Efficient and Low-Backaction Quantum Measurement Usinga Chip-Scale Detector
byEric I. Rosenthal
B.A., University of Pennsylvania, 2015
M.S., University of Pennsylvania, 2015

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2021
Committee Members:
Konrad W. Lehnert, Chair
Graeme Smith
John D. Teufel
Adam Kaufman
Joshua Combes

iiRosenthal, Eric I. (Ph.D., Physics)
Efficient and Low-Backaction Quantum Measurement Using a Chip-Scale Detector
Thesis directed by Prof. Konrad W. Lehnert

Superconducting qubits are a leading platform for scalable quantum computing and quantum
error correction. One feature of this platform is the ability to perform projective measurements
orders of magnitude more quickly than qubit decoherence times. Such measurements are enabled by
the use of quantum-limited parametric amplifiers in conjunction with ferrite circulators — magnetic
devices which provide isolation from noise and decoherence due to amplifier backaction. Because
these nonreciprocal elements have limited performance and are not easily integrated on chip, it
has been a long-standing goal to replace them with a scalable alternative. Here, we demonstratea solution to this problem by using a superconducting switch to control the coupling between aqubit and amplifier. Doing so, we measure a transmon qubit using a single, chip-scale device toprovide both parametric amplification and isolation from the bulk of amplifier backaction. Thismeasurement is also fast, high fidelity, and has 70% efficiency, comparable to the best that has beenreported in any superconducting qubit measurement. As such, this work constitutes a high-quality
platform for the scalable measurement of superconducting qubits.

iii

Dedication

To my parents, brother, and Mia

iv

Acknowledgements

I’m grateful to everyone who has supported and helped me over the course of my progression
into a scientist. I can’t express how lucky I am to be surrounded by so many thoughtful, encouraging
and brilliant people. A thorough acknowledgement would be longer than this dissertation, but I
want to specifically thank a few of you here.
First to my advisor Konrad: thank you for the dedication you took to advising me. Words
can’t express how grateful I am, both for the positive experience and for everything you taught me.
Doing it over again I wouldn’t hesitate to work for you.
To my undergraduate mentors including Matt LaHaye, Riichiro Saito and Alison Sweeney:
thank you for the opportunity to get started, for your patience, and for believing in me.
To my colleagues in the Lehnert lab: thank you for creating such a supportive environment.
I’m especially grateful to Ben Chapman, both for mentoring me and for being a positive role model,
both as a scientist and an individual. To Christian Schneider, working with you was an absolute
pleasure, thank you. To Ziyi Zhao, I look forward to seeing how you apply your creativity andbrilliance to the project going forward.
To my friends, fellow graduate students, all of the JILA staff, and the Boulder physics
community: I couldn’t have done it without you. No matter what, I could always count on yourhelp and camaraderie.
Finally to my family: thank you. To my parents and brother Greg, I appreciate that you
have always being there for me near or far. To my partner, Mia Froehling Gallier, thank you forthe love and support over the years.

Contents

Chapter
1 Introduction1

1.1

Motivation for quantum computing . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

The need for high quality measurement . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3

Quantum information with superconducting circuits . . . . . . . . . . . . . . . . . .

3

1.4

Introduction to superconducting qubit measurement . . . . . . . . . . . . . . . . . .

4

1.4.1

Introduction to quantum limited amplification . . . . . . . . . . . . . . . . .

6

1.4.2

Isolating qubits from measurement backaction . . . . . . . . . . . . . . . . . .

7

1.4.3

Measurement hardware areas for improvement . . . . . . . . . . . . . . . . .

9

1.5Comparison of this thesis to related work . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6

Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Oscillators
2.1

14Classical oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1

LC-oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.2

LC-oscillator with damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.3

LC-oscillator with damping by an external load . . . . . . . . . . . . . . . . . 17

2.1.4

LC-oscillator with parametric gain . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.5

LC-oscillator with nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.6

Coupled oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

vi
2.1.7
2.2

2.3

2.4

Parametrically coupled oscillators . . . . . . . . . . . . . . . . . . . . . . . . . 24

Quantum oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.1

Quantization of the electromagnetic field

2.2.2

A quantum simple harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . 25

2.2.3

Coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.4

Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Input-output theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3.1

A one-port cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.3.2

A two-port cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Qubit measurement
3.1. . . . . . . . . . . . . . . . . . . . 25

38

Formalism of quantum measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.1

Difference from classical measurement . . . . . . . . . . . . . . . . . . . . . . 38

3.1.2

POVMs and Kraus operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1.3

Measurement induced dephasing . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.4

Environment induced dephasing . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2

Dispersive readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3

Measurement efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4

3.3.1

Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.2

Linear measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3.3

Bifurcated measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Information Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.1

Accessible information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4.2

Information gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Josephson parametric amplifiers
4.1

54

Introduction to Josephson parametric amplifiers . . . . . . . . . . . . . . . . . . . . . 54

vii
4.2

Flux-pumped Josephson parametric amplifier design . . . . . . . . . . . . . . . . . . 56

4.3

Theory of operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.4

4.3.1

Full equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3.2

Simplified equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3.3

Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4.1

Tunability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.4.2

Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.4.3

Reset and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5 Superconducting switches

66

5.1

Review of current technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2

Tunable inductor bridge design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2.1

Lumped-element schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2.2

Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.3

Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.4

Switch networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6 The SIMBA

76

6.1

Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.2

Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.3

6.2.1

Metrics of performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.2.2

Version-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.2.3

Version-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.2.4

Quantum non-demolition fidelity . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.2.5

Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Current limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.3.1

Readout fidelity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

viii
6.3.2

Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.3.3

Excess backaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.3.4

Readout time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.3.5

Ease of calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7 Outlook93

7.1

Parametric coupling to reduce backaction . . . . . . . . . . . . . . . . . . . . . . . . 93

7.2

Autonomous feedback using a SIMBA . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.3

Utility of a readout cavity with tunable external coupling . . . . . . . . . . . . . . . 96

7.4

Superconducting switch from a Wheatstone bridge of capacitors . . . . . . . . . . . . 98

Bibliography

101

Appendix
A Introduction to Josephson junctions and SQUIDs

116

A.1 The Josephson relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
A.2 SQUIDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
A.3 Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
B SIMBA experimental schematic120

C Graph theory approach to circuit analysis

121

C.1 Calculating the admittance matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
C.2 Calculating the scattering matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

TablesTable
1.1

Comparison to related work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1

Important qubit states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2

Cavity coupling regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3

Summary of important systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1

Review of Josephson parametric amplifiers . . . . . . . . . . . . . . . . . . . . . . . . 55

5.1

Review of cryogenic microwave switches . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.1

Qubit readout performance metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.2

SIMBA performance summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.3

Sources of readout infidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.4

Sources of loss

6.5

Contributions to measurement time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Figures

Figure1.1

Simplified schematic of superconducting qubit measurement . . . . . . . . . . . . . .

5

1.2

The role of ferrite circulators/isolators . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.1

A simple harmonic oscillator

2.2

A simple parametric oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3

A nonlinear oscillator

2.4

Two coupled oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5

Coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6

A qubit from an anharmonic resonator . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.7

Scattering parameters of a cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1

Dispersive readout model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2

Readout infidelity vs. photon number . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3

The Helstrom bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4

Information gained by a qubit measurement . . . . . . . . . . . . . . . . . . . . . . . 52

4.1

Review of Josephson parametric amplifier (JPA) operation . . . . . . . . . . . . . . . 55

4.2

JPA optical micrograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3

Parametric cavity model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.4

Parametric oscillator dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.5

JPA characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

xi
4.6

Phase diagram for the pumped parametric cavity . . . . . . . . . . . . . . . . . . . . 63

4.7

Bifurcation reset and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.1

TIB lumped-element schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2

Balun and low-pass filter simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.3

TIB optical micrograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.4

TIB transmission characterization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.5

TIB power handling and switching speed characterization. . . . . . . . . . . . . . . . 74

5.6

TIB network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.1

Introduction to the SIMBA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.2

Qubit measurement using a SIMBA . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.3

Swap calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.4

TIB refect mode calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.5

SIMBA backaction, version-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.6

SIMBA efficiency, version-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.7

SIMBA characterization, version-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.8

Quantum non-demolition characterization . . . . . . . . . . . . . . . . . . . . . . . . 84

6.9

Measurement efficiency model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.1

A SIMBA operated using parametric coupling . . . . . . . . . . . . . . . . . . . . . . 94

7.2

Autonomous feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.3

Qubit multiplexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7.4

Superconducting switch from a Wheatstone bridge of capacitors . . . . . . . . . . . . 99

A.1 Introduction to Josephson junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
A.2 Introduction to SQUIDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
B.1 SIMBA characterization schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

xii
C.1 Graph theory approach to circuit analysis . . . . . . . . . . . . . . . . . . . . . . . . 122
C.2 Scattering parameters from an example circuit . . . . . . . . . . . . . . . . . . . . . 126

List of Symbols

Fundamental constants
h = 6.626 × 10−34 m2 kg/s
~ = h/2π = 1.055 × 10−34 m2 kg/s
e = 1.602 × 10−19 C
Φ0 = h/2e = 2.068 × 10−15 Wb
φ0 = Φ0 /2π = 3.291 × 10−16 Wb

Planck’s constant
reduced Planck’s constantcharge of an electron
magnetic flux quantum
reduced magnetic flux quantum

Chp. 2.1: Classical oscillators
Φ and qaAc, l and r
z0
ω0
H
κ
, Ω and θ
δ
γ
ξ
Ic
ζ
g and ∆
νbranch flux across an inductor and charge on a capacitor, respectively
field in a resonatorfield in a resonator, in the rotating wave approximation
capacitance, inductance and resistance, respectively
characteristic impedance of a resonatorresonance frequency
Hamiltonian, classical mechanics
loss-rate/linewidthmodulation amplitude, frequency and phase, respectively
detuning of a modulation frequency
parametric interaction ratesqueezing parameter
Josephson junction critical current
Kerr constant (parameterizes nonlinearity)
coupling rate and detuning between two resonators, respectively
amplitude of the parametric interaction between two detuned resonators

Chp. 2.2: Quantum oscillators

2Φ̂ and q̂âx̂ and ŷn̂σ̂zbranch flux and charge operators, respectively
field operator
quadrature operators
number operator
Pauli z-matrixD̂ (α)

displacement operator

Ĥ
ρ̂
|ni
|0i and |1i

Hamiltonian, quantum mechanics
density matrix
number eigenstate
qubit ground and excited state, respectively

|αi
ωq
χ
∆
α
δ

coherent state, such that α = |α|eiθ and θ is an angle in phase space
qubit frequency
readout resonator dispersive shift
detuning between superconducting qubit and readout resonatorsuperconducting qubit anharmonicity
detuning from resonator frequency

Chp. 3: Qubit measurement
Ax and Πx|ϕ0 i and |ϕ1 i|α0 i and |α0 i
T1
T2∗
Tφ
|ρ̂001 |η
P (x|y)Fr and F0
nr and nb
ρb
SNR
H
Iacc
Imeas

Krauss operator and POVM, respectively
ancilla systems entangled with a qubit
coherent states entangled with a qubit
qubit energy relaxation time
qubit dephasing (decoherence) time
qubit pure dephasing time
off-diagonal density matrix element after measurement (‘coherence’)
measurement efficiency
readout pulse amp (experimental units, e.g. voltage bias on a mixer)
probability of event x, given event y
readout fidelity and maximum readout fidelity, respectively
readout photon number and excess backaction photon number, respectively
dephasing due to excess backaction
signal-to-noise ratio
binary entropy
accessible information of a qubit
mutual information from a measurement

Chp. 4: Josephson parametric amplifiers3ωp , ωs and ωi
N
Iarraylarraylres
ϕs
ϕm

frequency of a pump, signal and idler modes, respectively
number of SQUIDs in an arraycurrent flowing through a SQUID arraylinear inductance of a SQUID array
total linear inductance of a resonator
amplitude of the static magnetic flux through a SQUID array
amplitude of the time-dependent magnetic flux through a SQUID array

Chp. 5: Superconducting switchesΦu and Φg

TIB bias flux applied in a uniform or gradiometric manner, respectively

Chp. 6: The SIMBA
FQN Dquantum non-demolition fidelity

Appendix A: Introduction to Josephson junctions and SQUIDs
I
Ic,1(c,2)
LJ
ψ1(2)
ϕ
ϕ1(2)
ϕext

current flowing across a Josephson junction / SQUID / arraycritical current of the first (second) Josephson junction in a SQUID
Josephson inductancesuperconducting order parameter on either side of a Josephson junction
phase drop across a Josephson junction / SQUID / array
phase drop across the first (second) Josephson junction in a SQUID
magnetic flux through a SQUID loop

Appendix C: Graph theory approach to circuit analysis
S and YyAn and mvνyµ and iµnpz0

scattering and admittance matrix, respectively
primitive admittance matrix
adjacency matrix
number of nodes and chords in a network, respectively
voltage on node νadmittance across and current flowing across chord µ, respectively
number of ports in a network
port characteristic impedanceChapter 1

Introduction

1.1

Motivation for quantum computingThis thesis is one small piece of the decades-long effort to advance information technology

using the laws of quantum mechanics. While classical computers have revolutionized society overthe past century, there are fundamental limits on the problems they can efficiently solve. Thisobservation was epitomized by the legendary physicist Richard Feynman, who said: “Nature isn’tclassical, dammit, and if you want to make a simulation of nature, you’d better make it quantum
mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy.” In the 1980’s [1,2], Feynman was one of the first to realize that nature, which is described by quantum phenomenonsuch as entanglement, cannot always be reasonably modelled by a physical system which is purely
classical. Since quantum computers are of course described by the laws of quantum mechanics, ina sense they represent the final frontier in computing technology. There are no known problems
which cannot be efficiently solved on a quantum computer but can be efficiently solved on someother type of physically realizable computer [3, 4].
Interest in quantum computing expanded from an intellectual curiosity to an object of potential use in the 1990s, when Peter Shor made two important discoveries. First, he invented analgorithm to factor large numbers in polynomial time, a task strongly believed to be exponentiallydifficult (NP-complete) on a classical computer [5]. The ability to efficiently factor large numbers
would allow for commonly used cryptography schemes to be easily broken. Secondly, Peter Shorand then others showed that error correction is possible on a quantum computer [6, 7, 8, 9]. The

2implication of this is that it is possible, in-principle, to physically realize a universal, fault-tolerant
quantum computer. Doing so in any way powerful enough to be generally relevant to society
remains an outstanding challenge.
The current status of quantum computers, deemed the NISQ era for ‘Noisy IntermediateScale Quantum’ by John Preskill [10], is perhaps analogous to that of classical computers in the
mid-20th century. There was a period when ideas from Alan Turing, John von Neumann and others
had made it clear that a ‘computer’ was a new class of machine, and rudimentary demonstrations
had even been built. If realizable at scale, such a ‘computer’ would be certainly be useful whenapplied to some specific problems, and perhaps would be useful to others in some unforeseen way.
There were competing ideas for how a computer should be built: computers based on mechanicalcomponents, vacuum tubes, etc. It was not for several decades until a transistor-based computer
emerged as the dominant architecture, and decades more until computers had spread to every
corner of the globe and revolutionized society. Likewise, despite the impressive progress that has
already been made in understanding and building quantum computers, nobody knows the best way
to create a scalable one in the long-term, if doing so in a broadly useful way is even possible, andif so, the scope of what it would be used for. These open questions create uncertainty but also
excitement.

1.2

The need for high quality measurement
In general, algorithms run on a quantum computers contain the following steps. First, thequbits — the basic units of quantum information —are prepared in a known state, for example theground state |0i. Next, a series of gates are applied, and finally, the qubits are measured. Thismeasurement is generally projective, meaning that the qubit post-measurement state is collapsed toeither the ground state |0i or excited state |1i. Ideally, all of the operations in an algorithm: state
preparation, single-qubit gates, multi-qubit gates and measurements, are performed with perfect
fidelity. Moreover, the qubit states should not change in some unknown or unspecified way between
steps.

3
In practice, errors occur. As the size and complexity of a quantum system increases, thefidelity of all operations in that algorithm must generally increase for it to yield a useful result. For
example, suppose a measurement operation works with 90% probability, but returns an error theother 10% of the time. The chance of having no errors when measuring 10 qubits is 0.910 = 0.35
which is generally tolerable; runs of an algorithm which contain errors can simply be discarded.
This strategy quickly becomes untenable in larger systems: for 100 qubits, the chance of having
no errors is 0.9100 = 2.6 × 10−5 , and for 1000 qubits it is 0.91000 = 1.7 × 10−46 . As the number ofoperations increases, fidelity must also increase so that an algorithm is not overwhelmed by errors.
Measurement performance is especially important for quantum error correction schemes,
which generally involve the measurement of ancilla qubits [11]. Conditional on these measurement
results, gates are applied to correct errors that have occurred in the data qubits. For such a
procedure to succeed, measurements must generally have near-unit fidelity and be much fasterthan qubit decoherence times. Since ancilla qubits are repeatedly measured, measurements should
be quantum non-demolition meaning that the qubit remains in its post-measurement eigenstate(rather than being kicked out of its two-level subspace, for instance). Measurement problems
generally reduce the effectiveness of any quantum error correction algorithm, and can easily cause
it to introduce more errors.
High quality qubit measurement is therefore essential to quantum computing and quantum
error correction in any hardware platform. Improving the measurement of superconducting qubits
motivates this thesis.

1.3

Quantum information with superconducting circuits
Superconducting systems falling within the scope of ‘circuit quantum electrodynamics’ (cQED)

are currently one of the most promising ways to create a scalable quantum computer [12, 13]. Thisapproach was pioneered at Yale in the mid-2000s [14, 15, 16]. In cQED, qubits are realized fromsuperconducting, nonlinear microwave frequency resonators. Qubits are controlled and measured
using microwave frequency pulses routed in and out of the base temperature of a dilution refriger-

4ator, whose millikelvin temperature keeps these qubits in their quantum ground state.
This approach is a promising route towards quantum computation because qubits can havegate speeds orders of magnitude faster than their coherence times, fast and high-fidelity, qubitspecific measurement, and the promise of scalability due to their compact size and relative ease of
fabrication. In other words, superconducting qubits arguably fulfill the DiVincenzo criteria [17],the features required to build a scalable quantum computer.
The transmon qubit [18], also developed at Yale in the mid-2000s, is an especially successful
type of superconducting qubit. Companies including Google, IBM, Rigetti Computing, and othersaround the world, are all trying to build quantum computers using transmons as their data qubit[19, 20]. Other superconducting architectures built around linear cavities still make use of transmons
for state preparation and measurement [21, 22].
Despite great promise, many challenges remain in implementing superconducting quantum
computers. Today, the largest systems are becoming prohibitively difficult to simulate on classicalcomputers (for example the recent demonstration of ‘quantum supremacy’ by Google [20]), but are
not close to solving problems generally relevant to society. To do so, virtually every performance
metric: e.g. coherence times, gate fidelities, measurement fidelities, must be improved by orders ofmagnitude. Moreover, there is the daunting technical challenge of fabricating and fitting increasingly many qubits and the hardware they require into a cryostat, and doing so in a way which
doesn’t degrade performance. Thousands of people around the world are working on all of these
problems, and parallel challenges in other quantum computing architectures.

1.4

Introduction to superconducting qubit measurement
To place this thesis in context we review the standard approach to superconducting qubit

measurement, an example of which is illustrated in Fig. 1.1. We discuss the advantages — andlimitations — of this approach, and conclude with a summary of the current state-of-the-art.
Superconducting qubits are generally engineered in a dispersive architecture [15], meaning

5
control and readout pulses

data acquisition

room temperatureattenuationand filtering

circulators/
isolators

noisier amp.
>> 1 photon noise

superconducting qubit
in readout cavity

replace with a single,
chip-scale device

parametric amp≲ 1 photon noise≲ 20 mK temperature

Figure 1.1: Simplified schematic of superconducting qubit measurement. A superconducting qubit is measured by scattering a microwave frequency pulse off of a readout cavity dispersively coupled to it. Here, this pulse is illustrated to enter the readout cavity through a weakly
coupled port (left side of the red box) and exit through a strongly coupled port (right side). The
pulse typically populates the cavity state with several photons. After leaving, the pulse is routed
through circulators/isolators to a quantum limited parametric amplifier. Circulators/isolators are
non-reciprocal devices which enforce unidirectional flow of information and prevent amplified signal
and noise from the amplifier chain from entering the readout cavity. The pulse is then routed outof the cryostat, through further amplification, and to a digitizer at room temperature. The workin this thesis replaces the setup boxed in green with a single, chip-scale device called a SIMBA (aSuperconducting, Isolating, Modular Bifurcation Amplifier) [23]. The SIMBA has state-of-the-art
performance and the promise of further improvement.

that the qubit is weakly1 coupled to a second, linear resonator. This resonator inherits somenonlinearity from the qubit, such that its resonance frequency shifts by ±χ/2π depending on if thequbit is in the ground or excited state. A typical transmon has χ/2π order 1 MHz, with a resonatorfrequency of order 10 GHz and a detuning between the qubit and resonator of 1 GHz or more.
An advantage of the dispersive architecture is that the qubit is well isolated from its environment, which is a source of loss and noise. Its only explicit connection to any dissipative load isthrough the external coupling of the readout cavity, which is already coupled weakly to the qubit.
The qubit can still be measured, however, by detecting microwave frequency photons scattered off
1

Meaning g/∆  1 with g the coupling between the resonator and qubit, and ∆ their detuning. See Chp. 2.1.6.

6of the cavity. An appropriate readout pulse will acquire a different phase and/or amplitude shiftdepending on if the qubit is in the ground state or excited state. This procedure is called ‘dispersive
readout’ [15, 24] and is discussed quantitatively in Chp. 2.2.4.3. For now, we care that the problem
of measuring the qubit state has been mapped to detecting this readout pulse.
High quality detection of the readout pulse is complicated by the fact that it typically contains less than 10 photons, an extremely weak signal by the standards of conventional microwave
electronics. Yet high-fidelity readout relies on the precise discrimination of this signal. To do so, thereadout pulse must first be routed to an amplifier which adds little noise compared to the amplitudeof the readout pulse. Superconducting parametric amplifiers are suitable for this purpose.

1.4.1

Introduction to quantum limited amplification

Realized by Carlton Caves in the 1980s, quantum mechanics sets a fundamental limit on thenoise that can be added by any linear amplification process [25, 26]. Analogous to the uncertainty
principle, the product of the noise added along each quadrature2 of the amplified field must be
greater than or equal to a minimum non-zero value.
In general there are two classes of quantum limited parametric amplifiers. An ideal phasesensitive amplifier can approach noiseless amplification of one quadrature of the electromagnetic
field, while adding increased noise to the other quadrature. An ideal phase-preserving amplifier
amplifies both quadratures equally, adding a quarter quanta of noise to each.
That quantum mechanics places a fundamental limit on amplification is of profound importance for the measurement of weak signals, for example the single photon level microwave pulses
used in superconducting qubit measurement. On one hand, there is a limit to how well any amplifier can possibly perform. Advantageously however, this limit is zero when using a phase-sensitive
amplifier to measure a single oriented along one quadrature, and only a half quanta of added noise
when using a phase-preserving amplifier.
Quantum limited amplifiers are readily engineered in superconducting systems by making
2

See Chp. 2.2.3 for a definition of ‘quadrature’.

7
use of the nonlinearity of the Josephson junction [27, 28, 29, 30, 31]. The basic theory behindthese amplifiers is discussed in Chp. 2.1.4, and the specific parametric amplifier used in this work
is discussed in Chp. 4.

1.4.2

Isolating qubits from measurement backaction

In addition to low noise measurement, it is important to keep a qubit well isolated in orderto maintain the delicate quantum information preserved in the qubit’s phase coherence. Dephasing3 will introduce errors into any quantum computing protocol. Photons coupled to the qubit
will dephase it, regardless of if these photons come from a measurement pulse or a noise source.
Exposure to too high a photon number can also kick a qubit out of its two-level subspace [32, 33].
Fortunately, it is possible to couple a qubit to a measurement chain without exposing it to extra
noise.
(a)
3

1

2

(c)
~20 mK noise

(b)

50

~4 K noise

readout
pulse

amplifiedreadoutpulse
parametric amplifier

Figure 1.2: The role of ferrite circulators/isolators. (a) Schematic of a 3-port circulator:
a signal incident on port 1, 2 or 3 is routed to port 2, 3 or 1, respectively. (b) Photograph of acommercial cryogenic isolator (a 4-port circulator, with matched terminations on two ports). (c)Illustration of a circulator used in conjunction with a parametric amplifier. A pulse incident on
the left port is routed to a 1-port parametric reflection amplifier. This pulse is amplified and thenrouted by the circulator to a measurement chain connected to the right port. The measurement
chain spits noise back to the circulator, which is dissipated in a matched load thermalized to thebase temperature of a cryostat (black square, equivalent to a 50 Ohm resistor). Noise consistent
with the base temperature of the cryostat is then routed back to the left port. The circulator
therefore protects anything connected to the left port (e.g. a readout cavity containing a qubit),from amplifier backaction and the added noise of the measurement chain.

3

See Chp. 3.1.3 and 3.1.4 for a definition of dephasing.

8
In a typical measurement, circulators and/or isolators4 protect the qubit from the excess
backaction caused by the measurement chain. Circulators/isolators are nonreciprocal devices,meaning that they enforce a unidirectional flow of information, Fig. 1.2a [34]. A circulator canroute a signal output from a qubit readout cavity into a parametric amplifier, Fig. 1.2c, and thenroute the amplified signal to the rest of a measurement chain and ultimately out of the cryostat,rather than returning to the qubit. This keeps the qubit from being exposed to the order onehundred photon amplified signal. Circulators and/or isolators also route the noise from the next
amplifier in the measurement chain to a cold bath, rather than back into the qubit-cavity system. For a standard HEMT (high-electron-mobility-transistor) amplifier, the temperature of thenoise it spits back to the base temperature of the cryostat is approximately 4 Kelvin or greater,corresponding to thermal state with a mean photon number of order ten or more at microwave
frequencies.
The commercial circulators/isolators used in cryogenic microwave experiments break Lorentz
reciprocity using ferromagnetic materials [34]. An example of one of these ferrite devices is shownin Fig. 1.2b. Magnetic shielding is included on the outside of this isolator to prevent the permanent
magnetic field it produces from disrupting any superconducting electronics, which can be sensitive
to magnetic fields on the order of a single flux quanta or less. Advantageously, commercial circulators/isolators are passive devices (requiring no external bias), have far more power handling thanis needed in any cryogenic experiment, and have octave bandwidth.
Use of ferrite circulators/isolators combined with parametric amplifiers works reasonably wellfor superconducting qubit readout. For example, superconducting qubit readout faster than 100
ns with greater than 99% fidelity has been demonstrated using such hardware [35], with higher
fidelities in longer measurements [36, 37]. However, there is a clear need for improvement as thescale and requirements of superconducting qubit systems increase.
4

An isolator is an N -port circulator with N − 2 ports connected to matched loads.

