSORB® to prevent thermal noise from entering the device

and better thermalize the signal [145]. These filters provide attenuation for high

frequency (up to 40 GHz according to data sheet) noise and have been reported to

improve qubit coherence time [135 (section 4.2.2)]. The filter, shown in figure 3.10(a),

is designed to have characteristic impedance of 50 Ω using ANSYS HFSS and the

measured transmission and reflection are shown in figure 3.10(b).

48


0 2 4 6 8 10 12 14Frequency (GHz)

−40

−30

−20

−10

0

Spa

ram

eter

(dB)

S21S11

(a) (b)

Figure 3.10: ECCOSORB® filter. (a) The filter consists of a copper recess filledwith ECCOSORB® CR-110 and covered by a lid. The dimensions of the filter aredesigned for 50 Ω characteristic impedance. (b) Measured transmission (𝑆21, blue)and reflection (𝑆11, red) at room temperature. The insertion loss is below 6 dB forthe typical frequency range (4 ∼ 8 GHz) of our devices and reflection is below 10 dB.

3.4 Measurement setup

The measurements in this thesis are performed in a dilution refrigerator where the

devices are cooled down to ∼ 20 mK. At such low temperature the niobium and

aluminum in the circuits are in the superconducting phase, and thermal fluctuations

are suppressed to a level well below the frequency range (≳ 5 GHz) for cQED devices.

The diagram of fridge wiring and instrument setup for a typical experiment is shown

in figure 3.11.

The baseband input signals are generated by a high-speed (Keysight M9330A, 1.25

GSamples/s) arbitrary waveform generator (AWG) and upconverted to microwave

regime through internal I/Q modulation in a vector signal generator (Keysight

E8267D, up to 20 GHz) . The I/Q imbalance, offset and skewness are adjusted to

minimize pulse distortion and carrier leakage. The dc control signals are generated

by voltage sources (YOKOGAWA GS200).

Inside the fridge, input signals are sent down to the device through stainless steel

semi-rigid coaxial wires and cryogenic attenuators are used at each stage to reduce49


AcqirisU1082A

Keysight33250A

Keysight E8267D

Keysight E8267D

YokogawaGS200

I

Q

ch1

ch2

Trig

Keysight E8257D

I

Q

ch1

ch2

300 K70 K

10 d

B

4 K

20 d

B

Still

10 d

B

MC

20 d

B

20 d

B

20 d

B

ch1

ch2

Trig20

GH

zEc

coso

rb Ecco

sorb

2 M

Hz

Ecco

sorb

2 M

Hz

RF

LO

I

Q

HEMT

MITEQ

SRS

Trig

YokogawaGS200

Device

KeysightM9330A

KeysightM9330A SC

wire

Dilution refrigerator

Attenuator Low pass flter Circulator Amplifer Power

splitter Mixer

Marki

Figure 3.11: Measurement setup for cQED experiments. MC: Mixing chamber.HEMT: High electron mobility transistor amplifier. SRS: Stanford Research Systemsamplifier. SC wire: Superconducting wire.

thermal noise coming from upper stages. The attenuators provide a resistive path

between the center and outer conductor of the coaxial wire, and allow heat to be

dissipated onto the plates of the fridge. The amount of attenuation is chosen to

be ∼ 𝑇hot/𝑇cold for each stage, as the power spectral density for thermal noise is

given by 4𝑘B𝑇 𝑅. For dc lines, low pass filters are used instead of attenuators below

4 K stage to avoid too much heat generation and ensure enough voltage to tune the

device. At the base stage, copper coaxial wires are used for interconnection and

ECCOSORB® filters are used to absorb thermal radiation noise. Additional low

50


pass filters with proper bandwidth are used depending on the frequency range of

the input and control signals. The output signal from the device are amplified by

a high electron mobility transistor (HEMT) amplifier at 4 K stage. Two circulators

are placed before the HEMT to prevent noise and reflection from entering the device,

and superconducting wires are used to avoid loss of signal before it is amplified.

Figure 3.12 shows photographs of loaded sample, microwave components and wiring

inside the dilution refrigerator.

4 K

Still

MC

100 mK

Sample &shield

Circulator

MC

100 mK

Copperwire

SSwire

SCwire

ECCOSORBfilter

Reflectivefilter

Attenuator

(a) (b)

Figure 3.12: Cryogenic setup for cQED experiments. (a) Dilution refrigerator andloaded sample. (b) Components and wiring at base stage of the fridge. MC: mixingchamber. SS wire: Stainless steel wire. SC wire: Superconducting wire.

51


Outside the fridge, the output signal is further amplified by a MITEQ ampli-

fier and downconverted to an intermediate frequency of 0 ∼ 25 MHz by a Marki

mixer. Finally the signal is pre-amplified by SRS amplifiers, acquired by a high-speed

digitizer (Acqiris U1082A, 8-bit, 1 GSamples/s) and processed by software. The gen-

erators and digitizer are clocked by an SRS 10 MHz rubidium frequency standard and

triggered by pulses from a low-speed AWG (Keysight 33250A, 200 MSamples/s).

More recently we studied the feasibility and limitation of direct synthesis for pulse

generation in our experiments using a 65 GSamples/s AWG and more details can be

found in [146].

52

Chapter 4

Suppression of Photon Shot Noise

Dephasing

In section 1.2 we discussed the importance of qubit lifetime in cQED devices, and fig-

ure 1.1(b) illustrates the remarkable improvement in relaxation time 𝑇1 and coherence

time 𝑇2 in the past two decades. However, the progress in 𝑇2 is slower than that in 𝑇1

and 𝑇2/𝑇1 ratios in cQED devices fall in the range between 0.5 and 1.5 [79, 147, 148].

Deviation from the theoretical limit of 𝑇2 = 2𝑇1 indicates dephasing mechanisms that

need to be understood and circumvented.

This chapter discusses the design and measurement of a TCQ device for the pur-

pose of suppressing photon shot noise dephasing. Section 4.1 explains the origin of

this dephasing mechanism and how tunable dispersive coupling can be utilized to

suppress it. In section 4.2 we discuss the tunability of the dispersive coupling rate 𝜒and in particular the criteria for 𝜒 to be near zero. Based on those insights, a TCQ

device is designed and calibrated, as is described in section 4.3 and 4.4. To enable effi-

cient measurement of qubit state in small 𝜒 regime, a readout method is developed in

section 4.5. This method is used in the coherence measurement in section 4.6, where

53

CHAPTER 4. SUPPRESSION OF PHOTON SHOT NOISE DEPHASING

we demonstrate 𝑇1 limited coherence time even in the presence of injected photon

noise.

4.1 Photon shot noise dephasing

The origin of photon shot noise dephasing is the dispersive coupling between the qubit

and the resonator. In section 1.2 we introduced the Jaynes-Cummings Hamiltonian

(1.1), and in the dispersive regime where |Δ| = |𝜔𝑞 − 𝜔𝑟| ≫ 𝑔, the interaction

between the qubit and cavity can be approximated using second order perturbation

theory (similar to the treatment in section 2.2),

𝐻/ℏ = 𝑞𝜎𝑧/2 + 𝑟𝑎†𝑎 + 𝜒𝑎†𝑎𝜎𝑧/2, (4.1)

where 𝑞, 𝑟 are the renormalized qubit and cavity frequencies, and 𝜒 is the dispersive

coupling rate.1 For a transmon coupled to a cavity, their expressions are [83]

𝑞 = 𝜔𝑞 + 𝑔2

Δ , 𝑟 = 𝜔𝑟 − 𝑔2

Δ − 𝛼, 𝜒 = 2𝑔2𝛼Δ(𝛼 − Δ), (4.2)

where 𝛼 is the anharmonicity of the transmon. The last term in (4.1) describes

the dispersive coupling between the qubit and cavity, and indicates a qubit state

dependent resonator frequency

𝜔↓𝑟 = 𝑟 − 𝜒/2, 𝜔↑

𝑟 = 𝑟 + 𝜒/2, (4.3)

where ↓ and ↑ denote the ground and excited states of the qubit. When a coherent

drive is applied to the resonator, the amplitude or phase of the response can be used

1The definition of 𝜒 in some literature (e.g. [83, 149]) differs by a factor of 2.

54


to distinguish the qubit state if 𝜒 is large enough. This is the basic mechanism for

the dispersive readout of qubits in cQED [24], as is illustrated in figure 4.1(a).

ffl

»

|↑⟩ |↓⟩

!

Tran

smiss

ion

(arb

.uni

ts)

!r + ffl=2 !r − ffl=2

0

10

20

30

40

50

6.14 6.16 6.18 6.14 6.16 6.18Frequency, ‌s [GHz]

Phas

esh

ift,ffi

[deg

]

‌a n = 1 n = 20

2‹‌HWHM

‌ac

(a) (b)

Figure 4.1: Dispersive coupling between qubit and cavity. (a) The resonator frequencyis qubit state dependent due to dispersive coupling. This mechanism is used for qubitstate readout in cQED devices. Figure adapted from [24]. (b) The qubit frequencyis dressed by cavity photons due to dispersive coupling and exhibits ac Stark shift asthe cavity photon number increases. Dephasing of the qubit can be deduced from thebroadening of its spectrum. Figure adapted from [150].

On the other hand, the qubit frequency is dressed by photons in the resonator

because of the same coupling mechanism

𝜔𝑞() = 𝑞 + 𝜒, (4.4)

where = ⟨𝑎†𝑎⟩ is the average photon number in the cavity. From (4.4) we can

see that changes in photon number will shift the qubit frequency and fluctuations in

cavity photon number will lead to dephasing of the qubit. When the photons come

from a coherent drive field, (4.4) corresponds to the ac Stark shift effect [150], and

the dephasing of the qubit can be seen in the broadening of its spectrum, as is shown

in figure 4.1(b). Even in the absence of a drive field, thermal photon fluctuations can

give rise to qubit dephasing. This mechanism, illustrated in figure 4.2, is known as

photon shot noise dephasing and has been recognized as one of the limiting factors

for qubit coherence.55


Dephasing

Coupling

Fluctuation

Qubit Cavity

Γffi

fflffza†a

»n

Figure 4.2: Photon shot noise dephasing. Thermal or quantum fluctuations in cavityphotons induce dephasing of the qubit through the dispersive coupling.

The photon shot noise dephasing mechanism has been studied both theoreti-

cally [149] and experimentally [151, 152], and the explicit expression for the dephasing

rate Γ𝜙 is derived in [148, 153],

Γ𝜙 = 𝜅2Re ⎡⎢

⎣√(1 + 𝑖𝜒

𝜅 )2

+ 4𝑖𝜒𝜅 − 1⎤⎥

⎦=

⎧⎨⎩

𝜅, 𝜒 ≫ 𝜅,4𝜒2

𝜅 ( + 1), 𝜒 ≪ 𝜅.(4.5)

From (4.5) we find that the parameters determining Γ𝜙 are photon decay rate

𝜅, photon population of the resonator, and the dispersive coupling rate 𝜒. Three

different approaches can be taken to suppress the dephasing correspondingly.

1. Reduce the photon number fluctuation rate, characterized by 𝜅.

2. Reduce the thermal photon population .

3. Reduce the frequency shift caused by each photon, characterized by 𝜒.

Most work in the past has adopted the first two strategies. In [152], the linear

dependence of Γ𝜙 on and 𝜅 in the regime 𝜒 ≫ 𝜅 is observed experimentally for

different cavity modes and the authors use high quality factor (high-𝑄) cavity (𝑄 ∼2×105) and low pass filtering to obtain 𝑇1 (𝑇2) = 48 (87) μs. In [148], copper is used

for a 3D cavity instead of aluminum to provide better thermalization and reduce ,

56


and 𝑇1 (𝑇2) = 70 (92) μs are achieved. In [154], the authors perform systematic study

to reduce the effective temperature of their devices and achieve 𝑇1 (𝑇2) = 80 (154) μs

by careful cryogenic engineering, e.g., proper filtering [152] and attenuation [155], etc.

Dynamical decoupling and spin-locking spectroscopy have also been used to identify

and mitigate different types of noise [156–159].

The third approach cannot be easily adopted for a transmon qubit. As can be

seen from (4.2), 𝜒 approaches zero in the limit of small 𝑔 or large Δ, where it becomes

hard to access and control the qubit. Moreover, the dispersive readout method no

longer works for vanishing 𝜒. In the next section we show that 𝜒 can be tuned to near

zero in a TCQ device to suppress photon shot noise. In section 4.5 we demonstrate

that efficient qubit control and readout can also be achieved in the zero-𝜒 regime.

