Topics in Quantum Computing Architecture...viii research. I also want to express my thanks to the...

Topics in Quantum Computing

Architecture

Thien Nguyen

Research School of Engineering

The Australian National University

A thesis submitted for the degree of

Doctor of Philosophy

College of Engineering and

Computer Science May 2019

© Copyright by Thien Nguyen 2018

All Rights Reserved

This page intentionally left blank.

To my family:

Dad Thang, Mom Phuong, and especially my wife Quyen.


Declaration

I hereby declare that except where specific reference is made to the work of others,

to the best of my knowledge and belief, the contents of this dissertation are original

and have not been submitted in whole or in part for consideration for any other degree

or qualification in this, or any other university. This dissertation is my own work or

partially in collaboration with others. This dissertation contains fewer than 100,000

words exclusive of appendices, bibliography, footnotes, tables and equations.

Thien Nguyen

May 2019


Acknowledgements

In no way would this thesis have been possible without the support of many people who

guided and encouraged me along the way. First and foremost, I am much indebted to

my supervisors, Prof. Matthew James, who has been incredibly supportive from even

before my arrival at the ANU and remained so throughout my study. Matt gave me total

freedom to pursue my interests and helped me connect with experts in the field. At the

same time, he was constructively critical of various aspects of my study and professional

development.

I am grateful to Prof. Lloyd Hollenberg for taking an interest in me and giving

me an opportunity to work with him on the interesting problems of quantum computer

architecture. My special thanks also go to Dr Charles Hill. Charles’ eye for detail and

technical expertise has helped me in refining my ideas into research projects.

In the course of my study at the ANU, I have had the opportunity to work with and

learn from incredibly smart people to whom I would like to express my gratitude. They

are Dr Yu Pan, Dr Guodong Shi, Dr Shibei Xue, Dr Zibo Miao, Dr Shuangshuang Fu,

Dr Michael Hush, and Dr Andre Calvalho. I appreciated every conversation which I had

with every one of you, through which I have expanded my knowledge and understanding

of the quantum field.

To my fellow PhD students, Alfred, Oliver, Jessica, Alex, James, and Ruvi, it was

great listening to your talks in our group meeting. Special thanks to those who always

accepted my invitation to give a speech when I was the coordinator.

During my PhD, I have had the opportunity to work as a research intern at the Atos

Quantum Lab in Paris, France. To this, I’d like to thank Dr Cyril Allouche, head of

the Atos Quantum research program, and Sophie Houssiaux, head of R&D, Big Data

and Security, at Atos for the invaluable experience which had a profound impact on my

viii

research. I also want to express my thanks to the whole Atos quantum lab members who

taught me a lot about quantum computing, supercomputers, and especially about French

culture.

I am grateful to my wife who, despite struggling with her PhD study, has always

provided me emotional support whenever I need. Special thanks to my family members

in Vietnam, who have supported me along the way.

Last but not least, to every participant of the Friday lunch Quantum Cybernetics

group meeting, it was pleasant sharing lunch and discussing a wide variety of topics

with all of you during last four years.

I also gratefully acknowledge the generous financial support I have received from

the ANU and the ARC CQC2T for my PhD study and research.

Abstract

Quantum computing, which is considered the next revolution of computing technology,

brings together theories of mathematics, physics, and computer science. Building a

quantum computer thus requires a synthesis of knowledge and skills from multiple

disciplines. In this thesis, we take a step toward bridging and connecting the full “stack”

of quantum computing technology which spans across theoretical foundations, hardware

architecture, software and simulation.

At the foundation level, we study one of the central problems of quantum computing,

namely quantum error correction, from a control-engineering perspective. This approach

not only complements the conventional coding-based interpretation but also provides a

potential pathway to designing self-correcting quantum computers. First, we analyse

the surface-code quantum error correction under a continuous feedback-based protocol.

Second, we study the fundamental question of self-correcting quantum systems using

the control method of reservoir engineering.

Next, we study the scalability of a generic surface code quantum computer based on

spin qubits such as quantum dots and donor atoms. Solid-state qubits (quantum dots and

donor atoms), especially those that are Si-based, share similarities in device structures

and manufacturing processes with advanced semiconductor CMOS industry. However,

scaling up those quantum devices present immense challenges related to connectivity.

By applying tools and methods from the semiconductor industry, we can concretely

estimate the routing limitation of a planar connectivity scheme.

Finally, as part of my PhD education at the Australian National University, I took

part in an industry-based research internship to develop a high-performance quantum

simulator. We provide the full description of the software architecture, implementation

details, and benchmarking results of the simulator.


List of Publications

This thesis is based on the following collection of papers which have either been

published in or submitted to peer-reviewed journals or conferences. Some of this work

was completed jointly with other researchers and selections from these joint papers to be

included in this thesis are only those that this author contributed significantly towards.

Journal Publications

1. Thien Nguyen, Charles D. Hill, Lloyd C. L. Hollenberg, and Matthew R. James.

Fan-out Estimation in Spin-based Quantum Computer Scale-up. Scientific Reports

7, Article number: 13386 (2017)

2. Yu Pan and Thien Nguyen. Stabilizing Quantum States and Automatic Error

Correction by Dissipation Control. IEEE Transactions on Automatic Control

(Volume: 62, Issue: 9, Sept. 2017) Page(s): 4625 - 4630

3. Thien Nguyen, Zibo Miao, Yu Pan, Nina Amini, Valeri Ugrinovski, and Matthew

James. Pole placement approach to coherent passive reservoir engineering for

storing quantum information. Control Theory Technology (2017) 15: 193.

4. Shibei Xue, Thien Nguyen, Matthew R. James, Alireza Shabani, Valery Ugri-

novskii, Ian R. Petersen. Modelling and Filtering for Non-Markovian Quantum

Systems. Pre-print: arXiv:1704.00986

5. Yi Li, Thien Nguyen, and Yu Pan. Deterministic Multi-Party Quantum Key

Distribution for Wireless Sensor Networks. In preparation.

xii

Peer-reviewed Conference Publications

1. Thien Nguyen, Charles D. Hill, Lloyd C. L. Hollenberg, and Matthew R. James.

Surface Code Continuous Quantum Error Correction Using Feedback. In Proceed-

ings of IEEE 54th Annual Conference on Decision and Control (CDC): Osaka,

Japan, December 2015

2. Yu Pan, Thien Nguyen, Zibo Miao, Nina Amini, Valeri Ugrinovski, and Matthew

James. Coherent observer engineering for protecting quantum information. In

Proceedings of the 35th Chinese Control Conference (CCC): Chengdu, China,

August 2016

3. Thien Nguyen, Zibo Miao, Yu Pan, Nina Amini, Valeri Ugrinovski, and Matthew

James. Pole placement approach to coherent passive reservoir engineering for

storing quantum information. In Proceedings of American Control Conference

(ACC): Seattle, WA, USA, July 2017

4. Shibei Xue, Thien Nguyen, and Ian R. Petersen. Feedback Control of a Non-

Markovian Single Qubit System. In Proceedings of the 11th Asian Control

Conference (ASCC) 2017: Gold Coast, QLD, Australia, December 2017

Table of contents

List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Quantum Technologies 1

1.2 Quantum Computing 3

1.2.1 Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.2 Quantum Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.3 Physical Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.4 Quantum Error Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Software and Programming 16

1.3.1 Quantum Programming Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3.2 Quantum Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4 Quantum Control Engineering 17

1.4.1 Quantum Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.4.2 Quantum Input-Output Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.4.3 Quantum Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.5 Outline 25

1.6 Contributions 28

xiv TABLE OF CONTENTS

2 Solid-state Spin Qubit Control Routing . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.1 Introduction 30

2.2 Routing Dimension Parameters 32

2.3 Solid-State Spin Qubit Unit Cell Model 33

2.4 Methods 39

2.4.1 Ring-by-ring Routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.4.2 Layer Optimisation Routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.5 Results 41

2.6 Summary 46

3 Continuous Quantum Error Correction . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.1 Quantum Errors 49

3.1.1 Quantum Error Correction Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1.2 Surface Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.1.3 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.1.5 Outline and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2 Model 56

3.2.1 Distance-2 Surface Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2.2 Continuous QEC in the SLH Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3 Methods 60

3.4 Results 61



3.5 Conclusions 65

4 Quantum Reservoir Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.1 Introduction 70

TABLE OF CONTENTS xv

4.2 Background 72

4.2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2.2 Linear Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2.3 Lyapunov Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.3 Quantum Error Correction by Dissipation Control 81

4.3.1 Scalability of Lyapunov Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.3.2 Error Correction Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.3.3 Definition of AQEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.3.4 Scalability of Dissipation Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.3.5 Automatic Quantum Error Correction by Dissipation Control . . . . . . . . . . . 92

4.3.6 Dissipation Control of 3-qubit Repetition Code States . . . . . . . . . . . . . . . . . 95

4.4 Decoherence Free Subsystem Synthesis 98

4.4.1 Open-loop Reservoir Engineering for DFS Generation . . . . . . . . . . . . . . . . 99

4.4.2 Coherent Feedback Reservoir Engineering . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.4.3 Special Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.4.4 Special Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.5 Concluding Remarks 117

5 Quantum Programming and Simulation . . . . . . . . . . . . . . . . . . . . . . . . 119

5.1 Quantum Programming Language 120

5.1.1 Quantum Assembly Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.1.2 High-level Programming and Data Processing . . . . . . . . . . . . . . . . . . . . . . 122

5.2 Simulation Engine 124

5.2.1 Overall Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.2.2 Technical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.2.3 Benchmark Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

xvi TABLE OF CONTENTS

6 Simulating Input-Output Quantum Systems with LIQUi|⟩ . . . . . . . 133

6.1 Background 134

6.1.1 Single-photon State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.1.2 Compound Gradient Echo Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.1.3 Engineering Dissipation Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.1.4 Trotter Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.2 Models and Results 142

6.2.1 Amplitude Damping Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.2.2 Toy Example: Single-Atom Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.2.3 Discretized Gradient Echo Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6.3 Conclusions and Future Work 146

6.3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.3.2 The Path Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.1 Where We Stand 149

7.2 The Way Forward 150

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

A Appendix Mathematical Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

A.1 General form of the QSDE for continuous error correction in Chapter 3 169

A.2 Proof of the scalability condition (4.44)-(4.45) 172

A.3 Proof of the condition (4.52)-(4.53) 173

B Appendix Feynman Path Integral Simulation With FPGA . . . . . . . 175

B.1 Method Overview 175

B.2 FPGA Implementation 177

B.3 Software Components 179

B.3.1 Host Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

TABLE OF CONTENTS xvii

B.3.2 FPGA Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183


List of figures

1.1 Qubit Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Solid-state qubit diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Quantum gate timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Quantum gate decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 Surface code layout diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.6 Quantum circuit to measure surface code syndromes . . . . . . . . . . . . . . . . . . . 131.7 Logical CNOT gate braiding diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.8 Logical CNOT gate by braiding holes in surface code . . . . . . . . . . . . . . . . . . 151.9 Active and passive QEC diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.10 Diagram of a typical quantum filtering set-up. . . . . . . . . . . . . . . . . . . . . . . . 201.11 SLH network connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.12 Diagram of beamsplitter in series with cavity QED. . . . . . . . . . . . . . . . . . . . 221.13 Sample QHDL code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.14 Sample QHDL schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.15 Measurement feedback block diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.16 Coherent feedback block diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.17 Overall architecture of a universal quantum computer. . . . . . . . . . . . . . . . . . 27

2.1 Routing parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2 Diagram of surface code lattice with embedded readout devices . . . . . . . . . . 342.3 Diagram of a generic surface code array unit cell . . . . . . . . . . . . . . . . . . . . . . 352.4 Metal 1 pitch scaling roadmap from ITRS . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.5 Ring-by-ring routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.6 Layer optimal routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.7 Illustration of 2-D qubit lattice surface gate fanout . . . . . . . . . . . . . . . . . . . . . 422.8 Fanout scalability at fixed interconnect length . . . . . . . . . . . . . . . . . . . . . . . . 432.9 Qubit fanout scalability in terms of number of routing layers . . . . . . . . . . . . . 452.10 Fan-out scalability vs. interconnect length for SSI and ECI schemes . . . . . . 46

3.1 Surface code layout diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2 Distance-2 surface code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.3 Surface code continuous error correction diagram . . . . . . . . . . . . . . . . . . . . . 58

xx LIST OF FIGURES

3.4 Syndrome estimation trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.5 Continuous feedback error correction fidelity comparison . . . . . . . . . . . . . . . 633.6 Distance-3 surface code diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.7 Time-domain simulation of distance-3 surface code under continuous QEC . 663.8 Fidelity vs. feedback strength plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.1 Potential function plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.2 Lyapunov function scalability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.3 State evolution simulation of repetition error correction code . . . . . . . . . . . . 974.4 Repetition error correction code under dissipative couplings . . . . . . . . . . . . . 984.5 Coherent feedback network for DFS generation. . . . . . . . . . . . . . . . . . . . . . 1004.6 Open loop setup for DFS generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.7 Coherent plant-observer network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.8 Coherent feedback network for DFS generation . . . . . . . . . . . . . . . . . . . . . . 1114.9 Two-cavity system diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.1 AQASM example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1225.2 AQASM QFT example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.3 PyAQASM QFT example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.4 Quantum simulator diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.5 Path integral diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.6 Quantum circuit simulation objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.7 Benchmark of Hadamard gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.8 Comparison with Liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305.9 Comparison with other simulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.1 QHDL design flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.2 Single-photon generating filter cascading realization. . . . . . . . . . . . . . . . . . . 1366.3 Gradient settings for GEM operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376.4 Engineered dissipation by direct coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396.5 Quantum circuit of cascading Hamiltonian unitary . . . . . . . . . . . . . . . . . . . . 1416.6 Discretized single-photon GEM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1426.7 Amplitude damping LIQUi|⟩ code snippet . . . . . . . . . . . . . . . . . . . . . . . . . . 1436.8 Histogram of photon detection events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1446.9 Photon absorption vs. photon bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

B.1 OpenCL implementation of complex arithmetic . . . . . . . . . . . . . . . . . . . . . . 177B.2 OpenCL implementation of multiplication and summation kernels . . . . . . . 180

List of tables

1.1 Types of experimental quantum bits (qubits) . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Hadamard gate implementation steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1 Gate count configurations for spin shuttling interconnect (SSI) and end control

interconnect (ECI) protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2 Minimum interconnect length for spin shuttling interconnect (SSI) and end control

interconnect (ECI) protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1 Fidelity and trace distance comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.1 Technical details of the quantum simulator . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.2 Memory capacity and bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126


Chapter 1

Introduction

Technology is anything that wasn’t around when you were born.