9
1.4.3

Measurement hardware areas for improvement

As superconducting qubits have developed, so has their measurement and the requisite hardware. Efforts toward fast, high-fidelity single-shot readout of superconducting qubits began shortly
after their advent, for example Refs. [27, 38, 39]. Today, such efforts include the improvement ofboth amplifiers and circulators/isolators, with the broader goal of improving measurement performance and scalability.
For over a decade, superconducting, microwave parametric amplifiers have been shown toachieve both high gain and added noise near the quantum limit. These approaches are often
based on resonant modes made nonlinear by the participation of a Josephson junction, for example
Josephson parametric amplifiers (JPAs) [28, 40, 29] or the Josephson parametric converters (JPC)[30, 41]. More recent research efforts have included improving bandwidth [31, 42, 43, 44, 45],
improving the power handling and ease of calibration [46, 47], and integrating these amplifiers moreclosely with superconducting qubits [48, 49, 50]. In particular, Josephson travelling wave parametricamplifiers (JTWPAs) [51, 31, 52, 53, 54] and related technology based on the nonlinearity fromkinetic inductance [55, 56] offer added noise that can approach the quantum limit combined with> 20 dB of gain over several GHz of bandwidth. Room for improvement remains and similar
technology can be engineered for other applications such as astronomical detectors [57]. Arguably
however, superconducting microwave amplifiers are an increasingly mature technology.
Improving the requisite circulators/isolators used in conjunction with parametric amplifiers is
a separate challenge [58]. Despite desirable features, ferrite devices have limited isolation, unavoidable loss, and centimeter-scale size. Limited isolation (typically specified at 18 dB per circulator)
requires that several circulators be used in front of each parametric amplifier, increasing loss between the amplifier and qubit. This lowers the efficiency of the measurement chain,5 meaning
greater loss before amplification or equivalently greater added noise during amplification. Ferrite
circulators cannot be readily integrated on-chip with superconducting qubits due to their permanent magnetic fields, and the separate packaging they require is an additional source of loss. Finally,5

See Chp. 3.3 for a precise definition of efficiency.

10
fitting so many bulky circulators/isolators at the base temperature stage of a cryostat is an eventual
bottleneck towards systems containing thousands or more individually measurable qubits.
In recognition of these problems, beginning in the mid-2010s the superconducting microwave
circuit community has attempted to engineer superconducting circulators/isolators [59, 60, 61, 62]
and/or directional amplifiers [63, 64, 65, 66]. Such devices must actively break Lorentz reciprocity
using various bias tones, rather than using permanent magnetic fields [67]. These efforts have beenparalleled by a (larger scale) telecom-focused effort to engineer active, chip-scale circulators using
conventional transistor technology and operated at room-temperature [68].
Despite the added requirement of bias tones and/or nonlinearity, active superconducting
circulators/isolators can in-principle outperform ferrites. Active circulators can have octave-scale
bandwidth and high isolation [43], combined with minimal loss [45]. However, engineering a superconducting circulator with such performance is a difficult. In particular, the device in Ref. [45]is a design for a low-loss, broadband superconducting circulator that requires long, low-loss delaylines and fast, superconducting crossover switches with large on/off ratios. Long delay lines andother types of broadband switches have already been developed (see Table 5.1), but combining
them together and demonstrating high-quality circulation in a superconducting system has not yet
been achieved.
Alternatively, excellent performance has been demonstrated in narrow band superconducting
circulators/directional-amplifiers [63, 64, 60, 65, 61, 62, 66]. Such devices contain three or moreresonant modes with nonlinearity introduced by Josephson junctions. Non-reciprocity and/or gain
can be realized by pumping the circuit at the difference and/or sum frequencies of these modes.
State-of-the-art superconducting qubit readout has recently been demonstrated with such devices[69, 70, 71]. Despite impressive performance, these approaches suffer from narrow bandwidth
limited by the high quality-factor of the modes, along with multiple requisite bias tones which must
be carefully synchronized. These limitations manifest in the fact that to-date, no superconducting
technology has been broadly adopted as an alternative to ferrites. All large-scale superconducting
quantum experiments today still provide isolation exclusively using ferrites.

11

1.5

Comparison of this thesis to related work
The work forming the bulk of this thesis, Ref. [23], is also motivated by a desire to replace

the ferrite circulators/isolators used in superconducting qubit measurements. Ref. [23] differs fromother approaches in that it does not try to engineer a circulator or isolator, and that it is notdesigned to operate continuously. Instead, both isolation and pulsed directional amplification isachieved by the coordinated operation of superconducting switches. These switches, combined witha parametric amplifier, are integrated into a single-chip scale devices called a ‘Superconducting
Isolating Modular Bifurcation Amplifier’ (SIMBA), discussed in detail in Chp. 6. Here, we willsummarize the SIMBA’s performance in comparison to other work.
In this comparison, we focus on the figures of merit reported in Table. 1.1: measurement
efficiency η, excess backaction nb , the maximum readout fidelity F0 , and the time it takes tocomplete the measurement.6

Qualitatively, measurement efficiency η is the fraction of readout

signal lost between the qubit and amplifier, assuming noiseless amplification (added noise also
decreases efficiency). ‘Excess backaction’ in Table. 1.1 refers to the number of photons a qubit is
exposed to due to measurement in excess of those in the readout. The presence of such photons
indicates imperfect isolation between the qubit and amplifier. Both efficiency and backaction are
discussed in detail in Chp. 3. Maximum readout fidelity is essentially the ability of the detector todiscriminate the state of the qubit. This is defined as F0 = 1 − p0 − q0 , where p0 and q0 are theerror probabilities of a projective measurement when the qubit is prepared in the excited or groundstate, respectively.
The highest measurement efficiency which has thus-far been reported in a circuit quantum
electrodynamics system is η = 0.80 in Ref. [49]. In Ref. [49], a qubit is dispersively coupled to areadout cavity which shares functionality as a phase-sensitive pumped parametric amplifier, withno non-reciprocal elements between the qubit and amplifier. The highest efficiencies that have
6
Another important metric is how quantum-non-demolition (QND) a measurement is (how well a qubit remains
in the measured eigenstate) [72]. Dispersive readout of superconducting qubits is understood to be QND [15, 24] andis not characterized in most of the references in Table. 1.1. Additionally, the amplifier bandwidth, ease of calibration,and requisite number of bias lines (along with the bandwidth of the bias signals), are also important considerations
from the standpoint of scalability.

12
Table 1.1: Comparison to related work. Recent demonstrations of superconducting qubit
readout in which the measurement efficiency η is characterized. Many recent experiments whichmake use of ferrite circulators/isolators have demonstrated maximum readout fidelities F0 of 0.95
or greater, readout times of several hundred nanoseconds, and measurement efficiencies between
η = 0.1 and η = 0.6. All works in this table use Josephson-junction based parametric amplifiers,
which are operated either in a phase-sensitive (S) or phase-preserving (P) manner. Note that(excepting Ref. [36]) the works labelled (P) generally use a definition of measurement efficiency
which goes to unity when using a phase-preserving amplifier, a different definition than that usedby works which use a phase-sensitive amplifier (see Chp. 3.3.1).

Reference

AmplifierEfficiency,ηRosenthal et al. (2021), [23]Lecocq et al. (2021), [71]Andersen et al. (2020), [73]Abdo et al. (2020), [70]Peronnin et al. (2020), [74]Andersen et al. (2019), [73]Abdo et al. (2019), [69]Touzard et al. (2019), [75]
Eddins et al. (2019), [49]
Heinsoo et al. (2018), [76]Bultink et al. (2018), [77]Eddins et al. (2018), [78]Walter et al. (2017), [35]Macklin et al. (2015), [31]Jeffrey et al. (2014), [36]SIMBA (S)FPJA (S)TWPA (P)JDA (P)TWPA (P)TWPA (P)JPC (P)SPA (S)JPA (S)TWPA (P)TWPA (P)JPA (S)
JPD (S)TWPA (P)JPA (P)

0.70
0.72
0.15 − 0.30
∼ 0.2
0.11
0.24
∼ 0.3
0.6
0.80
0.43 − 0.52
0.165
0.38
0.75
0.49
0.13

Excessbackaction
nb , (photons)

Readout
fidelity, F0

Readout
duration

0.66
0.07
N/A
0.002
N/A
N/A
0.01
N/A
1
N/A
N/A
N/A
N/A
N/A
N/A

0.955
0.97
0.978 − 0.994
0.92
0.95
0.987 − 0.992
0.9
0.978
N/A
0.936 − 0.988
N/A
N/A
0.992
0.967
0.998

265 ns
350 ns300 − 400 ns
1000 ns
220 ns200 ns200 ns
870 ns
N/A
250 ns
N/A
N/A
88 ns
100 ns140 nsyet been reported with considerable ferrite-based isolation between the qubit and amplifier areη = 0.75 in Ref. [35],7 η = 0.72 in Ref. [71] and η = 0.70 in Ref. [23]. In other experiments
using ferrite circulators/isolators before a parametric amplifier, the measurement efficiency falls
between η = 0.1 and η = 0.6. Ref. [71] is a chip-scale ‘FPJA’-style amplifier which synthesizes
directional amplification with the use of several microwave frequency pumps and the sum anddifference frequencies between several microwave modes. Ref. [23] introduces the SIMBA, thesubject of this thesis.
The technologies in Refs. [70, 23, 71] are all promising alternatives to ferrite circulators/isolators.
7
Note that the measurement in Ref. [35] (which uses ferrite circulators/isolators to provide isolation) calibrates
η differently than the other works in Table. 1.1 which potentially complicates a direct comparison. In Ref. [35], η isestimated by determining the power spectral density of a readout pulse using the ac-Stark shift of the qubit [16], andcomparing this to the power spectral density of a measured signal. Other references in Table 1.1 directly measure
the decoherence of a qubit due to a weak measurement (see Chp. 3).

13These devices result from years of preliminary work, but have only recently been developed tothe point of demonstrating high-quality readout of a superconducting qubit without any ferrite/circulators between the qubit and amplifier. SIMBAs have the advantage, compared to thedevices in Refs [69, 66, 70, 71], of requiring only one microwave bias tone and the promise of relative ease of calibration. The limitations of the SIMBA in Ref. [23] are well understood and notinherent to the concept itself (see Chp. 6.3). As such, SIMBA amplifiers based off of the device inRef. [23] are a promising avenue towards the widespread replacement of ferrite circulators/isolators
in superconducting quantum information experiments.

1.6

Thesis organization
The remainder of this thesis is organized as follows: Chp. 2 is a pedagogical introduction

to simple-harmonic oscillators in the context of superconducting circuits. This includes damping,parametric gain, and coupled resonators treated classically. It also includes introductions to quantum optics, superconducting qubits, and the formalism of input-output theory. Chp. 3 discusses
qubit measurement, beginning with a formal definition of measurement in quantum mechanics anddeveloping this to show how measurement-induced dephasing can be used to calibrate the efficiency
of a measurement. Chp. 4 overviews Josephson-parametric amplifiers (JPAs), the specific type of
amplifier integrated within a SIMBA. Chp. 5 describes superconducting switches based on a ‘tunable inductor bridge’ (TIB) design [79], also integrated into a SIMBA. Chp. 6 describes the SIMBA
itself and characterizes its use for superconducting qubit readout. Finally, Chp. 7 presents several
avenues for future research.

Chapter 2

Oscillators

This chapter begins with the canonical example of a classical, simple harmonic oscillator. This
simple model is extended to include loss, parametric gain, nonlinearity and coupling to anotheroscillator. A quantum description is then given, along with an introduction to superconducting
qubits. Finally, coupling of an oscillator to its environment is reviewed using the framework of
input-output theory. These examples are useful for understanding the work presented later in thisthesis.

2.1

Classical oscillators
This section deals with classical oscillators. Here ‘classical’ refers to oscillators described byscalar-valued variables which commute, such as position and momentum.1 The examples discussed
in this section are useful for building intuition for their quantum counterparts. In many systems,
classical dynamics alone are a sufficient description.

2.1.1

LC-oscillator

A simple harmonic oscillator is formed from an inductor-capacitor circuit, Fig. 2.1a. Following
from Maxwell’s equations its equations of motion are1Φ̇ = q,c11q̇ = − Φ.
lMore formally, classical coordinates satisfy the Poisson bracket. See Refs. [80, 81] for more information.

(2.1)

15
where c is a lumped-element capacitance and l is a lumped-element inductance. We have writtenthese equations using the time-dependent branch flux Φ(t) = Il (t) × l across the inductor suchRt
that Φ(t) = −∞ dt0 v(t0 ), where v(t) is the voltage drop across the inductor and Il (t) is the current
flowing through it. We also use the time-dependent charge q(t) = v(t) × c across the capacitor suchRt
that q(t) = −∞ dt0 Ic (t0 ) where Ic (t) is the current flowing through it. Kirchhoff’s current law gives
the relation Ic + Il = 0.
(a)
Φ(𝑡)

(b)

l

c

𝑞(𝑡)

(c)

c

l

r

l

c

lc

rext

Figure 2.1: A simple harmonic oscillator. (a) An LC-oscillator (‘LC’ named for inductor ‘L’
and capacitor ‘C’) can be made from an inductance l to ground, in parallel with a capacitance c
to ground. The branch flux Φ(t) across the inductor and charge q(t) on the capacitor plates will
oscillate in time at the circuit’s resonance frequency. (b) A resistor r adds loss. (c) Coupling toa transmission line of characteristic impedance rext also adds loss (the transmission line symbol isassumed to be terminated in a matched load, and is then equivalent to a resistor of value rext toground).

It is useful to transform these equations to normal mode form, such that the two first-orderdifferential equations describing the oscillator’s dynamics are complex conjugates of each other. To
do so, we introduce the variable a(t) and its complex conjugate a† (t), defined by the transformation2

1
a= √
2~




√
Φ
√ + i z0 q ,z0where we have introduced the characteristic impedance of the resonator z0 =

(2.2)
p
l/c.

After this transformation, the equation of motion for the circuit in Fig. 2.1 becomesȧ = −iω0 a,(2.3)

which has the solution a(t) = Ae−iω0 t , along with its complex conjugate. Here, A is a phasor, a2

Despite our introduction of ~, this analysis remains classical such that a and a† (or Φ and q) commute.

16complex number which specifies the amplitude and phase of oscillations in a frame rotating at theresonance frequency ω0 .
Notice that from the classical Hamiltonian H = Φ2 /2l + q 2 /2c, we find the energy stored in
the oscillator to be
H = ~ω0 A† A,

(2.4)

√
where ω0 = 1/ lc is the oscillator’s resonance frequency.

2.1.2

LC-oscillator with damping

Eq. 2.1 is modified by the inclusion of loss. In Fig. 2.1b, this is modelled as a resistor r to
ground in parallel with the inductor and capacitor. The equations of motion become1Φ̇ = q,c11q̇ = − Φ − q.
l
rc

(2.5)

Again using Eq. 2.2 to transform to normal coordinates, Eq. 2.5 becomesȧ = −iω0 a −κ
a − a†
2(2.6)

and its complex conjugate. Here, κ = (rc)−1 is a loss rate.
It can be useful to rewrite Eq. 2.6 in the rotating frame, taking a(t) = A(t)e−iω0 t suchthat ȧ = Ȧ − iω0 A e−iω0 t by the product rule. Here, A(t) is the now time-dependent phasor
describing the oscillator field in the rotating frame. Substituting these relationships into Eq. 2.6
gives Ȧ = − κ2 A + κ2 e2iω0 t A† . Neglecting the fast-rotating term e2iω0 t simplifies the equations of
motion toκȦ = − A2(2.7)

and its complex conjugate.
Eq. 2.7 has the solution A(t) = A0 e−κt/2 , where A0 is a phasor describing the system at
t = 0. The energy stored in a circuit is proportional to the modulus-squared of its field, which
exponentially decays at rate κ. It is typical to parameterize this decay by the quality factor,
Q=

ω0
.
κ

(2.8)

17
Physically, Q is the number of cycles it takes for energy in the oscillator to decay by e−1 .

2.1.3

LC-oscillator with damping by an external load

Loss may also be introduced by coupling to a dissipative load, for example the circuit inFig. 2.1c. Here, this dissipative load is modelled by a transmission line of characteristic impedance
rext which is terminated in a matched load, equivalent to a resistor to ground of the same value.
In Fig. 2.1c this load is connected to the resonator through an inductance but equivalent physics
arises for a capacitive coupling (or any high impedance).
In the limit of weak coupling l/lc  1 and in the rotating wave approximation, the equationsof motion for the field in the resonator in Fig. 2.1c become
κextȦ = −
A,2

rextκext =
l

 2
l
,
lc

(2.9)

and its complex conjugate. Notice that Eq. 2.9 takes exactly the same form as Eq. 2.7, but witha loss rate dependent on the small parameter lc /l describing how strongly coupled the resonator isto the dissipative load. Note that if the coupling inductor lc is replaced by a coupling capacitor cccc 2
then in the limit of cc /c  1, the external loss rate becomes κext = rext
.
l
c
2.1.4

LC-oscillator with parametric gain

The field in a resonator can be amplified by modulating the circuit’s resonance frequency ω0
at twice its natural value. Consider a harmonic oscillator now with a time-dependent inductancel(t) as in Fig. 2.2al(t) =

l0.
1 +  cos (Ωt + θ)

(2.10)

Here, l0 is a constant inductance, and Ω and θ are the frequency and phase of the modulation,
respectively. The modulation amplitude  is specified by the dimensionless parameter 0 ≤ || < 1.
The resulting equations of motion are identical to those in Eq. 2.5, but now with a timedependent inductance l(t):1Φ̇ = q,cq̇ = −

11
Φ − q.
l(t)rc(2.11)

18
We again transform Eq. 2.11 to normal coordinates using Eq. 2.2, where now z0 =p

l0 /c. Thisgivesȧ = −iω0 a −




κ
a − a† − iγ a + a† cos (Ωt + θ)2

(2.12)and its complex conjugate. Eq. 2.12 is identical to the damped harmonic oscillator in Eq. 2.6, but
now with an extra term proportional to γ = ω0 /2.
We consider modulation near twice the oscillator’s natural resonance frequency such thatΩ = 2ω0 + 2δ,(2.13)

where δ is a detuning. Then, we transform Eq. 2.12 to a frame rotating at half the modulation
frequency such that a(t) = A(t)e−iΩt/2 . Neglecting fast-rotating terms, the equations of motion
simplify toγκȦ = − A + iδA − i e−iθ A†
22(2.14)and its complex conjugate. For a boundary condition A(0) = 1 and Ȧ(0) = 0 at t = 0, maximum
parametric amplification or de-amplification is achieved when θ = −π/2 or θ = π/2, respectively.
This can be seen in Fig. 2.2b for the case when pump detuning and loss are both set to zero. Thisparametric process is phase-sensitive, as it depends on the modulation phase θ.
(a)

c

r

log(|𝐴|2)

l(𝑡)

(b)

2
1
0

amplification, 𝜃 = -𝜋/2

2

4

6

8

10

time (ns)-1-2de-amplification, 𝜃 = 𝜋/2

Figure 2.2: A simple parametric oscillator. (a) Modulating the frequency of a simple harmonic
oscillator near twice its natural resonant frequency ω0 can enable parametric gain. (b) Solutions
to Eq. 2.14 are plotted for δ = 0, loss κ = 1/rc set to zero, and γ/2π = 30 MHz, with a boundary
condition A(0) = 1 and Ȧ(0) = 0 at t = 0. The energy in the oscillator, proportional to |A(t)|2 ,is plotted on a logarithmic scale. This process is phase-sensitive: changing the phase of the initial
field A(t = 0) changes which angle θ causes amplification/de-amplification.

19
The Hamiltonian describing the parametrically modulated cavity in Fig. 2.2 is H(t) =Φ2 /2l(t) + q 2 /2c where now the inductance is time-dependent given by Eq. 2.10, or equivalently
H = ~ω0 a† a + Φ2  cos (Ωt + θ) /2l0 , where the second term describes parametric gain. TakingΩ = 2ω0 and going to a rotating frame, this simplifies to:
H = ~ω0 A† A +


2
~ 2
ξA + ξ ∗ A† ,2

(2.15)

where ξ = ω0 eiθ /4 = γeiθ /2 and fast-rotating terms are neglected. The second term on theright-hand side of Eq. 2.15 is called the squeezing Hamiltonian [82].

2.1.5

LC-oscillator with nonlinearity

In the model for parametric gain shown in Fig. 2.2b, energy in the pumped oscillator increases
to infinity. In any physical process, nonlinear effects will eventually prevent this from happening. In
superconducting systems this nonlinearity can be engineered using a Josephson junction, Fig. 2.3.
The relationship between current and branch flux across a Josephson junction is: I =Ic sin (2πΦ/Φ0 ), where Ic is the junction’s critical current (see Appx. A.1 for details) and Φ0 = h/2e
is the magnetic flux quantum. The equations of motion for the circuit in Fig. 2.3 are identical tothose of Eq. 2.5, but with an extra term describing current flowing across the Josephson junction:1Φ̇ = q,
c

l(𝑡)

Ιc

1
1
q̇ = − Φ − q − Ic sin
l
rc

l

c



2πΦ
Φ0


.

(2.16)r

Figure 2.3: A nonlinear oscillator. In this nonlinear RLC-oscillator, part of the inductancecomes from a Josephson junction (× symbol) parameterized by its critical current Ic . The Josephson
junction behaves as a nonlinear inductor (see Appx. A).

20
We transform Eq. 2.16 to normal mode form and expand in the limit where |Φ/Φ0 |  1 toget,ȧ = −iω0 a −


3
κ
ζ
a − a† + i a + a† ,2
3

(2.17)whereζ=(2e)2 z02
,
~ 8lJ

lJ =

Φ0
.
2πIc

(2.18)In Eq. 2.18, ζ parameterizes the nonlinearity of the oscillator and lJ is the Josephson inductance(the linear inductance associated with the Josephson junction). Note that in Eq. 2.18 and Eq. 2.19
the resonance frequency and characteristic impedance are modified from their values in previousp√sections to now be ω0 = 1/ lres c and z0 = lres /c, respectively, where lres = (1/l + 1/lJ )−1 isthe combination of both the Josephson inductance lJ and the other linear inductance l. Next, wetransform Eq. 2.18 to a frame rotating at the resonance frequency ω0 , giving
κȦ = − A + iζ |A|2 A
2(2.19)

along with its complex conjugate.
hiFinally, the oscillator’s Hamiltonian is H = Φ2 /2l + q 2 /2c + EJ 1 − cos 2πΦ
, where EJ =Φ0
Ic Φ0 /2π is the Josephson energy [81, 83]. Going to normal mode form and again expanding in thelimit where |Φ/Φ0 |  1 gives the Hamiltonian,
H = ~ω0 A† A −~ζ  † 2
AA ,2(2.20)

where fast-rotating terms are neglected, and the resonance frequency ω0 has been renormalized toaccount for the linear inductance of the Josephson energy.
Eq. 2.20 is identical to the Hamiltonian of a simple harmonic oscillator, Eq. 2.4, but with an
extra term added to account for the resonator’s nonlinearity.3 The meaning of the nonlinearity ζ,
also known as the Kerr-constant, becomes clear: the oscillator’s resonance frequency ω0 − ζa† a/2
is shifted by the modulus-squared of the field inside of it, times the Kerr constant.4
3

The Duffing oscillator is a system described by Eq. 2.20. See Refs. [84, 85] for a thorough theoretical treatmentin the context of superconducting circuits.
4
Note that here, the Kerr-shift is defined such that a positive value of ζ denotes a negative shift in the resonancefrequency.

21
2.1.6

Coupled oscillators

Many physical systems are described by two or more resonators coupled together. In thissection, we consider two RLC-oscillators coupled by an inductance lg as in Fig. 2.4a.5
Four coordinates are required to describe these two oscillators: ΦA (t), ΦB (t) the branch
fluxes across inductors lA and lB , respectively, and qA (t) and qB (t), the charges on the platesof capacitors cA and cB , respectively. Four coupled first-order differential equations describe therelationship between these parameters:
1
qA ,
cA


1
1
1
1
q̇A = −+ΦA −qA + Φ B ,
lA lg
rA cA
lg
Φ̇A =

(2.21)(2.22)

along with two analogous equations specifying Φ̇B and q̇B .
√We allow the bare resonant frequencies of the two oscillators ωA = 1/ lA cA and ωB =√1/ lB cB to be different, but for simplicity enforce their characteristic impedances to be equalppsuch that z0 =lA /cA =lB /cB . Doing so and specifying the loss rates of each resonatorκA = (rA cA )−1 and κA = (rB cB )−1 , the circuit equations of motion becomeq̇A = −

ωA
1
+
z0
lg

Φ̇A = ωA z0 qA ,(2.23)

1ΦB ,lg(2.24)

Φ A − κA qA +along with two analogous equations for Φ̇B and q̇B . We then transform to the normal coordinates


√
√
√
√
a = √12~ ΦA / z0 + i z0 qA and b = √12~ ΦB / z0 + i z0 qB , giving

 κ 

A
ȧ = −iωA a − ig a + a† − b − b† −a − a† ,
2

 κ 

B
ḃ = −iωB b − ig b + b† − a − a† −b − b† ,2(2.25)(2.26)and their complex conjugates. Here, we have defined the coupling rate g = z0 /2lg .
5Similar dynamics emerge if this coupling inductor is replaced by a coupling capacitor. Recall that at a given
frequency ω, there exists some capacitor cg whose impedance has the same magnitude as that of lg such thatjωlg = −1/jωcg .

22

l0

rA

energy (a.u.)

(b)

cA

lB

lA

cB

rB
6.10

1.0
0.8
0.6

|𝐴|

0.4

|𝐵|2

2

0.2
0.0

0

20

40
60
time (ns)80100eigenfrequencies (GHz)

(a)

(c)

6.05
2𝑔/2𝜋
6.00

(𝜔𝐴+𝑔)/2𝜋

𝜔𝐴/2𝜋

5.95

5.90

5.95

6.00
6.05
𝜔𝛣/2𝜋 (GHz)

6.10

Figure 2.4: Two coupled oscillators. (a) Two resonators labelled ‘A’ and ‘B’ are coupled by an
inductance lg . Forpsimplicity, p
we work in a limit where each oscillator has the same characteristic
impedance z0 = lA /cA = lB /cB . (b) Solutions to the equations of motion for this circuit
taking ∆ = 0 and g = z0 /2lg = 2π × 12.5 MHz, with loss set to zero such that κA = κB = 0 anda boundary condition at t = 0 of A(0) = 1, B(0) = 0, and Ȧ(0) = Ḃ(0) = 0. Here, A(t) and B(t)are the fields in respective resonators in the rotating frame. (c) Eigenfrequencies (black lines) with
ωA /2π = 6 GHz while sweeping ωB /2π. When ωB = ωA the ‘dressed’ eigenfrequencies are split by
2g/2π.

Next, we express the field in each oscillator in a frame rotating near their resonance frequencies
0

0 = ω + g and ω 0 = ω + g by making the transformations a(t) = A(t)e−iωA t and b(t) =
ωA
B
A
B
0

B(t)e−iωB t . We also define the detuning ∆ = ωA − ωB . Doing so and removing the fast-rotating
0  ∆ and ω 0  ∆), gives Ȧ = − κA A+igei∆t B and Ḃ = − κB B+ige−i∆t A,
terms (i.e. assuming ωAB
22

along with their complex conjugates. Here, A(t) and B(t) are phasors corresponding to the fields
in the ‘A’ and ‘B’ oscillators in their respective rotating frames. We are often interested in the‘degenerate’ case where ∆ = 0 in which these equations simplify toκA
A + igB,2
κB
Ḃ = − B + igA,
2
Ȧ = −(2.27)(2.28)and their complex conjugates. Solutions to Eqs. 2.27 and 2.28 are plotted in Fig. 2.4b. In this case,23energy swaps back and forth between the oscillators with a swap time of π/2g.
The Hamiltonian describing the circuit in Fig. 2.4a is the sum of the energy in either oscillator,
plus the energy stored in the coupling inductor:00
H = ~ωAA† A + ~ωB
B † B + ~g A† B + AB † ,(2.29)

where fast-rotating terms e.g. AB and A† B † are neglected. The final term on the right-hand sideof Eq. 2.29 is called the beamsplitter Hamiltonian.
It can be useful to consider the eigenfrequencies of Eqs. 2.25 and 2.26, plotted in Fig. 2.4c.
For the degenerate case of ω0 = ωA = ωB , these eigenfrequencies are ω0 + g ± g. Physically, these
two eigenfrequencies correspond to symmetric and anti-symmetric modes. In the symmetric mode,no current flows across the coupling inductor and the fields in both resonators oscillate in phase at
the ‘decoupled’ resonance frequency ω0 . In the antisymmetric mode, the fields in both resonators
oscillate out of phase at the higher frequency ω0 + 2g.