4.2 Dispersive coupling rate

The dispersive Hamiltonian for a TCQ coupled to a cavity is derived in (2.40). To

show the tunability of 𝜒, we recall the expression for the dispersive coupling rate

between a TCQ and a cavity derived in (2.39),

𝜒− = 𝜒1 + 𝜒2 = 2𝑔2−𝛼−

Δ−(𝛼− − Δ−) + 𝑔2+𝛼𝑐

Δ+(𝛼𝑐 − Δ+). (4.6)

In the above expression, the two terms 𝜒1,2 correspond to the contribution from the

two qubits (denoted by ±), and since we use the lowest two energy levels as the

computational basis, we focus on the dispersive coupling rate 𝜒− of the lower qubit.

The dependence of 𝜒− on Δ− is plotted in figure 4.3(b) together with 𝜒1,2. For the

purpose of suppressing photon shot noise dephasing, we are most interested in whether

𝜒− can be tuned to zero. The mechanism that makes zero dispersive coupling possible

is the sign change of 𝜒1 when the lower qubit is in the straddling regime [83, 160, 161],57


0

1

2

0

1

2

Cavity Qubit

2!r

!r

2!− − ¸−

!−

!r > !− − ¸−

!r < !−

−100 −50 0 50 100 150 200Δ−/2ı (MHz)

−2

−1

0

1

2

ffl/2ı

(MHz

)

ffl1

ffl2

ffl−

(a) (b)

Figure 4.3: Dispersive coupling rate for TCQ in the straddling regime. (a) In thestraddling regime, qubit-cavity detuning 𝜔− − 𝜔𝑟 is positive but smaller than theanharmonicity 𝛼−. (b) 𝜒1 (blue dashed line) and 𝜒2 (red dashed line) have oppositesigns when the lower qubit is in straddling regime (shaded area) and the upper qubitis far above the cavity, resulting in 𝜒− (blue solid line) crossing zero (blue circles).

where 0 < Δ− < 𝛼− [See figure 4.3(a)] and 𝜒1 becomes positive. If the upper qubit

is far above the cavity (Δ+ ≫ 𝑔+), 𝜒2 remains negative and the resulting 𝜒− will

cross zero because of the cancellation between 𝜒1 and 𝜒2. The requirement for zero

𝜒− can be summarized as

𝑔− ≪ Δ− < 𝛼−, Δ+ ≫ 𝑔+. (4.7)

We note that in a normal transmon qubit, the anharmonicity 𝛼 and coupling 𝑔 are

of the same order (several hundred MHz), so the first requirement in (4.7) cannot be

easily met. It is the TCQ’s ability to tune 𝑔− to much lower (< 10 MHz) that enables

the device to be in both straddling and dispersive regime. Furthermore, the ability to

tune 𝑔− and 𝜔− continuously makes possible the exact cancellation between 𝜒1 and

𝜒2.

We also use the black-box quantization method, introduced in section 2.3, to

verify the above analysis, as is shown in figure 4.4. The eigenenergies for one- and58


7.9

8.0

8.1

8.2!

/2ı

(GHz

)

|0−0+1⟩

|1−0+0⟩(a)

17.1

17.3

17.5

17.7 |1−1+0⟩(b)

35 36 37 38 39 40−2

−1

0

1

2

ffl−

/2ı

(MHz

) (c)

35 36 37 38 39 4015.7

15.9

16.1

16.3|1−0+1⟩

|2−0+0⟩|0−0+2⟩

EJ2/h (GHz)

!/2ı

(GHz

)Figure 4.4: Black-box quantization calculation for (a) single-excitation energy levels,(b) two-excitation energy levels, and (c) dispersive coupling rate of a TCQ. Themodel is shown in figure 2.3, and 𝐸𝐽1/ℎ is fixed at 46 GHz. Shaded area indicatesthe straddling regime and blue circle corresponds to the zero 𝜒− point used in theexperiment.

two-excitation states are calculated (to fourth order) and the dispersive coupling rate

is given by 𝜒− = 𝜔|1−0+0⟩ + 𝜔|0−0+1⟩ − 𝜔|1−0+1⟩, where we use the same labels for the

states as in section 2.2. The straddling regime is between the |1−0+0⟩ ↔ |0−0+1⟩and |1−0+1⟩ ↔ |2−0+0⟩ avoided crossings. The dispersive interaction between en-

ergy levels can be understood as pairwise level repulsions, and 𝜒− crosses zero in

the straddling regime because of the destructive interference between these repulsive

interactions. The small 𝑔− that is required in (4.7) can be seen in the small avoided

crossing between |1−0+0⟩ and |0−0+1⟩ in figure 4.4(a).

4.3 Device design

Based on the analyses in previous sections, we designed and fabricated a TCQ device

to study the effect of 𝜒− on photon shot noise dephasing. The device images are shown

59


in figure 4.5 and the measured parameters are listed in table 4.1. The parameters

are designed and tuned to ensure that the TCQ is in both the straddling regime

(Δ− = 𝜔− − 𝜔𝑟 = 2𝜋 × 110 MHz < 𝛼− = 2𝜋 × 129 MHz) and the dispersive regime

(𝑔− < 2𝜋 × 10 MHz ≪ Δ− = 2𝜋 × 110 MHz).

Input

Output

Cavity

TCQ

Flux bias 2

Flux bias 1

1 mm 200 µm 2 µm

Figure 4.5: TCQ device for suppression of photon shot noise. Optical micrographsshow a TCQ coupled to a readout cavity. The rf input and output ports are coupledcapacitively to the cavity and spiral inductors are used as low pass filters for the dcbias lines. Electron micrograph shows two Josephson junctions forming a SQUIDloop of size ≈ 5 × 5 μm2.

Table 4.1: TCQ device parameters.

Parameter Symbol ValueLower qubit frequency 𝜔−/2𝜋 7.25 GHz

Lower qubit anharmonicity 𝛼−/2𝜋 129 MHzUpper qubit frequency 𝜔+/2𝜋 9.80 GHz

Upper qubit anharmonicity 𝛼+/2𝜋 239 MHzCross anharmonicity 𝛼𝑐/2𝜋 358 MHz

Cavity frequency 𝜔𝑟/2𝜋 7.14 GHzCavity linewidth 𝜅/2𝜋 250 kHz

A few points are worth noting for the device design.

1. Flat edge islands, instead of interdigitated fingers (as is shown in figure 3.3) are

used to form the shunting capacitors for the qubit. This reduces the qubit’s

participation in lossy interfaces because there is less electric field concentrated

in the gap between islands [162–166].60


2. The Josephson energies of the two SQUIDs 𝐸max𝐽1,2 are chosen to ensure the state

|1−0+⟩ can be tuned into the straddling regime, i.e., 𝜔max− > 𝜔𝑟 + 𝛼−. The

hopping rate 𝐽 needs to be large enough (designed to be larger than 1 GHz) so

that the state |0−1+⟩ is far above the cavity in frequency. In addition, the two

magnetic fluxes are biased near the sweep spot (where 𝐸𝐽 reaches maximum)

to reduce the susceptibility of 𝑇2 to flux noise.

3. We use the end-coupled structure for qubit-cavity coupling, where the gap at the

end of the output side of the CPW resonator provides both the open-circuited

termination and the capacitive coupling to the qubit. Compared to the side-

coupled structure shown in figure 3.3, this design allows for other coupling

structures (e.g., bus cavities and qubit drive lines) to be added with low crosstalk

between each other [167, 168].

4. Spiral inductors are used in the flux bias lines to act as low pass filters [169].

4.4 Device calibration

After the device is designed and fabricated, we perform a series of calibrations on it

to demonstrate the tunability of 𝜒−.

The measurement setup for calibrating the device is discussed in section 3.4 and

shown in figure 3.11. The basic procedure and techniques for calibrating TCQ are

discussed in [110, 111, 109 (chapters 4 and 5)] and can be summarized as follows:

1. The mapping between the current flowing through the flux bias line and the

induced magnetic field can be calibrated through the periodic dependence of

cavity transmission on the magnetic flux biases [See figure 4.6(a)].

61


2. The qubit frequency as a function of flux biases can be measured using standard

spectroscopy method [170, 76 (section 5.1.2)].

3. The coupling rate between qubit and cavity can be measured from the vacuum

Rabi splitting when they are on resonance with each other [41, 110], as is shown

in figure 4.6(b).

From the above calibrations we can map out the correspondence between (𝜔−, 𝑔−)and the two flux biases (Φ1, Φ2) and focus on the region in the parameter space

where the requirements (4.7) are satisfied, as is shown in the yellow dashed ellipse in

figure 4.6(a).

−2 0 1 2Flux bias 1 (a.u.)

−2

−1

0

1

2

Flux

bias

2 (a

.u.)

(a)

0

1

2

3

4

5

6 Homodyne vo ltage (m

V)

7.6 7.7 7.8 7.9 8.0Frequency (GHz)

−1.0

−0.9

−0.8

−0.7

−0.6

Flux

bias

1 (a

.u.)

(b)

−50

−40

−30

−20

Transmission (dB)

−50−40−30−20

7.5 7.6 7.7−50−40−30−20

−1

! −>! r

! −<! r

Large g−

Small g−

Figure 4.6: Basic calibration for TCQ device. (a) Cavity transmission shows periodicdependence on the two magnetic flux biases and black dashed lines indicate one fluxquantum in each SQUID loop. The two directions of periodicity are not parallel withthe axes because of cross coupling between the flux biases and the SQUID loops. Theyellow dashed ellipse indicates where we operate the device to achieve zero 𝜒−. (b)The splitting in transmission when the TCQ is on resonance with the cavity showstunable 𝑔−. Insets correspond to horizontal cuts (dashed lines) where 𝑔−/2𝜋 = 55MHz (upper) and 𝑔−/2𝜋 = 22 MHz (lower).

We now proceed to discuss the tunability and calibration of 𝜒−. The tunability of

𝜒− can be seen from (4.6). The ability to vary 𝑔− in addition to 𝜔− in the TCQ allows

us to tune 𝜒− in a flexible way. Experimentally, we fix 𝜔−/2𝜋 = 7.25 GHz (Δ−/2𝜋 =

62


110 MHz) and tune 𝑔−/2𝜋 in the range of 0 ∼ 10 MHz. In this regime 𝑔+ and Δ+

(and therefore 𝜒2) vary relatively slowly compared to 𝑔−, so we can use 𝑔− as the

main control knob to tune 𝜒−: When 𝑔− is large enough, 𝜒1 > |𝜒2| and 𝜒− is positive;

As we tune down 𝑔−, 𝜒1 decreases and 𝜒− becomes negative when 𝜒1 < |𝜒2|.

7.248 7.249 7.250 7.251 7.252−1.2

−1.1

−1.0

−0.9

−0.8 (a)

g −in

crea

ses

7.248 7.249 7.250 7.251 7.252

(b)

−40

−20

0

20

40

Spectroscopy frequency (GHz)

Flux

bias

1 (a

.u.) Phase shift ( deg)

n = 0.2 n = 2.3

ffl− > 0

ffl− ≈ 0

ffl− < 0

Figure 4.7: Qubit spectroscopy for constant frequency 𝜔− and various coupling 𝜒−.(a) When = 0.2, the phase shift of the cavity transmission switches sign when𝜒− crosses zero. (b) When = 2.3, upward (downward) ac Stark shift of the qubitfrequency illustrates positive (negative) 𝜒−. The spectroscopy signal vanishes when𝜒− approaches zero.

The above scheme is performed in qubit spectroscopy measurements and the mea-

sured spectra for various 𝜒− are shown in figure 4.7. Standard qubit spectroscopy

measurement, where the phase shift of the cavity transmission is monitored and a

second spectroscopy tone sweeps around the qubit frequency, is repeated for different

combinations of the two flux biases (Φ1, Φ2) to map out the constant 𝜔− contour in

figure 4.7(a). Here, Φ1 is varied linearly and Φ2 (not shown in the figure) is deter-

mined by the condition that 𝜔−/2𝜋 remains 7.25 GHz when the intracavity photon

number is small. Along the contour, the phase shift of the cavity transmission

changes sign, indicating that 𝜒− crosses zero. For larger , the qubit frequency is

dressed by cavity photons and exhibits ac Stark shift [150] of 𝜒−, as is discussed in

section 4.1. The dressed qubit frequency in figure 4.7(b) shows clearly that 𝜒− can

63


be tuned to both positive and negative, from a few MHz down to below the cavity

linewidth 𝜅/2𝜋 = 250 kHz, which cannot be resolved in the qubit spectroscopy.