Alan Kay

1.1 Quantum Technologies

The discovery of quantum physics in the early 20th century has fundamentally changed

our understanding of the physical world at the microscopic scales. Quantum mechanics,

which said that objects could be in different states (superposition) at the same time and

correlate over long distance without any direct contact (entanglement), has since become

universally accepted as the most accurate description of physical systems.

The so-called second quantum revolution [37] has started a few decades ago pi-

oneered by both physicists and computer scientists. The novel ideas which brought

about this second wave of quantum revolution are, instead of avoiding quantum effects,

we could transform it from a theory for understanding nature into utilising them as

computing and engineering resources.

2 Chapter 1. Introduction

Quantum Mechanics Primers

• Quantum States (Superposition)

A quantum particle can be in two or more states at the same time. The

quantum state can be described mathematically by a Hilbert space, which is

described by vectors with complex coefficients.

• Composite Systems (Entanglement)

The joint state of multiple quantum systems thus can be described by a

tensor product of their Hilbert spaces. This tensor product results in an

exponential growth in the dimensionality of the composite system. Unlike

classical counterparts, quantum composite systems can be inseparable in

the sense that we can not describe them individually as separate compo-

nents. This is the principle of quantum entanglement which is one of the

fascinating quantum phenomena.

Quantum technologies will lead to significant advances in precision timing, sensors

and computation, have a substantial impact on the finance, defence, aerospace, energy,

infrastructure and telecommunications sectors. For example,

• Ultra-high-precision spectroscopy and microscopy, positioning systems, clocks,

gravitational, electrical and magnetic field sensors.

• Quantum-safe secure networks and quantum key distribution.

• Universal quantum computers: quantum algorithms, modelling materials, quantum

chemistry, biological systems.

One of the emerging quantum 2.0 fields, quantum computing [122], is a significant

breakthrough and paradigm shift in our intuition and understanding of computation. Sci-

entists and industries are developing quantum algorithms and applications which could

potentially speed advancements in materials science [8, 67], financial modelling [132],

machine learning and artificial intelligence [24, 39]. However, realising a quantum

computer is as challenging as the very problems it can solve. There are multitudes of

challenges from the device fabrication up to the software to run quantum algorithms.

Another essential field of quantum technology is quantum control [197, 198, 128, 36]


whose goal is to control physical systems whose behaviour is governed by the laws of

quantum mechanics.

We will explore fundamentals of quantum computing, quantum programming lan-

guage and simulation, and quantum control in greater details in the next sections.

1.2 Quantum Computing

Quantum computers make use of a quantum-mechanical phenomenon (e.g. superposition

and entanglement) that allows data to be represented as quantum bits (qubits) - these are

not constrained to conventional 0 or 1 binary values, but instead can be a superposition

of zero and one simultaneously. Hence, a set of qubits can represent exponentially more

values than their classical-bit counterparts1. This makes quantum computing a promising

platform which could potentially solve computational problems unmanageable for even

the most advanced conventional supercomputer.

To be qualified as a quantum computing platform, there are several necessary condi-

tions which are often summarised as the DiVincenzo’s criteria [34] for scalable quantum

computers. They are,

1. well-defined qubits: addressable and coherently manipulable,

2. initializable to a well-defined quantum state,

3. long coherence times relative to gate times of a universal set of gates, and

4. high quantum efficiency measurements.

Building a quantum computer, despite how challenging it is, is not in itself the

end goal of quantum computing research. Realizing quantum computing capability

demands that hardware fabrication and control efforts be augmented by the design and

development of quantum algorithms/software. The latter part has seen tremendous

progress being made in the last few decades following the seminar paper of Peter Shor

[157] which sparks the public interest in quantum computing. Despite not having a real

quantum computer, researchers have come up with a host of quantum algorithms to solve

cryptography, searching, and simulation problems.

1In this comparison, classical bits refer to the unit of information (bit) in information theory. Infact, classical probabilistic state spaces also scale exponentially in dimension, hence require specializedcomputational algorithms such as the Monte Carlo method.


Quantum Algorithms

• Shor’s algorithm [157, 158]

This is one of the first applications of quantum computers which states that

there exists an efficient way to factor a product of two prime numbers. This

has a huge implication since the difficulty of prime factorization problem is

the foundation of our public-key cryptosystem.

• Grover’s algorithm [62]

Grover’s quantum search algorithm for unstructured data is quadratically

faster than its classical counterpart.

• HHL algorithm [65]

Named after its authors, Harrow, Hassidim and Lloyd, HHL algorithm gives

us the capability to solve systems of linear equations. This paves the way to

transform optimization and machine learning applications to the quantum

domain.

• Quantum Simulation

This includes the fields of quantum chemistry, material science, and particle

physics. Quantum simulation utilizes the most accurate description of

properties and dynamics of systems at nano-scale. Potential applications

are catalyst design (e.g. carbon/nitrogen capture), drug development, and

superconductivity.

1.2.1 Qubits

A probabilistic classical bit can then be represented by a sum of these state vectors:

p |0⟩C + q |1⟩C, where p and q are real numbers in the unit interval that represent the

probability of finding that bit in either state, e.g. the bit has a probability p of being

in the |0⟩C state. We always have p+q = 1, and the state 12 |0⟩C + 1

2 |1⟩C represents a

uniformly random bit. Of course, when we measure the state, it can only be in the state

of 0 or 1 after measurement.


A quantum bit or qubit has a similar structure. It has two output states, |0⟩ and |1⟩,and a general qubit state is represented as a sum of vectors α |0⟩+β |1⟩, except now the

weights α and β are complex numbers so that |α|2 + |β |2 = 1.

The probability of the qubit being in state |0⟩ is given by |α|2, and similarly, |β |2

gives the probability of observing outcome |1⟩. Mathematically we represent qubit states

as vectors whose elements are the coefficients are α and β .

|ψ⟩= α |0⟩+β |1⟩ ≡

α

β

, α, β ∈ C (1.1)

Indeed, a qubit has many more states than a random (probabilistic) classical bit. This

can be demonstrated geometrically by the Bloch sphere: the states of a probabilistic bit

lie on the line between the north and south poles, while the states of a qubit occupy the

whole surface of the sphere. The basis states of |0⟩ and |1⟩ then reside on the north and

south poles.

Fig. 1.1 Qubit Bloch sphere: (left) A qubit state can be in any place on the surface of thesphere; (right) A random classical bit can only exist on the north-south poles line.

Mathematically, multi-qubit states are composed via the tensor product. Hence, the

state space of qubits scales exponentially with the number of qubits. Despite its huge

state-space, we can only ever extract n classical bits of information from an n-qubit

system. Therefore, quantum computers are suitable for applications studying global

properties of functions and data.


Experimentally, there are a wide-range of physical systems which can be used to

implement qubits. Some of the more prominent platforms are listed in Table 1.1. The

number of qubits quoted in Table 1.1 is based on the current fabrication technology.

In this thesis, one particular platform which will be studied in great details is the

donor-based qubit system (donor atoms). In fact, doping Si crystal with impurity has

been used in the semiconductor industry to fabricate MOS transistors for decades.

However, doping control down to single-atom level [155, 140, 178] is what enables

quantum computation. Donor atoms, like Phosphorous, have one extra valence electron

which is naturally confined by the atomic potential of the donor atom itself. Therefore,

the electron’s degrees of freedom, such as spin, can be utilised as quantum information

carrier (qubit). The pictorial illustration of a donor-based qubit is shown in Fig. 1.2,

which is similar to the original configuration proposed by Kane [87].

A AJ

P P

D

1Fig. 1.2 Diagram of the Phosphorous in Silicon donor spin qubit. The donor atom hasone extra electron whose spin is used as qubit. The surface A gate on top of the atomcontrols the hyperfine interaction which can be used together with a magnetic field toperform addressable single-qubit operations. The J gate in between two adjacent atomscontrols the inter-qubit coupling which is used to implement two-qubit gates.

1.2.2 Quantum Gates

Quantum algorithms are often described by sequences of quantum operations which

are quantum gates. Mathematically, quantum gates are complex matrices by which the

quantum state vector gets multiplied. This transformation by a single-qubit quantum

gate can be visualised by rotation on the Bloch sphere.

If classical algorithms are represented by boolean circuits, quantum algorithms are

described by a sequence of quantum gates which are referred to as quantum circuits.


Table 1.1 Types of experimental quantum bits (qubits)

Physical System Characteristics

Trapped ions [92, 114]Laser-controlled charged ions are used as qubits.Long coherence time but gate operations are also slow.Scalability issues w.r.t. laser control.

Coherence time: hoursGate fidelity: >99.9%Max number of qubits: 10-50

Superconducting resonators [33]Electrical LC oscillators with zero resistance.Electrically-controlled by microwave signals.Operates at cryogenic temperature.

Coherence time: 10-100 µsGate fidelity: >99.5%Max number of qubits: 50+

Quantum dots [101, 10]Gate-defined nano structures in semiconductormaterials.Small (nano-scale) qubits, electrically controlled.Using the same fabrication technology as classicalchips.Relatively few interacting qubits can be demonstrated.

Coherence time: µs - msGate fidelity: >99%Max number of qubits: 2-5

Donor atoms [48, 47]

Single dopant atom site in semiconductor.Both nuclear and electron spins can be used as qubits.Single-atom doping is challenging.

Coherence time: up to secs(nuclear spins)Gate fidelity: >99%Max number of qubits: 2-5

Diamond vacancies (NV centers) [192, 203]Nitrogen atom in diamond and a vacant lattice site.Laser-controlled and can operate at room temperature.Difficult to couple multiple NV qubits for quantumcomputing.

Coherence time: secsGate fidelity: >99%Max number of qubits: 5-10

Topological qubits [117, 2]Quasi non-local particles (majorana fermions).Can have extreme long coherence time due tonon-locality.Less developed theoretical and experimental capabili-ties.

Coherence time: unknownGate fidelity: unknownMax number of qubits: 1

There are complementary frameworks which can be used to describe quantum algorithms

(e.g. tensor networks and graphical flow diagrams [25]). However, the quantum circuit


framework is by far the most widely-adopted thanks to its analogy to the classical logical

boolean circuit framework.

A quantum gate or quantum logic gate is a fundamental quantum circuit operating

on a small number of qubits. They are the analogues for quantum computers to classical

logic gates for conventional digital computers. Quantum logic gates are reversible,

unlike many classical logic gates. Mathematically, we represent quantum logic gates

using unitary matrices.

For example, the Hadamard gate, which operates on a single qubit, is represented by

the matrix:

H =1√2

1 1

1 −1

. (1.2)

Another widely-used set of single-qubit gates are called Pauli gates whose matrices

are:

X =

0 1

1 0

, Y =

0 −i

i 0

, Z =

1 0

0 −1

. (1.3)

Given a single-qubit gate which is represented by a unitary matrix U , a controlled-U

gate is a two-qubit gate as follows:

C(U) =

1 0 0 0

0 1 0 0

0 0 u00 u01

0 0 u10 u11

. (1.4)

where

U =

u00 u01

u10 u11

. (1.5)

is the original gate matrix.


Universal Quantum Gates

A set of quantum gates is universal if any operation possible on a quantum

computer can be reduced to it.

For example, the set of the Hadamard gate (H), a phase rotation gate (by an

arbitrary angle), and the controlled-NOT (CNOT ) gate is universal.

1.2.3 Physical Implementations

Logical qubit operations (quantum gates) have to be experimentally implemented via

complex instrumental control schemes. For instance, to perform quantum operations on

donor electron spin qubits, we utilise the two available surface gates/contacts (A and J)

as shown in Fig. 1.2 together with the applied magnetic field. Then, the scheme of Hill

et al. [70] can be used to implement single qubit gates as well as the CNOT gate.

As an example, the Pauli X-rotation on solid-state spin qubits can be performed in

two steps. Firstly, the target qubit is rotated by 2π while all other qubits are rotated

2π+θ by applying voltage on their A-gates thus increasing their rotation speed. We have

created a relative rotation of θ as desired between the target qubit and all other “observer”

qubits. However, to correct the overall rotation, we need to perform a correction by

rotating all qubits by θ around X-axis. This time we apply voltage on all A-gates since

we want all qubits to rotate evenly. The procedure is depicted in Fig. 1.3.

A1

Jk

Ai6=1

Target: Rx(2π) ≡ I

Observer: Rx(2π + θ) ≡ Rx(−θ)Target: Rx(θ)

Observer: Rx(θ)

Fig. 1.3 Timing diagram to implement X-rotation on donor spin electron. A1 is theA-gate contact of the target qubit.

Similar techniques can be used to rotate the qubit around an arbitrary axis. Indeed,

the Hadamard is nothing but a π rotation around the axis m = i+k√2

(i, j, and k are basis

vectors), which can be implemented similarly as the X rotation. The steps to perform a

Hadamard gate is summarised in Table 1.2. For more information, see Hill et al. [70].


Table 1.2 Hadamard gate implementation steps

Step Target qubit Observer qubits

1 Rm(π)≡ H Rx(α)

2 Rx(2π)≡ I Rx(−α)

We can pretty much apply arbitrary single-qubit rotation using this technique. Two-

qubit gates, such as the CNOT gate, can be achieved by using the following decomposi-

tion,

CNOT = (H ⊗ I)exp(

iπI −X

2⊗ I −X

2

)(H ⊗ I)

= (H ⊗ I)[Rx

(π

2

)⊗Rx

(π

2

)]

× exp(

iπ

4X ⊗X

)(H ⊗ I).

(1.6)

The only two-qubit operation in Eq. (1.6) is exp(iπ

4 X ⊗X), which can be implemented

straightforwardly using the coupling H = J(σ ⊗σ) term, where σ are the Pauli spin

matrices (with eigenvalues ±1), as

exp(

iπ

4X ⊗X

)= (X ⊗ I)exp

(iπ

8σ ⊗σ

)

× (X ⊗ I)exp(

iπ

8σ ⊗σ

).

(1.7)

Using quantum circuit notation, we can express the sequence of gates to implement

a CNOT gate between two qubits as in Fig. 1.4, which is physically executed by pulsing

the A’s and J surface metal gates.

U(π8

)U(π8

)|a〉 H X X Rx

(π2

)H

|b〉 Rx(π2

)

|a〉

|a⊕ b〉

Fig. 1.4 Quantum circuit decomposition of a CNOT gate. U(

π

8

)= exp

(iπ

8 σ ⊗σ). Each

of the decomposed elements can be realized by properly-timed pulsing of the surfacegates (A and J).