2.1.6.1

Dispersive limit

We often consider the case where the oscillators are far detuned such that g/∆  1. First,
consider the lossless case such that κA = κB = 0. The eigenfrequencies λA and λB are then
g2
,
∆
g2
0
λ B = ωB
+ ,∆0
λA = ωA−(2.30)(2.31)

0 and ω 0 of either resonator are changedplus higher order corrections. The ‘bare’ eigenfrequencies ωAB

by the dispersive shift equal to ±g 2 /∆.
In many systems, one resonator is much lossier than the other. We now take κA = 0 andκB = κ to be nonzero, but in the limit where κ/∆  1. The now complex eigenvalues of the systemare:g 2 iκg 20
λ A = ωA
−
−
,
∆
2∆2

g2
g 2 iκ
0
λB = ωB+−
1− 2 ,∆
2∆(2.32)(2.33)

24
plus higher order corrections. Resonator ‘A’ therefore inherits an energy decay rate of κg 2 /∆2 due
to its dispersive coupling to the lossy resonator ‘B’.

2.1.7

Parametrically coupled oscillators

We now consider a time-dependent modulation of the coupling inductor such that: lg (t) =l0 / (1 +  cos (Ωt + θ)) as in Eq. 2.10. This results in the time-dependent parametric coupling
gp (t) = z0 /2lg (t), or
gp (t) = g + 2ν cos (Ωt + θ) ,(2.34)

where g = z0 /2l0 is the time-independent coupling of the previous section and ν = z0 /4l0 is a new
time-dependent coupling. For the remainder of this section we set θ = 0.
Eqs. 2.25 and 2.26 are now modified such that the static coupling g of the previous section
is replaced by the dynamic coupling gp (t) is given by Eq. 2.34. We make this substitution andtransform to frames rotating at the respective resonant frequencies of the cavities such that a(t) =A(t)e−i(ωA +2g)t and b(t) = B(t)e−i(ωB +2g)t . Neglecting the fast-oscillating terms yields the following
equations of motion when modulating at the difference frequency between the resonators:κAȦ = − A + iνB 
2
Ω=∆

κB

Ḃ = − B + iνA2

(2.35)and their complex conjugates. Related equations of motion are obtained when modulating at thesum frequency:

κA
†

Ȧ = − A + iνB 2Ω = 2ω0κB†Ḃ = − B + iνA2(2.36)and their complex conjugates. Modulating the coupling between two resonators at their difference
frequency, as in Eq. 2.35, turns on a coupling between the two resonators, just like the systemdescribed by Eqs. 2.27 and 2.28. Now, however, the cavity bare resonant frequencies can be
detuned by an arbitrary amount much greater than their bare coupling rate g. Modulation at thesum frequency, as in Eq. 2.36, turns on parametric gain.

25
Notice that the strength of the parametric interaction ν = g/2 is a function of the barecoupling g and the parametric modulation amplitude . Many physical systems are described by
the limit where ||  1, so that the strength of parametric interaction is weak compared to thedegenerate case (Eq. 2.27 and Eq. 2.28).

2.2

Quantum oscillators

2.2.1

Quantization of the electromagnetic field

In Sec. 2.1 the flux Φ and charge q, or likewise the fields a and a† , are scalar values which
commute and can simultaneously be zero [82, 86, 87]. In a quantum picture this is not true. Instead,these variables are replaced by analogous operators Φ̂, q̂ and â, â† which obey the commutation
relations [81],ihâ, â† = 1,hiΦ̂, q̂ = i~,(2.37)

where â and â† are the creation and annihilation operators. Note that Eq. 2.2 still describes the √√ 
normal mode transformation between these variables, such that â = √12~ Φ̂/ z0 + i z0 q̂ , wherez0 is the characteristic impedance of the resonator.

2.2.2

A quantum simple harmonic oscillator

The quantum mechanical Hamiltonian for a simple harmonic oscillator, e.g. Fig. 2.1a, is1Ĥ = ~ω0 â† â +.
(2.38)
2
Note the similarity between Eq. 2.38 and Eq. 2.4. In the quantum description, the fields are replacedby operators and there is an extra ‘zero-point energy’ ~ω0 /2.
A quantum simple harmonic oscillator has the eigenstates |ni such that Ĥ |ni = En |ni, wheren = 0, 1, 2, etc.. Here, the energy eigenvalues En = ~ω0 (n + 1/2) of the harmonic oscillator forma ladder of levels equally spaced by ~ω0 . The creation and annihilation operators and the number
operator n̂ = â† â act on the eigenstates |ni byâ |ni =√n |n − 1i ,â† |ni =√n + 1 |n + 1i ,n̂ |ni = n |ni ,(2.39)

26
with â |0i = 0.
The quantum state of the harmonic oscillator can in general be in a superposition of numberP2
eigenstates, such that |ψi = ∞n=0 cn |ni, where |cn | gives the probability of the oscillator being inP
2
number state |ni when measured. States are normalized such that ∞n=0 |cn | = 1.
2.2.3

Coherent states

2.2.3.1

Representation in the number basis

Of special importance are coherent states, denoted |αi [82, 86, 87]. Coherent states are
eigenstates of the annihilation operator such that â |αi = α |αi. From this property,
|αi =

∞
X
αn
2
√ e−|α| /2 |ni ,n!
n=0(2.40)

where α is a complex number specifying the amplitude and phase of the coherent state. The
probability that a coherent state will be measured in a particular eigenstate |ni follows a Poisson
distribution with a mean of |α|2 (the mean photon number). This probability distribution |hn|αi|2 =2|α|2n e−2|α| /n! is plotted in Fig. 2.5a for three different photon numbers. Notice that α = 0
corresponds to the oscillator being in state |n = 0i with unit probability. This special coherent
state is called the vacuum state.

2.2.3.2

Representation in the quadrature basis

It is often useful to describe the state of a simple harmonic oscillator in the quadrature basis,
rather than the number basis. Unlike the discrete number basis which is defined by eigenstates of
the number operator n̂ = â† â, the quadrature basis is a continuous basis defined by two orthogonal
quadrature operators,6
x̂ =


1
â + â† ,
2

ŷ =


1 
â − â† .
2i(2.41)

6
Note there are different conventions for how to define the quadrature operators. We follow the conventions of
Loudon [86] and Gerry & Knight [87], but Walls & Milburn [82] take x̂ = â + â† and ŷ = â − â† /i, yielding a
different uncertainty relation in Eq. 2.42, etc.

27
1.0
probability, |⟨𝑛|𝛼⟩|2(a)|𝛼|2 = 0

0.8

2 (b)
vacuum1|𝛼|2 = 0.63
|𝛼|2 = 2.4

0.6

2
1

𝜃

𝑦 0

0.4

𝛼 = |𝛼|𝑒𝑖𝜃

-1

0.2
0.0

(c)
displacedvacuum-1|𝛼|2 = 0

-2
0

1

2
3
4
5
number state, |𝑛⟩

6

0

-2

-1

-2

|𝛼|2 = 2.4
0
𝑥

1

2

-2

-1

0
𝑥

1

2

Figure 2.5:
Coherent states. (a) Number state probability distribution for three different
coherent states of mean photon number |α|2 . The state with |α|2 = 0 is the vacuum state. (b,c)
Quasiprobability distribution (the Wigner function, Eq. 2.44) of a coherent state with |α|2 = 0, and|α|2 = 2.4 at angle θ = π/4, respectively. The complex number α = |α|eiθ , illustrated by the arrowin (c), displaces the vacuum state from the origin to create the coherent state |αi. Note that here,θ is defined so that θ = 0 corresponds to a state pointing along the y-quadrature, and θ = π/2
corresponds to a state pointing along the x-quadrature.

These operators are analogous to the charge and flux operators in dimensionless units, and haveeigenstates |xi and |yi respectively such that x̂ |xi = x |xi and ŷ |yi = y |yi. They have the non-zerocommutation relation [x̂, ŷ] = i/2, yielding the uncertainty relation, 21(∆x) (∆y) ≥
.
4
2

2

(2.42)

Here, (∆x)2 is a variance (i.e. the square of a standard deviation ∆x) associated with an uncertainty
along the x-quadrature such that (∆x)2 = x2 − hxi2 . Eq. 2.42 implies that even in its lowest
energy eigenstate, the ‘vacuum fluctuations’ of the field in an oscillator are not zero, seen alreadyby the zero-point energy ~ω0 /2 in Eq. 2.38. For a coherent state, the fluctuations along eachquadrature are equal such that ∆x = ∆y = 1/2.
To express a coherent state in the quadrature basis, we make use of the displacement operator
D̂(α). Coherent states can also be defined as the displacement operator acting on the vacuum statesuch that†∗D̂ (α) = eαâ −α â ,|αi = D̂(α) |0i .

(2.43)

Notice that when |α| = 0 the displacement operator is the identity matrix, and so the oscillator

28
remains in the vacuum state.
A quantum state may be graphically represented in phase space by one of several quasiprobability distributions. Commonly used is the Wigner function, defined as [82, 86, 87]:
W (x, y) =

1
π~

Z ∞



x0
x0
0
dx0 x +
ρ̂ x −
eix y/~ ,22−∞(2.44)

where ρ̂ = |αihα| for a coherent state, which in Eq. 2.44 is represented in the quadrature basis, withx, y and x0 real variables. The Wigner function of a vacuum state |0i with photon number |α|2 = 0
is a Gaussian centered at the origin, Fig. 2.5b, and a non-zero α translates this Gaussian, Fig. 2.5c.
The meaning of the displacement operator thus becomes clear: coherent states are vacuum statestranslated from the origin to a mean position in phase space specified by the complex number α.
To determine the ‘wavefunction’ of a coherent state along the x-quadrature, we take the innerproduct hx|αi = hx| D̂ (α) |0i with α = |α|eiθ , defining the angle θ such that θ = 0 corresponds to astate pointing along the y-quadrature, Fig. 2.5c.7

Doing this computation and an analogous one

for the y-quadrature yields
2hx|αi = (2/π)1/4 e−i|α| sin(2θ)/2 e−[(x−|α| sin θ) −2ix|α| cos θ] ,22hy|αi = (2/π)1/4 e−i|α| cos(2θ)/2 e−[(y−|α| cos θ) −2iy|α| sin θ] .
2

(2.45)(2.46)

Given an ideal homodyne detector, the probability of measuring a coherent state to have quadrature
value x or y, respectively, is given by|hx|αi|2 =p22/πe−2(x−|α| sin θ) ,(2.47)|hy|αi|2 =p22/πe−2(y−|α| cos θ) .

(2.48)

R∞
R∞
Here, Eq. 2.47 and Eq. 2.48 are normalized such that −∞ dx |hx|αi|2 = −∞ dy |hy|αi|2 = 1, andR∞R∞
are related to the Wigner function by −∞ dy W (x, y) = |hx|αi|2 and −∞ dx W (x, y) = |hy|αi|2 .
Notice that both Eq. 2.45 and Eq. 2.46 are Gaussians with the single-quadrature standard√
deviation of 1/ 2 and the single-quadrature variance of 1/2. Eq. 2.47 and Eq. 2.48 are also
7This unconventional choice of angle is so that in Chp. 3, we measure along the conventional choice of thex-quadrature.

29
Gaussians with the single-quadrature standard deviation of 1/2 and the single-quadrature varianceof 1/4 as previously stated in Eq. 2.42. The classical limit emerges when |α|2  0, so that thephoton number is much greater than the variance of 1/4. Neglecting these quantum fluctuations,the coherent state is analogous to the phasor ‘A’ in Sec. 2.1, which is a point in phase space.

2.2.3.3

Two coherent states

Two coherent states |α0 i and |α1 i are not in general orthogonal: their inner product hα0 |α1 i =60 for finite values of α0 and α1 . However, two coherent states can be ‘more’ or ‘less’ orthogonal
dependent on the difference in phase space between α0 and α1 . The inner product between two
coherent states is straightforwardly computed by using the displacement operator in Eq. 2.43:
1

2

∗

2

hα0 |α1 i = h0| D̂† (α0 ) D̂ (α1 ) |0i = e− 2 (|α0 | −2α0 α1 +|α1 | ) . Let us consider the specific case of twocoherent states with the same amplitude but different phase, so that α0 = |α|eiθ and α1 = α0∗ =|α|e−iθ . We compute22|hα0 |α1 i| = e−2|α| sin θ ,(2.49)

where |α| sin θ = Re [α] is the projection of the phasor α along the x-quadrature, equal to the angle
θ as defined in Figs. 2.5b,c. From Eq. 2.49, we can see that the inner product of these two coherent
states goes to zero in the limit of |α| → ∞ as long as they are at different angles in phase spacesuch that sin θ 6= 0. The inner product between any two identical coherent states, including theinner product between the vacuum state and itself, is equal to 1.

2.2.4
2.2.4.1

QubitsIntroduction to qubits

Unlike the infinite dimensional Hilbert space of a harmonic oscillator, a qubit is described bya two dimensional Hilbert space with eigenstates |0i and |1i only. Unlike a classical bit which can√have value 0 or 1 only, a qubit can be in a superposition state, for example |ψi = (|0i + |1i) / 2

30with the density matrix ρ̂ = |ψi hψ|. More explicitly,|ψi = c0 |0i + c1 |1i ,∗∗
c0 c0 c0 c1 ρ̂ = 
,∗∗
c0 c1 c1 c1(2.50)

where c0 and c1 are complex valued coefficients with the property Tr (ρ̂) = |c0 |2 + |c1 |2 = 1.
The state of a qubit can be visualized using the Bloch sphere, Fig. 2.6a. A qubit’s density
matrix corresponds to a vector pointing from the origin to any position on the surface or interiorof the sphere. Points on the surface of the sphere are called pure states which have the propertyρ̂ = ρ̂2 . Any state with a nonzero component along the x or y-axes of the Bloch sphere describes aqubit in a superposition — a situation not allowed classically. States on the interior of the sphere
are mixed states corresponding to a probabilistic distribution of different pure states. State’s falling
along the z-axis of the Bloch sphere are ‘classical’ in that they correspond to the qubit being either
in the |0i or |1i state with some probability, equivalent to describing a weighted coin. For example,
the point at the center of the sphere is the maximally mixed state meaning that the qubit is either
in state |0i or |1i with equal probability — but not in any superposition between them. Some
important qubit states are summarized in Table. 2.2.4.1.
Table 2.1: Important qubit states. Manipulations of a qubit’s state can be deconstructed interms of the Pauli operators: σ̂x , σ̂y and σ̂z . Also useful are the qubit lowering operator σ̂− = |0i h1|
and raising operator σ̂+ = |1i h0|.
Name

Bra-ket notation

Density matrix, ρ̂Ground

|0i


1 0
0 0



0 0
0 1



Pure/mixed?
pure

Excited|1i

Maximally mixed

1
1
2 |0i h0| + 2 |1i h1|

1
2



1 0
0 1

mixed

Superposition√1 (|0i + |1i)
2

1
2



1 1
1 1

pure

pure

(a)

𝑧

31
𝑦
𝑥

|0⟩+|1⟩
‒
√2

(b)
|2⟩

energy

|0⟩

ℏ𝜔12
𝜙
-𝜋

|1⟩

ℏ𝜔01

|1⟩

harmonic osc.
ℏ𝜔21 = ℏ𝜔10|0⟩anharmonic osc.
ℏ𝜔21 = ℏ(𝜔10 + 𝛼)

-𝜋/2
0
𝜋/2
superconducting phase, 𝜙𝜋Figure 2.6: A qubit from an anharmonic resonator. (a) The quantum state of a qubit canbe represented on the Bloch sphere, for example
the ground state |0i, the excited state |1i, or the√particular superposition state (|0i + |1i) / 2. The density matrix ρ̂ can be represented as a vector
pointing from the origin to any point on or within the sphere. (b) The eigenstates of a harmonic
oscillator are an evenly spaced ladder of energy levels, here plotted overlaid onto a quadratic energy
potential vs. the superconducting phase φ across an inductor. Adding nonlinearity to the resonatorcan modify this potential, changing the spacing between energy levels such that ω21 = ω10 + α,
where here α is the anharmonicity, which is typically negative with 0.01 . |α/ω0 | . 0.1 forsuperconducting qubits. (Note the variable α, here referring to anharmonicity, is distinct from thesame notation used in the discussion of coherent states in Sec. 2.2.3.) The lowest two levels aretaken as a qubit with frequency ωq = ω10 . (Subplot (b) is inspired by diagrams in Refs. [81, 88].)

2.2.4.2

Superconducting qubits from anharmonic resonators

In superconducting systems, qubits are commonly created by using an anharmonic LCoscillator [88, 13].8

Recall that the eigenstates of the simple harmonic oscillator (Eq. 2.38)

are a ‘ladder’ of equally spaced energy levels. ‘Anharmonic’ means that the spacings between theseenergy levels are made to be unequal such that ω21 − ω10 = α,9 where the anharmonicity α 6= 0 is
typically negative for superconducting qubits [18]. The lowest two of these energy levels are taken
as the ground and excited state of the qubit. Because the spacing between these levels is now
unique, the qubit can be controlled by driving at this specific frequency difference ωq = ω10 . The
8
The single-photon-level nonlinearity required to achieve this anharmonicity is provided by a Josephson junction.
A Josephson junction is a non-superconducting link between two superconducting leads and functions as a nonlinear
inductor. Its energy is a cosinusoidal function of branch flux across it, rather than quadratic as for a linear inductor.
More information about Josephson junctions is provided in Appx. A.
9Note the symbol α is used separately for both the anharmonicity and coherent state amplitude.

32
qubit’s Hamiltonian is Ĥ = ~ωq σ̂+ σ̂− which is often shifted to1Ĥ = ~ωq σ̂z .
2(2.51)

Note that a qubit prepared in a superposition state will precess around the equator of the Bloch
sphere at frequency ωq . It is typical to work in a frame rotating at this frequency.
There are many different types of superconducting qubits. The ‘transmon’ qubit typicallyhas −α of several hundred MHz compared to a resonator frequency of ωq /2π typically between
5 and 10 GHz. Note that the ‘transmon’ regime is defined by the ratio of Josephson energy EJ(energy stored in the Josephson junction) to charging energy EC (energy stored in the capacitor)
such that EJ /EC ≈ 20 − 80. This ends up being a favorable tradeoff between reducing charge noiseon the capacitor while preserving anharmonicity [18, 89].

2.2.4.3

Qubit-cavity systems

Superconducting qubits are often coupled to linear cavities [15]. These cavities help keep thequbit well isolated from the environment and are also used for qubit measurement (see Chp. 3).
This qubit-cavity system is described by the Jaynes-Cummings Hamiltonian [90]:1Ĥ = ~ω0 â† â + ~ωq σ̂z + ~g â† σ̂− + âσ̂+ .
2(2.52)

The first term of Eq. 2.52 desribes a linear cavity with the zero-point energy neglected and thesecond term describes the qubit. The third term describes excitations being exchanged between thecavity and the qubit, parameterized by a coupling rate g. Note that the Jaynes-Cummings model
is within the rotating-wave approximation and ignores any dissipative processes.
Superconducting systems are often engineered in the dispersive limit of the Jaynes-Cummings
model [15, 13], meaning |g/∆|  1 where ∆ = ω0 − ωq is the detuning between the cavity andqubit. In this limit Eq. 2.52 simplifies to1
Ĥ = ~ω00 â† â + ~ωq0 σ̂z + ~χâ† âσ̂z .
2(2.53)

33
Here, ω00 = ω0 + g 2 /(∆ + α) and ωq0 = ωq − g 2 /∆ are the shifted frequencies of the resonator andqubit, respectively [18, 13], like the shifts of the eigenfrequencies of two dispersively coupled linear
cavities derived in Eqs. 2.30 and 2.31. The dispersive-shift χ is the change in the readout cavity
frequency dependent on if the qubit is in the ground or excited state. In the transmon regime thisis [88]:
 2
g
α
χ=
.
∆ ∆+α

2.3

(2.54)

Input-output theory
This section reviews input-output theory which describes the coupling between an oscillator

and an external environment. This treatment can be either classical or quantum [91, 92, 82, 93],
but the classical description is given here.

2.3.1

A one-port cavity

We first consider the case of a cavity with a single port, e.g. Fig. 2.1c. The port is parameterized by a coupling rate κ1 , along with some internal loss κint such that the cavity has a total energy
decay rate of κ = κ1 + κint . Input-output theory [91] solves the Heisenberg equation of motion
for this system including the incoming and outgoing fields ain,1 (t) and aout,1 (t), respectively, in
addition to the field a(t) inside the resonator. The equation of motion for this cavity is:ȧ = −iω0 a −√κ
a + κ1 ain,1 (t),2(2.55)

with the input-output relation:10
aout,1 + ain,1 =

√

κ1 a.

(2.56)

After taking the Fourier-transform of the input-output relations, the frequency-dependent
reflection coefficient (the scattering parameter) S11 (ω) = aout,1 /ain,1 is given by
S11 =
10

iδ + κ1 /2 − κint /2,−iδ + κ/2

(2.57)

It is a potential source of confusion that per convention, the field in the cavity a has different units from theincoming and outgoing fields ain and aout . This can be roughly understood as the incoming and outgoing fields arepropagating so |ain |2 and |aout |2 describe a flux in units of photons/s. The field in the cavity is not propagating andso |a|2 is expressed in units of photons.

34
where κ = κ1 + κint is the total loss rate (linewidth) of the cavity, and δ = ω − ω0 is the detuningfrom its resonance frequency ω0 .
Table 2.2: Cavity coupling regimes.
Name
UndercoupledCritically coupled
OvercoupledCriteria
κ1 < κ − κ1
κ1 = κ − κ1
κ1 > κ − κ1

Reflection at δ = 0<00>0

Phase wrap
0π
2πSolutions to Eq. 2.57 fall into three categories dependent on the relationship between an
external coupling rate κ1 and the additional loss rate κ − κ1 . In the under coupled case, a port’s
coupling rate is smaller than other contributions to the cavity’s loss, and the majority of incoming
radiation is reflected without interacting with the cavity. In the critically coupled case, κ1 =
κ − κ1 and on-resonance incoming radiation is entirely absorbed. Finally, in the over coupled case
κ1 > κ − κ1 , most on-resonance incoming radiation is also reflected but after interacting with thecavity, giving a different phase shift from the undercoupled case. These cases are summarized inTable 2.3.1.

2.3.2

A two-port cavity

Often we are interested in a cavity with two ports of coupling rates κ1 and κ2 , Fig. 2.7a.
This results in the term√

κ2 ain,2 (t) being added to the right-hand sides of Eq. 2.55, and a second

input-output relation analogous to Eq. 2.56 describing the field continuity at the second port.
After taking the Fourier-transform of these modified relations, the scattering parameters are
S11 = aout,1 /ain,1 and S21 = aout,2 /ain,1 are given byiδ + κ1 /2 − κ2 /2 − κint /2,
S11 =−iδ + κ/2

√

S21 =

κ1 κ2
,
−iδ + κ/2

(2.58)

where κ = κ1 + κ2 + κint . Solutions to S11 for the three cases in Table 2.3.1 are plotted in Fig. 2.7.
Such analysis can be extended to N -port cavities. In a cavity with N ports, κ = κint + ΣN
i=1 κiand the subscripts 1 and 2 in Eq. 2.58 can be replaced by any desired choice of ports (e.g. Sij

(a)
𝑎in,1
𝑎out,1

𝜅1

𝜔0

𝑎

35

𝜅int
𝜅2

(b) 1.0

undercoupled
critically coupled𝑎in,2𝑎out,2(c)overcoupled𝜋(d)Im{𝑆11}
1

arg{𝑆11}

|𝑆11|

0.8
0.6
0.4

0
-1

0

1

Re{𝑆11}

0.2
0.0

2
-2
0
detuning 𝛿/2𝜋 (MHz)

-𝜋

2
-2
0
detuning 𝛿/2𝜋 (MHz)-1

Figure 2.7: Scattering parameters of a cavity. (a) Input-output theory model for a 2-port
cavity. The field a inside the cavity interacts with incoming and outgoing fields ain and aout ,respectively, from either port. The cavity has a bare resonance frequency of ω0 , external coupling
rates of κ1 and κ2 , an internal loss rate κint , and a total loss rate κ = κ1 + κ2 + κint . Internal
loss is equivalent to a ‘fictitious’ third port with coupling rate κint , whose outgoing field cannot
be measured. (b-d) Reflection from port 1, Eq. 2.57, for three different parameter regimes. The
frequency response is a Lorentzian function. Here the ‘undercoupled’ case takes κ1 /2π = 0.1 MHz
and κ2 /2π = 1 MHz, the ’critically coupled’ case takes κ1 /2π = 0.5 MHz and κ2 /2π = 0.5 MHz,and the ‘overcoupled’ case takes κ1 /2π = 1.0 MHz and κ2 /2π = 0.2 MHz. All cases take κint = 0.
(b) Absolute value of reflection. (c) Phase of reflection. (d) Representation in the complex plane.

where i ∈ {1, 2, ...N } and j ∈ {1, 2, ...N }), but the form of Eq. 2.58 otherwise remains the same.

2.4

SummaryThis chapter introduces some important examples which are generally relevant to the field ofsuperconducting circuits and the work presented later in this thesis. In a ‘classical’ description, thestate of an oscillator is specified by the scalar fields a and a† which commute, and specify a single
point in phase space. In a ‘quantum’ description, the oscillator is instead described by operatorsâ and â† which do not commute, leading to an uncertainty relation and the state of the oscillator
being a ‘blob’ in phase space.
In the number basis, the energy eigenstates of simple harmonic oscillator form a ladderof equally-spaced eigenstates (‘rungs’ on the ladder). Nonlinearity added to the resonator can36
Table 2.3: Summary of different oscillators. Hamiltonians are listed
 in a quantum description
where â and â† , etc. are operators with the commutation relation â, â† = 1, and zero-point
energy is neglected. In a classical description, operators are replaced by scalar fields. Resonance
0 → ω , ω 0 → ω , ω 0 → ω and ω 0 → ω in thefrequencies are re-parameterized such that ωA
0
q
A
B
q
0
B
Hamiltonians listed here, which are written in the rotating-wave-approximation when appropriate.
Name

SectionHamiltonian

Physical example

harmonic
oscillator

2.1.1, 2.1.2,
2.2.2~ω0 â† â

superconducting
microwave cavity [94]parametricoscillator

2.1.4



2
~ω0 â† â + 21 ~ ξâ2 + ξ ∗ â†‘flux-pumped’
JPA [29, 95]

nonlinearoscillator

2.1.5~ω0 â† â − 12 ~ζ â† â

coupledoscillators

2.1.6



~ωA â† â + ~ωB b̂† b̂ + gâ† b̂ + g ∗ âb̂†

SIMBA +
readout cavity [23]qubit

2.2.4.1

1
2 ~ωq σ̂ztransmon [18, 89]

Jaynes-Cummings model,
dispersive limit

2.2.4.3~ω0 â† â + 21 ~ωq σ̂z + ~χâ† âσ̂zqubit + readout
cavity [14, 97]

2

JPA (ζ  κ) [28],aqubit (ζ  κ) [15]aA Josephson parametric amplifier (JPA) is a nonlinear cavity where the Kerr-constant ζ (frequency-shift perphoton) is typically much smaller than the linewidth κ, see Chp. 4. The JPA can generally be treated in theclassical limit; note that its amplified state can contain hundreds of photons, far larger than vacuum fluctuations. A
superconducting qubit (e.g. a transmon) is also an example of a nonlinear oscillator, but where ζ is generally muchlarger, so that the oscillator has a nonlinear response for even a single excitation and must be treated with a quantum
description. See Ref. [96] for analysis of the transition between these limits.

shift the spacing of these eigenstates so that a qubit can be formed from the lowest two of these
eigenstates. In superconducting qubits, this nonlinearity is provided by the Josephson junction,
which is nonlinear even at the single-photon level. In practice, superconducting qubits are often
dispersively coupled to a readout resonator, realizing the Jaynes-Cummings Hamiltonian in thedispersive limit.
To measure a resonator, a ‘port’ is introduced to couple the field in the cavity to an external
environment. This coupling is described using the formalism of input-output theory. The frequency

37
dependent response of a signal scattered from this port can then be used to measure parameters of
the cavity.
Not discussed in this chapter are systems of many oscillators, the effects of non-zero temperature, the fluctuation-dissipation theorem, squeezed light, and many other important examples
from quantum optics. Further information and more detailed discussion can be found in textbooks[82, 86, 87, 93] or review articles [24, 81, 88, 98].