7.140 7.141 7.142 7.143 7.1440

2

4

6

Hom

odyn

evo

ltage

(mV)

(a) ffl−/2ı = 1.75 MHz

|0−⟩ |1−⟩

7.140 7.141 7.1420

1

2

3(b) ffl− ≈ 0

|0−⟩|1−⟩

Without ı pulseWith ı pulse

Frequency (GHz)

Figure 4.8: Readout cavity transmission when TCQ is in ground (blue) or excited(red) state for (a) 𝜒−/2𝜋 = 1.75 MHz and (b) 𝜒− ≈ 0. The small peak near 7.1412GHz in (a) for the excited state is caused by relaxation to the ground state duringthe measurement.

We also use cavity transmission as another calibration tool for 𝜒−. In fig-

ure 4.8, the transmission amplitude of the cavity is plotted when the TCQ is in the

ground/excited state for large and small 𝜒−. When 𝜒− > 𝜅, two transmission peaks

separated by 𝜒−/2𝜋 = 1.75 MHz can be resolved, corresponding to different qubit

states; When 𝜒− ≈ 0, no shift in cavity transmission can be seen between the two

measurements, indicating 𝜒− < 𝜅 = 2𝜋 × 250 kHz.

4.5 Readout method

As 𝜒− approaches zero, so does the readout contrast because there is no dispersive

shift caused by the |1−0+⟩ state, as is illustrated in figure 4.7 near Φ1 = −0.95 and

in figure 4.8(b). To achieve efficient readout for small 𝜒−, we adopt a scheme that

utilizes a third state |1−1+⟩, as is shown in figure 4.9. In this scheme, we apply

64


a transfer pulse at frequency 𝜔+ − 𝛼𝑐 to the TCQ, inducing a |1−0+⟩ → |1−1+⟩transition, immediately before the readout pulse at frequency 𝜔𝑟. The |1−1+⟩ state

provides a measured dispersive shift of −1.2 MHz around the zero 𝜒− point and

can be used to indirectly read out the logical qubit state. We note that it is the

cross anharmonicity 𝛼𝑐 in a TCQ that allows us to drive |1−0+⟩ → |1−1+⟩ transition

without simultaneously driving |0−0+⟩ → |0−1+⟩. This type of readout using ancillary

levels in V-shaped systems is proposed in [171].

(a) (b)Preparation

Transfer

ReadoutPreparation

Transfer

!− + !+ − ¸c

!−

ffl = 0

ffl = 0

|1−1+⟩

|1−0+⟩

|0−0+⟩

!−

!+ − ¸c

!r

„1

„2

Figure 4.9: Readout scheme when 𝜒− ≈ 0. (a) A third state |1−1+⟩ which has nonzero𝜒 is used for indirect readout. (b) Two consecutive Gaussian pulses with 𝜎 = 16 nsand frequencies 𝜔−, 𝜔+ − 𝛼𝑐 are sent to drive qubit transitions for state preparationand transfer, followed by a readout pulse at the cavity frequency 𝜔𝑟.

To test this readout method, we prepare the TCQ to a state |𝜓⟩ characterized by

Rabi angle 𝜃1, i.e., |𝜓⟩ = cos(𝜃1/2)|0−0+⟩ + sin(𝜃1/2)|1−0+⟩, by sending a Gaussian

pulse at frequency 𝜔−. A transfer pulse with Rabi angle 𝜃2 is then applied as described

above, followed by the readout pulse. The widths of the Gaussian pulses are fixed at

𝜎 = 16 ns and 𝜃1,2 are controlled by the pulse amplitudes. It is worth noting that

when 𝜒− approaches zero, both 𝑔− and 𝑔+ remain non-zero, as can be inferred from

(4.6). This enables us to efficiently drive |0−0+⟩ ↔ |1−0+⟩ and |1−0+⟩ ↔ |1−1+⟩transitions using short pulses.

Figure 4.10(a) shows the measured homodyne signal at 𝜒− ≈ 0 as a function of 𝜃1

and 𝜃2. Rabi oscillations for both |0−0+⟩ ↔ |1−0+⟩ and |1−0+⟩ ↔ |1−1+⟩ transitions65


0.0 0.2 0.4 0.6 0.8 1.0„1/ı

0.0

0.2

0.4

0.6

0.8

1.0 (a)

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0„1/ı

0.0

0.5

1.0

1.5

2.0

2.5 (b)„2 = 0„2 = ı

„ 2/ı

Hom

odyn

evo

ltage

(mV)

Figure 4.10: Readout signal when 𝜒− ≈ 0. (a) Homodyne readout signal shows Rabioscillations for both |0−0+⟩ ↔ |1−0+⟩ and |1−0+⟩ ↔ |1−1+⟩ transitions, with Rabiangles 𝜃1,2 determined by the amplitudes of the two drive pulses. (b) Horizontal cutsat 𝜃2 = 0 and 𝜋 in (a). Maximum readout contrast (red crosses) is obtained aftertransferring |1−0+⟩ to |1−1+⟩ by a transfer pulse with 𝜃2 = 𝜋, in contrast to very lowreadout signal (blue dots) with no transfer pulse.

are observed, demonstrating coherent transfers between the quantum states. In the

absence of the transfer pulse, no visible contrast is observed because of small 𝜒−

[blue dots in figure 4.10(b)]. As the amplitude of the transfer pulse is adjusted to

𝜃2 = 𝜋, |𝜓⟩ is transferred to |𝜓′⟩ = cos(𝜃1/2)|0−0+⟩ + sin(𝜃1/2)|1−1+⟩, which gives

the maximum readout contrast [red crosses in figure 4.10(b)] and recovers the Rabi

angle 𝜃1. This method allows for the single qubit control and readout scheme to be

performed entirely with microwave pulses, and does not involve dynamical tuning of

𝜒− via fast flux biasing, which increases experimental complexity and might cause

unwanted qubit errors. It also allows low pass filtering to reduce the Purcell decay of

the qubit through the flux bias lines [172, 173].

4.6 Coherence measurements

Combining the tunability of 𝜒− and the readout method, we perform time domain

measurements for the qubit relaxation and coherence time. In addition to the stan-66


dard measurement setup, we use a noise source to study the influence of thermal

photon fluctuations on qubit dephasing. The pulse sequence and typical measured

result are shown in figure 4.11. White noise within the bandwidth 7.1375 GHz ± 5

MHz is injected to the device, populating the cavity but not the qubit, and its power

density determines the intracavity noise photon number 𝑛th.

Drive

Transfer

Noise Readout!r

!− − ¸c

!− Δt/2 Δt/2(ı/2)X (ı/2)ffi

(ı)X

ı

0 10 20 30 40 50Δt (µs)

0

2

4

6

Hom

odyn

evo

ltage

(mV) T2E = 20.15 µs

DataFit

(a) (b)

Figure 4.11: Measurement scheme for TCQ coherence time. (a) Pulse sequence formeasuring 𝑇2E consists of two 𝜋/2 pulses with varying delay Δ𝑡 in between and anecho 𝜋 pulse in the middle. External noise is injected to simulate the effect of thermalphotons and a transfer pulse is used for indirect readout in the small 𝜒− regime. (b)Typical measured and fitted data. The Ramsey-like fringes are introduced by varyingthe phase 𝜙 of the second 𝜋/2 pulse linearly with respect to Δ𝑡.

In figure 4.12(a), the measured 𝑇1 and 𝑇2E for 𝜒−/2𝜋 = 1.9 MHz are plotted

as a function of the injected noise power. A Hahn echo pulse is used in measuring

coherence time to eliminate slow dephasing processes caused by flux noise, etc. While

𝑇1 exhibits little dependence on the injected noise and remains 8 ∼ 11 μs, 𝑇2E drops

from 16 μs to 3.5 μs as the noise power increases, reflecting a photon shot noise limited

𝑇2E when 𝜒− is comparable to 𝜒 used in normal transmon devices. Figure 4.12(b)

shows the result for the same measurements when 𝜒− ≈ 0 and we obtain 𝑇1 = 10 ∼ 12μs and 𝑇2 = 20 ∼ 22 μs, and no reduction is observed in either 𝑇1 or 𝑇2E up to

𝑛th = 0.13. To quantitatively analyze the result, we use (4.5) to fit the measured 𝑇2E

data, shown in the red curves in figure 4.12. In figure 4.12(a), a single linear fitting67


parameter converts the output power of the noise source to the 𝑛th values in the 𝑥axis; In figure 4.12(b) we extract 𝜒−/2𝜋 = 22 kHz from the best fit to (4.5).

0 0.04 0.08 0.120

5

10

15

20

25

Qubi

tlife

time

(µs)

(a)

ffl−/2ı = 1.9 MHz

T1

T2E

Fit

0 0.04 0.08 0.12

(b)

ffl−/2ı = 22 kHz

Noise photon number

Figure 4.12: Qubit relaxation time 𝑇1 (blue crosses), coherence time 𝑇2E with Hahnecho (red dots) and fit to (4.6) (red curves) as a function of injected photon noise.(a) When 𝜒−/2𝜋 = 1.9 MHz, 𝑇2E drops from 16 μs to 3.5 μs with increasing photonnoise. (b) When 𝜒−/2𝜋 = 22 kHz, no drop in 𝑇2E is observed up to noise photonnumber 𝑛th = 0.13.

To estimate Γ𝜙, we assume a typical 𝑛th = 0.02 [148, 150]. The small 𝜒− leads

to Γ𝜙 = 0.25 kHz, corresponding to a photon shot noise limited 𝑇2E of 4000 μs. In

comparison, to achieve the same level of Γ𝜙 with the same 𝑛th in a high-𝑄 cavity

device would require 𝜅/2𝜋 = 2 kHz (𝑄 ∼ 3 × 106). The 𝑇1 of the device is Purcell

limited [172] because of the small detuning between the qubit and cavity, evidenced

by a measured 𝑇1 = 21 μs when tuning the qubit to 2.2 GHz below the cavity, and

can be improved by increasing cavity 𝑄 or engineering the cavity spectrum using

filters [173, 174].

Compared to other methods to suppress photon shot noise dephasing, our ap-

proach does not rely on very high-𝑄 cavities and 𝑇2 is limited by 𝜒−, which in princi-

ple can be tuned to zero. Gate operation and readout can be performed conveniently

without dynamical control of the qubit. The planar geometry also makes it easy to

68


integrate the device into larger cQED systems. The ability to access and measure

the quantum state while maintaining the high coherence makes the TCQ a promising

building block for the processing and storage of quantum information.

69

Chapter 5

Suppression of Qubit Crosstalk

In chapter 4 we demonstrated the application of tunable coupling in a single qubit de-

vice. To form more complicated and powerful devices, multiple qubits and resonators

are integrated together [42, 47, 59] and a universal set of quantum gates [11 (sec-

tion 4.5)] includes single-qubit gates and one type of two-qubit entangling gate [175],

e.g., controlled-NOT (CNOT), controlled-Phase (CPhase) or iSWAP gate, etc. To

implement those gates, interactions between qubits need to be turned on and off

in a controlled manner. This process is often accompanied by unwanted or uncon-

trolled couplings between qubits that can lead to crosstalk and deteriorate device

performance. The understanding, characterization and elimination of crosstalk have

become import and challenging tasks as the number of qubits scales up in cQED

experiments [176–178].

In this chapter we explore the possibility of utilizing tunable coupling to sup-

press static ZZ crosstalk in a multi-qubit system. In section 5.1 we review some

of the crosstalks and their origins in cQED systems, including static ZZ crosstalk.

Section 5.2 introduces the design of a two-qubit, two-coupler device where static ZZ

interaction can be tuned to zero. We then present experimental results for device cal-

ibration and crosstalk suppression using spectroscopy and randomized benchmarking

70

CHAPTER 5. SUPPRESSION OF QUBIT CROSSTALK

in section 5.3. Finally in section 5.4 we discuss the implementation of iSWAP gate

between two qubits using parametric modulation.

5.1 Qubit crosstalk

Crosstalk in a multi-qubit system refers to undesirable responses to external controls

and unwanted interactions between subsystems. For example, a drive field targeting

at one qubit can off-resonantly drive other qubits and excite them or cause frequency

shifts; A magnetic flux tuning the frequency of one qubit can affect nearby qubits due

to the mutual inductance between them. As the number of qubits increases, devices

need to be designed carefully to minimize those effects and the crosstalk needs to be

calibrated and compensated properly [See figure 5.1(a) for one example].