This type of quantum gate decomposition is a crucial aspect of the quantum software

stack which we will explore in Chapter 5. Since each qubit hardware architecture can

handle a specific set of elementary gates (satisfying the universality condition), it is the

task of the software stack to translate from an hardware-agnostic quantum computing

instructions into the actual physical gate operations which the underlying hardware

platform can support [109, QSh].

This is pretty much similar to what a compiler is doing when you wrote classical

computer codes, i.e. converting from text-based high-level programming languages

into assembly bytecodes for the particular target platform. Developing an effective tool-

chain for quantum programming can potentially improve the productivity of algorithm

development and also lower the entry barrier of quantum computing since most of

the complication of underlying physical hardware can be abstracted away and hence

developers can rely on a uniform computing model at the top of the stack.

1.2.4 Quantum Error Correction

The most prominent drawback of the technology is that qubits are inherently fragile.

Hence, they can only retain quantum information for a tiny amount of time. Interactions

within the system and with a noisy environment are typically the limiting factors of

coherence time: the duration during which the underlying qubit system retains its

quantum properties. Therefore, the best devices require extremely clean control signals

and cryogenic operation to reduce thermal noise.

Those random fluctuations will occasionally flip or randomise the state of a qubit,

potentially corrupting the computation. Hence, to create a “functional” quantum com-

puter, we need a quantum computer that is fault-tolerant in the sense that the quantum

computer must be able to detect failures so that they do not spread in space and time

during computation and hence can be corrected.

One of the most prominent contributions of fault-tolerant quantum computing re-

search is the invention and the continuous improvement of various quantum error

correction (QEC) codes [52, 53, 30, 148, 32, 161, 189, 43, 42, 162, 167, 164, 118],

which provide a potential pathway to achieving universal quantum computing.


The goal of quantum error correction is to use redundancy and correction to realise

logical qubits with improved error rates as compared with that of the elementary qubits.

The fault-tolerance threshold [99], which is the qubit failure probability below which

reliable quantum computation becomes feasible, is the standard measure of fault-tolerant

quantum computing, where error propagation is constrained such that error correction

protocols remain effective [54].

Among a wide variety of quantum error correction codes, the surface code [30, 94,

45, 189, 43] has stood out in terms of computational error threshold which is about two

orders of magnitude higher than that of conventional concatenated coding schemes. The

critical feature is that implementing the surface code requires a regular 2-D arrangement

of qubits, where neighbouring qubits interact with each other in a pairwise manner and

in parallel (see Fig. 1.5). Qubits are classified either as data qubits or syndrome (ancilla)

qubits according to their roles in the quantum error correction procedure. Each syndrome

qubit measurement fixes an eigensubspace of a stabilizer operator, which involves all

four neighbouring data qubits. Logical qubits are defined as topological defects on the

qubit lattice where syndromes are not measured. Thus, there are two types of logical

qubits, so-called smooth (Z-cut) and rough (X-cut) logical qubits. The code distance is

defined either by the perimeter of the defects or the distance between them, whichever is

smaller. Interested readers should consult Fowler et al.[43] for an in-depth review.

For surface code, there are only two types of syndromes: Z or X syndromes, which

stand for ZZZZ or XXXX operators acting on the four data qubits. The Z and X

syndromes are measured by performing a sequence of CNOT gates between the ancilla

and its four neighbouring data qubits as shown in the quantum circuit diagrams in Fig.

1.6a and 1.6b, respectively.

Logical qubits are defined as topological holes (defects) on the qubit lattice where

syndromes are not being measured. Thus, there are two types of logical qubits, so-called

smooth (Z-cut) and rough (X-cut) logical qubits. The code distance is then defined

either by the perimeter of the holes or the distance between them, whichever is smaller.

The reason is that an uncorrectable error occurs in surface code whenever physical

qubit errors form a continuous chain surrounding a hole or connecting two holes. It is

worthwhile mentioning that the boundary of the code lattice could also be considered


1 2 3 4

5 6 7

8 9 10 11

12 13 14

15 16 17 18

X X X

Z Z Z Z

X X X

Z Z Z Z

X X X

Fig. 1.5 Surface code layout: white circles represent data qubits; filled circles aresyndrome qubits (X stabilizers in green and Z stabilizers in yellow). Each internalstabilizer acts on four adjacent data qubits, while boundary stabilizers act on either twoor three data qubits.

|0〉 MZ

|a〉

|b〉

|c〉

|d〉(a) Z-syndrome measurement

|0〉 H H MZ

|a〉

|b〉

|c〉

|d〉(b) X-syndrome measurement

Fig. 1.6 Quantum circuit to measure surface code syndromes. The first line representsthe ancilla, while the following lines represent the surrounding data qubits.

as a superficial hole. Therefore, crossing the boundary from one end to the opposite

or connecting one hole to the boundary are also irreversible logical errors. Interested

readers should consult [43] for an in-depth explanation.

The bigger and farther apart the surface code holes are, the higher code distance

we can get. Therefore, the ability to fabricate a gigantic lattice of qubits is pivotal in

making the surface code to work. Besides, because the logical CNOT operation in the


surface code involves braiding a pair of holes as shown in Fig. 1.7, the actual distance

between two distance-d holes needs to be at least 3d. An example of the CNOT braiding

evolution [43] between an X-cut qubit and a Z-cut counterpart is illustrated in Fig. 1.8. In

the surface code layout, a pair of holes that are 3d apart will serve as a unit cell. Braiding

operations can be optimised at the architectural level with regard to the computation

time or lattice area as shown in [135]. At the micro-architectural level, on the other hand,

we need to be able to keep track of the holes’ locations as well as grow and shrink them

by turning off and on the respective syndrome qubits.

1st pair of holes

2nd pair of holes

Time

Fig. 1.7 Braiding diagram of a logical CNOT gate in surface code with time axis runninghorizontally.

The code distance is calculated by the intrinsic error rate of the underlying qubit

hardware as well as the threshold of the selected error correction code, as following [86]

εlogical ≥C1

(C2

εphysical

εthreshold

)⌊ d+12 ⌋

, (1.8)

where C1 and C2 are constants that are code-dependent. For surface code, according to

[46], the values for C1 and C2 are 0.13 and 0.61, respectively. We denote the error rate

per logical gate operation as εlogical , which depends on the target success probability

of the algorithm and the number of logical gate operations to complete it. An entire

algorithm success rate of εalgo will require εlogical ≈ εalgo/Nlogical , where Nlogical is the

number of logical gate operations required for the entire computation. The physical error

rate εphysical is the experimental fidelity of the qubits, which is compared against the

threshold of the surface code at εthreshold ≈ 1% to determine the minimum code distance


3d

(a) Original holes (b) Growth hole A

(c) Shrink hole A (d) Growth hole A

Fig. 1.8 Logical CNOT gate by braiding holes in surface code. The braiding is done byexpanding and shrinking a hole. In order to maintain the code distance d throughout thisprocess, the orginal holes need to be separated by at least 3d.

d. This is the minimum code size that can guarantee the algorithm to be completed with

high probability.

From this code distance estimation, we can asymptotically approximate the number

of physical qubits needed and therefore the expected classical resources. Remember

that a distance d logical qubit needs a hole covering approximately d2 physical qubits.

In addition, all the holes need to be 3d apart to make sure logical CNOT gates can be

readily performed between any pair of logical qubits. A simple calculation can be made


to estimate the number of physical qubits as follows.

Nphysical ∝ d2 × (4×√

Nlogical)2, (1.9)

in which, the number of logical qubits Nlogical is algorithm dependent, and the code

distance d is from the previous estimation step.

1.3 Software and Programming

1.3.1 Quantum Programming Languages

A quantum programming language is the means by which quantum algorithms, sub-

routines, or applications can be expressed in a human-comprehensible form yet can be

faithfully and consistently translated into quantum operation sequences which imple-

ments the high-level instructions across various hardware platforms. The abstraction

away from the specific hardware implementation is particularly important at this stage be-

cause we still have a handful of competing platforms which are very different in terms of

low-level operations. For application developers, a consistent and platform-independent

view of quantum computation is the key to innovation and productivity.

Most importantly, the programming language must obey the principles of quantum

computing, namely the no-cloning theorem and the reversibility (unitary) of quantum

operations. On top of that, the language itself needs to be able to cope with emerging

computing models, e.g. the mix of classical and quantum data in a quantum/classical

hybrid programming model.

Early quantum programming language proposals from academic researchers, for

example, QCL (Quantum Computation Language) [208], QGL (Quantum Gate Lan-

guage) [165], Scaffold [3], Chisel-Q [100], or Quipper [60], provide the foundation for

the quantum programming research community. Recently, thanks to the tremendous

interest from industry in quantum computing, we have seen the released of industry-

backed quantum programming languages and development kits/environment. Some of

the most popular are:

• LIQUi|⟩ from Microsoft, embedded in F#

• Q#, also from Microsoft


• OpenQASM and Quantum Information Software Kit (QISKit) from IBM

• Quil/Forrest from Rigetti

• AQASM from Atos

1.3.2 Quantum Simulation

Since functional quantum computers will not be available anytime soon, we need to be

able to simulate quantum algorithms on conventional computers to validate and debug

those algorithms. The simulation of quantum computers using first principle linear

algebra approach, i.e. matrix-vector multiplication, is strictly memory-bound. The

memory capacity required grows exponentially with the number of qubits involved in

the algorithms.

For reference, one of the largest quantum circuits that have been simulated using

this first principle approach is a 45-qubit simulation, which used 500 terabytes2 of

memory on a state-of-the-art supercomputer [81]. There are other various approaches

which could help reduce the memory requirement at the cost of computing time (space-

time trade-offs) or the accuracy of the simulation (e.g. tensor product approximation

methods).

The lack of access to real quantum computing hardware necessitates the use of

quantum simulators to test algorithms and programs. In addition, by deploying multiple

computation models (not just the universal first principle linear algebra approach) on the

simulator back-end, we could provide the best performance outcome for each particular

quantum circuit of interest.

1.4 Quantum Control Engineering

Quantum control engineering, which has been evolving in tandem with quantum com-

puting, plays a vital role in the design and realisation of quantum devices. The ideas

of using feedback control to stabilise naturally-unstable systems are the cornerstone of

classical control theory. Various control techniques can be deployed to autonomously

correct quantum error in the same manner as the conventional QEC schemes.

21 terabyte (TB) = 1,024 gigabytes (GB)


Observables, Measurement and Decoherence

• Observables and Measurement

The wavefunction is what defines the probability of finding (measuring) the

particle’s property (e.g. velocity or position) at a certain value.

When a measurement is made, the quantum state is settled (collapsed) into

one of the possibilities. The outcome of the measurement is randomised

according to the wavefunction (quantum state). However, after measure-

ment, the quantum state is completely determined, i.e. we will always get

the same measurement outcome if repeating that measurement.

• Decoherence

Since quantum superposition states collapse if measured, unintended in-

teractions with the surrounding environment can lead to the “leaking” of

quantum information.

In this section, I want to summarise the fundamental concepts of quantum control

and the role it may play in the entire fault-tolerant quantum computing system, or

“stack”. This can be visualised in the diagram in Fig. 1.9 where the quantum control

layer sits just above the hardware qubit implementation. This control layer is supposed

to provide fully-autonomous stability enhancement and error rejection to the underlying

qubit network before any active error correction actions are applied, e.g. measuring

syndrome qubits, decoding for potential errors, etc.

1.4.1 Quantum Noise

The dynamics of quantum states (ψ(t)) are often described in terms of the Schrödinger’s

equation. For open quantum systems, the dynamics are described by the Lindblad master

equation [19] (Markovian case) which has the following form:

ρ(t) =−i[H,ρ(t)]+∑L ∗L (ρ(t)) (1.10)


Quantum Error Correction Qubit Network

Passive Error Correction/Noise Cancellation

Active Error Correction

SyndromeMeasurement

Correction

Coherent/classical feedback control

Dynamical decoupling (open-loop)

Fig. 1.9 Combination of passive and active techniques for quantum error correction inwhich the passive (autonomous) layer improves the stability and/or error rejection tothe underlying qubit network before any active error correction actions are applied, e.g.measuring syndrome qubits, decoding for potential errors.

where ρ(t) is density matrix, H is the self-energy Hamiltonian operator, and the super-

operator L is defined as

L ∗L (ρ) = LρL† − 1

2L†Lρ − 1

2ρL†L. (1.11)

The coupling operator L in equations (1.10) and (1.11) describes the coupling (inter-

action) between the quantum system of interest and the external fields (environment/bath).

It is worth noting that we could in principle describe the system + bath as a whole using

the interaction Hamiltonian. This so-called quantum noise model [49, 141], which

describes the overall system in terms of plant and interacting fields (hence coupling

operators), is the foundation of the quantum feedback control field [198, 84, 205].

1.4.2 Quantum Input-Output Model

Emerging engineering problem of extracting information about the dynamical state of

quantum systems through measurement created the field of quantum filtering [18] which


is another form of the generalised continuous measurement formalism [198]. Front and

centre in this theory is the interaction between the quantum field and system-of-interest

as shown in Fig. 1.10 which in essence has two main effects. Firstly, information about

the system is gained by observing the field output which we will harvest using filtering.

This is the most common mechanism used in any metrology schemes. However, what

distinguishes quantum measurement is the inevitable back-action effect which dictates a

conditional or posterior state of the system consistent which an observed outcome.

Measure

afterinteraction

filter

measurementsignal estimate

beforeinteraction

B(t) Bout(t)

System

Y (t) X(t)

Fig. 1.10 Diagram of a typical quantum filtering set-up.

QHDL (Quantum hardware Description Language) is one of the very first Computer-

Aided-Design (CAD) tools for quantum systems built upon the concept of (S, L, H)

encapsulation. This formulation provides a common thread in the design and verification

flow from schematic capture to an HDL-like description and basic dynamical simulation

with both symbolic and numeric capabilities. Thanks to QHDL, a wide variety of

optical devices can be described in a systematic way that is ready for integration, such

as quantum optical logic gates [107], Set-Reset latch [105], or fully-coherent error-

corrected quantum memory [90].