Chapter 3

Qubit measurement

3.1

Formalism of quantum measurement
Qubits are the basic unit of quantum information, analogous to bits in classical computing.

Measuring the state of a qubit in a ‘high-quality’ way is essential to any quantum computing
protocol on any hardware platform. In this chapter, we precisely define qubit measurement andseveral important metrics describing its performance. We then show how these metrics can bequantified.

3.1.1

Difference from classical measurement

First, we briefly acknowledge the fundamental difference between classical and quantum measurement. A classical bit is in either the 0 or 1 state, both before and after any measurement. A√
qubit prepared in the state (|0i + |1i) / 2 is in a superposition state of ‘both’ |0i and |1i at thesame time (the ‘superposition state’ in Table. 2.2.4.1). This is Schrödinger’s cat being alive anddead, if you like, and is fundamentally different than the qubit being in either |0i or |1i (e.g. the‘maximally mixed’ in Table. 2.2.4.1). When a qubit prepared in a superposition state is projectively
measured, its state is collapsed onto either the |0i or |1i eigenstate — equivalent to ‘looking’ atSchrödinger’s cat and finding it either alive or dead.
What exactly this ‘quantum superposition’ means has been, and will continue to be, thesubject of a great deal of debate within the context of the philosophy of physics. Quantum superposition troubled Einstein [99], who famously objected to it saying “God does not play dice39
with the universe.” Going forward, we will ignore such debates and take measurement to be anaxiom of quantum theory, per the modern viewpoint described in John Preskill’s notes on quantum
information [100].

3.1.2

POVMs and Kraus operators

Recall that a quantum state is fully described its density matrix ρ̂ = |ψi hψ|.1

A quantum

measurement on this state is characterized by a positive operator-valued measure (POVM) Π ={Πx : x ∈ X }, which consists of a set of positive semidefinite operators Πx ≥ 0 labeled by thePmeasurement outcome x in some set X , and satisfying x∈X Πx = I. For each x one may choose
Kraus operators Ax such that Πx = A†x Ax . Note that the choice of Kraus operators is not unique.
Given a measurement outcome x, the POVM maps a quantum state ρ̂ to a new quantum state ρ̂0x
via the formula [101]ρ̂0x =
Since the POVM is complete,Ax ρ̂A†x.
Tr (Πx ρ̂)

(3.1)

P

x∈X Πx = I, the post-measurement state is then

ρ̂0 =

X

λx ρ̂0x ,

(3.2)

x

with probabilities λx = Tr (Πx ρ̂).
In other words, measurement of a quantum state ρ̂ returns a set of possible outcomes indexedby the measurement outcome x. Each measurement outcome (each different x) maps the initial
density matrix ρ̂ to a new density matrix ρ̂0x , Eq. 3.1. This mapping is specified by the Kraus
operators A†x and Ax (or the related POVM Πx = A†x Ax ), where each possible measurement value
x corresponds to a separate Kraus operator and POVM. The quantum state’s post-measurement
density matrix ρ̂0 , Eq. 3.2, is the mixed state which is the ensemble of density matrices corresponding to all possible measurement outcomes ρ̂0x , weighted by the probabilities λx of obtaining eachmeasurement.
1

See Sec. 2.2.2 for more information.

40
For example, consider a pre-measurement pure state c0 |0i + c1 |1i. After a projective measurement, this state will ‘collapse’ to either |0i with probability |c0 |2 or |1i with probability|c1 |2 = 1 − |c0 |2 . Since this measurement has two possible outcomes labelled by 0 or 1 there
are two Kraus operators A0 = |0i h0| and A1 = |1i h1|, or equivalently:




1 0
0 0
A0 = 
A1 = 
,
.
0 0
0 1
Applying Eq. 3.1 and Eq. 3.2 to compute the post-measurement density matrix ρ̂0 gives
2 c c∗
2
|c
|
|c
|
0
0 1
 0
 0

0
ρ̂ = 
 −→ ρ̂ = ,0|c1 |2c∗0 c1 |c1 |2

(3.3)

(3.4)

with measurement outcome probabilities of λ0 = |c0 |2 and λ1 = |c1 |2 .
If instead a quantum state is described in a continuous basis, its measurement can be described by a continuum of infinitely many Kraus operators. For example, measurement along thex-quadrature of a coherent state |αi, using an ideal homodyne detector, is described by the Kraus
operators A†x = Ax = |xi hx| corresponding to the POVM,
Πx = |xi hx| .

(3.5)

For these Kraus operators and the pre-measurement density matrix ρ̂ = |αi hα|, Eq. 3.1 and Eq. 3.2
give the post-measurement density matrix ρ̂0 = Σx λx |xi hx|, where magnitude λx = |hx|αi|2 =p22/πe−2(x−|α| sin θ) follows a Gaussian distribution, Eq. 2.47.
As we discuss later in Sec. 3.3.3, measurement of a quantum state may be done withadded noise and/or loss before detection, with imperfection quantified by the ‘measurement efficiency’ 0 ≤ η ≤ 1. While Eq. 3.5 gives the Kraus operators for an ideal homodyne detector, defined here to mean η = 1, the Kraus operators for a general homodyne detector areR∞
0 −(x−x0 )2 /4∆2η |x0 i hx0 | with POVM [102]:A†x = Ax = √ 1√−∞ dx e∆η 2π1Πx = q
2π∆2η

Z ∞
−∞

0 2

2

dx0 e−(x−x ) /2∆η x0

x0 .

(3.6)

41
Physically, ∆η =p
(1 − η) /4η is a standard deviation parameterizing the noise added by thedetector,2 equivalent to loss caused by a beamsplitter of transmissivity η between state preparation
and an ideal homodyne detector. In the limit of η → 1 corresponding to ∆η → 0, Eq. 3.6 reduces
to Eq. 3.5. Applying Eq. 3.1 and Eq. 3.2, using Eq. 3.6, gives the post-measurement density matrixp2ρ̂0 = Σx λx |xi hx|, where magnitude λx = 2η/πe−2η(x−|α| sin θ) follows a Gaussian distribution√ 
whose standard deviation is 1/ 2 η , taking a minimum value of 1/2 at η = 1. Such a measurement
gives the same signal-to-noise ratio as using an ideal homodyne detector to measure a coherent state
whose displacement |α| sin θ along the measured quadrature is first reduced by the factor

3.1.3

√

η.

Measurement induced dephasing

Notice from Eq. 3.4 that a qubit prepared in a superposition state, for instance taking c0 =√c1 = 1/ 2 for a qubit prepared along the equator of the Bloch sphere, is taken to a mixed state
after measurement. In other words the qubit is dephased by measurement. For the Kraus operatorsin Eq. 3.3 this measurement is projective, such that the post-measurement state ρ̂0 falls entirely
along the z-axis of the Bloch sphere.
For a different type of measurement, the post-measurement qubit state need not be entirely
dephased, or equivalently the post-measurement density matrix might still have some non-zero offdiagonal value. For example, consider the common scenario of a qubit prepared in a superposition
state and then entangled with a measurement system in the state |ϕ0 i or |ϕ1 i to create the bipartitestate1|ψi = √ (|0i |ϕ0 i + |1i |ϕ1 i) ,
2

(3.7)

with density matrix σ̂ = |ψi hψ| describing the joint state of the qubit and the measurement
system. For the POVM element Πx = I ⊗ |xi hx|, Eq. 3.1 gives the following state after obtaining
the measurement outcome x:σ̂x0 =
2

1 

2λx

2



|hx|ϕ0 i|hx|ϕ0 i hϕ1 |xi  ⊗ |xi hx| ,2
hx|ϕ1 i hϕ0 |xi|hx|ϕ1 i|

Note that the noise added by the detector is assumed here to follow a Gaussian distribution.

(3.8)

42
where λx = Tr (Πx σ̂). Summing Eq. 3.8 over all measurement outcomes as in Eq. 3.2 and takingP
the partial trace over the second system gives the average density matrix ρ̂0 = Tr2 ( x λx σ̂x0 ) of thequbit after measurement:3
ρ̂0 =





1

2

hϕ1 |ϕ0 i .
hϕ0 |ϕ1 i
1
1

(3.9)

In other words, the projective measurement on the ancilla via POVM element Πx transforms thepure qubit state into the mixed state given by Eq. 3.9.4 Qubit coherence (the off-diagonal density
matrix element of the qubit) will be reduced by the inner product hϕ1 |ϕ0 i, i.e. the extent to which|ϕ0 i and |ϕ1 i are orthogonal.

3.1.4

Environment induced dephasing

The concepts of measurement and dephasing can now be connected more generally. ‘Dephasing’ describes any process where a qubit prepared in a superposition state reduces to a mixed state,corresponding to its off-diagonal matrix elements ρ̂001 diminishing in magnitude.
In a projective measurement, as in Eq. 3.4, it is simple to think of this dephasing as happening
at a specific moment. Any physical qubit, however, will also naturally dephase over time as it
interacts with its environment. A qubit’s energy relaxation rate Γ1 = 1/T1 and dephasing rate
Γ2 = 1/T2∗ evolve its density matrix over time according to [81] 1+
ρ̂0 (t) = |c0 |2 − 1 −Γ te 1c∗0 c1 e−Γ2 tc0 c∗1 e−Γ2 t|c1 |2 e−Γ1 t,(3.10)

for a qubit starting in state c0 |0i + c1 |1i at time t = 0. The dephasing rate is the sum ofhalf the energy relaxation rate Γ1 = 1/T1 and the ‘pure’ dephasing rate Γφ = 1/Tφ such that
1/T2∗ = 1/2T1 + 1/Tφ . In the limit where the qubit’s energy relaxation rate T1 goes to infinity, then
Γ2 → Γφ .
3

The partial trace Tr2 (·) is defined as the unique linear operator satisfying Tr2 (X ⊗ Y ) = XTr(Y ) for any two
operators X and Y . When applied to a bipartite state, the partial trace Tr2 yields the marginal of the first constituent
system.
4
Note that Eq. 3.9 holds regardless of the value of measurement efficiency η, or equivalently when using the
POVM given in Eq. 3.6 rather than Eq. 3.5.

43
As Γφ t → ∞, the qubit has been ‘dephased’ into a mixed state falling along the z-axis of the
Bloch sphere. Assuming Γ1 = 0, the qubit’s post-measurement density matrix is not distinguishable
from that after a projective measurement, Eq. 3.4. Therefore measurement and dephasing are ina sense the same operation. Dephasing can be understood as the qubit being ‘measured’ due
to interaction with degrees of freedom in the environment, with the measurement results hidden
insofar as the environment cannot be probed to gain information about the qubit state.
In superconducting qubits dispersively coupled to readout cavities, pure dephasing is often
dominated by the presence of a thermal state in the readout cavity (as opposed to vacuum) [103,104]. When the mean thermal photon number in the readout cavity n̄th is much less than one, thepure dephasing rate due to thermal photons is [105, 106],Γφ =4κr χ2n̄th .
κ2r + 4χ2

(3.11)

Here, χ is the dispersive shift defined such that the readout cavity shifts by ±χ/2π dependent on
the qubit state (as in Eq. 2.53), and κr is the energy decay rate of the readout cavity.

3.2

Dispersive readout
In dispersive readout of superconducting qubits [15], it is typical to scatter a coherent state5

off of a readout cavity and then measure the amplitude and/or phase of the scattered pulse. A
coherent state in the readout cavity becomes entangled with the qubit to make the bipartite state1|ψi = √ (|0i |α0 i + |1i |α1 i) .
2

(3.12)

Note the similarity to Eq. 3.7, where |ϕ0 i = |α0 i and |ϕ1 i = |α1 i, illustrated in Fig. 3.1a. Thesestates have equal amplitudes |α| but are separated by a maximum angle 2θ = 2 arctan (2χ/κr ) in
quadrature space, Fig. 3.1a.6
From Eq. 2.45, a normalized coherent state in the quadrature basis has the property: hx|αi =
2 1/4 −i|α|2 sin(2θ)/2 −[(x−Re[α])2 −2ixIm[α]]ee, where Re[α] = |α| sin θ and Im[α] = |α| cos θ. Note thatπ
5
6

See Sec. 2.2.3 for a review of coherent states
The angle 2θ is found by computing arg [S21 (ω)] using Eq. 2.58 for δ = ±χ and κ = κr .

44√
in this definition, hx|αi is a Gaussian function with a standard deviation of 1/ 2, variance of 1/2,
p
2
and that |hx|αi|2 = 2/πe−2(x−|α| sin θ) . Therefore |hx|αi|2 , the modulus-squared of the coherent
state, has a standard deviation of 1/2 and a variance of 1/4, as per the convention in Refs. [86, 87].
From Eq. 3.9, the qubit after measurement is in a mixed state with coherence ρ̂001 = 21 hα0 |α1 i.
Evaluating this gives

ρ̂0 =

1

2

1

2
2
e−2|α| sin θ−2|α|2 sin2 θ

e

1



,

(3.13)

as in Eq. 2.49. As |α| → ∞, qubit coherence vanishes and Eq. 3.13 reduces to the density matrix ofa qubit prepared in a superposition state and then projectively measured. Eq. 3.13 also concludes
that dephasing goes to zero in the limit where θ → 0 such that χ  κr . This makes sense —a readout signal much more strongly coupled to the environment than to the qubit will dissipate
before becoming entangled with the qubit.
With no readout signal, |α| = 0, a qubit may still be dephased by the excess backaction ofa detector (for example, due to finite isolation between the qubit and amplifier). This backaction
is parameterized as another coherent state |αb i interacting with the qubit, where αb = |αb |eiθb .
Post-measurement qubit coherence is then modified to22|ρ̂001 | = ρb e−2|α| sin θ ,2(3.14)

2

where 0 ≤ ρb = 12 e−2|αb | sin θb = 12 e−2nb ≤ 1/2 is the dephasing in excess of that caused by thereadout pulse, or equivalently
1nb = − log (2ρb ) ,2(3.15)

where nb = |αb |2 sin2 θ is the mean photon number of this backaction. Here, we have assumed thatthe state |αb i caused by detector backaction entangles with the qubit and is measured either before
or after the readout pulse |αi, so that |αb i and |αi do not interfere with each other.

std. dev. = 1/√‒22|𝛼|sin(𝜃)1/√‒2(a) ∣𝛼1⟩

∣𝛼1′⟩

∣𝛼0′⟩

𝜃

|𝛼|
counts(b)√ 𝜂loss|𝛼|

𝑑1 + 𝑑0
𝑤00
amplifiedquadrature (a.u.)

∣𝛼0⟩

𝑥

𝑑1 + 𝑑0counts
(c)

𝑤1

45

𝑤1

threshold𝑤0
amplified0
quadrature (a.u.)

(d)

𝑒

𝑔

Figure 3.1: Dispersive readout model. (a) A coherent state of amplitude |α|, whose state|hx|αi hα|yi| is illustrated√ by a blob in phase space, is a Gaussian function with a single quadrature
standard deviation 1/ 2 (see Eq. 2.45 and Eq. 2.46). Loss before a detector scales the amplitude√of this coherent state by ηloss to create |α00 i and |α10 i, here assuming that η = ηloss . (b) Linear
measurement along the x-quadrature yields the cyan (pink) distribution if the qubit was preparedin the ground (excited) state. The height of these distributions are proportional to |hx|α00 i|2 and|hx|α10 i|2 , such that their standard deviations w0 and w1 represent the coherent state variance.
After noiseless parametric amplification, the experimentally measured signal-to-noise ratio (SNR)
is defined according to Eq. 3.17. Errors due to finite SNR (b) and preparation infidelity (c), are
not distinguishable when a linear measurement result is thresholded as shown in (d). This is true
regardless of whether the thresholding is due to use of a bifurcation amplifier, or due to postprocessing of a linear measurement. However, thresholding due to the non-unitary dynamics of abifurcation amplifier is fundamentally irreversible.

3.3

Measurement efficiency

3.3.1

Definition

As a qubit is dephased by measurement, classical information about the qubit state may also
be gained. As given by Eq. 3.14, the amount of dephasing — and therefore possible information
gain — is a function of the separation in phase space between coherent states |α0 i and |α1 i. In
practice, loss present between the qubit and detector will diminish this separation before |α0 i

46and |α1 i can be measured. Loss scales the measured coherent state amplitudes by |α| →√ηloss |α|,where 0 ≤ 1−ηloss ≤ 1 is the fraction of readout energy remaining immediately before amplification,
Fig. 3.1a. The total efficiency of a measurement chain, i.e. its measurement efficiency,7 is equalto η = ηloss ηamp , where 0 ≤ ηamp ≤ 1 is the efficiency of the amplifier chain (quantifying its added
noise). In Ref. [23], we measure η but can assume ηamp = 1 and therefore η = ηloss , because weuse a phase-sensitive parametric amplifier which in principle adds no noise along the amplified
quadrature.

3.3.2

Linear measurement

Loss before measurement transforms |α0 i and |α1 i to the states |α00 i and |α10 i, respectively,
Fig. 3.1a. The ability of a detector to discriminate |α00 i and |α10 i may be quantified by a signalto-noise ratio (SNR) [24]. We define SNR [75, 13] as the separation in phase space between these√states divided by their single quadrature standard deviation of 1/ 2. The expected SNR of ameasurement along the x-quadrature is therefore
SNR =p8η|α| sin θ.

(3.16)

Referring to Fig. 3.1b,c, this is equal to the experimentally measured value of|d1 + d0 |SNR = p 2,
w1 + w02(3.17)

where |d1 + d0 | is the separation between the two measured histograms (corresponding to thequbit in the ground/excited state), and w1 = w0 are the standard deviations of these histograms(corresponding to the coherent state variance).
Eq. 3.14 and Eq. 3.16 can be used together to quantify the efficiency of a linear measurement
η=

SNR2
,
−4 log (|ρ̂001 |/ρb )(3.18)

which is a function of the experimentally measurable quantities of |ρ̂001 |, ρb and SNR. For any readout amplitude |α|, measurement of SNR and dephasing |ρ̂001 |/ρb thus determines η. It is expedient
7

Note that measurement efficiency has also been referred to as ‘quantum efficiency’ or simply ‘efficiency’ in the
superconducting qubit literature [24, 36, 107, 76, 78, 77, 49, 75].

47
to simplify this determination by measuring |ρ̂001 | and SNR at different values of readout amplitude.
Consider a readout amplitude in the experimental units  proportional to |α| (e.g. voltage bias ona mixer). From Eq. 3.16, SNR will increase linearly with . From Eq. 3.14, qubit coherence isreduced as a Gaussian function of the readout amplitude . Therefore,
SNR = a,22|ρ̂001 () | = ρb e− /2σ ,(3.19)

where a is a constant of proportionality and σ is a Gaussian standard deviation. Substituting these
expressions into Eq. 3.18 and solving for η givesη=a2 σ 2.
2(3.20)

Eq. 3.20 gives the measurement efficiency in terms of σ and a, experimental quantities which are
determined from fits to measurements of |ρ̂001 | and SNR as functions of the readout amplitude .
We reiterate that in Eq. 3.20 and Ref. [23], we define measurement efficiency such that η = 1
when using an ideal, phase-sensitive amplifier, with no other loss or added noise introduced by thedetector.8

3.3.3

Bifurcated measurement

The measurement discussed in Ref. [23], however, does not use a linear amplifier. Pumping
a parametric amplifier into bifurcation is a non-unitary process which destroys information: allpossible input states are irreversibly mapped to two output states, illustrated by the distribution
in Fig. 3.1d. Finite SNR and preparation infidelity are indistinguishable after thresholding which
prevents a direct measurement of SNR for use in Eq. 3.18 and thus Eq. 3.20.
To understand the efficiency of a bifurcated measurement, we instead consider the qubit
8

Note that Ref. [77] derives an expression which is written identically to Eq. 3.20, but whose derivation contains
two differences from ours. First, Ref. [77] defines unit measurement efficiency to correspond to an ideal phasepreseving amplifier, such that 0 ≤ η ≤ 1/2 accordingly to our definition. Second, SNR is defined in Ref. [77] toequal (d1 + d0 ) /w (taking w1 = w0 = w, referring to our Fig. 3.1), which differs from our definition in Eq. 3.17. Ourdefinition of SNR is chosen to keep with the convention in Refs. [75, 13], and references therein. In practice, suchconflicting definitions can be a source of confusion, and it is important to make sure that the definitions for η and
SNR are consistent with the formula being used.

0.1

0.5

1

5

10

0
0.1
0.01
0.001
10-4
readout infidelity
𝑃(𝑒|0) + 𝑃(𝑔|𝜋)
10-6
10-8

48
|𝛼|2
readout photon number𝜂 = 0.1
𝜂 = 0.5
𝜂=1

10-10

Figure 3.2: Readout infidelity vs. photon number. Readout infidelity 1−Fr = P (e|0)+P (g|π)
represents the inability to distinguish the coherent states |α0 i and |α1 i, which are associated with
the qubit states |0i and |1i, respectively. A model for readout infidelity (see Eq. 3.24) is plotted vs.
readout photon number nr = |α|2 (assuming sin θ = 1 and F0 = 1). At |α|2 = 10 photons, the states|α0 i and |α1 i can in principle be discriminated with error probability of less than 10−9 , far exceeding
the performance threshold necessary for error correction. For η = 0.1, however, an error rate ofseveral percent is expected. Given that the readout photon number cannot be arbitrarily increased
due to unwanted nonlinearities [32, 33], high measurement efficiency is therefore important for highperforming readout.

readout fidelity [108]Fr = 1 − P (e|0) − P (g|π),(3.21)

where P (e|0) and P (g|π) are the error probabilities of measuring the qubit to be in the excited(ground) state, when the qubit is prepared in the ground (excited) state, respectively, such that0 ≤ Fr ≤ 1. Intuitively, readout fidelity is the probability of the detector to correctly discriminate
the state that the qubit has been prepared in: Fr = 12 (P (e|π) − P (e|0)) + 12 (P (g|0) − P (g|π)).
Readout fidelity is limited by both finite readout amplitude and preparation infidelity. Preparation infidelities, p0 and q0 , are defined as the probability of having prepared the qubit in the
excited (ground) state when trying to prepare it in the ground (excited) state, respectively. These
infidelities p0 and q0 define a maximum readout fidelity F0 = 1 − p0 − q0 .
To model the readout fidelity, we model our bifurcated measurement as a linear amplification
whose result (a measured value along the x-quadrature, Fig. 3.1) is thresholded by its sign, assigning
the qubit state to one of two distinct outcomes. Measurement error is the sum of correct state

49
preparation but incorrect assignment, and incorrect state preparation with correct assignment[108]:Z 0
P (e|0) =−∞Z ∞P (g|π) =

x α10

2

i

,

(3.22)+ (1 − q0 ) x α10

2

i

.

(3.23)hdx (1 − p0 ) x α00
h
dx q0

x α00

2

2

+ p0

0

Evaluating these integrals and plugging the results into Eq. 3.21 yields,
Fr = F0 erf

hpi
2η|α| sin θ .

(3.24)

Note that the argument of Eq. 3.24 is always positive, since 0 ≤ θ ≤ π/2 as defined in Fig. 3.1a.
Using Eq. 3.14, readout fidelity can be expressed in terms of the measurement induced dephasingq
0
Fr = F0 erf−η log (|ρ̂01 |/ρb ) .
(3.25)In order to finally determine η, we use Eq. 3.25 to solve for η in a similar manner to how we
obtained Eq. 3.20. Again, we consider readout amplitude in the experimental units of  ∝ |α|. From
Eq. 3.24, readout fidelity will increase as an error function with respect to , and qubit coherence
will still decrease as a Gaussian:Fr () = F0 erf [ν] ,22

ρ̂001 () = ρb e− /2σ .

(3.26)(3.27)

Where ν and σ are obtained from fits of experimental data, e.g. Fig 3c. Using Eq. 3.24 andEqs. 3.26 and 3.27 to solve for η givesη = 2ν 2 σ 2 .

(3.28)

Eq. 3.28 gives the measurement efficiency η in terms of σ and ν, quantities which can be experimentally determined from fits to measurements of |ρ̂001 | and Fr as functions of .

3.4

Information Efficiency
The methods described in the previous section are specific to either a linear or bifurcatedmeasurement. Alternatively, measurement of a qubit using any type of detector can be understood

50
in terms of information theory [109, 110]. In this section, qubit measurement is framed as thefollowing questions: given a qubit prepared in a given state, how much information could be learned
about the state given a specific type of detector? How much information was actually learned?
Where here “information” follows a precise definition developed from information theory [111].
This analysis can be useful if the specific mechanism for qubit readout measurement is unclear
or does not cleanly follow the models of Eq. 3.20 or Eq. 3.28. This analysis will also allow us toquantitatively compare the information gained by linear vs. bifurcated measurements.

3.4.1

Accessible information√
When a qubit prepared in a superposition state (|0i + |1i) / 2 is measured, it is collapsed
toward the eigenstate |0i or |1i. How projective, or how strong, the measurement is quantifies thiscollapse. For two quantum states |ϕ0 i and |ϕ1 i of a measurement system which are entangled with
the qubit as given by Eq. 3.7, this collapse is characterized by the ‘error probability’ r associatedwith discriminating |ϕ0 i and |ϕ1 i. Error here, given by the Helstrom bound [112], is a function of
the non-orthogonality of these states. For pure states |ϕ0 i and |ϕ1 i this minimum error probabilityq
is r = 21 1 − 1 − |hϕ0 |ϕ1 i|2 , which from Eq. 3.9 equals
1 1
r= −
2 2

q
1 − 4|ρ̂001 |2 ,(3.29)

with 0 ≤ r ≤ 12 . The Helstrom bound has a simple geometric interpretation: Fig. 3.3 illustrates aqubit density matrix before and after measurement in the z-basis, such that a superposition stateis partially projected onto |0i, with some remaining phase coherence |ρ̂001 |. The error probability ris the remaining uncertainty of the post-measurement qubit state, equal to the length of the green
vector.
Explicitly, consider the states |ϕ0 i and |ϕ1 i to be prepared with equal probability. We takethat a measurement of |ϕ0 i occurs with error probability r0 , and a measurement of |ϕ1 i with error
probability r1 . The probability of obtaining the measurement outcomes corresponding to |ϕ0 i or
|ϕ1 i are therefore s0 = (1 − r0 )/2 + r1 /2 and s1 = 1 − s0 , respectively. The quantities r1 /2s0 and

51

|0⟩
𝑟

^
𝜌´

1-𝑟

^
|𝜌´|
01

qubit after
measurement(example)𝜌^

1/2

qubit before
measurement|1⟩

Figure 3.3: The Helstrom bound. A qubit is prepared in a superposition state ρ̂ and nonprojectively measured
For a given measurement outcome, the post-measurement√ in the z-basis.
√
qubit state is |φi = 1 − r |0i + r |1i, with density matrix ρ̂0 = |φi hφ|. The projection of thisstate along the z-axis, 1 − r, quantifies the remaining uncertainty r about the qubit state. In
this example a projection toward |0i is illustrated, but a projection toward |1i occurs with equal
probability.

r0 /2s1 are the fractional likelihoods of error, given by Bayes’ theorem. This is generalizable to thecase where |ϕ0 i and |ϕ1 i are prepared with different probabilities, but here we consider only thecase of Eq. 3.7 for simplicity.
Measurement therefore gains information about the qubit state, quantified by the probabilities r1 /2s0 and r0 /2s1 . As in Ref. [110], this information is quantified by the mutual information,

I = 1 − s0 H

r1
2s0




+ s1 H

r0
2s1


,

(3.30)

where H(x) = −x log2 (x) − (1 − x) log2 (1 − x) denotes the binary entropy.
The accessible information Iacc is defined as the maximum possible mutual information I
which may be obtained by a measurement. This occurs when the measurement error probabilities
saturate the Helstrom bound, i.e., r0 = r1 = r and s0 = s1 = 1/2. Eq. 3.30 then simplifies to
Iacc = 1 − H(r).