CR 4 OnCR 4 Of

10 20 30

1

0Ramsey Delay (µs) Frequency (MHz)

0 1 2−1−2

Cross resonance

Ramsey DelayRamsey

Source device

q9cp89q8cp78q7cp67q6cp56q5cp45q4cp34q3cp23q2cp12q1

Targ

et d

evice

q1cp12 q2

cp23 q3

cp34 q4

cp45 q5

cp56 q6

cp67 q7

cp78 q8

cp89 q9

0.3%

0

Crosstalk mat rix-elem

ent

(a) (b)

Excit

ed st

ate

pop.

Figure 5.1: Crosstalk in multi-qubit systems. (a) In a nine-qubit device, the crosstalkbetween all flux biases and qubits needs to be calibrated and compensated. Thecrosstalk is caused by unwanted mutual inductance between those elements. Figurereproduced from [64]. (b) In a five-qubit device, cross resonant gates cause qubitfrequency shifts and echo pulses need to be added to remove the crosstalk. Figureadapted from [177].

Moreover, the crosstalk can involve correlation or entanglement between qubits.

In a multi-qubit system, an entangling gate between a “control” and a “target” qubit

is typically implemented by turning on the interaction between them for a certain71


amount of time [101, 179, 180]. If the control or target qubit is connected to other

qubits, this operation can activate unwanted interaction and entanglement with those

“spectator” qubits. For a concrete example, we examine the Hamiltonian for a two-

qubit system driven by two coherent tones, as is discussed in [176],

𝐻/ℏ = 𝜀1[𝜎𝑥1 + (𝑚12 − 𝜈1)𝜎𝑥

2 − 𝜇1𝜎𝑧1𝜎𝑥

2 + 𝑚12𝜇2𝜎𝑥1 𝜎𝑧

2]

+ 𝜀2[𝜎𝑥2 + (𝑚21 + 𝜈2)𝜎𝑥

1 + 𝜇2𝜎𝑥1 𝜎𝑧

2 − 𝑚21𝜇1𝜎𝑧1𝜎𝑥

2 ]

+ 𝜔1𝜎𝑧1/2 + 𝜔2𝜎𝑧

2/2 + 𝜁𝜎𝑧1𝜎𝑧

2/4.

(5.1)

Here 𝜔1,2 are the qubit frequencies, 𝜀1,2 are the driving amplitudes, 𝜇1,2 are the cross

resonance coupling rates that enable two-qubit gates [179], and 𝑚12,21 correspond to

crosstalk between drive 1(2) and qubit 2(1). The presence of crosstalk (characterized

by 𝑚12,21) not only causes unwanted single-qubit rotations but also induces two-

qubit interactions between control and spectator qubits. Figure 5.1(b) shows such a

situation where extra echo pulses need to be applied to spectator qubits to eliminate

the crosstalk [177].

Yet another type of crosstalk, represented by the last term in (5.1), exists in a

multi-qubit system. It describes cross-Kerr interaction (state dependent frequency

shift) between the two qubits characterized by the rate 𝜁 and gives rise to static ZZ

crosstalk. The origin of this crosstalk is the couplings between higher energy levels

of the qubits and its explicit expression is

𝜁 = 2𝑔21𝑔2

2 [ 1(Δ12 + 𝛼2)Δ2

1+ 1

(Δ21 + 𝛼1)Δ22

+ 1Δ1Δ2

2+ 1

Δ21Δ2

] , (5.2)

for two transmons dispersively coupled to a bus cavity [180], where Δ12 = 𝜔1 − 𝜔2 =−Δ21 is frequency difference between qubits, 𝑔1,2/Δ1,2 are coupling rates/detunings

between qubits and bus cavity, and 𝛼1,2 are anharmonicities of the qubits.

72


The static ZZ crosstalk is of particular concern in quantum computing devices

because of its following features.

1. It is an always-on crosstalk even in the absence of drive fields, and the 𝜁𝜎𝑧𝑖 𝜎𝑧

𝑗

term will accumulate phase errors between pairs of qubits.

2. It is a correlated crosstalk between pairs of qubits and its elimination requires

echo schemes on multiple qubits, which adds to the complexity of the pulse

scheme and control hardware.

3. It limits the fidelity of XX-type parity measurements, which is important in the

implementation of quantum error correction [60, 177].

5.2 Device design

As can be seen from (5.2), 𝜁 can be reduced by decreasing 𝑔𝑖 or increasing Δ𝑖, but this

approach will reduce the required coupling for two-qubit gate as well. We adopt an

approach that is similar to TCQ and utilize quantum interference effect to suppress

ZZ crosstalk. By introducing a tunable coupler in addition to the bus cavity, we show

that 𝜁 can be tuned to zero and demonstrate in section 5.4 that an efficient two-qubit

gate can be implemented when 𝜁 ≈ 0. A similar scheme has been proposed very

recently in [181].

Figure 5.2 shows the device, where two computational qubits (Q1, Q2) are coupled

to a coupler qubit (C−) and a bus cavity (C+). The Hamiltonian for the device is

𝐻/ℏ = ∑𝑖=1,2,±

(𝜔𝑖𝑎†𝑖 𝑎𝑖 − 𝛼𝑖

2 𝑎†𝑖 𝑎†

𝑖 𝑎𝑖𝑎𝑖) + ∑𝑖=1,2𝑗=±

𝑔𝑖𝑗(𝑎†𝑖 𝑎𝑗 + 𝑎𝑖𝑎†

𝑗). (5.3)

where the subscripts 1, 2, +, − correspond to the elements in figure 5.2, 𝜔𝑖, 𝛼𝑖 are

73


1 mm 200 µm

Fluxbias

Output 1

Output 2

Input 1

Input 2

10 µm

(a) (b)R1

Q1

C+ C−

R2

Q2

!1

!2

!+ !−

g1+ g1−

g2+ g2−

Q1

C+ C−

Q2

Figure 5.2: Tunable 𝜁 device for suppression of ZZ crosstalk. (a) Conceptualschematic. (b) Micrograph shows two qubits (Q1, Q2) coupled via a bus cavity (C+)and a coupler qubit (C−). The frequency of the coupler qubit can be tuned by anearby flux bias line and two readout resonators (R1, R2) are used to measure thestates of either qubit.

their frequencies and anharmonicities, and 𝑔𝑖𝑗 are the coupling rates between them.

Theoretical analysis (See appendix B for details) shows that the criteria for zero 𝜁 is

the bus cavity (coupler qubit) being above (below) both qubits in frequency and one

qubit being in the straddling regime of the other, i.e.,

𝜔− < 𝜔1,2 < 𝜔+, |𝜔1 − 𝜔2| < 𝛼1,2. (5.4)

Similar to the tunable 𝜒 in a TCQ, here the tunability of 𝜁 results from the interference

between the ZZ coupling caused by the upper (bus cavity) and lower (tunable qubit)

couplers, and zero 𝜁 can be achieved when the coupler qubit frequency is adjusted to

give complete destructive interference.

In principle all the components in the device can be made frequency-tunable

for maximum flexibility. We use fixed-frequency transmons as computational qubits

because they are insensitive to flux noise and have better coherence time. The bus

cavity can be replaced by a qubit to make the device more compact, but it is easier

to design and fabricate a CPW cavity with targeted resonance frequency. Tuning

74


and suppressing 𝜁 is realized by adjusting the frequency of the coupler qubit through

magnetic flux bias Φ,

𝜔−(Φ) = 𝜔max− √| cos(𝜋Φ/Φ0)|. (5.5)

We operate the device in the dispersive regime, where |Δ𝑖𝑗| ≫ 𝑔𝑖𝑗, to minimize

population leakage into the coupler qubit during iSWAP gate (See section 5.4 and

[182]) and decoherence of computational qubits induced by flux noise in the coupler

qubit.

5.3 Device calibration

The measurement setup for device calibration and later experiments is similar to

figure 3.11. The main difference is that we add an rf tone to the magnetic flux line

via a bias tee on the base stage of the fridge to enable flux modulation (See section 5.4)

in addition to dc flux bias.

4.9 5.0 5.1 5.2Frequency (GHz)

0.2

0.3

0.4

Flux

(Φ0)

Q1 Q2

C−

(a) 0

−10

−20

−30

−40

Phaseshift(deg)

−2.5 −2.0 −1.5 −1.0 −0.5(!− − !1)/2ı (GHz)

−0.2

0.0

0.2

0.4

0.6

“/2ı

(MHz

)

(b)DataFit

Q1

Q2

R1

ı/2 Δt ı/2

ı

Readout

Figure 5.3: Calibration for tunable 𝜁 device. (a) Spectroscopy for qubits and coupler.Red dashed lines are fit to (5.3). Black dashed lines correspond to 𝜔𝑖 + 𝑔2

𝑖+/Δ𝑖+ (𝑖 =1, 2) and 𝜔−(Φ). (b) ZZ coupling rate 𝜁 as a function of coupler frequency 𝜔−. Thevalue of 𝜁 (blue points) is obtained by cross Ramsey calibration, where the frequencyof qubit 1 is measured with/without sending a 𝜋 pulse to qubit 2. Inset shows thepulse scheme and red line is the theoretical result from numerical diagonalizing (5.3)using parameters in table 5.1.

75


We perform spectroscopy on both qubits while tuning the frequency of coupler

qubit to map out the magnetic flux dependence of the device, and the result is shown

in figure 5.3(a). When the coupler is brought into resonance with either qubit, avoided

crossing is observed and the corresponding coupling rate 𝑔1−, 𝑔2− can be extracted

by fitting the data to (5.3). The coupler frequency in a full flux quantum is measured

(not shown in the figure) using spectroscopy and the bus cavity frequency is measured

by monitoring the ac Stark shift of either qubit while sweeping the frequency of a

cavity populating tone. The measured or fitted device parameters are summarized in

table 5.1.

Table 5.1: Tunable 𝜁 device parameters.

Parameter Symbol ValueQubit 1 frequency 𝜔1/2𝜋 4.973 GHz

Qubit 1 anharmonicity 𝛼1/2𝜋 400 MHzQubit 1 relaxation time 𝑇 (1)

1 15.2 μsQubit 1 coherence time 𝑇 (1)

2E 4.2 μsQubit 2 frequency 𝜔2/2𝜋 5.163 GHz

Qubit 2 anharmonicity 𝛼2/2𝜋 400 MHzQubit 2 relaxation time 𝑇 (2)

1 12.1 μsQubit 2 coherence time 𝑇 (2)

2E 4.0 μsBus cavity frequency 𝜔+/2𝜋 7.036 GHz

Maximum coupler frequency 𝜔max− /2𝜋 7.18 GHz

Coupler anharmonicity 𝛼−/2𝜋 750 MHz(Qubit 1, bus cavity) coupling 𝑔1+/2𝜋 135 MHz(Qubit 2, bus cavity) coupling 𝑔2+/2𝜋 135 MHz

(Qubit 1, coupler) coupling 𝑔1−/2𝜋 95 MHz(Qubit 2, coupler) coupling 𝑔2−/2𝜋 95 MHz

The dependence of 𝜁 on the coupler frequency 𝜔−(Φ) is mapped out using cross

Ramsey measurement on qubit 1. The frequency difference in qubit 1 when qubit

2 is in the ground or excited state corresponds to 𝜁 = 𝜔|11⟩ − 𝜔|10⟩ − 𝜔|01⟩, and

the measured data is shown in figure 5.3(b). Based on the criteria (5.4) we tuned

76


the frequency of coupler qubit to be below both qubits and 𝜁 crosses zero at (𝜔− −𝜔1)/2𝜋 = −1.47 and − 0.75 GHz. The theoretical curve is calculated from the

numerical diagonalization of (5.3) using the parameters in table 5.1. The error bar

corresponds to the fitting error for the Ramsey data. Although the uncertainty in 𝜁is relative large mainly due to short 𝑇2 of the qubit, we show clear evidence of the

tunability and sign change of 𝜁, as well as qualitative agreement with theory. The

extraordinarily large error bar at (𝜔− − 𝜔1)/2𝜋 = −1.36 GHz is likely due to an

avoided crossing between higher energy levels. Cross Ramsey measurement on qubit

2 (not shown in the figure) shows similar value of 𝜁 but larger error due to shorter

𝑇2.