In the quantum input-output formalism [50], the stochastic evolution of an open

Markov quantum system driven by vacuum noise inputs (dA(t)′s) is given by the Hudson-

Parthasarathy Quantum Stochastic Differential Equation [79]:

dU(t) = −iHdt +(S− I)dΛ(t) (1.12)

+ ∑i

(dA†

i (t)Li −L†i SdAi(t)−

12

L†i Lidt

)U(t),

in which, the unitary evolution U(t) is defined on the combined space of the system

plant and coupling fields. Any system operator dynamics can be derived from that using

the relation X(t) = U(t)†XU(t). The resulting operator-based differential equations


are referred to as Heisenberg-Langevin equations and completely equivalent to the

Schrödinger picture master equation.

This Heisenberg-picture dynamical model can be parametrised conveniently by a

triple G = (S, L, H), where H is the internal Hamiltonian, L = Li is a set of coupling

operators (e.g. annihilation operators for amplitude damping), and S is a unitary input-

to-output scattering matrix (e.g. beam-splitters in quantum optics or quantum point

contacts in solid-state). This parametrisation scheme is often referred to as the SLH

quantum network theory [55, 56].

The network part of this model is what important since from the three parameters

and the given network topology we can compute the equivalent SLH model of the entire

network thanks to the two basic rules as shown in Fig. 1.11.

G1

G2

G1 G2

G1 G2

G2 C G1

Fig. 1.11 SLH network connections: (left) Concatenation and (right) Cascading.

Mathematically, these two connection rules can be expressed as:

G2 ⊞G1 =

S1 0

0 S2

,

L1

L2

,H1 +H2

, (1.13)

G2 ◁G1 = (S2S1,L2 +S2L1,H1 +H2 + ImL†2S2L1). (1.14)

This is the backbone of the QHDL toolbox, whereby network topology (can be in the

form of schematics or Verilog-style inputs as shown in the below example) is processed

symbolically to derive the overall network model. The system dynamics can then be

simulated by differential equation solvers.


Besides the bottom-up approach, we can also perform a top-down decomposition

in the SLH framework by the network synthesis theory [127]. Given an arbitrary SLH

model which may contain a large number of internal dynamical variables (optical modes

or qubits) and inputs/outputs, one can always faithfully identify a suitable collection of

one degree of freedom oscillator components and to connect them serially with proper

Hamiltonian interaction to build up the prescribed system model.

One recent development of the SLH modelling approach is the effort to extend its

application to a wide variety of input states besides the conventional vacuum inputs,

such as thermal field, single-photon and two-photon states [58, 125, 159, 57].

For demonstration purposes, considering a fundamental system of a beam-splitter

(M) and a cavity QED (C) [187] in series as diagrammatically shown in Fig. 1.12.

A1

A2

A2

B1

B1

γ

κ

Fig. 1.12 Diagram of beamsplitter in series with cavity QED.

Individually, each component can be described by its corresponding SLH parame-

ters [56]3:

M =

β −α

α β

,0,0

,

C =

I,

√κa

√γσ−

,Hc

,

Hc = ∆ f a†a+∆aσ+σ−+ ig(σ+a−σ−a†),

where κ and γ are the cavity and atomic decay rates, respectively; ∆ f and ∆a are the

detunings; g is the coupling constant between the field and atomic transition.3assuming instantaneous coupling


-- Structural QHDL generated by gnetlist-- Entity declaration

ENTITY BeamSplitterNetwork ISPORT (

A1 : in fieldmode;A2 : in fieldmode;Vacin : in fieldmode;B1 : out fieldmode;B2 : out fieldmode;Vacout : out fieldmode);

END MachZehnder;

-- Secondary unitARCHITECTURE netlist OF BeamSplitterNetwork IS

COMPONENT BeamsplitterGENERIC (

theta : real := theta_value);PORT (

In1 : in fieldmode;In2 : in fieldmode;Out1 : out fieldmode;Out2 : out fieldmode);

END COMPONENT ;

COMPONENT SingleSidedJaynesCummingsGENERIC (

kappa : real := kappa_value);gamma : real := gamma_value);g : real := g_value);Delta_a : real := Delta_a_value);Delta_f : real := Delta_f_value);

PORT (In1 : in fieldmode;VacIn: in fieldmode;Out1 : out fieldmode;UOut : out fieldmode);

END COMPONENT ;

SIGNAL A_1_in : fieldmode;SIGNAL A_2_in : fieldmode;SIGNAL A_1_out : fieldmode;SIGNAL A_2_out : fieldmode;SIGNAL B_1_out : fieldmode;SIGNAL Vac_in : fieldmode;SIGNAL Vac_out : fieldmode;

BEGIN-- Architecture statement part

B1 : Beamsplitter PORT MAP (In1 => A_1_in,In2 => A_2_in,Out1 => A_1_out,Out2 => A_2_out);

C1: SingleSidedJaynesCummings PORT MAP (In1 => A_1_out,VacIn => Vac_in,Out1 => B_1_out,UOut => Vac_out);

-- Signal assignment partA_1_in <= A1;A_2_in <= A2;Vac_in <= Vacin;B1 <= B_1_out;B2 <= A_2_out;Vacout <= Vac_out;END netlist;

Fig. 1.13 Sample QHDL code of the cavity-beamsplitter system.


We can easily translate the diagram in Fig. 1.12 into a functional schematic using

appropriate pre-defined components as shown in Fig. 1.14. This can then be rendered

into a QHDL description as listed in Fig. 1.13 which will specify the ports, components,

connections, as well as any user-defined parameters of the system. One can use this to

generate Heisenberg or Schrödinger type dynamical equations to perform simulation or

verification as desired.

In1

In2

Out1Out

2

(−)

BeamSplitterIn1

Out1

VacIn UOut

Cavity

A1

B1

A2

Vac

In

Vac

Out

B2 = B2 = A2

B1

Fig. 1.14 Sample schematic using QHDL schematic capture tool: A QED cavity isconnected in series with a beamsplitter.

1.4.3 Quantum Control

The quantum input-output model, as captured by the (S, L, H) parameter set, bears

similarities to the classical control theory. In particular, the output field channels carry

information about the state of the quantum system due to the couplings as described by

the L operators. This information can be probed by a controller to gain knowledge about

the plant, hence can control it to the desired state. This closed plant-controller loop is

the basis of quantum feedback control.

Depending on the nature of the controller used in the feedback loop, quantum control

can be classified into two main categories:

1. Measurement feedback

The output field is measured continuously and the classical measurement signal

(e.g. analog electrical signal or digitised bits) is processed by a classical controller

which could be a Digital Signal Processing (DSP) chip, a Field Programmable

Chip Array (FPGA), or an electronic circuit. The controller will then emit control

1.5 Outline 25

Meas.I/V

Filter

F

Est.: ρ

Target: ρ

+

Quantum Device

Fig. 1.15 Measurement feedback block diagram.

signals based on pre-defined control algorithms. A typical block diagram of a

measurement feedback setup is shown in Fig. 1.15.

2. Coherent feedback

In coherent feedback control, the output field is fed to another quantum systems

directly as input signals. One quantum system acts as the controller whilst the

other is the plant. No quantum information is “leaked” to the outside world4 and

thus quantum coherence is preserved in the loop. It is worth noting that besides

field coupling, we could also have direct Hamiltonian coupling between the plant

and controller. This can be described as an interaction Hamiltonian involves

the dynamical variables of both systems. Coherent quantum feedback control is

illustrated in Fig. 1.16.

1.5 Outline

This thesis will study quantum computing at various layers of the quantum “stack” as

depicted in Fig. 1.17, from hardware devices to the error-correction layer as well as

quantum algorithms and applications. More specifically,

• Chapter 2: we investigate the routing scalability of solid-state quantum computers

under 2-dimensional error-correction code, namely the surface code. By applying

the classical electronics know-how regarding interconnect routing to a ubiquitous 2-

D qubit array with independent gate control and readout fan-out, we will develop a

concrete procedure for scalability estimation, which is adaptable to a wide range of

4when forming the feedback loop


Quantum System

(plant)

Quantum

Controller

Quantum

Field

Quantum

Field

Direct Coupling

Fig. 1.16 Coherent feedback block diagram.

surface code implementations by adjusting the gate configuration and dimensional

parameters.

• Chapter 3: we look at quantum error correction from a control-engineering centric

standpoint whereby we apply continuous control method to error correction qubit

network. A measurement feedback scheme is used to correct quantum error

continuously in real-time for a surface code 2-dimensional qubit array. The

controller design in this chapter is considered classical since it acquires and

processes classical measurement signals to output control signals.

• Chapter 4: we extend the controller design for quantum memory and quantum

error correction into the quantum realm by using techniques of dissipation control

(reservoir engineering) and coherent feedback control. In this chapter, coherent

feedback control technique is used to synthesise perfectly-isolated quantum infor-

mation storage system, namely the decoherence-free subsystem (DFS). Given the

hardware overhead of quantum error correction regarding the number of qubits

depends on the quality of the underlying physical qubits, utilising control tech-

niques to improve qubit stability even before any active error correction operations

are applied as shown in Fig. 1.9 will potentially reduce the amount of overhead

needed for fault-tolerant quantum computing.

1.5 Outline 27

Quantum Hardware

Superconductors Ion Traps Quantum Dots Donors NV Centers Topological

Quantum Gate Operations

Quantum Error Correction

Quantum Software (Algorithms)

PhysicalQubits

Logical

Qubits

Fig. 1.17 Overall architecture of a universal quantum computer.

• Chapter 5: we discuss the features and the development of a quantum programming

language, namely the Atos Quantum Assembly Language (AQASM) which I was

involved during my research internship at their quantum research lab in Paris.

The new quantum software suite also includes a classical quantum simulator

based on Feynman path integral approach. The simulator was successfully tested

on high-performance computing (HPC) platforms and is complementary to the

linear algebra simulator. A combination of the two is shown to provide the best

performance while not sacrificing the universality of the simulator.

• Chapter 6: we develop a method to simulate quantum open systems based on

the quantum control input-output formalism on a commercial simulator, namely

Microsoft LIQUi|⟩. A combination of theoretical and practical solutions can

provide accurate and consistent simulation results when using a digital gate-based

quantum simulator instead of conventional solver-based simulators.

Lastly, Chapter 7 summarises the content of this thesis with a brief outlook on future

work on quantum control, quantum computing architecture, and quantum programming

language. The follow-up Appendices provides detail proofs and additional information

as referred to in the main text.


1.6 Contributions

The aims of this thesis are

(i) to study a scalability aspect (fanout routing) of fault-tolerant quantum computation

from an engineering perspective (Chapter 2),

(ii) to apply quantum control technology to improve the stability of qubit systems

which are the foundation of the quantum computing hierarchy (Chapter 3 and 4),

and

(iii) to develop practical solutions to enhance quantum software development (Chap-

ter 5) and to bridge the gap between the digital gate-based simulation model and

the dynamical open quantum system model which is the cornerstone of quantum

control theory (Chapter 6).

It represents the author’s attempt to contribute to knowledge through academic

research and industry engagement activities.

Specifically, Chapter 2, 3 and 4 are derived from published works in peer-reviewed

journals or conference proceedings.

Chapter 5 summarises the author’s research results accomplished during an industry

internship with Atos Quantum Lab in Paris, France. Not only was the quantum simulator

successfully commercialised, but a patent has also been granted5 for the simulation

method, demonstrating the originality of the work.

Chapter 6 is based on the report that the author produced as an entry to the worldwide

Microsoft Quantum Challenge in which he won the Grand Prize. In this work, the

author has “extended LIQUi|⟩’s capabilities by supplementing the existing Hamiltonian

simulator with an innovative gadget for simulating dissipation. This clever use of

amplitude-damping noise enables a quantum computer to be used to simulate open

quantum systems as well as closed systems, and has important applications in real-world

situations.”6, according to the judges from Microsoft.

5France patent No FR3064380, Cyril Allouche and Thien Nguyen, “Procede de Simulation, Sur unOrdinateur Classique, D’un Circuit Quantique”, September 28, 2018

6https://www.microsoft.com/en-us/research/blog/microsoft-quantum-challenge-results-are-in/

Chapter 2

Solid-state Spin Qubit Control Routing

Every person should have their escape route planned.

Simon Pegg

The need for quantum error correction has a profound implication on the scalability

of future quantum computers. For example, some of the significant problems which

need to be addressed are overhead and complexity in terms of the number of qubits, the

qubit layout design to maintain fault-tolerant properties, and the coordination between

quantum and classical software algorithmically to decode and correct errors as well as

perform the intended quantum algorithm on the quantum computer.

This chapter aims to study the fan-out routing scalability, i.e. the ability to route

planar electrical wires, of surface-code quantum error correction code. To deal with

this question, in this chapter we apply the well-developed routing techniques of modern

semiconductor chip design to solid-state qubit platforms. This chapter proposes a

parametrization scheme which models the qubit layout in terms of the surface gate count

and spacing dimension. The results strongly indicate a bottle-neck which requires novel

architecture designs or technological breakthroughs to provide long-term scalability of

surface code solid-state quantum computers.

30 Chapter 2. Solid-state Spin Qubit Control Routing

2.1 Introduction

Building a large-scale quantum computer which can solve classically intractable prob-

lems is a technologically daunting task. With their close connection to highly scalable

classical electronics [72] solid-state spin qubit platforms, such as donor-based qubits [87,

155, 178, 38, 47, 144, 150, 206, 116] and quantum dots [177, 64, 206, 182, 89, 183],

are emerging as promising candidates [10, 40] for scalable quantum computation. On

semiconducting materials, e.g. Si, SiGe, or GaAs, it is possible in principle to fabricate

a large number of interconnecting qubits for quantum information processing. However,

in designing such a large-scale solid-state quantum chip, there is still a gap between the

quantum computer architecture [133, 28, 73, 147, 32, 35, 108, 86, 71] and the physical

qubit device implementation [177, 64, 206]. Architectures necessarily must incorporate

fault-tolerant quantum error correction in order to perform quantum algorithms [122]

at the logical quantum gate level. The physical implementation generally deals with

individual qubits on the basis of physical quantum gate operations, initialisation, and

readout which are the foundation for higher level quantum logical operations. In the

middle ground, quantum computer micro-architectures [86, 180, 71] attempt to bridge

that gap by providing engineering solutions to issues such as classical control, fan-out

interconnects, and chip layout.