(3.31)

Note that Iacc is a function of only the post-measurement qubit density matrix |ρ̂001 |, on which r
depends via Eq. 3.29.

52

mutual info. (bits)

0.8

𝐼meas/𝐼acc

(a) 1

0.6

0.6
0.4
0.2

0
(b) 1
0.8
0.4
0.2
0

0

𝐼acc
bif
(bifurcated)𝐼meas
lin (linear)𝐼meas

bifurcatedlinear
0.4
0.8
1.2
1.6
readout photon number, |𝛼|22

Figure 3.4: Information gained by a qubit measurement. (a) Accessible information, Iacc
from Eq. 3.32, after the qubit is measured by a coherent state with amplitude |α| (dashed line, andassuming θ = π/2). Iacc grows with |α|, and (b) is always greater than or equal to the information
gain from either a bifurcated (blue) or linear (red) measurement. Notice that for even a linear
measurement (here meaning a homodyne detector with unit efficiency), Imeas remains less than
Iacc when 0 < |α|2 < ∞. For a description of other types of measurements which can saturate to
Imeas = Iacc within this range, e.g. a Dolinar receiver, see Refs. [113, 114, 115, 110].

3.4.2

Information gainbif
We can similarly determine the information attained by measurement. The information Imeas

gained from a bifurcated measurement (here meaning two possible outcomes) is:
1
1
bif
Imeas
= 1 − H [P (e|0)] − H [P (g|π)] .
22(3.32)

Here, we take that |ϕ0 i = |αi and |ϕ1 i = |−αi (assuming 2χ  κr such that θ = π/2, forsimplicity). Also for simplicity, we assume that preparation infidelity is zero, p0 = q0 = 0.
The probability of a measurement error for a bifurcated measurement, Eq. 3.23, therefore√
R0
reduces to P (e|0) = P (g|π) = −∞ |hα|xi|2 dx = 21 1 − erf 2|α| . This is plugged into Eq. 3.32
bif .
to model Imeas

Of course, a detector need not only give two measurements but instead can yield a continuum of outcomes. An example of this is a linear measurement such as homodyne detection of acoherent state which has been linearly amplified. To quantify information gain from a linear mea-

53
surement, consider discrimination between the pink and blue Gaussian distributions in Fig. 3.1b,which are prepared with equal likelihood of 1/2. If the qubit was prepared in the ground state(cyan distribution, Fig. 3.1), the probability of error for a measurement returning the value x isp2
rx0 = P (x|π) = 2/πe−2(x−|α|) . And if the pink distribution corresponding to the qubit in thep2
excited state was prepared, the probability of error is rxπ = P (x|0) = 2/πe−2(x+|α|) . The prob
ability of measuring the value x at all is the sum sx = 21 [P (x|0) + P (x|π)] = 12 rx0 + rxπ . At
each x, the information acquired from measurement is the entropy of the normalized measurementprobability, H rx0 /sx or H (rxπ /sx ). The mutual information from measurement is the integral ofthese conditional entropies over all values x, weighted by the likelihood sx of that measurement
outcome occurring [109, 110],
Z ∞

 0
 π 
rx
rx
dx H
+H
sx ,
2 −∞
sx
sx

1
lin
Imeas
=1−



(3.33)

bif
lin
bif
lin
where Imeas< Imeasfor |α| < ∞, seen in Fig. 3.4a. As |α| → ∞|, however, both Imeas
and Imeas

saturate to unity. This means that thresholding a linear measurement destroys information, but
only for the case of a weak or non-projective measurement.

Chapter 4

Josephson parametric amplifiers

4.1

Introduction to Josephson parametric amplifiers
Josephson parametric amplifiers (JPAs) are a type of superconducting parametric amplifierconsisting of a resonant cavity with added nonlinearity due to the participation of one or more
Josephson junctions [28, 29]. Such amplifiers have been developed over the past two decades
in conjunction with other superconducting microwave technologies, and are routinely used forsuperconducting qubit readout. This chapter gives a brief introduction to JPAs and then analyzes
the specific device used in Ref. [23].
In any parametric amplifier, a pump excitation at frequency ωp is converted into signal andidler excitations at frequencies ωs and ωi , respectively. Such a process can be described by either
4-wave or 3-wave mixing. In 4-wave mixing, two pump photons1 are converted to one signal andone idler photon (four photons total) such that 2ωp = ωs + ωi by energy conservation. In 3-wave
mixing, one pump photon is converted to one signal and one idler photon (three photons total)such that ωp = ωs + ωi .
JPAs can also be colloquially categorized as either ‘current-pumped’ or ‘flux-pumped’, basedon the manner in which the pump tone is operated. The nomenclature ‘current-pumped’ comes fromthe pump acting like a current source applied across the resonator’s nonlinear element, typicallyachieved by applying a pump field incident on the JPA port as illustrated in Fig. 4.1a. The pump is
1‘Photon’ assuming that the electromagnetic field is being amplified, but the theory is identical for any bosonicsystem (e.g. phonons).

55
Table 4.1: Review of Josephson parametric amplifiers (JPAs).
‘current-pumped’ JPA
4-wave mixing
pump near ω0linear amp example: Ref. [28]bifurcation amp example: Ref. [27]
𝑎pump
𝑎in
𝑎out

(a)

‘flux-pumped’ JPA
3-wave mixing
pump near 2ω0linear amp example: Ref. [29]bifurcation amp example: Ref. [116]

𝑎in
𝑎out

(b)

𝑎pump

Figure 4.1: Review of Josephson parametric amplifier (JPA) operation. (a) Example
schematic of a ‘current-pumped’ JPA. Signal, idler and pump photons enter and exit through asingle port on the resonator. Typically, a pump near the resonance frequency ωp ≈ ω0 provides
two photons which are converted to signal and idler photons at frequencies ωp ± δ where δ . κ, thelinewidth of the resonator. (b) Example schematic of a ‘flux-pumped’ JPA. Flux through a SQUID(or array of SQUIDs) participating in the resonator is modulated at twice the resonance frequency
ωp ≈ 2ω0 , creating signal and idler photons at (ωp ± δ) /2 and amplifying an incident field. In both(a) and (b), the resonator frequency ω0 can be tuned using an external dc flux often provided with
an off-chip superconducting coil (not shown). In both styles of amplifier and when operated in thehigh gain limit, the input field (ain , blue) is much weaker than the output field (aout , red), andboth are generally much weaker than the pump field (apump , green).

typically near the JPA resonance frequency in order to realize 4-wave mixing. Care must be taken
to route the reflected pump signal away from any sensitive devices (e.g. a qubit/cavity system).
The nomenclature ‘flux-pumped’ comes from modulating the flux through one or more SQUID
loops in order to modulate a resonator’s frequency at twice its natural frequency (3-wave mixing),
realizing phase-sensitive parametric gain as described in Sec. 2.1.4.
These concepts are summarized in Table 4.1 and Fig. 4.1. The JPA in Ref. [23] is a ‘fluxpumped’ device which we operate in bifurcation mode. We focus on the theory of operation of thisdevice for the remainder of this chapter. Further information on current-pumped JPAs is found in

56
Refs. [28, 84, 40, 85, 117, 118] and many references therein.
Finally, note that the bifurcation mode of operation qualitatively differs between currentpumped and flux-pumped devices. Bifurcation in current-pumped (4-wave mixing) devices is characterized by two stable states of different amplitude, as described by the physics of the driven
Duffing oscillator [84]. Bifurcation in flux-pumped devices is characterized by two stable states ofthe same amplitude, but opposite phase [85]. Both styles are referred to as ‘bifurcation amplifiers’
in the literature — a potential source of confusion.

4.2

Flux-pumped Josephson parametric amplifier design
The JPA (‘parametric cavity’) in Ref. [23] is a two-port device whose lumped element

schematic is given in Fig. 4.2a. A resonator is formed by an inductor (realized with a SQUID
array similar to that in the TIBs), with 0.43 pF capacitors to ground on either side. When the
SQUID array inductance is biased to be minimum, the array has an inductance of 0.66 nH. The
geometric inductance of the resonator is 0.52 nH.
To turn on parametric gain, flux through these SQUIDs is modulated at twice the JPA
resonance frequency using a microwave bias line (red, Fig. 4.2b), which contains an on-chip capacitor
to block dc-current. Current through a second flux bias line (containing low pass filters) can beused to change the uniform dc-flux through the SQUID array, tuning the JPA frequency. A fingered
capacitor (0.08 pF) is placed between the JPA and each TIB. This limits the coupling rate into/outof the JPA to κsp /2π = 52 MHz when a TIB is in its maximal transmit mode, and when the JPA is
tuned to 6.3 GHz.

4.3

Theory of operation
In this section we develop a theory of operation for the flux-pumped JPA used in Ref. [23].

We find that upon parametric amplification, the initial state is mapped to one of two specificlocations in phase space, which are characterized by high amplitude ( 1 photon) but opposite
phase. This classical analysis includes loss, parametric gain and nonlinearity, and expands upon

57

port
(b)

groundcapacitors

port
(a)

JPA
uniform
bias

SQUID array

LPF

LPF
groundport

groundgroundJPApump

0.5 mmport

Figure 4.2: JPA optical micrograph. (a) Lumped element schematic of the JPA (‘parametriccavity’) integrated within the SIMBA described in Ref. [23]. The JPA consists of a 2-port LCoscillator, whose tunable inductor is realized with an array of SQUIDs. The external coupling outof either port is tunable using TIBs (not shown). (b) False-color optical micrograph of this JPA.
The SQUID array (green) is at the center of the image, and is surrounded by interdigitated finger
capacitors (yellow), both to ground and connected to each port. Flux through the SQUID arraycan be controlled by two on-chip bias lines. One bias lines (red) is designed for the microwave
frequency parametric cavity pump signal, and the other (blue) is designed to apply a dc flux totune the parametric cavity frequency. Both bias lines contain low-pass filters (LPFs) realized with∼ 20 nH spiral inductors (see Sec. 5.2.2 for more information).

simpler examples found in Chp. 2.1. Note that we neglect coupling of the JPA to its external
environment (assuming the TIBs connected to both ports to be in reflect mode), and therefore
build our analysis upon the circuit in Fig. 4.3. As elsewhere in this thesis, zero temperature isassumed.

4.3.1

Full equations of motion

The equations of motion for the circuit in Fig. 4.3 are two coupled first order differential
equations governing the evolution of flux Φ(t) across the inductor l, and charge q(t) across the

58

𝜑ext(𝑡)2𝜋Φ(𝑡)/Φ0

×𝑁
𝐼𝑐

𝐼𝑐

𝑐

𝑙

𝑟

Figure 4.3: Parametric cavity model. A parametric amplifier is modelled as a parallel combination of a resistor r, capacitor c, linear inductor l, and array of N dc-SQUIDs. These SQUIDs
are assumed to be comprised of identical Josephson junctions with critical current Ic . The phasedifference across the SQUID array is 2πΦ(t)/Φ0 , where Φ(t) is the branch flux accross the arrayand Φ0 = h/2e is the magnetic flux quantum. The charge on the capacitor plates is q(t). Note that
this circuit has a slightly different topology from the circuit in Fig. 4.3, but results in equivalent
dynamics but with simpler equations of motion to initially write down.

capacitor c. From Kirchoff’s circuit laws, they are:qΦ̇ = ,cΦqq̇ = − −− Iarray ,
l
rc

(4.1)
(4.2)

where Iarray = 2Ic |cos (ϕext /2)| sin (2πΦ/N Φ0 ) is the current flowing through the SQUID array as
a function of flux across it (see Appx. A.3 for details).
It is useful to rewrite these equations in normal mode form (as in Sec. 2.1.1) given by,
1
a= √
2~




√
Φ
√ + i z0 q ,
z0

(4.3)

and its complex conjugate a† . Here, z0 is the characteristic impedance of the resonator. The circuit
equation of motion becomes,a − a† z0 ȧ = i −−a + a† −
2cz0
2l


r



κ
z0
Iarray −a − a† ,
2~
2

(4.4)

where κ = 1/rc.
Note that the linear inductance in Eq. 4.4 is comprised both of the inductor l and the linear
inductance of the SQUID array larray . We first consider a static external flux bias ϕext (t) = ϕs .
Combining Eq. 4.4 together with Eq. A.8, it follows that the SQUID array adds a ϕs -dependent

59
linear inductance larray in parallel to the inductor l. The resulting total linear inductance lres ofthe device becomes,
1larray=2π 2Ic cos (ϕs /2)
,
Φ0
N

1
lres

=

1
1
.
+
l
larrayThe corresponding values for the resonator impedance and resonance frequency are,
r
lres
1
z0 =
,
ω0 = √.
clres c

(4.5)

(4.6)

The resonance frequency ω0 is a 2π-periodic function of the static external flux bias ϕs . It assumes
a maximum at ϕs = 0 and a minimum at ϕs = ±π.
Next, we consider the parametric modulation of the SQUID critical current. This is achievedby a modulation of the external flux ϕext through each SQUID around its static bias value,
ϕext (t) = ϕs + ϕm cos (Ωt) ,Ω = 2ω0 + 2δ.

(4.7)
(4.8)

where ϕm is the modulation amplitude and the pump detuning δ is assumed to be small, such thatδ  ω0 .

4.3.2

Simplified equations of motion

Eq. 4.4, along with Eqs. 4.7 and 4.8 substituted into Eq. A.8, represent the full time-dependentequations of motion for the circuit in Fig. 4.3. These equations greatly simplify in certain realistic
limits.
First, we consider only weak excitations |2πΦ/Φ0 |  1, and in this limit we can expand thecurrent-phase relation of the SQUID arrays in Eq. A.8. We also assume that for the entire modulation period, the resonator remains detuned from its maximum frequency such that 0 < ϕext (t) < π,so that we may ignore the absolute value in Eq. A.8. Finally, we assume weak parametric modu
lation amplitude ϕm  1 so that cos(ϕext /2) = cos(ϕs /2) − (ϕm /2) sin(ϕs /2) cos(Ωt) + O ϕ2m .
In these limits, the circuit equation of motion then becomes,†ȧ = i −ω0 a + γ cos (Ωt) a + a3  κ 

ζ
†
+
a+a
−
a − a† .
3
2

(4.9)

60
Where we have defined an effective parametric drive amplitude γ and an effective nonlinearity ζ,γ=

ϕm tan (ϕs /2) z0,
4larray(4.10)

ζ=

(2e)2
z02
,
~ 8N 2 larray(4.11)

where e is the charge of an electron.
Finally, we transform Eq. 4.9 to the rotating frame such that a = Ae−iΩt/2 and ȧ =Ȧ − iΩA/2 e−iΩt/2 , where A is the cavity field amplitude in the rotating frame. Averaging
over fast oscillations removes explicit time-dependence, and the equation of motion simplifies to,γκȦ = iδA + i A† + iζ|A|2 A − A.
2
2

(4.12)

Solutions to Eq. 4.12 are plotted in Fig. 4.4a,b for the parameters used in Ref. [23].

4.3.3

Phase diagram

We now examine stable solutions to Eq. 4.12. For absent or small parametric pumping,γ ≤ κ, Eq. 4.12 has only the trivial solution A = 0. However, for γ > κ, Eq. 4.12 admits nontrivial steady-state solutions, a phenomenon known as parametric oscillation. One finds two stable,
nontrivial steady states A = |A|eiθpo to Eq. 4.12, both with the same amplitude,
1
|A| =
ζ
2




1p 2
2
−δ +
γ −κ ,
2(4.13)

but with π-shifted phases which are determined by sin (2θpo ) = κ/γ. Here |A|2 is the steady-state
photon number in the resonator. From Eq. 4.13, we can see that these states appear only for γ > κpand δ < δth = 21 γ 2 − κ2 . In the parameter regime γ > κ and |δ| < δth the trivial state A = 0 is
instable, while for γ > κ and δ < −δth it coexists as a stable state with the parametric oscillation
states [85, 116, 119]. The bistable region in parameter space is graphically illustrated in Fig. 4.4c
for the parameters used in Ref. [23].

(b)

400

10

300
200

-20

100

𝐴(0) = -1

0

0

40

80
120
time (ns)

-10

(c)

𝐴(0) = 1
10

-10
-20

20

Re[𝐴]

100
gain rate 𝛾/2𝜋 (MHz)photon number |𝐴|

2

(a)

61

Im[𝐴]
20

80

bi-stable

60 tri-stable

𝜅/2𝜋

40
20
0

-40

-20
0
20 40
detuning 𝛿/2𝜋 (MHz)Figure 4.4: Parametric oscillator dynamics. (a) The modulus squared of the field in a parametric cavity |A(t)|2 , in the rotating wave equation (solution to Eq. 4.12). Using a boundary conditionof A(0) = 1 or A(0) = −1 and Ȧ(0) = 0 at t = 0, and a parametric gain rate of γ/2π = 68 MHz,
loss rate of κ/2π = 52 MHz, pump detuning of δ/2π = 0 and nonlinearity of ζ/2π = 49 kHz as inRef. [23]. The field in the oscillator exponentially increases until it reaches a steady-state value ofapproximately 400 photons. (b) Solutions to the same simulation plotted in the complex plane.
For an initial state near the origin in phase space, the state of the oscillator will be pulled toward
one of two attractors (black stars), which are π out-of-phase from each other and represent stable
solutions to Eq. 4.12. The gray dashed line approximately specifies which initial conditions are
mapped to which attractor. The closer κ is to γ, the more directly the oscillator approaches theattactor in phase space. (c) Phase diagram for the parametric oscillator. When gain exceeds loss
(γ > κ) and detuning
p of the pump frequency from twice the resonator frequency is small enough1
such that |δ| < 2 γ 2 − κ2 (Eq. 4.13), the oscillator is bistable as in (a) and (b). The color scale
indicates the steady-state photon number |A|2 with gold or blue indicating higher or lower photon
numbers, respectively. For large enough negative detunings, the oscillator can become tri-stablewith a third attactor at the origin [85, 116, 119]. For γ ≤ κ and/or large enough positive detunings(white region), the dynamics have only one attractor at the origin representing the trivial solution
A(t → ∞) = 0. Asymmetry in the steady-state amplitude as a function of detuning is controlled
by the sign of the Kerr constant, and the tri-stable region occurs on the side of the bi-stable region
for which the steady state amplitude is higher (here the red detuned side). At high enough gain
rates and field amplitudes, higher order nonlinearities will change this phase diagram and can leadto greater numbers of stable states [85, 120].

4.4

Performance

4.4.1

Tunability

We frst characterize the JPA within a SIMBA by setting TIB2 to transmit mode and measuring in reflection off of TIB2. Doing so in Ref. [23], we find that the JPA frequency is tunable
between approximately 4 and 7 GHz (Fig. 4.5), a similar range over which the TIB is designed to62

uniform flux bias

(a)

probe

reflection (dB)

40 (c)
pump on30

pump off

20
10

35 dB

0
6.107

6.109 6.111 6.113
frequency (GHz)

6.115

3
uniform flux bias current (mA)parametric pumpreflectionangle(rad.)

(b)

2

𝜋
1
0
0

-1
-2
-3

4

5

6
7
frequency (GHz)-𝜋

Figure 4.5: JPA characterization. (a) The JPA within the SIMBA is characterized when TIB1
is set to reflect mode and TIB2 is set to transmit mode. Reflection off of TIB2 is measured. (b) The
JPA resonance frequency is tuned by changing a uniform flux through its SQUID array, appliedby current through its on-chip dc bias line (blue, Fig. 4.2b). (c) When the JPA is pumped nearresonance at the correct amplitude, it operates as a linear amplifier [95]. In the example shown
here, the JPA is tuned to give greater than 20 dB of gain over a 2.2 MHz range, with a maximumof 35 dB of gain.

operate. We emphasize that the SIMBA itself may therefore be tuned to operate over a several
GHz frequency range.
Note that the flux tuning curve shown in Fig. 4.5b was taken when sweeping current throughan on-chip bias line rather than through an off-chip coil. The rest of the data reported in Ref. [23]was taken in a separate cooldown, using an off-chip coil to tune the JPA frequency rather thanthe on-chip bias line. This functionality was temporarily removed out of concern for low-frequency
noise in the on-chip bias line, which can cause the bifurcated amplifier state to become unstable,
creating a source of readout error. With proper low frequency filtering, however, no off-chip coil isneeded to operate the SIMBA reported in this work.
While in Ref. [23] we operate the JPA/SIMBA in the bifurcation regime, it can also be
operated as a linear amplifier. This is tested in Fig. 4.5c, in which a low amplitude probe tone ofvariable frequency is reflected from one port of the SIMBA. When the parametric pump is turned

0

0.5

2

0.4 (c)
0.3

0.5
0

6.25

6.3
6.35
1/2 pump freq (GHz)

6.4

(d)

𝐼

0.1

1.5
1

63
𝑄

0.2

0
pump amplitude (V)pump amplitude (V)(a) 2.5

1

variance (a.u.)variance (a.u.)

0.4 0.8 1.2
pump amplitude (V)

(e)

𝑄
𝐼

(b)
1.2
0.8

(f)

𝑄

0.4
0

𝐼

6.33
6.34
6.35
1/2 pump freq (GHz)

Figure 4.6: Phase diagram for the pumped parametric cavity. (a) We examine the behaviorof the pumped parametric cavity by computing the variance of the distribution of single-shotmeasurements in the I/Q plane, i.e. the sum of I and Q variances. This is plotted as a function ofpump half-frequency and pump amplitude (in units of voltage bias on a double balanced mixer used
to modulate the pump amplitude). Note that for each measurement the state of the parametriccavity was near vacuum, and then the pump was pulsed on (no readout pulse to seed the cavity
state). (b) A higher resolution sweep of the boxed region in (a), and (c) a vertical linecut (thedashed white line in (b)) of variance vs. pump amplitude at the half-pump frequency used for qubit
readout. The star indicates the pump amplitude used in Ref. [23]. (d) At this operating point, theoscillator latches into one of two stable steady states. Each black dot is a single shot measurement.
(e) At a negative detuning and higher pump amplitude, we can see three stable states, one at theorigin [119]. (f) At even higher pump amplitudes, the oscillator jumps at random between two
quasi-stable states, averaging to the distribution shown.

on near the JPA resonance frequency, linear gain of up to 35 dB is observed (red line).

4.4.2

Bifurcation

In general, noise and nonlinear terms in the resonator equations of motion will lead to more
exotic behavior than predicted by the model in Sec. 4.3, which assumes ‘small’ nonlinearity taking
the limit |Φ/Φ0 |  1. Such behavior can complicate calibration of a SIMBA and can limit its
performance. To understand any such effects, we map the region in parameter space where robust
bifurcation is observed, Fig. 4.6. To do so, we make single-shot measurements of the pumped
parametric cavity with TIB2 in transmit mode, TIB1 in reflect mode, and no readout or qubit

64
pulses. The variance of the set of these measurements in the I/Q plane is plotted as the color axis
in Fig. 4.6a,b. Variance above the background level indicates a pumped parametric cavity state
which is no longer vacuum, including the presence of parametric oscillation. Stable bifurcation isobserved when pumping near twice the bare cavity resonance frequency, Fig. 4.6b. The region ofgain deviates from the parabolic region predicted by Eq. 4.13, however. This deviation is understoodto result from higher order nonlinearities in the cavity equations of motion, which can occur when
pumping at high enough amplitudes such that the cavity resonance frequency is no longer modulated
linearly with flux [85]. This effect can lower the steady state photon number from that predicted
by Eq. 4.13 [116].

4.4.3

Reset and stability
sweep delay

(a)
pump

pump

TIB2

TIB2

1

digitize(100 ns)1

digitize(100 ns)

correlated,
𝑃(0|0) + 𝑃(1|1)

0.5

0

0

digitize(100 ns)(d)

probability(c)digitize(100 ns)

sweep delay(b)

anti-correlated,
𝑃(0|1) + 𝑃(1|0)
100
200
delay (ns)

0
0

10

20 30 40delay (𝜇s)50Figure 4.7: Bifurcation reset and stability. The bifurcated state of the pumped parametriccavity is measured twice, with a variable delay between measurements. Here, the measurementsof ‘0’ and ‘1’ correspond solely to the state of the pumped parametric cavity, with TIB1 set toreflect mode for the entire sequence. (a) During the delay, the pump is turned off and TIB2 is set
to reflect mode. (b) During the delay, the pump remains on and TIB2 remains in transmit mode.
(c) When the pump is turned off, the second measurement quickly becomes uncorrelated with thefirst. (d) When the pump remains on, however, the state in the cavity remains exceedingly stable.
Data points in both plots are the mean of 1024 single-shot measurements.

We also characterize the stability and reset time of the bifurcated state at the operating point

65
used in qubit readout in Ref. [23]. Ideally, the pumped state in the bifurcation amplifier should bestable while pumped, and quickly decay when the pump is turned off. These qualities are tested by
the measurements shown in Fig. 4.7. While TIB2 is set to transmit mode, the parametric cavity
is pumped so that it latches into one of its two bistable states. While doing so, the state in theparametric cavity is digitized over two 100 ns intervals (the same duration used in our readout
scheme). These intervals are separated by a delay in which the JPA pump is turned off and
TIB2 is set to reflect mode. When sweeping this delay, the second measurement is initially highly
correlated with the first. This indicates that the bistable state of the first measurement has yet todissipate and is seeding the second measurement. After approximately 100 ns without pumping,
however, correlation between the first and second measurements has nearly vanished, indicatingthat the pumped state in the JPA has decayed to near-vacuum. When the pump and TIB state
are not changed during the variable delay however, the bifurcated state is extremely stable: thetwo measurements are greater than 99.9% correlated for any delay between them, with a maximum
measured delay of 50 µs.