To further characterize the effect of 𝜁 on qubit crosstalk, we adopt the simul-

taneous randomized benchmarking protocol from [176]. Randomized benchmarking

(RB) [183, 184] is a convenient and useful tool for characterizing error-rate for quan-

tum gates. The protocol consists of applying random sequences of Clifford gates to

the qubit followed by a final Clifford gate to bring the state back to |0⟩ and measuring

the resulting qubit state. As the number of gates 𝑁 increases, the probability of the

qubit being in the ground state 𝑃(|0⟩) decreases exponentially from the ideal value

of 1 due to gate error and decoherence. The above process is repeated for different

random sequences and the Clifford gate error rate 𝑟 is extracted from fitting the

averaged result to 𝑃(|0⟩) = 𝐴(1 − 2𝑟)𝑁 + 𝐵. The advantages of the RB protocol in-

clude the robustness against state preparation and measurement (SPAM) errors and

the modest scaling of required gate operations and measurements as the number of

qubits increases. Experimentally, the Clifford gates are decomposed into products of

primary single-qubit gates (see appendix C for details) and the primary gate fidelity

is given by 𝐹 = 1 − 𝑟/𝑁𝑔, where 𝑁𝑔 = 1.875 is the average number of primary gates

per Clifford gate for our decomposition scheme.

77


0.4

0.5

0.6

0.7

0.8

0.9

1.0(a) Qubit 1, “ ≈ 0

FI = 0.99863

FS = 0.99858

IndividualSimultaneous

(b) Qubit 1, “/2ı = 2.26 MHz

FI = 0.99857FS = 0.98704

0 100 200 300 400 500 6000.4

0.5

0.6

0.7

0.8

0.9

1.0(c) Qubit 2, “ ≈ 0

FI = 0.99846

FS = 0.998370 100 200 300 400 500 600

(d) Qubit 2, “/2ı = 2.26 MHz

FI = 0.99822

FS = 0.98665

Number of Clifford gates

P(|0

⟩)

Figure 5.4: Qubit crosstalk characterization. Individual (red) and simultaneous (blue)RB are performed on qubit 1 (upper) and 2 (lower), when the coupler qubit is tunedto give small (left) and large (right) 𝜁, and the corresponding primary gate fidelities𝐹𝐼 and 𝐹𝑆 are extracted. For small 𝜁 the difference 𝐹𝐼 − 𝐹𝑆 is within 0.01 %. Forlarge 𝜁 the difference is more than 1.15 %.

The red curves in figure 5.4 shows the RB results for both qubits. The pulses used

for qubit gates are of Gaussian shape truncated at 4𝜎 with 𝜎 = 6.4 ns. Derivative re-

moval via adiabatic gate (DRAG) [185–187] is used to reduce phase error and leakage

to higher energy levels. For individual RB, gate fidelity of 𝐹𝐼 > 99.8 % is achieved

for both qubits. The coherence limited fidelity can be estimated from an exponential

decay with a time constant of 𝑇1 and the result is

𝐹 = 1 − 1 − exp (−𝑁𝑔𝜏/𝑇1)2𝑁𝑔

= 99.89 %, (5.6)

78


where 𝜏 = 4𝜎 = 25.6 ns is the duration of a primitive gate and we use a typical

𝑇1 = 12 μs. The measured fidelity is close to the coherence limit and the deviation is

likely due to the short 𝑇2 of the qubits.

Simultaneous RB extends the above protocol by applying random Clifford gate

sequences to both qubits at the same time and the difference in gate fidelity between

individual (𝐹𝐼) and simultaneous (𝐹𝑆) RB provides a figure of merit for addressability

and crosstalk [176]. The effect of 𝜁 on qubit crosstalk is characterized in simultaneous

RB, as is shown in the blue curves in figure 5.4. When 𝜁/2𝜋 < 100 kHz, 𝐹𝐼 − 𝐹𝑆 is

less than 0.01 %, indicating that crosstalk is suppressed to a level below the gate error

for this device. By contrast, when 𝜁/2𝜋 = 2.26 MHz the gate fidelity drops by more

than 1.15 %, and crosstalk becomes the dominant source of gate error. The ability to

tune 𝜁 across zero allows us to operate in an optimal configuration to suppress static

ZZ crosstalk and provides a useful tool to study the impact of crosstalk on device

performance.

5.4 Parametric coupling and iSWAP gate

As mentioned at the beginning of this chapter, to realize a universal quantum gate

set we need a two-qubit entangling gate in addition to single qubit operations. For

our device we utilize parametric modulation to implement iSWAP gate.

We operate the device in the dispersive and small-𝜁 regime, and if we assume the

bus cavity and coupler qubit are in their ground states, the effective Hamiltonian for

the two qubits is

𝐻/ℏ =2

∑𝑖=1

(𝑖𝑎†𝑖 𝑎𝑖 − 𝛼𝑖

2 𝑎†𝑖 𝑎†

𝑖 𝑎𝑖𝑎𝑖) + 𝐽(𝑎†1𝑎2 + 𝑎1𝑎†

2), (5.7)

79


where

𝐽 = ∑𝑗=±

𝑔1𝑗𝑔2𝑗2 ( 1

Δ1𝑗+ 1

Δ2𝑗) , (5.8)

is the rate for the effective exchange coupling mediated by couplers, Δ𝑖𝑗 = 𝜔𝑖 − 𝜔𝑗

are the detunings between qubits and couplers, and 𝑖 = 𝜔𝑖 + ∑𝑗 𝑔2𝑖𝑗/Δ𝑖𝑗 are the

qubit frequencies dressed by the couplers.

When the magnetic flux threading the SQUID loop of coupler qubit is modulated

around Φ = Θ at frequency 𝜔Φ, phase 𝜙 and amplitude 𝛿,

Φ(𝑡) = Θ + 𝛿 cos(𝜔Φ𝑡 + 𝜙), (5.9)

the coupler frequency 𝜔− and exchange rate 𝐽 will have corresponding time depen-

dence through the relations (5.5) and (5.8). When the modulation amplitude 𝛿 is

small, we can expand 𝐽(Φ(𝑡)) to first order in 𝛿,

𝐽(Φ(𝑡)) ≈ 𝐽(Θ) + 𝛿 𝜕𝐽𝜕Φ∣

Φ=Θcos(𝜔Φ𝑡 + 𝜙) (5.10)

Substituting (5.10) into (5.7), and going to the rotating frame of the qubit frequencies,

we obtain the effective Hamiltonian for the two qubits,

𝐻/ℎ ≈2

∑𝑖=1

−𝛼𝑖2 𝑎†

𝑖 𝑎†𝑖 𝑎𝑖𝑎𝑖 + 𝐽(Θ) (𝑎†

1𝑎2𝑒𝑖∆12𝑡 + 𝑎1𝑎†2𝑒−𝑖∆12𝑡)

+ 𝛿 𝜕𝐽𝜕Φ cos(𝜔Φ𝑡 + 𝜙) (𝑎†

1𝑎2𝑒𝑖∆12𝑡 + 𝑎1𝑎†2𝑒−𝑖∆12𝑡) .

(5.11)

From (5.11) we can see when the flux modulation frequency is resonant with the qubit

detuning,

𝜔Φ = |Δ12| = |1 − 2|, (5.12)

80


the effective exchange coupling between the two qubits becomes stationary in the

rotating frame, whose rate is determined by the modulation amplitude 𝛿 and the

derivative of 𝐽 with respect to flux Φ,

𝐻int/ℏ = 𝛿2

𝜕𝐽𝜕Φ (𝑎†

1𝑎2𝑒−𝑖𝜙 + 𝑎1𝑎†2𝑒𝑖𝜙) . (5.13)

Intuitively, the coupling is turned on when the parametric modulation matches the

energy difference between the two qubits and brings them effectively into resonance.

Importantly, the effective coupling strength depends on the derivative of 𝐽 with re-

spect to Φ, and in our device 𝛿 ⋅ 𝜕𝐽/𝜕Φ can be on the order of MHz for moderate

modulation amplitude 𝛿 [See figure 5.5(b)] despite small 𝜁. Efficient two-qubit gate

can therefore be implemented using parametric modulation.

Qubit 1

Flux

Cavity 1,2Readout

ı

!Φ, ffi, Δt

−0.44 −0.43 −0.42 −0.41 −0.40Flux (Φ0)

−6

−5

−4

−3

−2

−1

0

J/2ı

(MHz

)

2‹

(Θ, J(Θ))

2‹ @J@Φ

(a) (b)

Figure 5.5: Parametric modulation for iSWAP gate. (a) Pulse sequence for iSWAPgate. One of the qubits are excited by a 𝜋 pulse to prepare the system in state|10⟩. Flux modulation is then turned on for a certain duration Δ𝑡, with frequency𝜔Φ and phase 𝜙, before the qubits are measured. An iSWAP gate corresponds tothe modulation parameters that lead to the state 𝑖|01⟩. (b) Flux modulation scheme.Due to the non-vanishing slope 𝜕𝐽/𝜕Φ at Φ = Θ, periodic modulation on flux biasleads to parametric coupling between qubits.

This type of parametric coupling is widely used in cQED [103, 188]. The reso-

nance condition is determined by the modulation frequency, which can be controlled

81


very accurately in experiments. Moreover, interactions between different parts of a

device can be turned on in a multiplexed way [189], abnormal interaction such as

blue-sideband can be realized to stabilize qubit states [190], and the phases of the

modulation can also be adjusted to implement nontrivial interactions [191].

The Hamiltonian (5.13) is a natural implementation for the iSWAP gate [182, 192–

194], when the modulation is turned on for Δ𝑡 = 𝜋/(𝛿 ⋅ 𝜕𝐽/𝜕Φ) with phase 𝜙 = 𝜋.

𝑒−𝑖𝐻∆𝑡/ℏ = 𝑈iSWAP =⎛⎜⎜⎜⎜⎜⎜⎝

1 0 0 00 0 𝑖 00 𝑖 0 00 0 0 1

⎞⎟⎟⎟⎟⎟⎟⎠

. (5.14)

To test the above analysis, we apply flux modulation to the device with varying

frequency and duration for initial qubit state |10⟩, followed by measurements of qubit

population. The dc flux bias Θ is chosen to ensure low ZZ crosstalk based on the

calibration in section 5.3. The pulse scheme is shown in figure 5.5(a) and the result

is shown in figure 5.6(a). The exchange oscillations between states |10⟩ and |01⟩show Chevron pattern and agree with the effective exchange interaction (5.13). The

40 140 240 340Duration (ns)

180

185

190

195

200

Freq

uenc

y(M

Hz) Qubit1

population

(a)

0.20.30.40.50.60.70.80.9

0 100 200 300 400 500 600Duration (ns)

0.0

0.2

0.4

0.6

0.8

1.0

Qubi

tpop

ulat

ion

(b)Q1 Q2

Figure 5.6: iSWAP gate between two qubits. (a) The qubit population 𝑃(|1⟩) showsChevron pattern as the frequency and duration of flux modulation are varied. (b)Maximum population exchange between qubits is achieved when the frequency of themodulation matches their detuning, 𝜔Φ/2𝜋 = 192 MHz, and the duration is 90 ns.

82


effective detuning is Δ = 𝜔Φ −(2 −1) and maximum population exchange happens

for Δ = 0 and duration of 90 ns, as is shown in figure 5.6(b).

To ensure the above operation implements the iSWAP gate, we apply the same

pulse sequences to initial qubit states |00⟩ and |11⟩ and observe no change in popu-

lation other than 𝑇1 relaxation. The modulation phase 𝜙 is determined by replacing

the 𝑋𝜋 on qubit 1 in figure 5.5(a) with 𝑋𝜋/2 and measuring maximum 𝑌 projection

of qubit 2 after flux modulation,

𝑈iSWAP [(|0⟩ − 𝑖|1⟩) ⊗ |0⟩] = |0⟩ ⊗ (|0⟩ + |1⟩). (5.15)

In the derivation of (5.13) we ignored higher order terms in 𝛿 cos(𝜔Φ𝑡). One of the

effects of those terms is qubit frequency shift during the iSWAP gate, as is discussed

in [182, 189], and needs to be calibrated and compensated by additional 𝑍 gates on

both qubits [195].