One key advantage of the surface code is its nearest-neighbor interaction scheme

which scales favorably over the concatenation approach. However, this scheme also re-

quires a two-dimensional qubit layout and parallel control. In terms of micro-architecture

considerations, one must account for (a) the spatial/geometrical requirements of a 2D

nearest-neighbor interacting qubit array, and (b) the temporal/control requirements of

parallel/synchronous QEC operations. Broadly, one can identify two approaches. In the

ubiquitous independent control model, each quantum element (qubit, gate, interconnect,

readout) are controlled independently. In principle, this approach has the highest density

of quantum control gates each of which must be carefully characterised and timed to

allow for parallel operation across the qubit array (in a number of steps which does

not depend on the array size). At the other extreme, in the distributed control model

introduced in Hill et al. [71], a high degree of multiplexing allows sufficiently large

groups of qubits to be controlled and readout with the required parallelism.

2.1 Introduction 31

Some authors have attempted to address the problem in the independent control

approach by assuming the qubit lattice can be broken into smaller sparsely linked 2D

arrays [181], however, such tiling schemes in general present significant difficulties in

implementing the full range of logical operations required by the surface code. We

instead focus on the spatial/geometrical challenge of fabricating and scaling up of the

full monolithic surface code under the assumption of the independent (non-distributed)

control model in order to compare with the distributed control scheme. Quantum

interconnect protocols to reduce the qubit density are encapsulated in our study by

adding extra coupling surface gates which drive the transport protocols, and assuming

operational errors can be accommodated in the QEC protocol. In terms of gate density,

the generalised quantum interconnect model effectively captures most interconnect

mechanisms by adjusting the number of control gates per interconnect channel.

Under our generalised independent control model, we apply known techniques in

interconnect routing to analyse the geometrical scaling problem of surface code control

fan-out. We consider two types of solid-state spin qubits: atomically confined qubits

(such as phosphorus donors in silicon) [87, 155, 178, 47, 144, 116] and electrostatically

confined quantum dot qubits [177, 64, 206, 182, 89, 183]. In the non-distributed indepen-

dent control approach, every qubit on the surface code lattice has its own separate control

and readout structures that need to be fanned out. The qubit geometry is parametrised by

a universal unit cell which can be used to represent both donor-based and quantum dot

implementations including the quantum interconnects to neighboring cells by adjusting

the number of gates in the unit cell. Other dimensional parameters are selected based on

experimental and technological considerations. We must also stress that the scalability

of 2D spin qubit arrays depends on multiple factors, not just the control fan-out which

we study in this chapter. In particular, one must also address the various control issues

such as parallelizability, synchronisation, control characterisation, and cross-talk as well

as the overall thermal budget given the system will be required to operate at cryogenic

temperatures.


2.2 Routing Dimension Parameters

In order to perform the routing analysis, we need to define the geometry of the wiring

and via pads. In particular, planar routing on each metal layer depends on the dimension

of the metal wires and the spacing between vias. These geometric parameters, which are

shown in Fig. 2.1, can be defined as followings:

• Pitch (p): the spacing between two neighboring pads after redistribution

• Pad diameter (d): the diameter of the pads

• Line width (w): the width of wires

• Line spacing (s): the spacing between wires or wires and pads

• Grid channel: the routing space available between two horizontal or vertical pads.

Its routing capacity is calculated by:

C = ⌊ p−d − sw+ s

⌋ (2.1)

• Diagonal channel: the routing space available between diagonal pads. In a square

array, its routing capacity is calculated by:

D = ⌊√

2p−d − sw+ s

⌋ (2.2)

d

w

s

grid channel

diagonal channel

p

Fig. 2.1 Routing parameters: pad pitch (p), wire width and spacing (w and s), paddiameter (d). The grid and diagonal channels are also indicated.

The smaller the wires, the better fanout scaling can be achieved. However, nar-

row and closely-spaced interconnects also tend to compromise the signal integrity,


especially at high frequency. In this work, to provide upper-bound for the scalability,

we assume minimal wiring dimension of width(w) = 5 nm and spacing(s) = 25 nm.

Similar nanoscale wires have been fabricated in the lab for nanowire structures [191].

This wiring dimension assumption is also consistent with the International Technology

Roadmap for Semiconductors (ITRS) projection that by 2020 mainstream semiconductor

manufacturing will reach nanowire diameter of 5 nm. We assume that the via contact

diameter will double the wire width, i.e. d = 10 nm. Regarding the interconnect pitch

(p) used for routing, as shown in (2.3), when we implement longer interconnect chains

between qubits, the pitch will be extended.

2.3 Solid-State Spin Qubit Unit Cell Model

In this analysis, we will consider solid-state spin-based quantum computer platforms

with a model that encompasses both donor qubits and quantum dot qubits. To construct

a basic unit cell model for the micro-architectural fan-out routing analysis, the low-level

physics of the quantum devices, as well as high-level quantum computing architecture,

can be abstracted by making the following assumptions:

• Generalised quantum spin interconnects between neighboring qubits,

• Dedicated single-shot spin readout for every qubit,

• Single-sided metallization routing.

• Uniform interconnect dimension and spacing.

The first assumption regards the mechanism by which the qubit-qubit interaction is

realised. In principle, we could only implement direct spin-spin coupling, e.g. by spin

exchange or dipole couplings, however, direct spin couplings require stringent spacing

between qubits which restricts the control and readout routing. By adding interconnects

between qubits, we have some flexibility in arranging the qubits and thus can analyse

the fanout scalability accordingly. Secondly, we assume each qubit in the array has

its own spin read-out device which is usually a Single Electron Transistor (SET [115])

or equivalent [27, 184, 51, 77, 149]. This assumption may appear to be more than

necessary since neighboring qubits can share a common readout device by using some

forms of readout multiplexing, for example, the schemes presented in [124, 71, 12].

However, for our generic fan-out analysis, this serves as a baseline scenario from which


we can straightforwardly adapt to other cases by modifying the gate count per qubit

to reflect other specific configurations with readout multiplexing. We assume that the

metal routing layers are built on a single side of the substrate. This is the predominant

routing technology used by the semiconductor industry. Lastly, we assume that the

feature size of interconnect wires is consistent between metallization layers. A pictorial

representation of the qubit array structure with dedicated readout devices is shown in

Fig. 2.2.

1 2 3 4

5 6 7

8 9 10 11

12 13 14

15 16 17 18

X X X

Z Z Z Z

X X X

Z Z Z Z

X X X

Data Qubits Syndrome Qubits Readout

Fig. 2.2 Diagram of surface code lattice with embedded readout devices. There aretwo types of qubits: data qubits and syndrome qubits (X and Z types). Neighboringqubits can interact with each other in order to perform CNOT gates. In this model, eachqubit has its own readout device. Dashed lines (black) represent quantum interconnectsbetween neighboring qubits. Dotted lines indicate qubits to which readout devices areassociated.

Regarding the interconnect protocols, while there seems to be a plethora of coherent

spin transport/coupling mechanisms [17, 23, 61, 199, 163, 68, 110, 176, 175, 12], for

the purpose of our fanout analysis, the main factor to consider is the number of additional

control gates versus interconnect length. We, therefore, consider two broad categories:

(i) gate count grows linearly with the interconnect length, and (ii) gate count is fixed

and independent of the interconnect length. For instance, SWAP-based interconnect and

spin shuttling protocols [12, 11] belong to the first category since we need surface gates


along the channel to execute the quantum operations for spin swapping or shuttling. On

the other hand, protocols such as CTAP [61, 73, 111], capacitive coupling via floating

gate [176], spin chain [17, 23, 199], microwave line coupling [186, 29, 123], electric

dipole coupling [175], and surface acoustic wave spin transport [163, 68, 110] are some

examples of the second category because in those protocols we only need to have some

additional transport control gates at the ends of the interconnect not along the channel.

In what follows, we will use the terms spin shuttling interconnect (SSI) and end control

interconnect (ECI) for those two interconnect categories, respectively. The overall length

of the interconnect is L (for interconnect schemes based on qubit chains we equivalently

describe the interconnect length in terms of the number of nodes, Nnodes).

READOUT

Nr

Nq

Nc

Nc

Fig. 2.3 Diagram of a generic surface code array unit cell. Each qubit (circle) has acertain number of surface gates (Nq) to define qubit confinement potential and to performsingle-qubit rotations. Between any pair of neighboring qubits, we have Nc couplinggates that are used to control qubit interconnect coupling. At the center of the cell, wehave a readout device that has Nr surface gates.

In terms of physical qubit implementation, we categorise the surface metal gates

that need to be fanned out for controllability and readout into three categories: qubit

confining and control (Nq), interconnect coupling control (Nc), and readout (Nr). The

types of physical spin qubits considered are primarily classified by the confinement

mechanism, i.e. either via an atomic Coulomb potential (e.g. donors) for which we

assume Nq = 1, or via electrostatic gates (e.g. quantum dots), for which we assume

Nq = 3. Since we assume spin coupling based interconnects, the center-to-center distance

(pitch) between qubits needs to be sufficiently small. We use the qubit-qubit pitch of 20


nm and 50 nm for Coulomb-confined and electrostatically-confined qubits, respectively.

Using the above gate classification, the surface code lattice can be decomposed into unit

cells, each of which contains one qubit and one readout device as shown in Fig. 2.3.

When partitioning the surface code lattice as shown in Fig. 2.2, there are four equivalent

interactions, namely along the north-east, north-west, south-east, or south-west direction.

For example, the unit cell in Fig. 2.3 is a south west participation scheme where the

interconnects and readout device on the bottom left of a qubit are associated with that

qubit for analysis purposes.

The latest International Technology Roadmap for Semiconductors (ITRS) [196],

which dictates the cadence of the semiconductor industry, predicts a wire pitch (distance

between two neighbouring wires) of around 20nm for the next ten years as shown in

Fig. 2.4. Indeed, 24nm-pitch copper interconnect has been recently demonstrated [200,

13] to be production-worthy using the next generation lithography process. Smaller

nanowires can be fabricated up to atomic precision using laboratory equipment [191].

Hence, the parameters which we use in this study are not only relevant to the laboratory

experiments but also to future quantum devices fabricated by commercial foundries.

2015 2020 2025

20

40

60

80

Year

Metal

1Pitch

(nm)

ITRS 2013Production Data

Fig. 2.4 Metal 1 pitch scaling roadmap from ITRS 2013 [196]; production data is takenfrom 22nm and 14nm nodes [9, 119].


As indicated, we will categorise the interconnect protocols into two groups: SSI,

where the number of interconnect control gate count grows linearly with interconnect

length and ECI with a fixed number of interconnect surface gates regardless of intercon-

nect length. The gate count assumptions for these two scenarios are listed in Table 2.1.

The surface contacts are placed directly on top of the qubits, interconnect rails and

Table 2.1 Gate count configurations for spin shuttling interconnect (SSI) and end controlinterconnect (ECI) protocols. The interconnect node count (Nnodes) is the number ofintermediate qubit nodes along the interconnect channel.

Interconnect type SSI ECI

Nc 4×Nnodes 4

Nr 3 3

readout devices. In order to facilitate routing, these gate contacts are then redistributed

into a square grid array. The surface code qubit array can then be assembled by placing

unit cells next to each other, thus forms a regular global square grid array used for

fan-out routing.

It is worth noting that qubits (dots or donors) along the interconnect rails in Fig. 2.3

are not counted as physical qubits in the following analyses. Only the corner qubit of

the unit cell which can act as a data or syndrome qubit in the surface code (Fig. 2.2) is

accounted for as a physical qubit in the scalability study. In fact, several ECI schemes

that we mentioned earlier do not require intermediate qubit nodes at all, e.g. microwave

or capacitive coupling. For this scheme, the absolute interconnect distance is the only

relevant parameter.

For ECI and SSI schemes that involve qubit chains, an important issue may arise

which is the loss of qubits during transfer/coupling (due to operational errors or per-

manent manufacturing defects). While acknowledging that there are quantum error

correction methods and techniques [15, 161, 69, 164, 118] to mitigate qubit loss, this

aspect of qubit connectivity is outside the scope of our considerations here. Therefore,

we assume the feasibility of reliable quantum interconnects in order to focus on the issue

of fan-out routing scalability.


The contact pitch after redistribution is related to the interconnect length (L) by the

following inequality:

pRL

≤ unit cell dimension(L)√Ntotal(L)

, (2.3)

where Ntotal is the total number of gate contacts in a unit cell and pRL

is the contact

pitch at the redistribution layer (RL). This total gate count may or may not depend

on the interconnect length. On the other hand, the unit cell dimension is proportional

to the interconnect length L. We can clearly see that by increasing the length of the

interconnect (L), the contact pitch after redistribution is extended since the denominator

is either constant or growing on the scale of square root of L while the nominator grows

linearly with L. In principle, larger pitches will benefit the global fan-out routing as

more interconnect routing space is created. This is explained in the Methods section

where we describe the routing parameters and the two commonly-used fan-out routing

algorithms.

At the redistribution layer the dimension parameters are d = 10 nm, w = 5 nm, and s

= 25 nm whilst the contact-contact pitch equals to the redistributed pitch computed by

(2.3). However, there is a minimum contact-contact pitch which needs to be satisfied,

namely pmin = d + s =35 nm. Thus, there is a lower bound on the interconnect length

to space the contacts sufficiently according to (2.3). The worst-case scenario occurs in

the SSI scheme for Coulomb-confined qubits because of their tight qubit-qubit spacing

and increasing number of coupling gates with interconnect length. We can estimate the

minimum interconnect length by using equation (2.3) in conjunction with the gate count

data in Table 2.1 and the qubit pitch assumption, e.g. for the 20 nm case we have

pRL

≈ 20nm×Nnodes√1+4Nnodes +3

> 35nm, (2.4)

which requires a minimum interconnect length (min(Nnodes)) of 14 nodes (280 nm).

Following the same procedure, we can derive the minimum interconnect length for

all configurations in terms of Nq configurations and interconnect schemes as shown in

Table 2.2.

2.4 Methods 39

Table 2.2 Minimum interconnect length in terms of chain node-count and absolutedistance for spin shuttling interconnect (SSI) and end control interconnect (ECI) proto-cols. Qubit-qubit pitch is 20 nm for atomically confined qubit (Nq = 1) and 50 nm forelectrostatically confined qubit (Nq = 3).

min(Nnodes)/Linterconnect SSI ECI

Nq = 1 (20 nm) 14 / 280 nm 5 / 100 nm

Nq = 3 (50 nm) 3 / 150 nm 3 / 150 nm

2.4 Methods

2.4.1 Ring-by-ring Routing

Fig. 2.5 Conventional ring-by-ring routing approach: the outermost ring of unconnectedpads are connected first, then inner rings are connected using grid channels of the outerring until their capacity exhausted. This procedure is then repeated on upper metal layers.The left is the routing on the first metal layer. Similarly, the middle one is the routingon the second layer, and so on. The overall procedure is depicted in the right diagramwhere each ring denotes the remaining pads after each layer of metallization.