Chapter 5

Superconducting switches

5.1

Review of current technologies
Over the past two decades, superconducting microwave circuits have emerged as a promising

platform for quantum information technology. This has catalyzed recent interest in superconducting
switch/coupler technologies. Different types of couplers have already been used to entangle adjacentqubits [121], entangle spatially distant qubits [122] and for qubit readout [23], for example. The
two qubit gates in Google’s recent demonstration of ‘quantum supremacy’ [20] are a particular
style of coupler designed to carry out fast, high-fidelity gates between adjacent qubits. Other
couplers [79, 123, 124] are instead designed to operate as more general purpose switches, for example
engineered to connect to a 50 Ohm transmission line on each port. As the field of superconducting
quantum information continues to grow, so will the need for switches/couplers or all types.
A selection of recently demonstrated cryogenic, microwave switches is reviewed in Table 5.1.1
Many of the devices in Table 5.1 achieve a tunable coupling by changing the inductance of a
Josephson junction or SQUID. On/off ratios of 50 dB or more have been demonstrated [23], with agreater than 40 dB on/off ratio over several GHz of instantaneous bandwidth [124] and nanosecond
switching speeds. Finally, the devices in Table 5.1 have power handling of better than −100 dBm,high enough to be integrated with a Josephson parametric amplifier operated in the high gain limit.
1

Note that there are many superconducting switches/couplers not listed in this Table 5.1. For example, Refs. [125,122, 126, 127, 128, 124] make use of superconducting couplers of a similar style to those in Refs. [121, 129, 130]. The
couplers in these works are used in conjunction with qubits on the same chip, rather than designed as stand alone
switches. Finally, tunable coupling can also be obtained without any explicit switch or coupler device, for example
via a parametric interaction (Sec. 2.1.7) as in Refs. [131, 132, 74, 133] and many others.

67
Table 5.1: Review of cryogenic switches. A selection of recent cryogenic switches which havebeen characterized at microwave frequencies. The values reported from each reference are generally
approximate, and whose precise definitions may differ across references. In the ‘Technology’ column,‘JJ’ refers to Josephson-junction based superconducting switches, ‘KI’ to kinetic-inductance based
switches, and ‘Comm.’ refers to commercial, mechanically actuated switches. Dissipation per
switching event is not characterized in most Josephson junction based devices, but is expected tobe negligible.

Reference

Tech.

On/offratio

Bandwidth

Speed

Power
dissipated

Power
limit

Rosenthal et al. (2021), [23]Chang et al. (2020), [124]Pechal et al. (2016), [134]
Chapman et al. (2016), [79, 135]
Naaman et al. (2016), [123, 136]
Wagner et al. (2019), [137]Potocnik et al. (2020), [138]Radiall: #R583423141 (2018)

JJ
JJ
JJ
JJ
JJ
KI
CMOS
Comm

50 dB
40 dB
30 dB
40 dB
20 dB
10 dB
35 dB
60 dB

4-7 GHz
4-8 GHz
7.1-7.3 GHz
4-8 GHz
9-11 GHz
2-8 GHz0-10 GHz
0-18 GHz

15 ns
10 ns
7 ns
15 ns
20 ns
0.4 ns
1 ns
25 ms

N/Aa
N/A
N/A
N/A
N/A
200 nW
36 µW
200 mW-98 dBm-83 dBm-90 dBm-85 dBm-100 dBm
N/A
N/A+50 dBmaRef. [23] reports 0.05 photons of excess backaction on a qubit due to a switching event.

The Josephson-juntion based superconducting switches in Table 5.1 can also be understoodin comparison to other cryogenic switch technologies.
Commercial cryogenic switches, for example part number ‘R583423141’ from Radiall, havegreater bandwidth, higher on/off ratios and higher power handling than other cryogenic switches
have been demonstrated to-date. However, commercial cryogenic switches work using the reconfigurable position of a mechanical component, whose switching mechanism is actuated by a large
applied current. The switching speed is far slower (25 ms) than the nanosecond time-scales required in superconducting qubit experiments. Switching also heats up the base temperature of acryostat due to the large required current (e.g. 62.5 mA specified at room temperature). Finally,
commercial cryogenic switches are bulky, non-superconducting and lossy.
Another class of cryogenic microwave switches are chip-scale CMOS technologies [139, 140,138]. These switches overcome the bulkiness problem and can have favorable bandwidth comparedto superconducting switches [138]. However, they also dissipate heat when switching although

68
far less than commercial switches. For example, the multiplexer in Ref. [138] dissipates 36 µW
when operating, raising the cryostat base temperature from 10 mK to 32 mK. Additionally, suchswitches are not superconducting and so have more significant internal loss than Josephson-junction
based alternatives. Cryogenic CMOS technologies are arguably an excellent candidate for analog
signal processing within a cryostat, but have yet to be demonstrated to be closely integrable withhigh-performance superconducting qubits.
Finally, superconducting switches may also be built based in the high kinetic inductance ofsuperconducting nanowires [137, 141]. Such switches have relatively high power handling (approximately -60 dBm), but have not yet demonstrated on/off ratios as high as the Josephson-junction
based technologies in Table 5.1.
In conclusion, Josephson-junction based superconducting switches are currently a favorable
choice for the sensitive signal routing of single-photon level microwave photons at millikelvin temperatures. These switches are especially useful for integration with superconducting qubits, as they
can have high on/off ratios with low loss and low power dissipated when switching.

5.2

Tunable inductor bridge design
The work in this thesis relies heavily on ‘tunable inductor bridge’ (TIB) style superconducting

couplers. The TIB design was introduced in 2016 in Ref. [79] and improved for the work in Ref. [23].
The TIB itself is a Wheatstone bridge of tunable inductors realized with SQUID arrays, and draws
inspiration from microwave mixers [34] and the Josephson ring modulator [30, 142]. This section
describes the TIB design and then performance.

5.2.1

Lumped-element schematicCentral to the TIB is a Wheatstone bridge of tunable inductors, realized with SQUID arrays.
Conceptually, the TIB can be thought of as a superconducting analog to a microwave mixer, withdiodes replaced by SQUID arrays. As with a mixer, the TIB functions as a microwave switching/modulation element where symmetry of a Wheatstone bridge allows for high-performance,

69
broadband operation. In particular, the process of preserving vs. breaking the symmetry of thebridge allows for transmission through the TIB to be tuned by a far greater ratio than its constituent
inductors can be tuned.
(b)
port





port

port

+

+

quarter wavelength

(c)

+𝑣/2
2 balanced
3 port-𝑣/2(d)Φ𝑢 + Φ𝑔unbalanced 𝑣 1
port

...

...

...

...

Φ𝑢 - Φ𝑔

(a)
port

open

Figure 5.1: TIB lumped-element schematic. A tunable inductor bridge (TIB) style superconducting switch is formed from a Wheatstone bridge of SQUID arrays. Ports can be connected tothis bridge by either (a) coupling to the differential mode across the north-south nodes (labelled‘a’ and ‘c’) and east-west nodes (labelled ‘b’ and ‘d’), as in the device in Refs. [79, 135]. Or (b), by
coupling to one opposing pair of nodes deferentially and another node in a single-ended way, as in
the device in Ref. [23]. (c) Differential coupling can be achieved with a Marchand balun [143] which
consists of a half-wave resonator coupled to two quarter-wave resonators, for a wavelength at thedesign center frequency (approximately 6 GHz in Refs. [79, 135, 23]). Assuming ideal performance,a signal of amplitude v at the node labelled ‘1’ (the ‘unbalanced’ port) will cause a response of
amplitudes ±v/2 at the nodes labelled ‘2’ and ‘3’. The differential mode across these ports is the‘balanced’ port. (d) The Wheatstone bridge is constructed by twisting the SQUID arrays intoa figure-eight geometry. The flux through the SQUID arrays is a combination of a background,uniform flux Φu , and a gradiometric flux Φg . The gradiometric flux is controlled by the current ina bias line running through the center of the figure-eight (pink arrow). The uniform flux can becontrolled by an off-chip superconducting coil, and/or an on-chip bias line threading around theoutside of the figure-eight (not drawn).

70
A lumped-element schematic of a TIB is shown in Fig. 5.1. A Marchand balun (explained inFig. 5.1c and a simulation shown in Fig. 5.2a) couples the left port of the TIB to the differential
voltage across the top and bottom nodes of the bridge (nodes labelled ‘a’ and ‘c’ in Fig. 5.1a,b). No
signal can couple between the two ports when the bridge is balanced, meaning that all four bridge
inductors have equal value. To see this, consider an oscillating signal of amplitude v applied at
the left port of the lumped-element circuit in Fig. 5.1a,b. Voltage at the top and bottom nodes ofthe bridge (labelled ‘b’ and ‘d’) will oscillate with amplitude ±v 0 , respectively, creating an effective
ground at the right port and therefore the TIB will reflect (the amplitude v 0 ≤ |v|/2 will in generaldepend on the operating frequency, choice of capacitors, etc). If instead the bridge is imbalanced,
as drawn, the symmetry of the bridge is broken so that the right port does not see an effective
ground, and thus transmission can be nonzero and in fact near unity. Capacitors are added tomatch the circuit over a desired frequency range.
0

𝑆𝛥

(a)

-10

-10

𝑆11

-20

-20
-30

-30

𝑆𝛴

-40

-40
-50
4

(b)

transmission (dB)scattering parameter (dB)

0

balun
4.5

5

5.5 6
6.5 7
frequency (GHz)

7.5

8

-50
0

low-pass filter
12

3

4 5 6 7frequency (GHz)

8

9

10

Figure 5.2: Balun and low-pass filter simulation. (a) Simulated scattering parameters of
the Marchand balun shown in Fig. 5.3. S11 is√reflection off of the unbalanced port of the balun(labelled ‘1’ in Fig. 5.1c). S∆ = (S21 − S31 ) / 2 is the differential mode transmission coefficient,√referring to the balanced port of the balun labelled in Fig. 5.1c), and SΣ = (S21 + S31 ) / 2 is thecommon mode transmission coefficient. Transmission through the differential mode is near unity
within the design bandwidth of between 4 GHz and 7 GHz. (b) Simulated trasmission through thegradiometric bias line low-pass filter (LPF) shown in Fig. 5.3, with a 50 Ohm port placed on eachside of the filter. Transmission goes to unity as frequency approaches zero, allowing the < 350 MHz
frequency components of the TIB bias signal in Ref. [23] to pass uninhibited. Transmission drops
below -20 dB above 4 GHz. The filter’s self-resonance frequency is approximately 8 GHz.

71
5.2.2

LayoutWhen constructing a TIB, it is essential to preserve the symmetry of the Wheatstone bridge
while still being able to balance or imbalance it. To do so, it is expedient to twist the bridge afigure-eight geometry, and tune the balance or imbalance with a single on-chip bias line running
through the center of the bridge, (pink arrow, Fig. 5.1d). This bias line a gradiometric flux ±Φg
into the SQUID arrays on opposite sides of the bridge. At the same time, all the arrays see an
identical uniform background flux Φu . The inductance l+ and l− of the thick and thin inductors in
Fig. 5.1 becomes [79],l+/− =cosl 0Φu ±Φg
2φ0

 .

(5.1)

Where φ0 = ~/2e is the reduced magnetic flux quantum, with e the charge of an electron. The
inductance l0 = N φ0 /2Ic is determined by the Josephson junctions critical currents. In Ref. [23],Ic = 5 µA and the number of SQUIDs per array is N = 20, so that l0 = 0.66 nH. As long as
Φu /2φ0 is not a multiple of π/2, applying a nonzero Φg will imbalance the bridge such that l+ 6= l−
and transmission will be nonzero. The greatest imbalance is achieved when Φg /2φ0 = π/4 andΦu /2φ0 = π/4 + nπ, where n is an integer.
A false-color optical micrograph of a TIB is shown in Fig. 5.3. Its bias lines (for a gradiometricflux, red, and uniform flux, blue, Fig. 5.3a) contain low-pass filters (LPFs), realized with ∼ 20 nH
spiral inductors. These filters limits microwave power coupling out of the bias line. A numericalfinite-element simulation, Fig. 5.2b, indicates that this inductor has a self-resonance frequency ofapproximately 8 GHz (note that in this style of inductive filter, a higher inductance will generally
lead to a lower self-resonance frequency). Further simulations indicate that transmission from a
microwave port of the TIB out a bias port is generally smaller than −40 dB between 4 and 8
GHz with inclusion of this LPF, but as high as −20 dB without it. At the operational frequencyof 6.34 GHz, transmission from a microwave port of a TIB out of its gradiometric bias port issimulated to be between −39 dB and −47 dB (the exact value changes slightly depending on whichof the two microwave ports is used, and whether the TIB is in transmit or reflect mode).

72

port

0.5 mm(a)groundMarchandbalun
groundLPF

TIB grad.
bias
LPF

TIB
uniform
bias

capacitor

groundgroundSQUID array

(b)

LPF

port

Figure 5.3: TIB optical micrograph. (a) False-color optical micrograph of a TIB integrated
within the SIMBA used in Ref. [23], following the lumped-element schematic in Fig. 5.1b. The
Wheatstone bridge (green, center of the image) is connected to capacitors (yellow), and on one
side a Marchand balun (purple). The balance or imbalance of the bridge is controlled with a single
bias line (red, coming from the left) which applies a gradiometric flux. A uniform flux is applied
from a second bias line (blue, from the right). Both bias lines contain low-pass filters (LPFs) which
prevent microwave signal in the TIB from leaking out of these lines. (b) Zoomed-in image of oneof the SQUID arrays.

5.3

Performance
Performance of the TIB used in Ref. [23] is shown in Fig. 5.4. Transmission through asingle TIB is measured while sweeping the gradiometric bias flux Φg . Near Φg = 0 the TIB isin ‘reflect mode’, meaning transmission is near zero. When changing to Φg 6= 0, the Wheatstonebridge becomes imbalanced and the TIB is changed to a ‘transmit mode’. Transmission is shownin Fig. 5.4c for both an example transmit mode and an example reflect mode, corresponding tohorizontal linecuts of the data in Fig. 5.4b, at the gradiometric flux bias specified by the blackarrows. By changing Φg , the on/off ratio can be tuned to greater than approximately 50 dB at any
frequency between 4 and 7.3 GHz.
The bandwidth of the TIB is set by the octave bandwidth of its Marchand balun, here de-

73

(a)

sweep
cryo

TIB gradiometric bias (flux quanta)(b)transmission (dB)transmission (dB)
0

1

-10
transmit-20

reflect

0

-30
-40
-1
0

transmit 60

-10

50

-20

40

-30

30

-40

20

-50
-60

reflect
4567frequency (GHz)

10
0
8

-50
maximum on/off ratio (dB)(c)

network
analyzer

Figure 5.4: TIB transmission characterization. (a) A single TIB is measured in transmission
while sweeping its gradiometric flux bias, Φg . (b) Measurement of transmission vs. Φg and frequency. Transmission is normalized to the maximum overall transmission. (c) Transmission vs.
frequency at an example transmit mode (Φg ∼ π/4) and reflect mode (Φg ∼ 0), for uniform fluxΦu ∼ π/8. At the reflect mode, transmission is lower than -30 dB over several GHz. The maximum
on/off ratio that can be achieved at any given frequency is plotted in red, and is better than 50
dB between 4 to 7.3 GHz. The reflect mode data and on/off ratio are smoothed with a 100 MHz
moving filter.

signed to have a center frequency near 6 GHz. Bandwidth is also affected by the maximum bridge
imbalances, l+ /l− ∼ 4 [79, 135, 61].2
width.3

In general, a higher maximum imbalance improves band-

The bandwidth and center frequency are also affected by choice of matching capacitors,

here ∼ 0.5 pF.
2

The maximum possible bridge imbalance is set by the SQUID array geometry, geometric inductance, andasymmetry of the Josephson junctions [83].
3This can be understood as a consequence of the Bode-Fano criteria (Chp. 5.9 of Ref. [34]).

74
Note that the TIB in Ref. [23] has a reported power handling of approximately -98 dBmalong with an approximately 15 ns swithing speed, comparable to previously demonstrated TIBs

0

-10

(a)

homodyne voltage (a.u.)transmission normalized
to the low power limit (dB)

[79, 135]. See Fig. 5.5 for additional measurements of the TIB power handling and switching speed.

4 GHz
5 GHz
6 GHz
7 GHz

-20
-100

-90
-80
-70
on-chip power (dBm)

-60

1
0.8

(b)
bypassing TIB/croystat
through TIB/croystat

0.6
0.4
0.2
0
-10

0

10
20
time (ns)30

40

Figure 5.5: TIB power handling and switching speed characterization. (a) Power handling:
Transmission through a TIB set to transmit mode, as a function of estimated on-chip power.
Transmission at each frequency is normalized to unity in the low power limit, but in general is lessthan unity due to reflections from the device. (Note that the rise in transmission at 7 GHz likely
does not represent gain, but instead reflection diminishing in magnitude.) The onset of nonlinearity
is observed between −84 dBm at 4 GHz and −73 dBm at 7 GHz. Note that this is higher thanthe −98 dBm reported in Ref. [23]. (b) Switching speed: a square wave pulse (generated at 1.2
Gs/s) is applied to the TIB gradiometric bias line. Transmission at 6 GHz is measured throughthe TIB (black, solid line), using a homodyne measurement, and is compared to the same squarewave measured directly (red, dashed line). A 350 MHz low-pass filter is placed on the TIB bias line
(black line, only), limiting the switching speed. Note that (b) was taken with a slightly difference
device/design than used in Ref. [23].

5.4

Switch networks
TIBs can also be combined into networks with more complicated connectivity for example a

2x1-TIB (a single-pull double-throw ‘SPDT’ geometry). To construct a 2x1-TIB, two 1x1-TIBs are
combined on-chip with a T-junction as in Fig. 5.6 and Ref. [124].
In order for the network to be matched when one TIB is set to reflect mode, the phase of thesignal reflected from a TIB must be taken into consideration. Transmission through the network
will always be zero if reflection from one or both TIBs look like a short at the reference plane of the
T-junction. To engineer a TIB in reflect mode to instead look like an open at this reference plane,

75

(a)
2

3
TIB1

TIB2
1

(b)

open

2

✅

TIB1
1

(c)

short

2

❌

TIB1
1

Figure 5.6: TIB network. (a) A three-port switch network can be constructed using two-TIBsconnected by a T-junction. Depending on the bias of the TIBs, a signal incident on port-1 can berouted either to port-2, port-3 or reflected. Appropriate electrical length can be placed between the
TIBs so that a TIB in reflect mode reflects close to an open at the reference plane of the T-junction,at the design frequency, as illustrated in (b). If instead a TIB in reflect mode looks like a short at
the reference plane of the T-junction (c), a signal incident on port-1 will be reflected even when
the other TIB is set to transmit mode.

appropriate electrical length can be added between the TIBs and the T-junction between them. In
general, the amount of electrical length required will depend on the specific properties of the TIBs
in the network and the design frequency.
This example can be extended to create larger switch networks. To do so and maintainhigh performance, it is crucial to understand the scattering parameters of the individual TIBs
both when in transmit mode and in reflect mode. Constructing an N xM or even N x1 networkof superconducting switches, where N  1 and/or M  1, remains an outstanding challenge.
Doing so while maintaining high-quality performance and ease of calibration would have general
use within large scale superconducting qubit systems.

Chapter 6

The SIMBAThe SIMBA (‘Superconducting Isolating Modular Bifurcation Amplifier’) is a 2-port parametric amplifier with a tunable external coupling on each port, Fig. 6.1. The SIMBA is designed
for superconducting qubit readout, Fig. 6.2. As in conventional dispersive readout [15], a pulse
is first sent into the weakly coupled port of a two-port readout cavity, where it acquires a qubit
state-dependent phase shift. TIB1 is then set to transmit mode for a duration chosen to fully swap
this pulse into the parametric cavity, which has previously been tuned near resonance. Next, thefield inside the parametric cavity is amplified by parametric modulation at twice the resonance frequency, Sec. 2.1.4. In this work, we choose to pump the parametric cavity into the bistable regime[85, 116]: a non-unitary process in which the cavity latches into one of two bistable states with
opposite phase but large, equal amplitudes, Sec. 4.3. Readout is achieved by seeding the parametriccavity state with the probe tone, such that the post-measurement qubit state is correlated withthe latched state of the parametric cavity [144, 119]. To learn the measurement result outside of
the cryostat, TIB2 is set to transmit mode, coupling the amplified state to a standard cryogenic
microwave measurement chain.

6.1

Calibration
The following steps are required to calibrate superconducting qubit readout using a SIMBA:(1) Tune the JPA frequency to the readout cavity frequency, Fig. 6.3a.
(2) Sweep the JPA pump amplitude/frequency to give desired gain/bifurcation, Fig. 4.6.

77

parametric cavity pump
HEMT
uniform⨯5 amplifier
dc-biasSIMBA(a)same circuit board
on-chip(c)

TIB dc-bias
4⨯6.5 mm2 chip

(b)
TIB2

TIB1
version 1

parametric cavity

TIB1

TIB2
on-chip

version 0 parametric cavity

Figure 6.1: Introduction to the SIMBA. (a) A transmon qubit in a 3-dimensional readout
cavity is measured using a SIMBA (‘Superconducting Isolating Modular Bifurcation Amplifier’).
(b) The SIMBA is comprised of a two-port parametric cavity with a tunable-inductor-bridge (TIB)
style coupler on each port. In this thesis, ‘version-1’ refers to the device in Ref. [23], which isconstructed on one chip using the illustrated geometry. (c) In a preliminary demonstration ‘version0’, the SIMBA is constructed using a 1-port resonator combined with a network of TIBs connected
to a T-junction, as in Sec. 5.4.

data
swap

probe
65 ns

amplify

20 ns 80 ns

100 nsreadouttransmit mode
reflect mode

TIB1
amplifier pump
TIB2

time
meas.

data acq.

reset (100 ns)

excited state probability

(b) 1

(a)

data
fit 𝑇2* = 15.5 𝜇s

0.8
0.6
0.4
0.2
0

0

4

8
12
16
Ramsey delay (𝜇s)20

Figure 6.2: Qubit measurement using a SIMBA. (a) A probe tone is sent into the readout
cavity, swapped into the parametric cavity, and then amplified. The amplified state is then coupled
to a standard cryogenic measurement chain and digitized. Cyan (pink) histograms correspond tosingle-shot measurements when the qubit has been prepared in the ground (excited) state. (b)Example measurement of a transmon qubit’s coherence time using a version-1 SIMBA. Two π/2pulse are separated by a variable ‘Ramsey’ delay (the x-axis) and then the qubit is projectively
measured using the sequence in (a). Contrast of the measured excited state probability is limited
primarily by state preparation errors, in particular the π/2 pulses not being precisely calibrated inthis example.

(3) Choose the readout pulse amplitude/frequency. Specify the readout pulse phase compared

78
to the JPA pump tone.
(4) Sweep the duration for which TIB1 is set to transmit mode, Fig. 6.3b. Choose the minimum
duration which maximizes readout fidelity.
(5) Fine-tune TIB reflect modes to minimize backaction (Fig. 6.4) and maximize the measurement efficiency and readout fidelity (Fig. 6.7c).
The first three steps are generally true of any readout scheme which uses a tunable, narrow band andphase-sensitive parametric amplifier. The final two steps are specific to the SIMBA. From knowledgeof the qubit, SIMBA and measurement setup, the parameters in all steps may be reasonably
estimated so that near-optimal readout can be calibrated quickly.
(b) 1transmission
0 (dB)

0.07

-10

0.06

-20
-30

0.05
6.31

6.35
6.39
frequency (GHz)-40

readout fidelity, 𝐹𝑟

uniform dc-bias(flux quanta)(a)readout
TIB1
pump
TIB2

0.8
0.6

sweep

0.4
0.2
0

0

100

200
300
400
500
TIB1 transmit duration (ns)600

Figure 6.3: Swap calibration (using a version-1 SIMBA). (a) A uniform background magnetic
flux is swept while probing the readout cavity in transmission with both TIBs in transmit mode.
The avoided crossing shows the parametric cavity tuning through the readout cavity. To operate
a SIMBA, this uniform flux bias is set so that the readout and parametric cavities are minimally
detuned. (b) Readout fidelity Fr (the ability of a measurement to distinguish the qubit eigenstate,[108]), is plotted vs. the duration for which TIB1 is set to transmit mode within the measurement
sequence. Oscillations with a period of 40 ns indicate coherent swapping of a readout pulse between
the readout and parametric cavities.

6.2

Characterization

6.2.1

Metrics of performance

Important metrics to describe superconducting qubit readout are summarized in Table 6.2.1.
These metrics include: excess backaction nb , the number of readout photons a qubit is exposedtransmission -30
(dB)
0.6
(a)
sweep
TIB1
bias

0

-20

-10

0

(b)
sweep
TIB2
bias

-0.6
6.32

6.34
6.36
frequency (GHz)

6.32

6.34
6.36
6.38
frequency (GHz)2𝑛𝑏 (effective photons)

TIB grad. bias (flux quanta)

(c)

qubit
readout
TIB1
pump

𝜋/2

backaction

𝜋/2

proj. meas.

79
sweep

TIB2

1
0.6
0

-0.04
0
0.04
TIB1 reflect mode grad. bias (flux quanta)

Figure 6.4: TIB refect mode calibration (using a version-1 SIMBA). Transmission through thereadout cavity when the gradiometric bias on (a) TIB1 and (b) TIB2 is swept with the gradiometricbias on the other TIB set to ∼ 0.3. The JPA is tuned near resonance with the readout cavity in bothcases. The presence of a reflect mode (transmission < -30 dB in this measurement) is seen by thenull in transmission near zero gradiometric bias. (c) Excess backaction nb is measured by insertinga measurement sequence with zero readout amplitude into a Ramsey sequence, as in Fig. 3. Excess
backaction is measured as a function of the gradiometric bias on TIB1, swept near reflect mode.
The gold square indicates the operating point used to calibrate the measurement efficiency, for thedata shown in Fig. 6.7.

to from detector backaction, the measurement efficiency η which quantifies the overall loss andadded noise in a detector, and the maximum readout fidelity F0 which quantifies how well thequbit state can be discriminated. To characterize nb and η we use the framework of measurementinduced dephasing [24], which is reviewed in Chp. 3. Important for quantum error correction is
how rapidly the measurement and digitization is completed, and also how quantum non-demolition(QND) a measurement is, meaning to what degree the qubit’s post-measurement state is correlatedwith the measurement result [145, 72]. Finally, considerations such as ease of calibration andnumber/bandwidth of bias tones are also important when trying to scale any measurement scheme.

6.2.2

Version-0

First, we present data from a ‘version-0’ demonstration of the SIMBA (Fig. 6.1c), preliminary
to the ‘version-1’ device and experiment in Ref. [23]. In version-0, measurement efficiency andreadout fidelity were lower than in version-1, and calibration was complicated by issues with trapped

80
Table 6.1: Qubit readout performance metrics.
MetricDescription

Relevant formula

More informationFr

readout fidelity

1 − P (e|0) − P (g|π)

Ref. [108]F0maximum readout fidelity

F0 erf|ρ̂001 |post-measurement coherenceρb e−2( nr ) = ρb e− /2σnr

readout photons (effective)a|α|2 sin2 θSec. 3.2

nb

excess backaction photons (effective)− log (2ρb ) /2

Sec. 3.2

ηmeasurement efficiency

2σ 2 ν 2

Sec. 3.3

FQN D

quantum non-demolition fidelity

1
2 [P (e2 |e1 ) + P (g2 |g1 )]Refs. [145, 72]√2ηnr = F0 erf [ν]

√

2

2

2

Sec. 3.3.3
Sec. 3.2

a‘Effective’ photon numbers nr and nb refer to the photon number of a coherent state scaled by sin2 θ, whereθ = arctan (2χ/κr ) describes the angle of the coherent state in phase space, defined as in Fig. 3.1.

flux in the parametric cavity. However, excess backaction was significantly lower than in version-1,
excess backaction due to TIB switching was negligible, and quantum non-demolition fidelity wasnear unity and limited by the qubit’s T1 time. The performance of version-0 serves as a proof-ofprinciple demonstration for some of the metrics which can be improved upon in version-1.
Excess backaction due to TIB switching is characterized in Fig. 6.5a,b. The qubit’s energy
relaxation and coherence times are measured while TIB1 is square-wave modulated between transmit and reflect mode. (In the example in Fig. 6.5b, the coherence time is measured using an echo
sequence: π/2 pulse, delay, π pulse, then another delay and π/2 pulse.) The qubit’s T1 and T2echo
times are not dramatically affected by the square-wave modulation rate up to Ω/2π. Measurementof T2echo ∼ 4 µs when square-wave modulating TIB1 at 100 MHz indicates less than . 6 × 10−4
photons of backaction per switching event.
The isolation provided by TIB1 can also be characterized by measuring the qubit’s coherencetime while the amplifier pump is turned on during the delay step. (In Fig. 6.5c,d, the coherencetime is measured using a Ramsey sequence: π/2 pulse, delay, then π/2 pulse.) In the measurement

𝜋

𝜋/2

qubit
readout
TIB1
amplifier pump16(b)time (𝜇s)12𝜋/2proj. meas.