4.85 4.90 4.95 5.00 5.05Frequency (GHz)

0.2

0.4

0.6

0.8

Flux

(Φ0)

(a)

−1.0

−0.5

0.0

1.0 Transmission

(dB)

4.85 4.90 4.95 5.00 5.05Frequency (GHz)

0.2

0.4

0.6

0.8

Flux

(Φ0)

(b)

0.0

0.1

0.2

0.3

Qubitpopulation

Figure 5.7: Thermal population of coupler qubit. (a) Measured spectroscopy for qubit1 shows splitting (indicated by arrows) due to the thermal population in the couplerqubit. (b) Numerical simulation shows the thermally dressed qubit frequency 𝜔|11−⟩ −𝜔|01−⟩. The thermal population is set to 0.05, corresponding to effective temperatureof 78.8 mK. The colormaps in both figures are saturated to better visualize thermallydressed frequencies.

83


We also assumed that the coupler qubit is in the ground state throughout the

flux modulation process. However, in the experiment we find a noticeable amount

of thermal population in the coupler qubit, evidenced by a splitting of frequency in

either qubit’s spectrum near the avoided crossing with the coupler qubit. The ther-

mally dressed frequency for qubit 1 (𝜔|11−⟩ − 𝜔|01−⟩) can be seen in the measured

spectroscopy, as is shown in figure 5.7(a). Figure 5.7(b) shows the numerical sim-

ulation, and comparison with measured data results in estimated effective coupler

temperature of ∼ 80 mK. The thermal excitation in the coupler changes the detuning

1 − 2 and leaves behind a fraction of ∼ 10 % qubit population during the iSWAP

operation, as can be seen in figure 5.6(b). We are investigating better thermalization

techniques, post selection methods and quantum optimal control schemes to further

optimize the iSWAP gate.

84

Chapter 6

Conclusion and Outlook

In this thesis, we develop two applications of tunable dispersive coupling devices in

cQED. The tunability arises from the quantum interference of interactions between

states and is used to suppress photon shot noise dephasing in a single qubit and

static ZZ crosstalk between two qubits. Both applications provide promising building

blocks for cQED systems. Through the proper engineering of dispersive interaction,

we demonstrate that tunable coupling not only provides extra control knobs for cQED

devices but also can be exploited to mitigate unwanted interactions.

Photon shot noise dephasing has been identified as one of the limiting factors for

coherence time of state-of-the-art cQED devices [79, 152, 158], and research efforts

have been made to improve coherence time through better thermalization [154, 155,

159] and using high-𝑄 cavities [196]. Our device in chapter 4 is compatible with

those approaches and robust against typical level of thermal photon noise. It can also

potentially relax the requirement for high-𝑄 cavities to enable fast qubit readout,

because larger 𝜅 leads to smaller Γ𝜙 in the small 𝜒 regime according to (4.5). The

main dephasing source is flux noise, which causes fluctuations in qubit frequency and

deviation of 𝜒− from zero. As our readout scheme does not involve dynamic flux

control, the flux bias lines can be heavily filtered to reduce noise.

85

CHAPTER 6. CONCLUSION AND OUTLOOK

The main limitation of the TCQ device is the requirement of being in the strad-

dling regime, and due to the weak anharmonicity in transmon this is a strong con-

straint. The small detuning between qubit and cavity also leads to Purcell-limited

𝑇1. By adopting other types of superconducting qubits with stronger anharmonicity,

such as flux qubit [160] and fluxonium [197], we will be able operate at larger detuning

and optimize device performance in a more flexible way (e.g., incorporating on-chip

Purcell filters [173, 174]).

The requirement of two flux biases adds to the complexity of design and control

when integrating the TCQ to larger systems. A simpler “half TCQ” design, where one

of the two SQUID loops is replaced by a single junction, will make the device more

scalable and easier to control. In this case we no longer have independent tunability

of the qubit frequency and coupling rate, but zero 𝜒− can still be achieved if the fixed

frequency transmon satisfies 𝜔1 − 𝐽 ≲ 𝜔𝑟 − 𝛼. When 𝜔2(Φ) is tuned near resonance

with 𝜔1, the device will be in straddling regime with small 𝑔−. The control signals

reduce to one dc flux bias and two microwave drives, which is comparable to other

tunable coupling schemes [101, 190] and no dynamic flux control is needed in the

operation.

The potential of long coherence time makes TCQ a promising device for quantum

memory, and the ability to tune 𝜒 could be used for systematic and quantitative

studies on the properties of photon shot noise (e.g., using the TCQ as a probe for the

noise spectrum [156]).

Static ZZ crosstalk in state-of-the-art devices is below 100 kHz and is not the

limiting factor for current device performance [60, 182]. This matches our result in

chapter 5, where individual and simultaneous RB show similar gate fidelities. Mitigat-

ing ZZ crosstalk will become more important as qubit coherence time keeps improving,

and especially when logical qubits contain XX-type stabilizers. Moreover, zero ZZ

86

CHAPTER 6. CONCLUSION AND OUTLOOK

coupling can also be realized in a singled coupler device, as is shown in appendix B.

Direct qubit-qubit coupling can also be utilized to destructively interfere with indirect

qubit-coupler-qubit coupling to achieve zero crosstalk, as is proposed in [181]. Those

approaches will make the device more compact and easier to be integrated into larger

systems.

The iSWAP gate for the tunable-𝜁 device in chapter 5 needs to be improved and

we have identified thermal population in the coupler qubit as the main source of gate

error. On the hardware level, better attenuators [155] and/or on-chip filters could

be used to better thermalize the device. On the software level, quantum optimal

control [198] can be used to extend parametric flux modulation to more complicated

pulse schemes to improve gate fidelity. Mølmer-Sørensen gate [199] for trapped ions

is insensitive to thermal phonon population in the vibrational modes, and physical

insights into the mechanism might be used for the design of a better iSWAP gate.

Finally, the ability to tune 𝜁/2𝜋 from less than 100 kHz to a few MHz can be used to

realize a controlled-Phase gate. After a proper two-qubit gate has been implemented,

two-qubit RB [200] can be performed to extract the average gate fidelity.

The two experiments in this thesis demonstrate that tunable coupling based on

quantum superposition and interference is a useful resource to improve device per-

formance in cQED. We hope these results and insights will inspire more innovative

research in the future.

87

Appendix A

Fabrication Recipes

A.1 Photolithography

C-plane (0001) sapphire substrates with 500 μm thickness are purchased from CrysTec

GmbH, and 200 nm of niobium is sputtered onto the substrates by STAR Cryoelec-

tronics.

Chip cleaning

• Soak in Baker PRS-1000 for 1 hour.

• Sonicate in Baker PRS-1000 for 1 minute.

• Sonicate in methanol for 1 minute.

• Sonicate in isopropanol (IPA) for 1 minute.

• Rinse with deionized (DI) water and dry with nitrogen.

Spin coating photoresist

• Shipley S1811 photoresist is coated at spinning rate 5500 revolution per minute

(rpm) for 60 seconds. Ramping rate of 10000 rpm/s helps reduce edge bead.88

APPENDIX A. FABRICATION RECIPES

• Bake at 105 for 1 minute on hotplate.

Exposure

The pattern is directly written onto the chip using Heidelberg DWL 66+. Its 10/5/2

mm write head provides 2.5/1.0/0.6 μm resolution.

• 5 mm write head: Optical focus −76, filter 10 %, intensity 85 %.

• 2 mm write head: Optical focus −50, filter 25 % + 10 %, intensity 50 %.

Development

• Use Shipley MF 319 developer and develop for 1 minute.

• Rinse with DI water and dry with nitrogen.

A.2 Niobium etching

South Bay PE2000 plasma etcher. Ar 400 sccm, SF6 200 sccm, rf power 50 W.

Etching rate is 0.5 nm/s and etching time is 8 minutes.

A.3 Electron beam lithography

E-beam lithography is performed using Elionix ELS-125.

Spin coating e-beam resist

• Spin MMA EL13 at 5000 rpm (ramp rate 500 rpm/s) for 1 minute.

• Bake at 175 for 2 minutes on hotplate.

• Spin PMMA A3 at 4000 rpm (ramp rate 500 rpm/s) for 1 minute.89


• Bake at 175 for 30 minutes on hotplate.

Anti-charging evaporation

40 nm of aluminum is evaporated on the PMMA/MMA resist stack to prevent electron

charge accumulation due to the insulating nature of the sapphire substrate.

Exposure

• Doses: Dwell time 1 μs/dot for PMMA (peak dwell time ∼ 1.5 μs/dot after

proximity error correction) and 0.32 μs/dot for MMA undercut.

• Parameters: Beam current 1 nA, aperture 60 μm, write field 500 μm, dots per

field 50000.

Development

• Remove the aluminum anti-charging layer by placing the chip in Shipley MF319

developer for 5 minutes.


• Develop in 1:3 methyl isobutyl ketone (MIBK):IPA for 50 seconds.

• Dip into IPA for 10 seconds.


A.4 Aluminum evaporation

Descum

An argon ion etch serves as a descum process before the evaporation.90


• Parameters: Acceleration voltage −50 V, anode voltage 150 V, emitter current

130 mA. Anode voltage can be increased to as high as 400 V for more aggressive

descum. Etching time is 30 seconds for each evaporation angle.

Evaporation

• Fill cold trap with liquid nitrogen.

• Evaporate titanium at rate 0.3 nm/s for 1 minute and wait for 2.5 minutes to

reduce chamber pressure to ∼ 2 × 10−8 mbar.

• Evaporate 30 nm of aluminum at rate 0.5 nm/s with tilt angle 40° and rotation

angle 0°.

• Oxidize in 15 %/85 % O2/Ar mixture at 10 mbar for 1 ∼ 10 minutes, depending

on the targeted Josephson energy.

• Evaporate 100 nm of aluminum at rate 0.5 nm/s with tilt angle 40° and rotation

angle 90°.

• Oxidize for 10 minutes to passivate the aluminum surface.

Liftoff

• Soak in acetone at 75 for 1 hour.

• Sonicate in acetone for 10 seconds.

• Dip into IPA for 1 minute.

• Dry with nitrogen.

91

Appendix B

Derivation for ZZ Coupling Rate

B.1 Hamiltonian

We start from the Hamiltonian (5.3),

𝐻 = 𝐻0 + 𝑉

= ∑𝑖=1,2,±

ℏ(𝜔𝑖𝑎†𝑖 𝑎𝑖 − 𝛼𝑖

2 𝑎†𝑖 𝑎†

𝑖 𝑎𝑖𝑎𝑖) + ∑𝑖=1,2𝑗=±

ℏ𝑔𝑖𝑗(𝑎†𝑖 𝑎𝑗 + 𝑎𝑖𝑎†

𝑗). (B.1)

and denote the eigenstates and eigenenergies by |𝑛1𝑛2𝑛+𝑛−⟩ and 𝐸𝑛1𝑛2𝑛+𝑛−. The

detunings Δ𝑖𝑗 are the differences between unperturbed, single-excitation energy levels,

e.g., ℏΔ1+ = 𝐸(0)1000 − 𝐸(0)

0010, etc. The ZZ coupling rate 𝜁 between qubit 1 and 2

(assuming the couplers are in ground state) is given by

ℏ𝜁 = 𝐸1100 − 𝐸1000 − 𝐸0100 (𝐸0000 = 0 for all orders). (B.2)

The sign convention of 𝑔𝑖𝑗 needs to be consistent for the bus cavity and the coupler

qubit. As the two computation qubits are coupled to the half-wavelength bus cavity

at either ends [See figure 5.2(b)], 𝑔1+ and 𝑔2+ have opposite signs if we define the

92

APPENDIX B. DERIVATION FOR ZZ COUPLING RATE

qubit phases in the natural way. In this convention 𝑔1− and 𝑔2− have opposite signs

as well. By redefining 𝜎±2 → −𝜎±

2 , we have the same sign for all 𝑔𝑖𝑗 in the calculation

(See table B.4).

B.2 Second order terms

Second order corrections are given by Eq. (11) in [201],

𝐸(2)n = ∑

n′

′ 0⟨n|𝑉 |n′⟩0 0⟨n′|𝑉 |n⟩0𝐸(0)

n − 𝐸(0)n′

, (B.3)

where n labels the energy levels and the primed sum is over n′ ≠ n terms.