A ring-by-ring router will work as follows:

1. Connect the outermost pads directly,

2. Use the grid channels between outermost pads to route internal pads on a ring-by-

ring basis,

3. When all the grid channels are exhausted, move up to an upper metal layer and

repeat step 1 and 2 until all pads are routed.

This approach is very intuitive, as shown in Fig. 2.5. However, the major drawback

of this scheme is that its boundaries are quickly shrinking layer-by-layer (as illustrated by

the smaller and smaller dotted squares on the rightmost diagram in Fig. 2.5). Therefore,

the routing capacity also decays as we proceed to higher and higher layer. This results in


a higher number of metallization layers required as compared to the layer optimisation

scheme.

2.4.2 Layer Optimisation Routing

A second widely used scheme for escape routing is the so-called triangular routing [190]

that is depicted in Fig. 2.6.

Fig. 2.6 Metal layer optimal routing approach: the routing procedure is performed byproceeding triangularly inward. In this way, it can deploy the diagonal channels, whichalways have higher routing capacity and take advantage of empty spaces resulted frompreviously routed pads. The left diagram shows the pads that are routed in the first layer.The middle is the routing on the next layer. The right is the overview of this routingapproach.

In contrast to the intuitive ring-by-ring approach, this scheme was derived as a

maximum flow optimisation problem whereby the opening space left by routed pads in

lower layers are utilised maximally, as shown in the middle diagram of Fig. 2.6. This

resulted in a minimal number of layers required to route all the pads.

An n×n array will require at least k layers of routing, where k is the smallest integer

that satisfies the below inequality [190]:

−2(D+1)(D+2)k2 +[4(D+1)n−10D+8C]k ≥ n2, (2.5)

where C and D are the grid and diagonal capacities in Eq. (2.1) and (2.2), respectively.

2.5 Results 41

2.5 Results

Generally, in order to supply the electrical signals to the control gates or the readout

devices to perform qubit readout, each and every gate needs to be fanned out to connect

to the classical control systems. In conventional nanoelectronics, this is achieved by

overlaying the qubit array with many metal lines on several layers. Electrical connections

from these metal lines to the surface gates are made by vertical conducting “vias”. The

unique advantage of Si-based solid-state quantum platforms is the compatibility with the

classical CMOS electronics, whereby both can be integrated onto the same silicon chip.

Classical electronics can be fabricated outside the surface code qubit lattice as shown

in Fig. 2.7. At the bottom layer lies the semiconducting material substrate in which

qubits are realised and controlled by surface gates. Therefore, we need to fan the surface

gates out to the peripheral classical electronics area where classical processing tasks

are performed. Under the generic model considered here, regardless of the interconnect

protocols, the gate contacts/vias are redistributed into a square-grid array before global

fan-out routing is performed, as illustrated by metal routing layers shown in Fig. 2.7.

The fan-out scalability of 2D qubit arrays is examined by looking at the number of

routing layers required for complete routability. As shown in Fig. 2.7, multi-layer routing

can potentially provide unlimited fan-out capacity if we let the number of metal layers

be unbounded. However, in practice it is imperative to keep the number of metallization

layers to the absolute minimum - usually in the range of 10-15 layers for the most

advanced semiconductor products [119]. The technological and economic challenges

associated with fabricating many layers of nano-scale interconnects are going to be

similar for the various solid-state quantum computing approaches. In the following

analyses, we stretch to a 20 routing layer limit to benchmark the fan-out scalability of the

various quantum interconnect schemes. We will adopt two standard routing algorithms

from classical electronics, namely the ring-by-ring and the layer optimisation algorithms,

which are described in detail in the Methods section.

First, we look at the raw differences between the two routing algorithms at a fixed

interconnect length. The triangular routing (layer optimisation) algorithm is the most

efficient way to fan-out all contacts in terms of the required number of layers (see

Methods). This is shown in Fig. 2.8, where we examine both ring-by-ring and layer


Fig. 2.7 Illustration of 2-D qubit lattice surface gate fanout using multiple metal routinglayers. The bottom layer is a semiconductor material (Si or GaAs) with top gates forcontrol and readout. On the same substrate lies classical integrated electronics usedfor signal generation, multiplexing, and sensing. In order to bring connections to thesurface gates, multi-layer routing is needed. After surface gates are redistributed intoa square-grid array of contacts, as shown in the first metal layer, the fan-out routingprocedure is carried out layer-by-layer using a specific routing algorithm. This figuredemonstrates ring-by-ring routing, which requires three metal layers for this particulargrid array. More sophisticated routing algorithms can be implemented using EDA(Electronic Design Automation) tools.

optimal routing solutions for the SSI and ECI protocols for L = 300 nm (which satisfies

the minimum interconnect length (14 nodes, 280 nm) for the atomically-confined SSI

scheme). It is worth noting that we use the same interconnect length to compute the

fan-out for electrostatically-confined qubits (Nq = 3). Because the qubit-qubit distance

is different, the number of qubit nodes in the interconnect chain varies across different

qubit configurations in the bar chart comparison (Fig. 2.8, right).

We observe a factor of 5 to 8 increase in the number of routable qubits by using

the optimal router across most of the scenarios (except for SSI, Nq = 1) as shown in

the right chart of Fig. 2.8. This highlights the fact that for large-scale qubit integration

the use of design-automation tool suites is important to achieve better routing solutions

2.5 Results 43

101 102 103 104 105

Number of physical qubits

100

101

102

103

Num

ber

ofla

yer

s

20-layer limit

Number of routing layers comparison for Nq = 1 (L=15, 300nm)

SSI Ring-by-ring

SSI Optimal

ECI Ring-by-ring

ECI Optimal

Nq = 1 Nq = 3 Nq = 1 Nq = 30

1000

2000

3000

4000

5000

6000

7000

8000

Num

ber

ofphysi

calqubits

End Control Interconnect Spin Shuttling Interconnect

27 27 58464

Number of qubits routable with 20 layers (L=300 nm)

Ring-by-ring

Optimal

Fig. 2.8 Comparison between different interconnect protocols and routing methodsat fixed interconnect length: (left) Plot of the number of routing layers vs. numberof physical qubits for atomically confined qubits (Nq = 1) at L = 300nm; and (right)scalability comparison between electrostatically confined qubits (Nq = 3) and atomicallyconfined qubits (Nq = 1) using the same number of routing layers (20) and interconnectlength (L = 300nm, 15 nodes for Nq = 1 and 6 nodes for Nq = 3). Dimension parametersare (see Methods): d = 10nm, w = 5nm, and s = 25nm. The red dashed horizontal line onthe left figure represents the technological limit of 20 metal layers that can be fabricatedreliably and economically on a semiconductor substrate.

compared to more intuitive and direct methods such as the ring-by-ring method. One

exception is in the case of SSI scheme for atomically-confined qubits (Nq = 1) where

both routing methods result in the same number of routable qubits. The reason is that

at this interconnect length (L = 300nm), the contact-contact pitch after redistribution

(eq. (2.3)) is barely above the minimum metal-metal pitch requirement, thus no escape

routing (wires between pads) is allowed.

Another point which can be seen from Fig. 2.8 is the advantage of the ECI scheme

over its SSI counterpart in terms of fan-out scalability. This naturally stems from the fact

that ECI protocols require far fewer surface gates than their SSI counterparts (Table 2.1).

We will later investigate the scaling differences between the two schemes in details by

looking at various interconnect lengths. The difference in qubit-qubit spacing manifests

itself in the opposite trend observed in Fig. 2.8 bar chart: while atomically-confined

qubits are the clear winner in the ECI scheme, the opposite is true if the shuttling

interconnect scheme is assumed. This can be explained by the minimum interconnect

length data in Table 2.2, i.e. while Nq = 1 has shorter minimum interconnect requirement

in the ECI scheme, it has a much longer minimum interconnect length in the SSI scheme.


The scalability difference in the ECI scheme between Nq = 1 and Nq = 3 is narrowed

significantly if we use optimal routing for electrostatically-confined qubits (from 2x

to about 25% different). In all the following analyses, we will only consider triangle

(layer optimised) router since this will better reflect the realistic engineering solution.

To highlight the effect of even more confining gates (e.g. double-dots as qubits), we also

perform the analysis for a hypothetical case of Nq = 5.

All the scenarios that we consider so far are homogeneous in the sense that corner

qubits and interconnect qubit nodes are of the same type. For SSI scheme, we can

implement a hybridisation protocol in which atomically confined qubits are used as

surface code physical qubits, while electrostatically confined qubits are utilised for

interconnect coupling. By doing this, we can achieve the best of both worlds for the SSI

scheme, namely minimising Nq and maximising qubit distance. This approach is only

effective for SSI scheme since for ECI the number of interconnect gates is constant.

Fig. 2.9 shows the fan-out scalability in terms of routing layers (optimised router) for

both the SSI and ECI protocols with interconnect length of 300 nm, 450 nm, and 600 nm

(15/20/30 nodes and 6/8/12 nodes for Nq = 1 and Nq = 3/5 or SSI hybrid, respectively).

In the top graphs, the fan-out scaling of atomically confined qubits (Nq = 1) is analysed

in detail to provide a reference and the horizontal line represents the limit of 20 metal

layers as previously explained. Other qubit configurations (Nq = 3/5 and SSI hybrid)

are compared to this reference in the bottom graphs.

An obvious conclusion which can be drawn from both the left charts in Fig. 2.9 is

that the SSI protocol does not provide a consistent fan-out scaling benefit as compared

to its ECI counterpart, as there is no clear trend in terms of the number of routable

qubits vs. interconnect length. The main contributing factors to this fluctuating trend

are the opposite effects of redistributed pitch extension, the increasing number of gates

per unit cell and the granularity of the routing problem (only full routing channels are

considered). On the other hand, ECI protocols provide a monotonic improvement in

terms of the number of integrated qubits vs. interconnect length because the interconnect

length (thus metal pitch) is Nc-independent. The order of routability vs. interconnect

length for different qubit configurations is reserved for both SSI and ECI schemes. While

the former interconnect scheme favours electrostatically-confined qubits due to their

2.5 Results 45

101 102 103 104 105


100

101

102

103N

um

ber

ofla

yer

s

20-layer limit

Number of routing layers for SSI scheme (Nq = 1)

L = 15 (300nm)

L = 20 (400nm)

L = 30 (600nm)

101 102 103 104 105


100

101

102

103

Num

ber

ofla

yer

s

20-layer limit

Number of routing layers for ECI scheme (Nq = 1)

L = 15 (300nm)

L = 20 (400nm)

L = 30 (600nm)

L = 300 nm L = 400 nm L = 600 nm0

100

200

300

400

500

600

700

Num

ber

ofphysi

calqubits

20 20

100

Number of qubits routable with 20 layers - SSI

Nq = 1

Nq = 3

Nq = 5

Hybrid

L = 300 nm L = 400 nm L = 600 nm0

10000

20000

30000

40000

50000

Num

ber

ofphysi

calqubits

Number of qubits routable with 20 layers - ECI

Nq = 1

Nq = 3

Nq = 5

Fig. 2.9 Qubit fanout scalability in the cases of interconnect protocols of (left) SSI and(right) ECI. (Top) Number of routing layers vs. number of physical qubits for atomicallyconfined qubits (Nq = 1) under different interconnect lengths; and (bottom) numberof routable qubits comparison between electrostatically confined qubits (Nq = 3 andNq = 5), atomically confined qubits (Nq = 1), and hybrid SSI (donors as qubits and dotsas shuttling nodes). Dimensional parameters are: d = 10nm, w = 5nm, and s = 25nm.The red dashed horizontal line on top charts represents the technological limit of 20metal layers that can be fabricated reliably and economically on a solid-state substrate.

long qubit-qubit spacing, the later scheme suits atomically-confined qubits a little bit

better thanks to the reduced number of confining gates needed. The hybrid SSI scheme

outperforms both of its homogeneous SSI counterparts but noted that the best it can

achieve is still an order of magnitude less than that of the ECI scheme.

To assess the fan-out scalability of interconnect protocols over extreme length scale,

we extend the interconnect length further (up to 100 intermediate nodes for Nq = 1,

i.e. 2 µm). The result is shown in Fig. 2.10 for the SSI and ECI schemes. This

analysis provides a concrete example to the scaling bottleneck of the SSI protocols in 2D

qubit lattice implementation (only routable up to about 103 qubits for electrostatically

confined qubits and about 200 for atomically confined qubits). The maximum number

of routable qubits is saturating over long SSI interconnect length for both types of qubits.


0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

SSI interconnect length (µm)

0

200

400

600

800

1000

1200

Num

ber

ofphysi

calqubits

Number of qubits routable with 20 layers vs SSI interconnect length

Nq = 1

Nq = 3

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

ECI interconnect length (µm)

0

100000

200000

300000

400000

500000

600000

700000

Num

ber

ofphysi

calqubits

Number of qubits routable with 20 layers vs ECI interconnect length

Nq = 1

Nq = 3

Fig. 2.10 Fan-out scalability vs. interconnect length for (left) SSI and (right) ECIschemes. The inter-qubit interconnect lengths are given in absolute unit (µm). Thenumber of interconnect qubit nodes can be inferred by noting that the qubit-qubit distanceis 20 nm for Nq = 1 and 50 nm for Nq = 3.

Multiplexing schemes for SSI control, e.g. Ref [86, 71], will improve the scalability of

these approaches to a certain extent. On the other hand, the ECI protocol can scale up

(quadratic) to an order of 105 qubits over that length scale. Again, as we have already

seen in Fig. 2.9, there is an incremental improvement in terms of scalability at the same

interconnect length when using electrostatically self-confined qubit structures due to

their gate count efficiency. Further steps can be taken to estimate the number of logical

qubits feasible based on the level of error correction required, namely the code distance.

The latter depends on multiple factors such as the gate fidelity, total number of gates

in the algorithm of interest, and the level of output accuracy required. The analysis

in ref [43] provides estimates of the qubit resource required for surface code quantum

computation.

In addition to quantifying the spatial requirements for scale-up, as we have here,

there are undoubtedly other aspects which may restrict the scalability of solid-state

qubit integration, namely control timing, signal integrity, thermal budget, testability, and

manufacturability . Nevertheless, being able to model and extrapolate the limit of each

of the scaling bottlenecks is good engineering practice.