(d)𝑇1𝑇2echo
20
40
60
80
100
TIB1 modulation freq., Ω/2𝜋 (MHz)

𝜋/2

qubit
readout
TIB1
amplifier pump

14

4

0

(c)

TIB2

8

0

sweep TIB1
reflect bias

echo delay

coherence time, 𝑇2* (𝜇s)square-wave
modulate TIB1
at Ω/2𝜋(a)12Ramsey delay

𝜋/2proj. meas.

81

TIB2
pump off during Ramsey delay

10
8
6
4
2
0
-0.4

pump on during
Ramsey delay

-0.2
0.0
0.2
0.4
TIB1 reflect bias, Φ𝑔/Φ0 × 10-3

Figure 6.5: SIMBA backaction, version-0. (a,b) Measurement of a qubit’s energy decay
time T1 and coherence time T2echo measured with an echo sequence. (Coherence time measured
using a Ramsey sequence was less than 1 µs in this experiment.) These measurements are taken
while square-wave modulating TIB1 during the Ramsey delay at rate Ω/2π. Neither T1 or T2echo
are dramatically changed by the modulation frequency, bounding any destructive effects due tomodulation. See Fig. S15 of Ref. [23] for similar data using SIMBA version-1, where backaction from
TIB1 modulation is small per switching event but more noticeable. In both versions, modulating
TIB2 had a negligible effect on the coherence time. (c,d) The isolation provided by TIB1 canbe measured by pumping the parametric cavity (here at the same pump amplitude as used ina projective measurement) while sweeping the reflect mode bias on TIB1. Away from optimalreflect bias, the qubit is increasingly dephased by pump backaction. At the optimal bias of TIB1’s
reflect mode, however, the qubit’s coherence time of T2∗ ∼ 10 µs was unaffected by pumping theparametric cavity during the delay. This bounds of n̄b ≤ 0.03 steady-state cavity photons due toamplifier backaction. Note than the x-axis value is scaled so that Φg /Φ0 = 0 corresponds to thecalibration of maximum isolation, which is slightly offset from zero bias voltage. See Fig. 6.4c for asimilar experiment using version-1. Note that data in (b) and (d) were taken in separate cooldowns,
with more careful filtering and radiation shielding in (d), resulting in a longer qubit coherence time.

shown in Fig. 6.5d, when the TIB1 reflect bias is set optimally no difference can be measured
between the qubit’s ∼ 10 µs coherence time when the parametric cavity is pumped during thedelay, or not. This places a bound of n̄b . 0.03 photons in the readout cavity due to pumping
the parametric amplifier. In a different data set taken with the same device and qubit, Fig. 6.6, asimilar value of nb = 0.02 photons is measured along with efficiency of approximately η = 44% andbackaction

qubit
readout
TIB1
pump on

𝜋/2

TIB2

proj. meas.

post-meas. coherence, |𝜌^01´|√𝑛𝑟 ∝ 𝜖𝜋/2

1𝐹0 = 0.86

0.4 (b)0.8

readout fidelity, 𝐹𝑟

(a)

82

0.5

0.3

fidelity fit,
𝜂 = 0.44

0.2
0.1
0

0.6
0.4
0.2

0

0.4
0.8
1.2
1.6
readout pulse amplitude, √𝑛𝑟 ∝ 𝜖

02Figure 6.6: SIMBA efficiency, version-0. (a) Excess backaction and measurement efficiency
are characterized by inserting a variable measurement into a Ramsey sequence, as in Fig. 6.7. (b)Results, with the JPA pump remaining on during the variable strength measurement within the
Ramsey sequence. Fitting data to models, Table. 6.2, yields measurement efficiency of approximately η = 0.44 and maximum readout fidelity F0 = 0.86. For a variable strength measurement of√zero readout amplitude ( nr = 0), ρb = 0.482 ± 0.004 corresponding to nb = 0.018 ± 0.002 effective
photons of excess backaction.

maximum readout fidelity of approximately F0 = 86%.

6.2.3

Version-1

In the SIMBA version-1, Ref. [23], we characterize readout to be low-backaction, high fidelity,and high efficiency. Excess backaction is found from ρb = 0.141±0.002 (left-most data point, ‘pumpon’ data, Fig. 6.7c, uncertainty represents plus/minus one standard deviation). Using Eq. 3.15, thiscorresponds to nb = 0.63 ± 0.01 effective photons of excess backaction; about one quarter of thenproj= 2.4 effective photons used in a projective measurement (the maximum value on the xr
axis of Fig. 6.7c), and far less than the ∼ 150 photons in the pumped state of the parametriccavity (Ref. [23], supplementary material Sec. IV.E). Next, we find ν and the maximum fidelity
F0 = 95.5% ± 0.3% by fitting Fr vs. readout amplitude (red data, Fig. 6.7c) to Eq. 3.26. Finally
we obtain σ from a fit of the ‘pump off’ data (cyan) to Eq. 3.27, and therefore determine η =70.4% ± 0.9% using Eq. 3.28.
This fit excludes the first four data points, which level off more quickly than predicted suchthat excess backaction includes 0.05 ± 0.01 effective photons caused solely by actuating the TIBs

𝜋/2

backaction

excited state prob.

(b)

1

𝜋/2

(c)

TIB2

pump on,
𝜖=0

no backaction

0.5

0

0.6

pump
off, 𝜖 = 0
0.8
1
1.2
Ramsey delay (𝜇s)

1.4

83

0.5

1 𝐹0 =
0.95

pump off
0.4

0.8

0.3

fidelity fit,
𝜂 = 0.70

0.6

0.2

fidelity prediction,
𝜂=1

0.4
0.2

0.1
0

𝜌𝑏 = 0.14
0readout fidelity, 𝐹𝑟

qubit
readout
TIB1
pump (on/off)proj. meas.
post-meas. coherence, |𝜌^01´|√𝑛𝑟 ∝ 𝜖(a)pump on0.5
1
1.5
readout pulse amp, √𝑛𝑟 (photon number1/2)

0

Figure 6.7: SIMBA characterization, version-1. (a) Post-measurement qubit coherence |ρ̂001 |is obtained by inserting a variable measurement into a Ramsey sequence, exposing the qubit tobackaction. The ratio of the amplitude of the measured Ramsey fringes to the amplitude of thosemeasured without this backaction (nothing inserted into the Ramsey sequence) equals 2|ρ̂001 |. (b)Excess backaction is determined by inserting a ‘measurement’ with zero readout amplitude. Postmeasurement coherence after excess backaction with the parametric cavity pump on (indigo) andoff (cyan), are compared to a case with no backaction (no readout pulse, pump, or TIB switching inserted in the Ramsey sequence, violet). (c) Post-measurement coherence |ρ̂001 | (left y-axis)
and readout fidelity Fr (right y-axis, red data points), are measured while sweeping the readout√
amplitude nr of a variable strength measurement. As in (b), |ρ̂001 | is measured both with theparametric pump turned on or off during the variable measurement sequence (indigo or cyan data
points, respectively). Post-measurement coherence with the parametricpump turned on, but in√the absence of readout photons, is specified by ρb = |ρ̂001
nr = 0 | and determines the excess
backaction nb = − log (2ρb ) /2. Measurement efficiency η is determined by a comparison between
measurement-induced dephasing and readout fidelity while sweeping readout amplitude.

For√nr & 0.2 our models for both |ρ̂001 | and Fr generally fall within the 95% confidence interval ofthe measurement, and so only these points are used to obtain efficiency. This choice conservatively
affects the reported efficiency; including the first four points, or instead fitting the ‘pump on’ data,
returns a larger value for η. We note that in this experiment, η must be limited to 78% or lessbased on an independent measurement of loss in the parametric cavity and a model for how thisloss affects the efficiency (Ref. [23], supplementary material Sec. VI.A). This dephasing process is
not captured by our model, and may result from a noise source on the parametric cavity side of
TIB1 (Ref. [23], supplementary material Sec. V.B).

84

0.5
~ 𝜋/2
qubit

probability

0.4
0.3

proj. meas.

0.2
3.5%

0.1
0

delay
proj. meas.

version-0, 𝐹𝑄𝑁𝐷 = 96.5%
version-1, 𝐹𝑄𝑁𝐷 = 89.4%

𝑃(𝑔2|𝑔1) 𝑃(𝑒2|𝑔1) 𝑃(𝑔2|𝑒1) 𝑃(𝑒2|𝑒1)
measurement outcomes

Figure 6.8: Quantum non-demolition characterization. A qubit is prepared in a superposition
state and projectively measured twice, with a delay between measurements much longer than the
SIMBA reset time of 100 ns (Fig. 4.7). Measurement outcomes are far more likely to be correlated
than anti-correlated, indicating that the measurement is quantum non-demolition. Red data is fromthe SIMBA version-0 (Fig. 6.1c), using a delay of 500 ns and a measurement time of 260 ns. Bluedata is from the SIMBA version-1 (Fig. 6.1b and Ref. [23]), using a delay of 1 µs and a measurement
time of 265 ns. QND-fidelity, Eq. 6.1, is FQN D = 96.5% using version-0 and FQN D = 89.4% using
version-1. In version-0, QND-infidelity is dominated by relaxation to the ground state during thedelay step, with a probability consistent with the qubit’s energy relaxation time of T1 ∼ 12 µs.

6.2.4

Quantum non-demolition fidelity

We also characterize the degree to which qubit readout using a SIMBA is quantum nondemolition (QND), meaning that the qubit remains in its measured eigenstate [145, 72]. Low
power dispersive readout of superconducting qubits is understood to be QND [15]. To test thisusing a SIMBA, we prepare the qubit in a superposition state, and then projectively measure it
twice with a delay between measurements much longer than the reset time, Fig. 6.8.
QND fidelity is defined as:
FQN D =1
[P (e2 |e1 ) + P (g2 |g1 )] ,
2

(6.1)

and is diminished by the error probabilities P (g2 |e1 ) = 1 − P (e2 |e1 ) and P (e2 |g1 ) = 1 − P (g2 |g1 )indicating decay from the excited to ground state between measurements, and excitation from theground to excited state between measurements, respectively. Note that in this definition of QND
fidelity taken from Ref. [72], FQND = 0.5 when the results of the two subsequent measurements are85
uncorrelated, meaning that measurement outcomes P (g2 |g1 ) = P (g2 |e1 ) = P (e2 |g1 ) = P (e2 |e1 ) =1/4 occur with equal probability.
In both experiments shown in Fig. 6.8, probability P (g2 |e1 ) of decay from the excited to
ground state between measurements is approximately consistent with the qubit’s T1 time and thedelay between measurements. The rate of spontaneous excitation from the ground to excited state
P (e2 |g1 ) may be indicative of a high parametric cavity temperature when the TIBs are disconnected.
In version-1, this is potentially related to the dephasing measured when modulating TIB1, (Fig. S15
of Ref. [23]). Note that in Ref. [23], prior to any qubit measurement both TIBs are set to transmit
mode for a duration much longer than T1 , as this improved readout fidelity seemingly by reducing
the residual qubit excited state population.

6.2.5

Summary of results

Readout performance to-date using a SIMBA is summarized in Table 6.2. In both versions readout is fast, high fidelity and quantum non-demolition. Moreover it is also efficientand low-backaction, indicating that the SIMBA is a successful alternative to using ferrite circulators/isolators before the first amplifier.
Table 6.2: SIMBA performance summary.
Parameter
Maximum readout fidelity, F0
Excess backaction, nb
Measurement efficiency, η
QND-fidelity, FQN D
Measurement time

Version-086%
0.02 photons
44%
96.5%
260 nsVersion-1 [23]95.5%a0.66 photons
70.4%
89.4%
265 nsaFidelity as high as 97% measured during the same cooldown as Ref. [23], but at a slightly different calibration
than in Fig. 6.7 using a higher JPA pump amplitude, leading to greater excess backaction.

In particular, the demonstration in Ref. [23] combines state-of-the-art measurement efficiency
and considerable isolation from amplifier backaction such that nb ∼ nproj/4. The measurementrefficiency of previous superconducting qubit readout schemes have been limited to η = 80% [49],and less when providing any isolation before a parametric amplifier [35, 95, 75, 71] (see Table 1.1).

86
Near-unit measurement efficiency after future improvements would allow for near-complete accessto the information extracted from a quantum system. Additionally, the SIMBA is chip-scale,compatible with scalable fabrication procedures including the use of through-silicon-vias [146], andrequires only one microwave control tone to operate. The SIMBA is therefore a favorable choicefor high-quality and scalable superconducting qubit measurement.

6.3

Current limitations
We now discuss the limitations of the qubit readout using the SIMBA version-1 (Fig. 6.1band Ref. [23]). With reasonable changes to the qubit and SIMBA as outlined in this section, we
estimate that it will be straightforward to achieve maximum readout fidelity of F0 ≥ 99%, excess
backaction of nb ≤ 0.02 photons and efficiency of η > 90%, for a measurement and digitization
procedure of ≤ 100 ns.

6.3.1

Readout fidelity

Readout infidelity results from a combination of state preparation errors, qubit decay during
measurement, and inability of the parametric cavity to correctly latch to the appropriate state.
Estimates of these errors are given in Table 6.3, for the characterization in Ref. [23].
Table 6.3: Sources of readout infidelity. Estimates for SIMBA version-1, Ref. [23].
Error mechanism
State preparation
Qubit decay before measurement
Parametric cavity latching failure
Inability to discriminate bifurcated state

Estimated probability
2.3%
1.3%a
1.1%b
< 0.1%

aConsistent with a 265 ns readout time and T1 ∼ 9 µs.
Estimated from the difference in F0 = 95.5% measured in Ref. [23] compared to the maximum value of F0 = 96.8%
measured using the same device, during the same cooldown, but with a higher parametric pump amplitude.
b

Readout fidelity can be improved to near-unity by improving state preparation due to better
thermalization, using a qubit with a longer T1 time, and reducing any latching failure by increasing

87
the JPA pump amplitude (at the potential cost of greater excess backaction). Readout fidelity
has also been shown to be increased by careful shaping of the readout pulse [35], not yet tried
using a SIMBA. Readout fidelity of approximately 99.9% in 100 ns is expected given state-ofthe-art transmon qubit performance of T1 ∼ 100 µs, and a residual thermal photon population ofn̄th ≤ 2 × 10−4 as in Ref. [104].

6.3.2

Efficiency

A SIMBA used for qubit readout can be modelled as two cavities connected by a timedependent coupling g(t). Here, we assume the cavities to each have the degenerate bare resonant
frequency ω0 . Inefficiency is caused by loss rates κr and κp in the readout and parametric cavities,
respectively, along with any inefficiency in the amplifier itself. For phase-sensitive gain with noadditional added noise, we might expect the measurement efficiency to be approximately: η ∼g0 / (g0 + κr + κp ). In Ref. [23], g0 /2π = 12.5 MHz, κr /2π = 0.44 MHz and κp /2π = 4.0 MHz (seeTable 6.4), and such an estimate gives η ∼ 0.74 — close to the measured value of η = 0.70.
This estimate can be made more precise by numerically solving a classical input-output modelfor the readout cavity + SIMBA system (see Sections 2.1.6 and 2.3). Consider the classical, timedependent readout cavity and parametric cavity fields, A(t)eiω0 t and B(t)eiω0 t , respectively, whichrotate at bare cavity resonance frequency ω0 (assuming no detuning between the cavities). These
cavities are coupled with a time-dependent coupling g(t). Loss in the readout cavity is the sum ofw
intw
an internal loss rate κintr along with loss from a weakly coupled port κr such that κr = κr + κr .

Using input-output formalism (Sec. 2.3), this port interacts with an input field AI eiω0 t and output
field AO eiω0 t . In a frame rotating at ω0 , the Heisenberg-Langevin equations of motion for thissystem simplify to,pκw
r A =AI + AO ,p
κr
A + ig(t)B + κw
r AI ,2κp
Ḃ = − B + ig(t)A.
2
Ȧ = −

(6.2)
(6.3)
(6.4)

88
Table 6.4: Sources of loss. Estimates of loss in the readout and parametric cavities, from theSIMBA version-1, Ref. [23]. Total loss rates are extracted from measurement. Contributions areapproximate and obtained from a combination of measurement, simulation and values reported in
the literature. See Table S3 of Ref. [23] for more information.
Mechanism
Coupling to microwave ports
Internal dissipation
Coupling to cable modes
Coupling out of bias lines
Total loss rateab

Readout cavity, κr /2π
8 kHza∼ 30 kHz∼ 400 kHz< 1 kHz440 kHz

Parametric cavity, κp /2π∼ 2.5 MHzb
∼ 1 MHz∼ 400 kHz80 kHz4.0 MHzThrough a weakly coupled port (in addition to the strongly coupled port connected to the SIMBA).
Estimated through TIB2 when in reflect mode. Note this rate is 52 MHz when TIB2 is in transmit mode.

Eqs. 6.2-6.4 are solved with the initial conditions A = B = 0 and Ȧ = Ḃ = 0 at time t = 0,
and with the cavities initially decoupled, i.e. g(t) = 0. Beginning at t = 0, an input field with aGaussian profile of standard deviation s and amplitude maximum at time τ1 /2 is incident on thereadout cavity until time τ1 . At τ1 , the input field is turned off and a coupling g(t) = g0 is turnedon until τ2 . These boundary conditions and tunable interactions are described by the piecewisefunctions,

AI (t) =g(t) =
e−(t−τ1 /2)2 /2s2 ,




0,




0,

0 < t < τ1




g0 ,

τ1 < t < τ2 .

0 < t < τ1
(6.5)
τ1 < t(6.6)

Time τ2 corresponds to the time at which the coupling g(t) is turned off, and when phase-sensitive
parametric gain is turned on in the parametric cavity. In this model, gain is assumed to instantaneously exceed loss such that the amplifier efficiency is assumed to be unity.
To estimate the fraction of readout power lost before τ2 , we first numerically solve Eqs. 6.26.4 using the definitions in Eq. 6.5 and Eq. 6.6, while setting the cavity internal loss rates to zero(κr − κw
r = 0 and κp = 0). Doing so, we compute the energy in the parametric cavity immediately

0.8𝑔(𝑡) = 0𝑔(𝑡) = 𝑔0(c)incident field
𝐴𝐼(𝑡)

0.6
0.4
0.2
0

|𝐵(𝑡)|2 (a.u.)(b)

without internal loss

0.6

𝜅𝑟𝜔

0.4
0.2
0

𝜂𝑚𝑜𝑑𝑒𝑙= 0.7

with internal loss

0.8

𝐴(𝑡)
𝜅𝑟 – 𝜅𝑟𝜔

0

20

𝑔(𝑡)

𝐵(𝑡)
𝜅𝑝

40
time, 𝑡 (ns)simulated meas efficiency, 𝜂𝑚𝑜𝑑𝑒𝑙|𝐴(𝑡)|2 (a.u.)

(a) 1

60

𝜏1

80

𝜏2

89
1

20 ns swap time
10 ns swap
time

0.9

0.8

0.7
0

(1)

(2)

(3)

(4)

(5)

loss model

𝜅𝑟/2𝜋

𝜅𝑝/2𝜋

(1)

440 kHz

4.0 MHz(2)

330 kHz

3.0 MHz(3)

220 kHz

2.0 MHz(4)

110 kHz

1.0 MHz(5)

25 kHz0.25 MHz

Figure 6.9: Measurement efficiency model. (a,b) Model of energy vs. time in the readout
cavity (a), and parametric cavity (b), which is proportional to |A(t)|2 and |B(t)|2 , respectively. This
simulation uses Eqs. 6.2-6.4 with the parameters κw
r /2π×8 kHz, κr /2π×440 kHz, κp /2π = 4.0 MHz,s = 13 ns, τ1 = 65 ns, τ2 = τ1 + 20 ns and g0 /2π = 12.5 MHz, chosen to emulate the experiment in
Ref. [23]. Solid green lines result from a simulation which includes internal loss in both cavities,
while the dashed black lines result from a simulation with this loss turned off, κr − κw
r = κp = 0.
Simulated measurement efficiency ηmodel is the ratio of energy remaining at time τ2 with/without
internal loss included in the model. The time τ2 corresponds to when the parametric pump isturned on in experiment. (c) Simulated measurement efficiency for five different models of loss in
the readout and parametric cavities. The blue bar in model (1) is the simulation shown in subplots
a,b, corresponding the the experiment in Ref. [23]. Measurement efficiency is expected to improveto η ≥ 0.9 for the readily realizable parameters of model (4).

before amplification to be E0 = ~ω0 |B(τ2 )|2 . Next, we choose κr 6= 0 and κp 6= 0, and re-simulate
Eqs. 6.2-6.4 to compute the energy in the parametric cavity before amplification including thisloss, E 0 . Modelled measurement efficiency is the ratio of these two numbers, ηmodel = E 0 /E0and depends on κp , κr and g0 . We experimentally measure the parametric cavity loss rate tobe κp /2π = 4.0 ± 0.2 MHz with both TIBs in reflect (Fig. S19 or Ref. [23]). We also measureg0 = 12.5 MHz from the data shown in Fig. 6.3a, and η = 0.70 ± 0.01 from the measurement in
Fig. 6.7. We therefore compute ηmodel = 0.70 by using these parameters and estimating the loss
rate of the readout cavity to be κr /2π = 440 kHz.

90
6.3.3

Excess backaction

Non-zero excess backaction indicates residual coupling between the readout and parametriccavities even when the TIB between them is nominally in reflect mode. The nb = 0.66 photons
of backaction reported in Ref. [23] are estimated to correspond to transmission of −26 dB throughTIB1. This is worse than expected given the 50 dB on/off ratio characterized in a single TIB,
Fig. 5.4. The discrepancy may result from an alternative transmission path which bypasses the
TIB when integrated into a SIMBA. An example of such an effect would be a cable mode whichis coupled to both the readout and parametric cavities, such that the TIB is not as fine-tuned
to reflect well at the cable mode frequency. Such an effect could be eliminated by improving thedevice packaging so that the SIMBA and qubit are separated by much less than a wavelength at thefrequency of interest. Non-zero excess backaction could also result from too low a power handlingof the TIB compared to the amplified state in the parametric cavity; this can be improved byincreasing the number of SQUIDs per array in the TIBs and also changing their critical currents.
While the precise contributions to excess backaction in the version-1 SIMBA demonstration(Fig. 6.7) are difficult to determine, excess backaction can in-principle be much lower than the
nb = 0.66 photons (0.63 effective photons) reported in that work. For example, the version-0
SIMBA demonstration (Fig. 6.6) saw nb = 0.018 ± 0.002 effective photons of excess backaction∼ 4 effective photons in a projective readout pulse. Moreover, this was donecompared to nprojrat an operating point with approximately η = 44% and readout fidelity of F0 = 86% (Fig. S21 ofRef. [23]). This proof-of-principle demonstration gives confidence that with further work, excess
backaction can be dramatically lowered while maintaining high efficiency and readout fidelity.

6.3.4

Readout time

Readout using a SIMBA can be significantly sped up without detriment to performance. In
principle, a faster readout should also be higher fidelity, due to less chance of a bit flip error from
qubit decay before/during measurement.

91
Table 6.5: Contributions to measurement time. Note that the experiment in Ref. [23] has
2χ/2π = 1.93 MHz, g0 /2π = 12.5 MHz and the SIMBA has a reset time of 100 ns. The ‘estimatedafter improvement’ column assumes 2χ/2π = 11 MHz, g0 /2π = 25 MHz and κp /2π = 1 MHz.
systemdemonstration in Ref. [23]estimated after improvement

readout pulse
65 ns
15 ns

swap
20 ns
10 ns

JPA ring-up
80 ns
20 ns

digization
100 ns
25 ns

total
265 ns
70 ns

Readout using a SIMBA differs from standard dispersive readout because the readout cavity
external coupling rate is now tunable. Advantageously, this rate can be made large during themeasurement to allow for fast readout, but can otherwise be tuned close to zero so that the qubit’s
T1 time is not Purcell limited [147].
For optimal readout using a SIMBA, it is desirable to minimize loss in the readout cavity
such that 2χ  κr . In this limit, a readout pulse of appropriate frequency and a duration of
approximately 1/2χ will be phase-shifted by ±π/2 in phase space, dependent on the qubit state.
The dispersive shift alone therefore sets the readout duration when using a SIMBA, not the ratioof 2χ/κr as in conventional dispersive readout (assuming κr is dominated by external loss througha strongly coupled port) [15]. The experiment in Ref. [23] uses 2χ/2π = 1.93 MHz, which can besignificantly increased to speed up the readout, while still remaining in the dispersive limit.
The swap time between the readout and parametric cavities can also be sped up simply byincreasing the bare coupling rate between them. Doing so should also increase efficiency, Fig. 6.9c.
Additionally in Ref. [23], the 80 ns ring-up time for the pumped parametric cavity to reach astable, bifurcated state was in-principle far longer than necessary, and can be reduced to several
nanoseconds by pumping the JPA harder. This results in a higher amplitude state in the pumped
parametric cavity, also reducing the time required to digitize this state at room temperature, albeit
at the cost of more excess backaction if the TIB isolalation is imperfect.
In conclusion, all steps of the SIMBA readout procedure may be sped up, Table. 6.5. Complicated trade-offs arise between the length of different readout steps and the various metrics forreadout performance, but we anticipate a total measurement time of less than 100 ns is achievable

92
with readily attainable improvements to the experiment in Ref. [23].

6.3.5Ease of calibration

Finally, ease of calibration is an important factor to consider with regards to scalability.
Calibration of qubit readout when using a SIMBA is not significantly more complicated than useof a conventional Josephson parametric amplifier, but can still be simplified.
In particular, in Ref. [23] the parametric amplifier performance (for example, the region
where it bifurcates as a function of pump frequency and amplitude), is dependent on the uniform
bias flux seen by both TIBs. This is because the SQUID arrays within the TIBs participate
somewhat in the parametric amplifier mode. In general this is undesirable; it requires extra carewhen choosing the uniform magnetic flux applied to the device. This participation can be easily
lowered by reducing the coupling between the JPA and TIBs, at the cost of a slower swap time.
More difficult but perhaps more useful, the superconducting switch element can be redesigned toparticipate minimally in the parametric cavity resonance.

Chapter 7

OutlookA great challenge of quantum information science is to isolate qubits extremely well fromtheir environment while still being able to couple them strongly to other qubits and a measurement
apparatus. State-of-the-art performance in any platform requires careful understanding and balanceof such tradeoffs. In superconducting systems, tunable coupling elements are a natural way tocircumvent this roadblock.
Toward this goal, in this thesis we develop superconducting microwave switches (TIBs),
which are used to engineer a ‘SIMBA’-style directional amplifier [23]. Using a SIMBA, we demonstrate state-of-the-art readout of a superconducting qubit while eliminating any ferrite circulators/isolators before amplification. Since the limitations of the current generation of SIMBAs arewell understood and not inherent to the concept itself, there is a clear path towards using an optimized SIMBA to improve upon current superconducting qubit readout performance in a scalable
way. Furthermore, the switches themselves are versatile tools with utility beyond our demonstrationof qubit measurement.
We conclude by presenting several promising avenues for future research.