Table B.1: Second order terms for ZZ coupling.

n n′ Contribution

|1000⟩ |0010⟩ 𝑔1+𝑔1+Δ1+

|0001⟩ 𝑔1−𝑔1−Δ1−

|0100⟩ |0010⟩ 𝑔2+𝑔2+Δ2+

|0001⟩ 𝑔2−𝑔2−Δ2−

|1100⟩ |0110⟩ 𝑔1+𝑔1+Δ1+

|0101⟩ 𝑔1−𝑔1−Δ1−

|1010⟩ 𝑔2+𝑔2+Δ2+

|1001⟩ 𝑔2−𝑔2−Δ2−

From table B.1 we find 𝜁(2) = 𝐸(2)1100 − 𝐸(2)

1000 − 𝐸(2)0100 = 0, so we need to go to

fourth order to calculate 𝜁.

93


B.3 Fourth order terms

Fourth order corrections are given by Eq. (18) in [201],

𝐸(4)n = ∑

m,p,q

′ 𝑉nm𝑉mp𝑉pq𝑉qn𝐸nm𝐸np𝐸nq

− ∑m

′ |𝑉nm|2𝐸2

nm∑

p

′ |𝑉np|2𝐸np

, (B.4)

where 𝐸nm = 𝐸(0)n − 𝐸(0)

m and 𝑉nm = 0⟨n|𝑉 |m⟩0. The two types of terms in (B.4)

are denoted by (I) and (II) and listed in table B.2 and table B.3.

Table B.2: Type (I) fourth order terms for ZZ coupling.

n m p q Contribution

|1000⟩ |0010⟩ |0100⟩ |0010⟩ 𝑔1+𝑔2+𝑔2+𝑔1+Δ1+Δ12Δ1+

|0010⟩ |0100⟩ |0001⟩ 𝑔1+𝑔2+𝑔2−𝑔1−Δ1+Δ12Δ1−

|0001⟩ |0100⟩ |0010⟩ 𝑔1−𝑔2−𝑔2+𝑔1+Δ1−Δ12Δ1+

|0001⟩ |0100⟩ |0001⟩ 𝑔1−𝑔2−𝑔2−𝑔1−Δ1−Δ12Δ1−

|0100⟩ |0010⟩ |1000⟩ |0010⟩ 𝑔2+𝑔1+𝑔1+𝑔2+Δ2+Δ21Δ2+

|0010⟩ |1000⟩ |0001⟩ 𝑔2+𝑔1+𝑔1−𝑔2−Δ2+Δ21Δ2−

|0001⟩ |1000⟩ |0010⟩ 𝑔2−𝑔1−𝑔1+𝑔2+Δ2−Δ21Δ2+

|0001⟩ |1000⟩ |0001⟩ 𝑔2−𝑔1−𝑔1−𝑔2−Δ2−Δ21Δ2−

|1100⟩ |0110⟩ |0020⟩ |1010⟩ 2𝑔1+𝑔2+𝑔1+𝑔2+Δ1+(Δ1+ + Δ2+ + 𝛼+)Δ2+

|0110⟩ |0020⟩ |0110⟩ 2𝑔1+𝑔2+𝑔2+𝑔1+Δ1+(Δ1+ + Δ2+ + 𝛼+)Δ1+

|0110⟩ |0011⟩ |1001⟩ 𝑔1+𝑔2−𝑔1+𝑔2−Δ1+(Δ1+ + Δ2−)Δ2−

|0110⟩ |0011⟩ |0101⟩ 𝑔1+𝑔2−𝑔2+𝑔1−Δ1+(Δ1+ + Δ2−)Δ1−

(Continued on next page)94


Table B.2: (continued)


|0110⟩ |0011⟩ |1010⟩ 𝑔1+𝑔2−𝑔1−𝑔2+Δ1+(Δ1+ + Δ2−)Δ2+

|0110⟩ |0011⟩ |0110⟩ 𝑔1+𝑔2−𝑔2−𝑔1+Δ1+(Δ1+ + Δ2−)Δ1+

|0110⟩ |0200⟩ |0110⟩ 2𝑔1+𝑔2+𝑔2+𝑔1+Δ1+(Δ12 + 𝛼2)Δ1+

|0110⟩ |0200⟩ |0101⟩ 2𝑔1+𝑔2+𝑔2−𝑔1−Δ1+(Δ12 + 𝛼2)Δ1−

|0101⟩ |0011⟩ |1001⟩ 𝑔1−𝑔2+𝑔1+𝑔2−Δ1−(Δ1+ + Δ2−)Δ2−

|0101⟩ |0011⟩ |0101⟩ 𝑔1−𝑔2+𝑔2+𝑔1−Δ1−(Δ1+ + Δ2−)Δ1−

|0101⟩ |0011⟩ |1010⟩ 𝑔1−𝑔2+𝑔1−𝑔2+Δ1−(Δ1+ + Δ2−)Δ2+

|0101⟩ |0011⟩ |0110⟩ 𝑔1−𝑔2+𝑔2−𝑔1+Δ1−(Δ1+ + Δ2−)Δ1+

|0101⟩ |0002⟩ |1001⟩ 2𝑔1−𝑔2−𝑔1−𝑔2−Δ1−(Δ1− + Δ2− + 𝛼−)Δ2−

|0101⟩ |0002⟩ |0101⟩ 2𝑔1−𝑔2−𝑔2−𝑔1−Δ1−(Δ1− + Δ2− + 𝛼−)Δ1−

|0101⟩ |0200⟩ |0110⟩ 2𝑔1−𝑔2−𝑔2+𝑔1+Δ1−(Δ12 + 𝛼2)Δ1+

|0101⟩ |0200⟩ |0101⟩ 2𝑔1−𝑔2−𝑔2−𝑔1−Δ1−(Δ12 + 𝛼2)Δ1−

|1010⟩ |0020⟩ |1010⟩ 2𝑔2+𝑔1+𝑔1+𝑔2+Δ2+(Δ1+ + Δ2+ + 𝛼+)Δ2+

|1010⟩ |0020⟩ |0110⟩ 2𝑔2+𝑔1+𝑔2+𝑔1+Δ2+(Δ1+ + Δ2+ + 𝛼+)Δ1+

|1010⟩ |0011⟩ |1001⟩ 𝑔2+𝑔1−𝑔1+𝑔2−Δ2+(Δ1+ + Δ2−)Δ2−

|1010⟩ |0011⟩ |0101⟩ 𝑔2+𝑔1−𝑔2+𝑔1−Δ2+(Δ1+ + Δ2−)Δ1−

|1010⟩ |0011⟩ |1010⟩ 𝑔2+𝑔1−𝑔1−𝑔2+Δ2+(Δ1+ + Δ2−)Δ2+

|1010⟩ |0011⟩ |0110⟩ 𝑔2+𝑔1−𝑔2−𝑔1+Δ2+(Δ1+ + Δ2−)Δ1+

(Continued on next page)

95




|1010⟩ |2000⟩ |1010⟩ 2𝑔2+𝑔1+𝑔1+𝑔2+Δ2+(Δ21 + 𝛼1)Δ2+

|1010⟩ |2000⟩ |1001⟩ 2𝑔2+𝑔1+𝑔1−𝑔2−Δ2+(Δ21 + 𝛼1)Δ2−

|1001⟩ |0011⟩ |1001⟩ 𝑔2−𝑔1+𝑔1+𝑔2−Δ2−(Δ1+ + Δ2−)Δ2−

|1001⟩ |0011⟩ |0101⟩ 𝑔2−𝑔1+𝑔2+𝑔1−Δ2−(Δ1+ + Δ2−)Δ1−

|1001⟩ |0011⟩ |1010⟩ 𝑔2−𝑔1+𝑔1−𝑔2+Δ2−(Δ1+ + Δ2−)Δ2+

|1001⟩ |0011⟩ |0110⟩ 𝑔2−𝑔1+𝑔2−𝑔1+Δ2−(Δ1+ + Δ2−)Δ1+

|1001⟩ |0002⟩ |1001⟩ 2𝑔2−𝑔1−𝑔1−𝑔2−Δ2−(Δ1− + Δ2− + 𝛼−)Δ2−

|1001⟩ |0002⟩ |0101⟩ 2𝑔2−𝑔1−𝑔2−𝑔1−Δ2−(Δ1− + Δ2− + 𝛼−)Δ1−

|1001⟩ |2000⟩ |1010⟩ 2𝑔2−𝑔1−𝑔1+𝑔2+Δ2−(Δ21 + 𝛼1)Δ2+

|1001⟩ |2000⟩ |1001⟩ 2𝑔2−𝑔1−𝑔1−𝑔2−Δ2−(Δ21 + 𝛼1)Δ2−

Table B.3: Type (II) fourth order terms for ZZ coupling.

n m p Contribution

|1000⟩ |0010⟩ |0010⟩ − 𝑔1+𝑔1+Δ2

1+Δ1+

|0010⟩ |0001⟩ − 𝑔1+𝑔1−Δ2

1+Δ1−

|0001⟩ |0010⟩ − 𝑔1−𝑔1+Δ2

1−Δ1+

|0001⟩ |0001⟩ − 𝑔1−𝑔1−Δ2

1−Δ1−

|0100⟩ |0010⟩ |0010⟩ − 𝑔2+𝑔2+Δ2

2+Δ2+

|0010⟩ |0001⟩ − 𝑔2+𝑔2−Δ2

2+Δ2−

(Continued on next page)96



n m p Contribution

|0001⟩ |0010⟩ − 𝑔2−𝑔2+Δ2

2−Δ2+

|0001⟩ |0001⟩ − 𝑔2−𝑔2−Δ2

2−Δ2−

|1100⟩ |0110⟩ |0110⟩ − 𝑔1+𝑔1+Δ2

1+Δ1+

|0110⟩ |0101⟩ − 𝑔1+𝑔1−Δ2

1+Δ1−

|0110⟩ |1010⟩ − 𝑔1+𝑔2+Δ2

1+Δ2+

|0110⟩ |1001⟩ − 𝑔1+𝑔2−Δ2

1+Δ2−

|0101⟩ |0110⟩ − 𝑔1−𝑔1+Δ2

1−Δ1+

|0101⟩ |0101⟩ − 𝑔1−𝑔1−Δ2

1−Δ1−

|0101⟩ |1010⟩ − 𝑔1−𝑔2+Δ2

1−Δ2+

|0101⟩ |1001⟩ − 𝑔1−𝑔2−Δ2

1−Δ2−

|1010⟩ |0110⟩ − 𝑔2+𝑔1+Δ2

2+Δ1+

|1010⟩ |0101⟩ − 𝑔2+𝑔1−Δ2

2+Δ1−

|1010⟩ |1010⟩ − 𝑔2+𝑔2+Δ2

2+Δ2+

|1010⟩ |1001⟩ − 𝑔2+𝑔2−Δ2

2+Δ2−

|1001⟩ |0110⟩ − 𝑔2−𝑔1+Δ2

2−Δ1+

|1001⟩ |0101⟩ − 𝑔2−𝑔1−Δ2

2−Δ1−

|1001⟩ |1010⟩ − 𝑔2−𝑔2+Δ2

2−Δ2+

|1001⟩ |1001⟩ − 𝑔2−𝑔2−Δ2

2−Δ2−

97


B.4 Final result

The final result for 𝜁 is

𝜁 = 𝐸(4)1100 − 𝐸(4)

1000 − 𝐸(4)0100

= 2𝑔21+𝑔2

2+Δ1+ + Δ2+ + 𝛼+

( 1Δ1+

+ 1Δ2+

)2

+ 2𝑔21−𝑔2

2−Δ1− + Δ2− + 𝛼−

( 1Δ1−

+ 1Δ2−

)2

+ (𝑔1+𝑔2+Δ1+

+ 𝑔1−𝑔2−Δ1−

)2

( 2Δ12 + 𝛼2

− 1Δ12

)

+ (𝑔1+𝑔2+Δ2+

+ 𝑔1−𝑔2−Δ2−

)2

( 2Δ21 + 𝛼1

− 1Δ21

)

+ [𝑔1+𝑔2− ( 1Δ1+

+ 1Δ2−

) + 𝑔1−𝑔2+ ( 1Δ1−

+ 1Δ2+

)]2 1

Δ1+ + Δ2−

− ( 𝑔21+

Δ21+

+ 𝑔21−

Δ21−

) ( 𝑔22+

Δ2++ 𝑔2

2−Δ2−

) − ( 𝑔22+

Δ22+

+ 𝑔22−

Δ22−

) ( 𝑔21+

Δ1++ 𝑔2

1−Δ1−

) .

(B.5)

To verify this result we compare it to two special cases in the literature.