2.6 Summary

If the advancement in solid-state spin qubit fabrication and control follows that of their

classical counterparts, the number of integrated qubits will soon reach the threshold

2.6 Summary 47

where scaling up becomes the next bottleneck. As the quantum network gets larger and

larger to cope with real-world applications, the amount of routing required to provide

control access to surface gates will soon become the limiting factor. By applying the

classical electronics know-how regarding interconnect routing to a ubiquitous 2-D qubit

array with independent gate control and readout fan-out, we have provided a concrete

procedure for scalability estimation, which is adaptable to a wide range of surface code

implementations by adjusting the gate configuration and dimensional parameters. This

estimation procedure is important for large-scale quantum processor design process

where we need to identify at the very early stages the required specifications (so-called

“landing zones” in classical electronics design) regarding quantum interconnect length

and fidelity, back-end metal interconnect dimensions and the number of fan-out layers.

For architectures where each qubit has its own dedicated control lines and readout

device, we have analysed fan-out scenarios associated with two categories of quantum

interconnects, namely spin shuttling interconnects (SSI) and end control interconnects

(ECI) with high and low gate densities respectively. Both interconnect models help

extend the contacts/vias pitch through redistribution, which potentially aids the fan-out

routing procedure. However, SSI protocols result in a poorly scalable situation since the

added interconnect control gates outweigh the pitch scaling benefit. On the other hand,

ECI protocols provide a more consistent fan-out scaling trend with interconnect length,

however, relatively long interconnects (greater than several microns) are required to scale

the system to the million qubit level, where issues such as interconnect fidelity, charac-

terisation and operation time, will affect the error rate and surface code error correction

performance negatively. Above all, the errors induced in the quantum interconnect must

be correctable by the QEC protocol. Multiplexing schemes [74, 12] alleviate the gate

density bottleneck to some extent, with fully distributed control schemes [71] providing

scalability without the need for quantum interconnects.


Chapter 3

Continuous Quantum Error

Correction

Almost any decision is better than no decision – just keep moving.

Danielle LaPorte

Fighting decoherence in quantum mechanical systems, due to their inevitable cou-

pling to the surrounding environment, often requires a combination of multiple ap-

proaches such as fabrication techniques, decoupling pulses, quantum feedback control

methods and coding schemes. In this chapter, we will not target a particular quantum

computing architecture but instead focus on the feedback and control scheme for a

promising quantum error correcting coding scheme, namely the surface code. The

approach we chose is an extension of the continuous quantum error correction scheme

proposed by Ahn and co-workers for simple codes [4], where continuously acquired

syndrome measurement signals are used to correct the errors in real-time via Hamiltonian

feedback.

3.1 Quantum Errors

The merits of quantum error correction are often evaluated concerning a specific error

model. In conventional error models, the Pauli X , Y , and Z errors are applied probabilis-

tically to qubits at discrete time intervals. For example, a noise model M is a sequence

50 Chapter 3. Continuous Quantum Error Correction

of quantum operations St(ρ) on the Hilbert space (density matrix, ρ) of the quantum

computer. These error operations are indexed by the time-step (t) at which each of them

is applied.

A general quantum operation, ρ 7−→ S(ρ), is a linear, trace preserving, and com-

pletely positive map1.

Theorem 3.1.1 [Kraus (operator-sum) decomposition]

S is a completely positive trace preserving map iff

S(ρ) = ∑k AkρA†k , with ∑k A†

kAk = I

These are common quantum error channels [122] that are used to model decoherence

in quantum computers:

1. Dephasing channel:

S(ρ) = (1− p)ρ + pZρZ

2. Depolarizing channel:

S(ρ) = (1− p)ρ +p3(XρX +Y ρY +ZρZ),

which can also be expressed using the representation:

S(ρ) = (1− 4p3)ρ +

4p3

(I2

).

Note: the identity operator (I) is the balanced mixture, i.e.I2 = ρ

4 +14XρX + 1

4Y ρY + 14ZρZ

3. General Pauli channel

S(ρ) = (1− px − py − pz)ρ + pxXρX + pyY ρY + pzZρZ

4. Amplitude damping channel

S(ρ) = A0ρA†0 +A1ρA†

1,

1This is equivalent to the Master equation formulation in (1.10).


where A0 =

1 0

0√

1− γ2

and A1 =

0 γ

0 0

.

In the above models, the parameter p represents the error probability at each time

step. Hence, we refer to this as the discrete error model. In physical quantum systems

such as atoms, photons, or spins, however, the amplitude and phase of a qubit will often

fluctuate continuously when it is interacting with the environment or other quantum

systems.

It is worth noting that the parity measurement in quantum error correction scheme

provides a discretisation of the set of errors. Hence, the discrete model of QEC is capable

of modelling general continuous decoherence processes. However, continuous errors

can also happen to parity measurements themselves which makes the fully digitised

picture an oversimplification.

3.1.1 Quantum Error Correction Code

A universal approach in information theory to deal with noise-corrupted data is to intro-

duce auxiliary qubits to carry away extra entropy transferred from the noisy environment

into the system. In essence, we need to encode the data (logical qubits) into a larger

codeword that has extra qubits. These added qubits will assist us in identifying the most

probable candidate error, of several possible errors, occurred, so that corrective action

can be taken to preserve the stored quantum information.

Among many quantum error correction (QEC) schemes, stabilizer codes [53] are pre-

dominantly the most effective and well-studied category, to which the surface code [43]

also belongs. A stabilizer code is described by an Abelian (mutually commuting) set of

Pauli operators whose simultaneous +1-eigenspace defines the code space. This set is

called “stabilizer generators” (S ). Stabilized code words (L ) are defined as:

L = |ψ⟩ ∈ (C2)⊗n : P |ψ⟩= |ψ⟩ ,∀P ∈ S .

Therefore, stabilizers act trivially on the code and can be measured simultaneously.

Their eigenvalues form an error syndrome that can help identify the error coset where

the actual error resides. By applying an effective decoding policy (such as minimum

weight matching), the most probable error candidate will be chosen to be corrected.


To build a large scale quantum computer, an error correction code is required through

which the error can be made arbitrarily small. In this regard, not all stabilizer codes scale

equally due to their specific construction. Some codes may become more susceptible

to noise on a larger block than the others. This figure of merit is characterized by

the accuracy threshold which is the bound of error rate that the code can tolerate for

effective scaling. Among all stabilizer codes, surface code has been proven to be one of

the best-performing codes with a threshold two orders of magnitude higher than other

error correction codes [45] along with many desirable features that we will explain in

the next section.

3.1.2 Surface Code

The surface code is a type of topological QEC code where qubits are laid out in a

2-dimensional lattice. This planar layout, in combination with nearest-neighbor interac-

tions, makes it a compelling alternative to the conventional stabilizer codes which often

require long-range interactions.

Similar to other stabilizer codes, a surface code is fully characterized by its stabilizer

generators. The stabilizer generators of the surface code come in two types: the so-called

X and Z stabilizers, as depicted in Fig. 3.1.

1 2 3 4

5 6 7

8 9 10 11

12 13 14

15 16 17 18

X X X

Z Z Z Z

X X X

Z Z Z Z

X X X

Fig. 3.1 Surface code layout: white circles represent data qubits; filled circles aresyndrome qubits (X stabilizers in green and Z stabilizers in yellow). Each internalstabilizer acts on four adjacent data qubits, while boundary stabilizers act on either twoor three data qubits.

These local stabilizers are mutually commuting by construction (X and Z stabilizers

either share none or two qubits) and thus form a valid set of stabilizer generators


for a QECC. These stabilizers fix a set of states, the so-called “quiescent” states, the

simultaneous eigenstates of all generators. A codeword is also a quiescent state with all

+1 eigenvalues.

Error decoding is based on the topological properties of the lattice whereby a bit-flip

(X) is detectable because it causes the change of the two neighboring Z syndromes; a

phase-flip (Z) by the change of the two neighboring X syndromes and a bit-and phase-flip

(Y ) by both types of syndrome. This is a simplistic explanation for surface code decoding

where we skip details about space-time correlations to detect qubit vs. measurement

errors and the standard minimum weight perfect matching algorithm used for error

decoding. Fowler et al. provides an excellent review for interested readers [43].

The code distance of a surface code can be quickly inferred from lattice dimension.

When using the optimal lattice structure [75], an N ×N (data qubit) lattice forms a code

of distance N. By definition, a distance-d code can correct arbitrary ⌊d−12 ⌋-qubit errors.

Thus, a distance-of-3 code is the smallest universal error correction code that can correct

any single qubit error.

3.1.3 Previous Work

QEC and fault tolerant quantum computing have been studied extensively in the last two

decades mostly focusing on the discrete time scheme. Discrete QEC bridges the gap

between the conventional computer science and the emerging quantum computing field.

However, it relies on the fact that we can do projective measurements and instantaneous

corrections. Therefore, research has been done regarding the dynamical representation

of the QEC process where errors, syndrome measurement, and correction are performed

continuously with finite strengths.

The very first framework for the continuous QEC was laid down by Paz and

Zurek [142] where continuous error correction is modelled as an infinitesimal limit

of syndrome measurement and error correction. The resulting master equation is of

the type of a cooling process where the ancillas are continuously “cooled” from the

entropy acquired by errors. Sarovar and Milburn used this cooling approach in the

3-qubit flip code [153]. Oreshkov and Brun studied continuous quantum error correction

for non-Markovian decoherence using the model of a bit-flip code [131]. Later, Hsu


and Brun also developed a method for continuous-time quantum error correction for any

[[n,k,d]] quantum stabilizer code [78].

This chapter on the other hand is inspired by the measurement feedback scheme that

was first developed by Ahn et al. for a simple 3-qubit code [4]. Later, elaboration on

this continuous scheme was done using classical signal filtering techniques [152], and

instantaneous emission syndromes [5].

Classical hybrid control techniques were also proposed for quantum continuous error

correction, in which the Wonham filter was used for error state probability tracking [179].

A filter dimension reduction scheme was also derived [21] for the Wonham filter with

Hamiltonian feedback. Mabuchi described an optimal hybrid controller for the 3-qubit

flip code [106]. [151] implements a continuous-time error correction protocol for

stabilizer codes and discusses some subtleties of routing for the accumulation of multi-

qubit syndromes and how that interacts with the subsystem nature of some stabilizer

codes.

Previously-mentioned control techniques are all referred to as measurement-based as

they involve quantum to classical signal conversion. The other type is coherent quantum

feedback control whose feedback loop only contains quantum mechanical devices. A

coherent feedback scheme for the 3-qubit error correction was shown in [90], in which

the feedback loop forms a fully autonomous self-correcting quantum memory without

the need for classical processing.

In terms of the topological error correction, discrete error correction schemes were

investigated by Fowler et al. [44] and Stephens [162], where an accuracy threshold was

derived for the surface code. This proved that the surface code is an outstanding candidate

for large scale quantum computer architecture. More sophisticated computational

simulations [174] were recently done for the minimal surface code of distance 3 under

various error models in the discrete framework.

A comprehensive review of QEC is presented in the book [99]. A more focused

review on error correction for quantum memory is presented in [166] where continuous

QEC techniques are also summarized.


3.1.4 Contributions

The aim of this chapter is two-fold. Firstly, we want to introduce continuous QEC by

measurement feedback for topological codes such as the surface code. This is done

using fully quantum mechanical representations in terms of stochastic master equations

and quantum stochastic differential equations. It provides a complementary view to the

conventional semi-classical circuit model approach of surface code error correction. In

order to keep the simulations numerically tractable, we also chose a reduced 2×2 lattice

containing only four data qubits affected by a limited error model. This can be regarded

as the smallest 2-D structure that still exhibits the non-trivial topological properties

which we want to investigate. Secondly, we have put this feedback scheme in the SLH

framework that is emerging as a useful tool for control engineering.

3.1.5 Outline and Notations

After the general introduction in Section 3.1, we will proceed to describing our surface

code model in Section 3.2 that includes the description of the qubit network, dynamical

equations, and the SLH model. Section 3.3 will detail the feedback policy for continuous

error correction followed by simulation results in Section 3.4. Section 3.5 will conclude

the chapter with some final words and avenues for future research.

In this chapter, Pauli operators σx, σy, σz are also represented by capital letters X ,

Y , and Z, respectively. To shorten the notation, tensor product signs are omitted, i.e.

X1X2 ≡ X1⊗X2, and identity operators (I) are assumed in all places where no other Pauli

operator applied, such as X1X4 ≡ X1 ⊗ I ⊗ I ⊗X4.

The commutator between operators is denoted by [A,B] = AB−BA, whilst anti-

commutator is A,B = AB+BA. Hermitian conjugate of A is A†. The Heisenberg-

picture evolution of an observable X is denoted by Xt = jt(X).

We will also use a couple of common superoperators such as the dissipation (D),

Lindblad generator (LL,H), and the homodyne measurement (H ), which are defined as:

D [L]ρ = LρL† − 12(L†Lρ +ρL†L),

LL,H(X) = i[H,X ]+D [L]X ,


H [L]ρ = Lρ +ρL† −ρTr[Lρ +ρL†].

3.2 Model

In this section, we will first introduce the distance-2 surface code block that will be

used to demonstrate the continuous feedback scheme. Then, we will briefly review key

concepts of the SLH dynamical model before deriving the detail model of our surface

code in the continuous measurement feedback scheme.

3.2.1 Distance-2 Surface Code

The surface code lattice that we will consider in the following sections comprises of

four data qubits and five syndrome measurements in a 2×2 configuration as shown in

Fig. 3.2. The code stabilizer generators are Z1Z2Z3Z4, X1X2, X1X3, X2X4, and X3X4.

1

2 3

4

Z

XX

XX

Fig. 3.2 Distance-2 surface code

This code lattice has a code distance of two which is insufficient to correct arbitrary

error such as the general depolarizing error: ρ 7→ (1− p)ρ + p3 (XρX +Y ρY +ZρZ).

Indeed, X errors cannot be localized by this lattice since all four X errors have the same

syndrome. However, single-qubit Y and Z errors can be identified by combining the five

syndrome measurements.

Unlike discrete QEC, the continuous syndrome measurement does not require a

physical qubit but only a field interacting with the relevant data qubits with some specific

coupling parameters. The output fields will carry both information about the syndrome,

3.2 Model 57

as well as additional noise. This information can be used to estimate the original

syndrome. This input-output process as well as the filter and estimation will be described

by the quantum stochastic differential equations.