7.1

Parametric coupling to reduce backaction
While the 0.66 photons of excess backaction reported in Ref. [23] is low compared to theseveral photon readout pulse and the hundreds of photons in the pumped parametric cavity, there
is significant room for improvement in performance. Easier calibration is also desirable; for example,

94

modulate coupling at (𝜔𝑟 - 𝜔𝑝)/2𝜋

coupler 1
weakly coupledport

coupler 2

to measurementchainparametriccavity at 𝜔𝑝/2𝜋transmon + readout cavity at 𝜔𝑟/2𝜋

Figure 7.1: A SIMBA operated using parametric coupling. As in Ref. [23], a superconducting
qubit is dispersively coupled to a readout cavity with a tunable coupler on its output. This coupler
is shared by a second cavity which can be parametrically modulated and can be connected to ameasurement chain. Unlike in Ref. [23], the parametric cavity is at a different frequency than thereadout cavity. Coupling is turned on by modulating the first coupler at the difference frequency
between these cavities. With this approach it is much easier to obtain and calibrate good isolation
between the readout and parametric cavities, as they are naturally detuned in addition to being
decoupled when the modulation is turned off.

the TIB gradiometric bias voltage for optimal isolation can be slightly offset from zero, Fig. 6.4,
due to small but unavoidable asymmetry in the circuit layout. It is desirable to have low excess
backaction without having to carefully calibrate this bias for every device.
One way to circumvent these problems is to engineer the parametric cavity to be at a different
frequency from the readout cavity (different by much greater than the linewidths of either cavity),
then turn on a beamsplitter interaction between them by parametrically modulating their couplingat this difference frequency, Section. 2.1.7. Isolation is therefore provided by the cavities being fardetuned, in addition to the coupler between them being tuned near reflect mode when interaction
is not desired.
In such a system, the coupling rate g(t) between two oscillators is modulated in time suchthat g(t) = g0 (1 +  sin(∆t + θ)) where ∆ = |ωr − ωp | is the difference frequency between the two
oscillators, g0 is a bare coupling rate, and  is a modulation amplitude. In the limit   1, thismodulation couples the two oscillators at rate g0 , Sec. 2.1.7. Improved isolation therefore generally
comes with a slower swap step, to the extent that modulation remains in this weak limit.
Alternatively, the parametric cavity could be rapidly tuned in and out of resonance with the95readout cavity in order to swap their states. A swap could be implemented in this manner bysending a square shaped voltage pulse into the uniform flux bias lines already implemented in the
SIMBA. Careful tuning of this pulse might be required but in-principle, such a procedure would
have the advantage of strong coupling during the swap, combined with isolation due to cavity
detuning when the swap is over. Such an approach could even be implemented without a tunable
coupling between the readout and parametric cavities.

7.2

Autonomous feedback using a SIMBA
As operated in Ref. [23], the SIMBA stores the qubit measurement result as a classical

bit of information in the pumped state of the bifurcation amplifier. This bit can be used forclassical information processing, such as in a feedback loop as part of a quantum error correction
algorithm. Current state-of-the-art feedback loops [73, 148] are limited by control electronics atroom temperature, but also the round-trip time for signals to travel in and out of a cryostat. Whenusing a SIMBA, however, the qubit measurement result is stored at the base temperature stageof a dilution refrigerator. This classical bit of information can in-principle be used to implement
fast feedback loops which avoid the round trip travel time between the base of the cryostat and adigitizer at room temperature.
To realize this advantage, we propose a scheme for a rapid feedback loop where a qubit isselectively π-pulse based on a measurement result, Fig. 7.2, similar to the concept in Ref. [149]. In
this approach, a carefully calibrated cancellation pulse shifts the state in the parametric cavity to
be either near-vacuum or high-amplitude depending on the measurement result. (Such a procedure
could also be achieved by using the tri-stable state of the parametric cavity to correlate the stateof the qubit with either a high-amplitude state or vacuum [119].) Next, coupling between theparametric cavity and readout cavity is turned back on, shifting the qubit frequency via the acStark shift [16], conditioned on the measurement result. Finally, a pulse is applied to the qubit
which is now effective only if its frequency has not been shifted. The pulse is therefore conditionalon the measurement result.

96

(a)
step 1:
measurethe qubit(b)
step 2:
displace theparametriccavity stateparametric cavity state
qubit state|1> or |0>

qubit state|1> or |0>

or

or

(c)
step 3:
shift the qubit freq
dependent on the meas outcomeorto readout cavity(d)step 4: pulse
the qubit at its original freq

qubit state|0> or |0>

Figure 7.2: Autonomous feedback. Fast, conditional π-pulse using a SIMBA. (a) The result ofa qubit measurement is stored in the bifurcated state of a parametric cavity. (b) The parametriccavity pump is turned off and the parametric cavity state is displaced by applying a bias tone ofappropriate frequency, amplitude and phase. The measurement result is therefore either correlatedwith near vacuum or a high amplitude state. (c) Next, TIB1 is set to transmit mode to swap thestate in the parametric cavity into the readout cavity, shifting the qubit frequency, or not, dependent
on the measurement result. (d) Finally, a π-pulse is applied to the qubit, which will only be effective
if the qubit frequency has not been shifted (correlated with vacuum in the parametric cavity, andone particular result from the first measurement).

If this procedure works with high fidelity and calibration is not prohibitively difficult, sucha procedure could readily be extended to multi-qubit networks. In other words, a Bloch-sphererotation can be applied on one qubit in a network conditional on the measurement result of adifferent qubit. Such a procedure, if effective, could be useful for improving the speed of feedback
loops used for quantum error correction.

7.3

Utility of a readout cavity with tunable external coupling
Even with a conventional measurement setup, it is generally useful to have a superconducting

qubit readout cavity with a tunable external coupling. This coupling can be set to zero until

97(a) Q1

(b)

Q2
Q3
Q4

Q1
Q2
Q3
Q4
time

Figure 7.3: Qubit multiplexing. (a) Example mutliplexing architecture using a SIMBA. A
SIMBA can be constructed using an N × 1 superconducting switch at the input (with N = 4
shown), constructed from TIBs. (b) With an adequate reset time between measurements, thequbits can be time-domain multiplexed using the same SIMBA. (Square pulses in this schematic
represent a full measurement, digitization and reset procedure).

measurement, when it can be turned on to allow for fast readout [74].
More precisely, the energy decay time of a superconducting qubit coupled dispersively to areadout cavity is limited by the Purcell effect to be 1/T1 = (g/∆)2 κr where g is the coupling rate
between the qubit and cavity, ∆ is the detuning between them, and κr is the linewidth of the readout
cavity [150, 15, 151] (see Sec. 2.1.6.1). Ideally, the linewidth of a readout cavity is dominated byits coupling to the measurement chain (taking any weak port coupling rate or internal loss to be
negligible). The dispersive shift χ is also a function of g and ∆ (see Eq. 2.54) such that larger
values of κr and χ, necessary for faster readout, lower the Purcell limit on a qubit’s T1 time.
This detrimental effect can be inhibited by using a ‘Purcell filter’, essentially a high-pass filteron the external port of a readout cavity which blocks transmission out of the cavity at the qubit
frequency [152]. However, there are still limits to how well any filtering scheme can work [34], andfiltering becomes less effective as χ increases. The same benefits of a Purcell filter can be achieved
using a readout cavity with a tunable external coupling, an approach which is just beginning to be
explored [153, 74, 23].
Moreover, readout cavities with a tunable external coupling can be generally useful for multiplexing [79]. Such a feature can be used to reduce frequency crowding as different resonators
may in-principle be placed at the same frequency and time-domain multiplexed, as illustrated inFig. 7.3. This concept can be extended by constructing larger switch networks from TIBs, Sec. 5.4,

98
or from other styles of couplers [124].

7.4

Superconducting switch from a Wheatstone bridge of capacitors
One non-ideal feature of the TIBs in Refs. [79, 23] is their dependence on the uniform1

magnetic flux. Recall that there is no transmit mode for certain values of this uniform flux
including when it is zero (see Eq. 5.1).
A potential solution to this problem is introduced in Ref. [23] by engineering on-chip ‘uniform’bias lines surrounding the Wheatstone bridges in these TIBs, different from the ‘gradiometric’ biasline used to imbalance the bridge. The uniform bias lines are designed so that current flowing
through them will thread the same amount of magnetic flux through all SQUIDs in a bridge,
allowing this flux to be tuned to an appropriate value without use of an off-chip coil. However,these uniform bias lines can be a source of noise/loss, and furthermore add both to requisite wirecount and the complexity of device calibration.
In recognition of these problems, here we present a design for a superconducting switch which
is not susceptible to any external magnetic field. This concept is illustrated in Fig. 7.4, and is stillbased on breaking/preserving the symmetry of a Wheatstone bridge. Unlike the TIBs developedin this thesis, however, the design in Fig. 7.4 is based on a Wheatstone bridge of capacitors. These
capacitors have fixed value, but any tunable impedance placed at the north/south nodes of thebridge can still be used to imbalance it.
For example, tunable inductances can be realized with arrays of Josephson junctions. Whenthe junction arrays are not biased, from symmetry of the circuit the Wheatstone bridge is balanced
and the device is in reflect mode, Fig. 7.4a. When current is applied through one array, however,symmetry is broken and the bridge becomes effectively imbalanced realizing a transmit mode.2
Capacitors are added to match the network and also break any superconducting loops. Finally, low
1‘Uniform’, meaning a flux which is the same through all SQUIDs.
In fact, symmetry is broken as long as a different current is applied to the two arrays, causing them to havedifferent inductances. Therefore to imbalance the bridge it is not necessary to apply any current through the second
bias line. This line is included in the design, however, to preserve the symmetry of the balanced circuit.
2

(b)transmit modecurrent > 0bias line

balun

2
2

balun

2

-10
-20
-30
-40

bias line𝛪bias/𝛪c = 0.9
𝛪bias/𝛪c = 0

-50
-60
3

bias line𝛪bias/𝛪c = 0.99

0

1

1

99

(c)

transmission (dB)current = 0(a)bias linereflect mode
on-chip LPF

4

5
6
7
8
frequency (GHz)9

Figure 7.4: Superconducting switch from a Wheatstone bridge of capacitors. The bridge
can be balanced/imbalanced by applying current through arrays of Josephson junctions placed at
its north/south nodes and connected to ground. The inductance of these arrays can be changed byapplying current through them, using on-chip bias lines with low-pass filters on them to prevent
microwave power from leaving the circuit. (a) Illustration of the bridge in reflect mode. No current
is applied through either bias line and the brige is balanced, causing it to reflect. (b) The bridge isimbalanced when current is applied through one bias line, leading to transmission between port 1
to port 2. (c) Numerical simulation of a device based on the design in (a,b). The simulated devices
uses the same baluns and low-pass filters as in the SIMBA in Ref. [23], with all capacitors equal
to 1 pF and unbiased junction arrays equal to 0.6 nH inductors. Bias current changes the array
inductance according to Eq. A.3. Applying a bias current of Ibias /Ic = 0.9 gives -6 dB or greaterof transmission between approximately 4 GHz and 6 GHz, with an on/off ration of greater than 35
dB over this range.

pass filters are added to the bias lines in order to prevent microwave frequency signal leaking outof them.
Performance of a preliminary version of this design is simulated in Fig. 7.4c. This is anumerical simulation of a chip layout using similar parameters as those of the TIBs in Ref. [23]. An
on/off ratio of greater than 40 dB with transmission above -6 dB is simulation between 4 GHz and 6
GHz, when biasing the bridge at Ibias /Ic = 0.9. While this performance is not an improvement upon
the TIBs in Ref. [23], it does captures the salient features of high on/off ratio and large bandwidth.
The parameters of the simulated device still have a great deal of room for optimization.
The advantage of the design in Fig. 7.4 is that it preserves the bandwidth and high on/off
ratio of a bridge circuit while not being susceptible to an external magnetic flux. Additionally, it
requires only a single on-chip bias line which can in-principle require a much lower bias current

100than the flux-biased TIB designs presented earlier in this thesis, therefore providing a lower heatload to the cryostat.
A disadvantage of using Josephson junction arrays as a tunable impedance is that the bias
current must be a large fraction of the junction’s critical current in order to significantly imbalance
the bridge (see Eq. A.3 and Fig. A.1d). A high bias current can potentially limit the match andpower handling of such a device, although these features can in general be less important thanon/off ratio for the proposed use as a tunable external coupling on a qubit readout cavity. Suchproblems could potentially be avoided, however, by replacing the arrays of Josephson junctions withasymmetric SQUIDs [46]. The loops of such SQUIDs can be twisted in a figure-8 or ‘asymmetricallythreaded SQUID’ geometry [154], to eliminate any performance dependence on a uniform external
magnetic field.

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Appendix A

Introduction to Josephson junctions and SQUIDs

A.1

The Josephson relations
A Josephson junction, Fig. A.1, is formed by a non-superconducting region between two

superconducting leads [83, 155]. The time-dependent current I(t) flowing through the Josephson
junction and the superconducting phase ϕ(t) = 2πΦ(t)/Φ0 across it are governed by the Josephson
relations:
I = Ic sin (ϕ) ,

(A.1)

Φ0 ∂ϕ
,
2π ∂t

(A.2)

v=
where Φ(t) =Rt

00−∞ dt v(t ) the ‘branch flux’ across the junction, v(t) the voltage across it, Φ0 =h/2e is the magnetic flux quantum and Ic is the junction’s critical current. The critical current
parameterizes the dynamics of the Josephson junction, and generally depends on the geometry andmaterial of its non-superconducting region.
From Eqs. A.1 and A.2, a Josephson junction acts like an inductor whose inductance L0J
depends on the current flowing across it according to,
LJ
L0J = q.
1 − (I/Ic )2

(A.3)

Where
LJ =

Φ0
2πIc

(A.4)

117
is the Josephson inductance. The solution to Eq. A.3 is plotted in Fig. A.1d. As the applied current
I approaches the critical current Ic , the Josephson inductance diverges. For I ≥ Ic the junction
behaves as a resistor and the Josephson relations no longer apply.
𝜓1

𝜓2

(a)
superconductor(Al)

dielectric(Al2O3)superconductor(Al)
1 micron(b)junction𝜑 = 𝜓1 - 𝜓2

(c)
𝐼

4
(d)

𝐿𝐽′/𝐿𝐽
3

2

1

0.0

0.2

0.4

0.6

0.8

1.0

𝐼/𝐼𝑐

Figure A.1: Introduction to Josephson junctions. (a) A Josephson junction is formed by anon-superconducting region (often a dielectric) between two superconducting leads. The absolutevalue of the superconducting order parameter (illustrated by the thick black line) evanescently
decays inside the junction, with the decay constant determined by the junction’s critical current Ic(dependent on the junction geometry and material). The superconducting phase difference acrossthe junction ϕ = ψ1 − ψ2 , where ψ1 and ψ2 are the phases on either side of the junction, is givenby the Josephson relations, Eq. A.1 and A.2. (b) Scanning electron microscope (SEM) imageof a Josephson junction, fabricated by the author using the common ‘double angle evaporation’technique [156]. The approximately 150 nm × 150 nm region at the center of the image consists of athin layer of aluminum oxide ( Al2 O3 ) sandwiched between two layers of aluminum. (c) Josephson
junctions are schematically illustrated by the ‘×’ symbol. (d) Josephson inductance L0J , as afunction of the current I applied across a single junction, divided by the critical current Ic .

A.2

SQUIDsA SQUID or ‘Superconducting QUantum Interference Device’ is formed by one or more

Josephson junctions placed in a superconducting loop. A SQUID behaves as a single Josephson
junction whose critical current is dependent on the magnetic flux threaded through its loop.
To understand this, consider a SQUID formed by two junctions with critical currents Ic,1
and Ic,2 , arranged in parallel as in Fig. A.2a, and where the linear inductance of the loop itself is118
neglected. Current I applied across the SQUID will divide into the currents I1 and I2 across sideof the loop such that I = I1 + I2 . From Eq. A.1,
I = Ic,1 sin ϕ1 + Ic,2 sin ϕ2 ,(A.5)

where ϕ1 and ϕ2 are the superconducting phase drops across each junction. To ensure the superconducting order parameter remains single valued around the loop, these phase drops are related
to the magnetic flux piercing the loop ϕext by:
2πn = ϕ2 − ϕ1 − ϕext ,(A.6)

where n is an integer.
𝜑1
(a)

(b)

𝐼1

𝐼
𝐼2
𝜑2

𝜑ext

junction

10 microns

(c)
loop

upper layer

lower layer

Figure A.2: Introduction to SQUIDs. (a) Two Josephson junctions in parallel form a SQUID(Superconducting QUantum Interference Device). Here, I = I1 + I2 is the current applied acrossthe SQUID, with I1 and I2 the currents across each arm. The superconducting phase drops acrossthe junctions, ϕ1 and ϕ2 , are related to the magnetic flux ϕext piercing the loop by Eq. A.6. (b)SQUIDs may be constructed in series to form an array. (c) Optical micrograph of an array of
SQUIDs used in the TIBs of Ref. [23]. (For further information on fabrication of these SQUIDs,
see Section 5.2 and Appendix B of Ref. [157]).

In a ‘dc-SQUID’ (the style used through this thesis), the two Josephson junctions are designed
to be identical such that Ic,1 = Ic,2 = Ic . Following from Eq. A.5 and Eq. A.6, the dc-SQUID
behaves like a single Josephson junction whose critical current is 2Ic |cos (ϕext /2)|, or equivalently,

119whose Josephson relation is modified to:
I = 2Ic cos

ϕ

ext



2

sin (ϕ) .

(A.7)

Here, Ic is the critical current of a single Josephson junction, and ϕext (t) = 2πΦext (t)/Φ0 is theexternal magnetic flux applied through each SQUID loop, Φext (t), scaled by the magnetic flux
quantum. In general, this flux may be time-dependent. Note that Eq. A.7 neglects the selfcapacitance of the Josephson junctions, the linear inductance of the loop, the inductance of the
SQUIDs themselves, and any loss.

A.3

ArraysA single junction or SQUID can be replaced by an array of many in series, Fig. A.2b,c.

Denoting the superconducting phase difference over an array of identical SQUIDs as 2πΦ/Φ0 , thephase difference across each individual SQUID equals 2πΦ/N Φ0 , where N is the number of SQUIDs.
From Kirchhoff’s current law, the common-mode current through each individual SQUID equals
the current I running through the entire SQUID array. The relationship between current flowing
through an array of SQUIDs and the phase across it is therefore
I = 2Ic cos

ϕ

ext

2



sin

ϕ
N

.

(A.8)

From Eq. A.8 we see that the effective nonlinearity of the SQUID-array is reduced relative toa single SQUID. In particular, the ratio of the leading nonlinear to the linear term in the expansionof the current-phase relation around Φ = 0 scales with 1/N 2 .

Appendix B

SIMBA experimental schematic

qubit control(𝜔𝑞/2𝜋 = 4.51 GHz)

parametric cavity pump
(2×𝜔𝑟/2𝜋 = 12.68 GHz)

readout and local oscillator(𝜔𝑟/2𝜋 = 6.34 GHz)∑arb. wfm. generator

data acquisition

uniform dc
flux control
TIB1
bias

∑

TIB2
bias

Miteq

1.9
MHz

HEMT

cryoperm canecco.

spiralinductor
LPFs

~3 cm
TIB1
transmon +3d aluminum
cavitybackground coil

TIB2
JPA

SIMBA chip

ecco.
optional on-chip
uniform flux bias

20 nH

ecco.
20 nH

20 nH

ecco.

18 mK

350
MHz

5-portcirculator

ecco.

350
MHz

4K

doubleisolatorallcryogenic
attenuators
20 dB

300 K

Figure B.1: Schematic for the experiment in Ref. [23].

Appendix C

Graph theory approach to circuit analysis

The scattering matrix S describes the response of a signal exiting port α of a network, given
a signal incident on port β. More precisely, its elements Sαβ are defined as the ratio of a phasorvβ+ incident on port β of a network to a phasor vα− outgoing from port α [34]:
Sαβ =

vα−
,
vβ+

(C.1)

assuming that no other ports are being driven (such that vγ+ = 0 for γ 6= β).1 The scattering matrix
is a useful tool for characterizing circuit performance, but can be cumbersome to analytically derivefor complicated circuits such as the superconducting switches developed in this thesis. Here, we
review a simple, formulaic method for using graph theory to analytically calculating the scatteringmatrix of an arbitrary linear circuit.
A linear network is completely described by the following information:(1) The connectivity of its nodes.2(2) The impedance/admittance between connected nodes.
(3) Which nodes are connected to ports, and the characteristic impedance of each port.
Points (1) and (2) fully determine the admittance matrix Y. The scattering matrix can be computedfrom Y combined with information about the ports of the network (3).
1Note that the diagonal elements of the scattering matrix are ‘reflection coefficients’ and the off-diagonal elementsare ‘transmission coefficients’.
2
Nodes are regions of equipotential.

122
8
(a)

3

𝑖1
1

1

3

𝑖3

𝑖8

𝑖6
6

5
𝑖4

5
4

port

𝑖2

6
4

(b)

2

1

2

6
3

4

port

𝑖5
port
7

2

5

𝑖7

Figure C.1: Graph theory approach to circuit analysis. (a) An example network. Nodes(pink circles) are equipotential regions at potential vν where ν ∈ {1, 2, ..., 6}. The m chords in the
network are connected by impedances yµ where µ ∈ {1, 2, ..., 8} (gray squares). The chords havecurrents iµ running across them. (b) A lumped element diagram for a tunable inductor bridge(TIB) circuit (Chp. 5) which realizes a superconducting switch. This circuit is an example of the
network in (a), where labels 1, 2, ..., 6 correspond to the same nodes as in (a), and where nodes 1,
2 and 3 are specified to connect to ports.

C.1

Calculating the admittance matrix
Elements of the admittance matrix Yαβ are defined as the current iα at port α when driving

port β with voltage vβ [34]:
Yαβ =

iα
,
vβ

(C.2)

when all other ports are short-circuited (vγ = 0 for γ 6= β). Here, we compute the admittancematrix using graph theory.
Consider a circuit with n nodes each at potential vν , where ν ∈ {1, 2, ..., n}. These nodes areconnected by m chords. Each chord contains an admittance yµ 3 where µ ∈ {1, 2, ..., m}, and has
a current iµ flowing across. The direction corresponding to positive current flow across each chord
can be arbitrarily chosen, but must be kept consistent throughout the analysis. The admittancematrix Y is [158, 159]:
Y = AT y A.
3

(C.3)

The admittance of a circuit element is the inverse of impedance, and generally depends on the frequency ω. For
example, the admittance of an inductor l is 1/jωl, the admittance of a capacitor c is jωc and the admittance of a
resistor r is 1/r.

123Here, y is the ‘primitive admittance matrix’, an m × m diagonal matrix whose frequency dependent
elements are the admittances ym across chords of the network,y1 0 . . . 0 0 y2 . . . 0
y=.
.. . .
.
 ..
. ..
.


0 . . . 0 ym






.





For example, for the network in Fig. C.1b the primitive admittance matrix is:


0
0
0
0
0
0 
jωc 0
 0 jωc
0
0
0
0
0
0 




 0
0 1/jωl1
0
0
0
0
0 






 0
0
0
1/jωl2
0
0
0
0 


y=
,


 0
0
0
0
1/jωl1
0
0
0 




 0
0
0
0
0
1/jωl2 0
0 






 0
0
0
0
0
0
jωc 0 




0
0
0
0
0
0
0 jωc

(C.4)

(C.5)

where we have assumed all capacitors have equal capacitance c, thin inductors (connecting nodes
3 and 6, and nodes 4 and 5) the have inductance l1 , and thick inductors (connecting nodes 4 and6, and nodes 4 and 5) to have inductance l2 . In Eq. C.3, the ‘adjacency matrix’ A is an m × n
matrix which describes the circuit topology, i.e. which pairs of nodes are connected. For example,

124
the adjacency matrix for the circuit in Fig. C.1 is:


0
0 −1
1 0 0
0 1 00 −1 0 0 0 −1 0
0
1






0 0 0 −1 0

1


A=
.


0 0 0 −1 1
0




0 0 −1 0
1
0






0 0 0

1
0
0




0 0 1
0
0
0

(C.6)

The simplicity of Eq. C.3 is that both y and A may be easily written down by inspecting a circuit
diagram.
To understand the adjacency matrix, let’s first consider some of the different rows in Eq. C.6:
the first row refers to a current i1 flowing from node 1 to node 6 across admittance y1 , with thecurrent direction specified by the direction of the arrow in the network diagram in Fig. C.1a. The
second row refers to current i2 flowing from node 2 to node 5 across admittance y2 . The final
row refers to current i8 flowing from node 3 to ground, etc. Now let’s consider the columns: forexample, the third column has −1 as its 3rd and 6th elements, and +1 as its 8th element. Thismeans node 3 (specified by the third column) has currents i3 and i6 flowing into it, and current i8
flowing out of it. As another example, consider the final (6th ) column: its first element is −1 andits 3rd and 4th elements are +1, meaning that node 6 has current i1 flowing into it and currents
i3 and i4 flowing out of it. In this manner we can construct the adjacency matrix A based on
inspection of a network, or equivalently, A fully describes the connectivity of this network.

C.2

Calculating the scattering matrix
The scattering matrix S follows from the admittance matrix. To determine the scatteringmatrix, we construct the network such that nodes 1 through 1 ≤ np ≤ n are connected to ports of125characteristic admittance 1/z0 (the inverse of the usually defined characteristic impedance z0 ).4
The next np + 1 through n nodes of the network are not connected to any port, or equivalently,
are connected to ports of characteristic admittance of zero (characteristic impedance approaching
infinity).
The scattering matrix of the network is [34]:

S=

1
M−Y
z0



1
M+Y
z0

−1
,

(C.7)

where M is an n × n matrix whose upper-left np × np block is an identity matrix and all otherblocks are zero. For the example in Fig. C.1b:

1


0



0
M=

0



0


0



0 0 0 0 0

1 0 0 0 0

0 1 0 0 0.

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

(C.8)

In conclusion, the scattering matrix for an arbitrary network can be calculated by constructing
the admittance matrix using graph theory, Eq. C.3, and then plugging the result into Eq. C.7. For
example, the scattering parameters for the circuit shown in Fig. C.1b are shown in Fig. C.2 for two
different choices of parameters.

4Here, we assume all ports have the same characteristic admittance. See Chp. 4 of Ref. [34] for analysis wherethese values are different.

126

1.0

1.0

(a)

0.8
|𝐒𝛼𝛽|

𝐒13
𝐒23
𝐒33

0.6
0.4

0.8
0.6
0.4
0.2

0.2
0.0

(b)

4

5

6
7
frequency (GHz)

8

0.0

4

5

6
7
frequency (GHz)8

Figure C.2: Scattering parameters from an example circuit. (a) Scattering parameters of
the circuit in Fig. C.1b where all capacitors have equal value of 1 pF, all inductors have an equalvalue of 1 nH, ports 1, 2 and 3 are attached to nodes 1, 2 and 3, respectively, and all ports have acharacteristic impedance of 50 Ohms. From symmetry of the circuit, the transmission from port
3 to port 1 and from port 3 to port 2 have equal magnitude at any frequency. (Connecting thebalanced ports of a balun to ports 1 and 2 realizes a TIB in reflect mode, where the bridge isbalanced.) (b) Scattering parameters where thick/thin inductors in Fig. C.1b have different valuesof 2 nH and 0.5 nH, respectively, but other parameters are the same as in (a). (Connecting thebalanced ports of a balun to ports 1 and 2 realizes a TIB in transmit mode, where the bridge isimbalanced).