1. Two transmons coupled to one cavity.

This corresponds to 𝛼+ = 0, 𝑔1− = 𝑔2− = 0, so (B.5) reduces to (We omit the

subscript “+” as there is only one coupler)

𝜁 = 2𝑔21𝑔2

2Δ1 + Δ2

( 1Δ1

+ 1Δ2

)2

+ (𝑔1𝑔2Δ1

)2

( 2Δ12 + 𝛼2

− 1Δ12

)

+ (𝑔1𝑔2Δ2

)2

( 2Δ21 + 𝛼1

− 1Δ21

) − 𝑔21

Δ21

𝑔22

Δ2− 𝑔2

2Δ2

2

𝑔21

Δ1

= 2𝑔21𝑔2

2 [ 1(Δ12 + 𝛼2)Δ2

1+ 1

(Δ21 + 𝛼1)Δ22

+ 1Δ1Δ2

2+ 1

Δ21Δ2

] ,

(B.6)

and reproduces the result in [180] (where 𝜁 is defined as 𝜔10 +𝜔01 −𝜔11 so there

is a sign difference).

2. Two cavities coupled to a two-level qubit.98


This corresponds to 𝛼+ → ∞, 𝛼1 = 𝛼2 = 𝑔1− = 𝑔2− = 0, so (B.5) reduces to

𝜁 = (𝑔1𝑔2Δ1

)2 1

Δ12+ (𝑔1𝑔2

Δ2)

2 1Δ21

− 𝑔21

Δ21

𝑔22

Δ2− 𝑔2

2Δ2

2

𝑔21

Δ1

= −2𝑔21𝑔2

2Δ1 + Δ2

Δ21Δ2

2,

(B.7)

and reproduces Eq. (3.30) in [202] (where the detunings are defined as Δ1(2) =𝜔𝑞 − 𝜔1(2) so there is a sign difference).

To show the possibility of zero ZZ interaction, we calculate 𝜁 for different param-

eter configurations using (B.5), and the results are shown in blue curves figure B.1.

The red curves are numerical results from diagonalizing (B.1) and calculating 𝜁 using

(B.2). The parameter configurations for each plot are listed in table B.4.

−1.5

−1.0

−0.5

0.0

0.5

1.0(a)

0.2

0.0

0.2

0.4

0.6(b)

Numerical diagonalizationPerturbation theory

0.60 0.65 0.70 0.75 0.80 0.85 0.90−30−25−20−15−10−5

05

(c)

−2.0 −1.8 −1.6 −1.4 −1.2 −1.0−0.8

−0.6

−0.4

−0.2

0.0(d)

(!− − !2)/2ı (GHz)

“/2ı

(MHz

)

Figure B.1: ZZ coupling rate calculated from perturbation theory (blue) and numer-ical diagonalization (red). The parameters are listed in table B.4.

99


Table B.4: Parameter configurations for ZZ coupling rate calculation in figure B.1.

Figure Configuration Parameters (2𝜋⋅MHz)(a) Qubits far apart, 𝛼1 = 𝛼2 = 350, 𝛼− = 750,

one coupler in between. Δ12 = 1500, 𝑔1− = 𝑔2− = 140.(b) Qubits in straddling regime, 𝛼1 = 𝛼2 = 350, 𝛼− = 750, 𝛼+ = 0,

one coupler above, Δ12 = 250, Δ+2 = 1800,one coupler below. 𝑔1+ = 𝑔2+ = 160, 𝑔1− = 𝑔2− = 140.

(c) Qubits in straddling regime, 𝛼1 = 𝛼2 = 350, 𝛼+ = 750,one coupler above. Δ12 = 250, 𝑔1+ = 𝑔2+ = 120.

(d) Qubits out of straddling regime, 𝛼1 = 𝛼2 = 350, 𝛼− = 750, 𝛼+ = 0,one coupler above, Δ12 = 450, Δ+2 = 1800,one coupler below. 𝑔1+ = 𝑔2+ = 160, 𝑔1− = 𝑔2− = 140.

From figure B.1 we find that there are several configurations that result in zero

𝜁. We choose the configuration in (b) because the two qubits are close to each

other in frequency and have stronger interaction strength compared to (a), which can

potentially lead to fast two qubit gates in addition to zero 𝜁. In configuration (c)

zero 𝜁 happens at relatively small detuning Δ−2/2𝜋 = 634 MHz, which increases the

susceptibility of the qubits to flux noise in the coupler.

100

Appendix C

Single-qubit Clifford Gates

The single-qubit Clifford gates and their primary gate decomposition used for RB

experiments in section 5.3 are listed in table C.1.

Table C.1: Single-qubit Clifford gates.

Name Unitary matrix Primary gatedecomposition

C1 ( 1 00 1 ) 𝟙

C2 exp (−𝑖𝜋4 𝜎𝑥) =

√2

2 ( 1 −𝑖−𝑖 1 ) 𝑋𝜋/2

C3 exp (−𝑖𝜋2 𝜎𝑥) = ( 0 −𝑖

−𝑖 0 ) 𝑋𝜋C4 exp (−𝑖3𝜋

4 𝜎𝑥) =√

22 ( −1 −𝑖

−𝑖 −1 ) 𝑋−𝜋/2C5 exp (−𝑖𝜋

4 𝜎𝑦) =√

22 ( 1 −1

1 1 ) 𝑌𝜋/2C6 exp (−𝑖𝜋

2 𝜎𝑦) = ( 0 −11 0 ) 𝑌𝜋

C7 exp (−𝑖3𝜋4 𝜎𝑦) =

√2

2 ( −1 −11 −1 ) 𝑌−𝜋/2

C8 exp (−𝑖𝜋4 𝜎𝑧) =

√2

2 ( 1−𝑖 00 1+𝑖 )

𝑋𝜋/2𝑌𝜋/2𝑋−𝜋/2𝑋−𝜋/2𝑌−𝜋/2𝑋𝜋/2𝑌𝜋/2𝑋−𝜋/2𝑌−𝜋/2𝑌−𝜋/2𝑋𝜋/2𝑌𝜋/2

C9 exp (−𝑖𝜋2 𝜎𝑧) = ( −𝑖 0

0 𝑖 ) 𝑋𝜋𝑌𝜋𝑌𝜋𝑋𝜋

(Continued on next page)

101

APPENDIX C. SINGLE-QUBIT CLIFFORD GATES

Table C.1: (continued)

Name Unitary matrix Primary gatedecomposition

C10 exp (−𝑖3𝜋4 𝜎𝑧) =

√2

2 ( −1−𝑖 00 −1+𝑖 )

𝑋𝜋/2𝑌−𝜋/2𝑋−𝜋/2𝑋−𝜋/2𝑌𝜋/2𝑋𝜋/2𝑌𝜋/2𝑋𝜋/2𝑌−𝜋/2𝑌−𝜋/2𝑋−𝜋/2𝑌𝜋/2

C11 exp (−𝑖 𝜋2

√2(𝜎𝑥 + 𝜎𝑦)) =

√2

2 ( 0 −1−𝑖1−𝑖 0 )

𝑋𝜋/2𝑌𝜋/2𝑋𝜋/2𝑋−𝜋/2𝑌−𝜋/2𝑋−𝜋/2

𝑌𝜋/2𝑋𝜋/2𝑌𝜋/2𝑌−𝜋/2𝑋−𝜋/2𝑌−𝜋/2


√2(𝜎𝑥 − 𝜎𝑦)) =

√2

2 ( 0 1−𝑖−1−𝑖 0 )

𝑋𝜋/2𝑌−𝜋/2𝑋𝜋/2𝑋−𝜋/2𝑌𝜋/2𝑋−𝜋/2𝑌𝜋/2𝑋−𝜋/2𝑌𝜋/2𝑌−𝜋/2𝑋𝜋/2𝑌−𝜋/2


√2(𝜎𝑥 + 𝜎𝑧)) =

√2

2 ( −𝑖 −𝑖−𝑖 𝑖 ) 𝑌−𝜋/2𝑋𝜋

𝑋𝜋𝑌𝜋/2


√2(𝜎𝑥 − 𝜎𝑧)) =

√2

2 ( 𝑖 −𝑖−𝑖 −𝑖 ) 𝑌𝜋/2𝑋𝜋

𝑋𝜋𝑌−𝜋/2


√2(𝜎𝑦 + 𝜎𝑧)) =

√2

2 ( −𝑖 −11 𝑖 ) 𝑋𝜋/2𝑌𝜋

𝑌𝜋𝑋−𝜋/2


√2(𝜎𝑦 − 𝜎𝑧)) =

√2

2 ( 𝑖 −11 −𝑖 ) 𝑋−𝜋/2𝑌𝜋

𝑌𝜋𝑋𝜋/2


√3(𝜎𝑥 + 𝜎𝑦 + 𝜎𝑧)) = 1

2 ( 1−𝑖 −1−𝑖1−𝑖 1+𝑖 ) 𝑋𝜋/2𝑌𝜋/2

C18 exp (−𝑖 2𝜋3

√3(𝜎𝑥 + 𝜎𝑦 + 𝜎𝑧)) = 1

2 ( −1−𝑖 −1−𝑖1−𝑖 −1+𝑖 ) 𝑌−𝜋/2𝑋−𝜋/2


√3(𝜎𝑥 − 𝜎𝑦 + 𝜎𝑧)) = 1

2 ( 1−𝑖 1−𝑖−1−𝑖 1+𝑖 ) 𝑌−𝜋/2𝑋𝜋/2


√3(𝜎𝑥 − 𝜎𝑦 + 𝜎𝑧)) = 1

2 ( −1−𝑖 1−𝑖−1−𝑖 −1+𝑖 ) 𝑋−𝜋/2𝑌𝜋/2


√3(𝜎𝑥 + 𝜎𝑦 − 𝜎𝑧)) = 1

2 ( 1+𝑖 −1−𝑖1−𝑖 1−𝑖 ) 𝑌𝜋/2𝑋𝜋/2


√3(𝜎𝑥 + 𝜎𝑦 − 𝜎𝑧)) = 1

2 ( −1+𝑖 −1−𝑖1−𝑖 −1−𝑖 ) 𝑋−𝜋/2𝑌−𝜋/2


√3(−𝜎𝑥 + 𝜎𝑦 + 𝜎𝑧)) = 1

2 ( 1−𝑖 −1+𝑖1+𝑖 1+𝑖 ) 𝑌𝜋/2𝑋−𝜋/2


√3(−𝜎𝑥 + 𝜎𝑦 + 𝜎𝑧)) = 1

2 ( −1−𝑖 −1+𝑖1+𝑖 −1+𝑖 ) 𝑋𝜋/2𝑌−𝜋/2

102

APPENDIX C. SINGLE-QUBIT CLIFFORD GATES

A few comments are worth mentioning for an RB experiment.

1. The primary gates are chosen as 𝟙, 𝑋±𝜋/2, 𝑌±𝜋/2, 𝑋𝜋, 𝑌𝜋 and 𝑋𝜋 (𝑌𝜋) can

be replaced randomly by 𝑋−𝜋 (𝑌−𝜋) in a sequence.

2. The primary decomposition for a Clifford gate might differ from its unitary

matrix by a factor of −1. This does not affect the measured result as it is a

global phase.

3. For each Clifford gate there might be multiple primary gate decompositions and

all possible decompositions with the shortest length are listed above. In an RB

experiment they can be interchanged randomly.

103

Appendix D

Publications and Presentations

• G. Zhang, Y. Liu, J. J. Raftery, and A. A. Houck. Suppression of photon shot

noise dephasing in a tunable coupling superconducting qubit. npj Quantum Inf.

3, 1 (2017).

• J. Raftery, A. Vrajitoarea, G. Zhang, Z. Leng, S. J. Srinivasan, and A. A.

Houck. Direct digital synthesis of microwave waveforms for quantum comput-

ing. arXiv:1703.00942 [quant-ph] (2017).

• G. Zhang, S. Srinivasan, Y. Liu, and A. A. Houck. Tuning the dispersive

coupling and suppressing measurement-induced dephasing in a superconducting

qubit. APS March Meeting, Denver, Colorado, 2014.

• G. Zhang, Y. Liu, J. J. Raftery and A. A. Houck. Suppression of photon

shot noise dephasing in a tunable coupling superconducting qubit. APS March

Meeting, Baltimore, Maryland, 2016.

• G. Zhang and A. A Houck. Parametric coupling and suppression of crosstalk

between two superconducting transmon qubits. APS March Meeting, New Or-

leans, Louisiana, 2017.

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