3.2.2 Continuous QEC in the SLH Framework

Our continuous QEC feedback network will be described in the Heisenberg-picture

using the Hudson-Parthasarathy quantum stochastic differential equation (QSDE) [79],

which is also known as the input-output formalism [49] in the physics literature. The

QEC network is encapsulated by the SLH parametrization [56], which comprises of

scattering matrix S, coupling vector L, and Hamiltonian H. Unless otherwise stated, we

assume there is no scattering between quantum fields, so S = I.

In the continuous QEC scheme, our surface code is coupled to two separate groups

of channels. The first is a collection of error channels (LE) which depend on the chosen

error model. For generic balanced depolarizing channels, the error coupling vector for a

collection of N qubits with per-qubit error rate of γ is:

LE =

√γ

3σ( j)i

j∈[1,N]

i=x,y,z.

This corresponds to a per time step error probability of γdt for each qubit.

As noted before, we will use a limited error model for our distance-2 surface code

that only includes Y and Z errors but not X . There are therefore a total of eight error

channels coupling to the surface code lattice in Fig. 3.2. Despite having separate channels

for errors, we will assume that all of these are unobservable. Indeed, if we can access the

error channels, there will be no need for syndromes as errors can be corrected directly.

In order to measure the error syndromes, we need to create five syndrome field

channels with equal coupling strength, denoted by κ:

LS =√

κ [Z1Z2Z3Z4,X1X2,X1X3,X2X4,X3X4]T .


SurfaceCodeArray Estimator Controller

dAS,LS

dAE ,LE dAE,out

dAS,out S(t) F(t)

Fig. 3.3 Block diagram of surface code continuous error correction in the SLH framework:the surface code array is coupled to two groups of channels, namely error channels (dAE)and syndrome channels (dAS). The coupling strengths are LE and LS, respectively. Thesyndrome outputs are measured to estimate the syndrome conditional expectation values(S(t)). The Hamiltonian feedback (F(t)) is a function of the syndrome estimators.

The unitary evolution of the surface code network can be derived using the Hudson-

Parthasarathy QSDE:

dU(t) =

dA†E(t)LE −L†

EdAE(t)−12

L†ELEdt

+dA†S(t)LS −L†

SdAS(t)−12

L†SLSdt

U(t),

(3.1)

where we assume no internal dynamics (H = 0). The QSDE for an arbitrary observable

X on the qubit network is given by:

d jt(X) = jt(LLE(X)+LLS

(X))dt

+ jt([L†E ,X ])dAE(t)+ jt([X ,LE ])dA†

E(t)

+ jt([L†S,X ])dAS(t)+ jt([X ,LS])dA†

S(t).

(3.2)

The measurement signals are taken to be of the forms of homodyme detection on the

syndrome field channels, i.e. dY (t) = dAS(out)(t)+dA†S(out)(t).

Assuming that the input fields are in vacuum state, the five output equations can be

explicitly derived for the surface code lattice in Fig. 3.2:

dYi(t) = 2√

κ jt(Si)dt +dWi(t), (3.3)

3.2 Model 59

where dWi(t) = dASi(t)+dA†Si(t) is equivalent to a classical Wiener process2, that is a

Gaussian distributed random variable with zero mean and variance of dt.

By observing the output field dYi(t), we can write down the optimal estimate of an

system observable using quantum filtering techniques:

dπt(X) = πt(LLE

(X)+LLS(X))

dt (3.4)

+ ∑iπt(LSi,X)−2πt(LSi)πt(X)dWi(t),

where we have used the Hermitian property of syndrome operators (LSi = L†Si) to simplify

the equation. The notation πt(X) stands for the conditional expectation of the observable

X given all measurement records up to time t.

So far, we have not made any simplification by using assumptions about the error

model or observable operators. Despite using a limited error model for our distance-2

lattice, we derive the filtering estimation equations for general depolarizing noise as this

will definitely be used for larger lattice with full error correction capability.

Denote gi the set of syndrome generators, by using (3.4) we can get their filtering

equations:

dπt(gl) =−(4w)γ3

πt(gl)dt

+2√

κ

k

∑i=1

πt(glgi)−πt(gl)πt(gi)dWi,

(3.5)

where w denotes the Pauli weight of the parity operator gl .

The filtering equations for syndrome measurements are non-linear due to the appear-

ance of high-order terms like πt(gl)πt(gi). Also, they expand beyond the initial stabilizer

generator set because of operator product terms like πt(glgi). However, they will form a

finite set of equations which include all operators generated by the group of syndrome

generators.

Having the estimation of the syndrome state, we can proceed to the next step to

build an estimation-based feedback controller to correct the surface code lattice as

diagrammatically shown in Fig. 3.3.

2assuming homodyne detection scheme


3.3 Methods

The most widely used feedback strategy is to use an additional Hamiltonian to control

the system of interest based on information acquired at the outputs. As the errors are

occurring, the feedback Hamiltonian will be used to rotate the surface code back to the

code space continuously. Therefore, codeword fidelity will be preserved.

Each type of error requires a corresponding correction Hamiltonian, thus our feed-

back Hamiltonian has the following form:

F(t) =4

∑i=1

λzi (t)Zi +

4

∑i=1

λyi (t)Yi (3.6)

where λzi (t) and λ

yi (t) are the real-time feedback terms that will depend on the estimated

syndrome signatures.

The topological structure of the code allows us to infer the error by identifying two

neighboring syndrome values which change their signs from +1 to -1. We can use that

logic to define the feedback policy as:

λz1 =

λ032

(1− X12)(1− X13)(1+ X24)(1+ X34)(1+ Z1234), (3.7)

λz2 =

λ032

(1− X12)(1+ X13)(1− X24)(1+ X34)(1+ Z1234), (3.8)

λz3 =

λ032

(1+ X12)(1− X13)(1+ X24)(1− X34)(1+ Z1234), (3.9)

λz4 =

λ032

(1+ X12)(1+ X13)(1− X24)(1− X34)(1+ Z1234), (3.10)

λy1 =

λ032

(1− X12)(1− X13)(1+ X24)(1+ X34)(1− Z1234), (3.11)

λy2 =

λ032

(1− X12)(1+ X13)(1− X24)(1+ X34)(1− Z1234), (3.12)

λy3 =

λ032

(1+ X12)(1− X13)(1+ X24)(1− X34)(1− Z1234), (3.13)

λy4 =

λ032

(1+ X12)(1+ X13)(1− X24)(1− X34)(1− Z1234), (3.14)

where we have dropped the time variable for brevity. The maximum feedback strength

λ0 is assumed to be time-independent and equal for all feedback terms. The hats over

the operators represent the conditional expectations, i.e. X12 = πt(X1X2).

3.4 Results 61

The Z errors are detected by X syndrome measurement alone, while Y errors are

detected by combining X syndromes with the central Z syndrome. For example, a Z

error on qubit number 1 will change both X1X2 and X1X3 signs, thus turning on the

feedback term λz1 . On the other hand, a Y error on the first qubit will also change the

center Z syndrome therefore we got a minus sign in the last term.

The stochastic differential equations are solved numerically using an Euler solver.

The time step, measurement and feedback strength will be normalized to the decay rate

(γ). A time step of order 10−5 is used for the solver which yields good convergence over

wide range of measurement and feedback strengths.

Unless otherwise noted, the default number of Wiener noise realisations is 104,

which will be averaged over. We use codeword fidelity, defined by Tr[ρ0ρ(t)], as the

primary figure of merit to assess the performance of our feedback scheme. A fidelity of

1 signals perfect match, while 0 means orthogonal states.

Also, to judge the continuous scheme against the conventional discrete error cor-

rection, we will also look at the correctable overlap projection which is defined as the

combination of the original state and all single-error subspaces:

ΠC = ρ0 +∑i, j

σ( j)i ρ0σ

( j)i . (3.15)

The expectation of this projector at time t gives us an upper bound estimate of the

recoverable fidelity if a discrete error correction cycle is to be performed at that time [4].

3.4 Results


Using our surface code model as described above, we can simulate the evolution of

the qubit array stochastically using numerical methods. A sample simulation run for

one realization of the noise process, also known as trajectory, is shown in Fig. 3.4. We

can clearly see the variation in our syndrome estimates due to noise. However, it also

indicates a clear transition from +1 to -1 of X1X3, X3X4, and Z1Z2Z3Z4 generators.

By inferring from the trajectory, we suspect that a Y3 error is the most probable error.

However, we need some quantitative measures to assess the claim that it is indeed the


0.00 0.02 0.04 0.06 0.08 0.10

Time (1/γ)

−1.0

−0.5

0.0

0.5

1.0Sy

ndro

me

Val

ueSurface Code Syndrome Estimation

Z1Z2Z3Z4

X1X2

X1X3

X2X4

X3X4

Fig. 3.4 Syndrome estimation for a single trajectory showing +1 to −1 transitions fromthree syndrome operators: X1X3, X3X4, and Z1Z2Z3Z4. This indicates a Y error mayoccur to qubit 3. Dashed line represents the point in time selected for the fidelity andtrace distance comparison in table 3.1.

case. Nielsen and Chuang have outlined two effective measures which are the trace

distance and fidelity [122]. The first measures how far apart the two states are, whilst the

latter measure the opposite. In our simulator, we can extract the actual state at the time

right after the transition as denoted by dotted line in Fig. 3.4. We call this inferred state

ρ and we will compare it against the initial state ρ0 as well as the Y3-rotated version

of it (ρ ′0 = Y3ρ0Y3). The comparison results are summarized in table 3.1. This shows

that the latter time state (ρ) is much more matched to the Y3-rotated initial state than to

the ρ0 itself. We can then apply our feedback correction policy laid out in Section 3.3.

Measure (ρ0,ρ) (ρ ′0,ρ)

Trace Distance: D(ρ,σ) = 12Tr|ρ −σ | 0.999 0.069

Fidelity: F(ρ,σ) = Tr√

ρ1/2σρ1/2 0.018 0.998

Table 3.1 Fidelity and trace distance comparison

The performance of our continuous error correction against no correction as well as

correctable overlap (discrete correction bound) is shown in Fig. 3.5.

The continuously corrected fidelity is clearly much better than no correction at all,

as the feedback loop is correcting the errors in real-time while they are occurring. More

3.4 Results 63

0.0 0.1 0.2 0.3 0.4 0.5

Time (1/γ)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Fide

lity

Fidelity Comparison

No FeedbackFeedbackCorrectable Overlap

Fig. 3.5 Comparison between continuous feedback error correction fidelity vs. nofeedback fidelity and correctable overlap. Feedback enhances the code fidelity over longperiod of time. The cross-over between no feedback correctable overlap and feedbackfidelity indicates continuous error correction outperforms its discrete counterpart ifcorrection period is longer than that time.

importantly, over long period of time3, the continuous error correction scheme even

outperforms its discrete counterpart. Note that in order for the discrete scheme to become

more preferable, we need to perform correction cycles frequently enough while still

keep the idealized projective measurement and fast correction assumptions.


To simulate a generic error model, we extend the surface code lattice to a code distance

of three. The qubit lattice is shown in Fig. 3.6.

In the following simulation, we will introduce depolarisation noise source to each

qubit. The noise coupling operators have the form:

L(i)e =

γ

3(σ

(i)x +σ

(i)y +σ

(i)z ), (3.16)

in which we assume the noise strength is the same for all qubits and the noise is balanced

among all Pauli operators. These assumptions can be relaxed when considered specific

physical systems.

3in terms of error correction cycles


7 8 9

4 5 6

1 2 3

Z X

X ZZ

X

Z

X

Fig. 3.6 Diagram of distance-3 surface code: white circles are data qubits, green circlesare X-stabilizer channels, and yellow circles are Z-stabilizer channels.

The Hilbert space of this qubit network is quite large. It requires a complex matrix

of the size of 29 ×29 to represent the density matrix. Fully stochastic master equation

simulation at this scale, as performed for the distance-2 lattice, demand significant

computing resources. Thus, to simplify the simulation, we can convert the stochastic

master equation into its counterpart stochastic Schrödinger equation. For a system of

size N, the latter only requires 2N complex elements. The drawback of this approach is

that we are unable to capture inefficient detection since it will induce mixed states.

A stochastic Schrödinger equation for our scheme has the following form:

d |ψ(t)⟩ = −iH |ψ⟩dt +12

(⟨L+L†⟩L−L†L− 1

4⟨L+L†⟩2

)|ψ⟩dt (3.17)

+ (L− 12⟨L+L†⟩) |ψ⟩dW.

We can verify the equivalence between this and the previous stochastic master equa-

tion by using: dρ = d(|ψ⟩⟨ψ|) = d(|ψ⟩)⟨ψ|+ |ψ⟩d(⟨ψ|)+ d(|ψ⟩)d(⟨ψ|) following

Ito rule with the convention: dt2 = 0 and dW 2 = dt.

The surface code in Fig. 3.6 is a degenerate code, which means that more than

one errors are associated with a specific syndrome signature. Nevertheless, errors with

3.4 Results 65

the same syndrome will constitute a stabilizer operator, thus acts trivially on the code.

Therefore, we can correct the error up to modulo of stabilizer operator, e.g. Z1 and Z4

errors have confounding syndrome signature as Z1Z4 is a stabilizer, but correcting one

of them is enough to preserve codeword fidelity.

Using the feedback policy similar to the distance-2 case, we can simulate the perfor-

mance of the continuous QEC for the surface code under general depolarisation noise

as in (3.16). The average code-word fidelity over time of the surface code is shown

in Fig. 3.7, which includes 1000 random noise realisations each. In all the range of

feedback rates considered, the feedback loop, in the long run, has always delivered

some improvement in fidelity. This proves the robustness of the scheme over feedback

parameters.

Fig. 3.7 Time-domain simulation of the distance-3 surface code under continuous QEC.Each curve is the average fidelity over 1000 stochastic trajectories with a specificfeedback correction strength.

From the above simulation, we can also see that the fidelity performance varies

considerably with regard to the feedback strength. This indicates an optimisation

opportunity of the parameter to deliver the best performance. This can be considered as


a classic feedback control problem where there is an optimum operating point. Too weak

or too strong feedbacks are both harmful to the performance as can be seen in Fig. 3.8.

0.0 0.2 0.4 0.6 0.8 1.0

λ/κ

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Fide

lity

Fidelity Plot vs. Feedback Strength

Fig. 3.8 Comparison of the final fidelity of the surface code under varying feedbackstrength.

The steep increase in the fidelity shows the significant gain when we have the feed-

back loop in place. Also, the feedback is relatively weak with regard to the measurement.

This makes perfect sense since the controller needs to average over long, noisy syndrome

records to get an accurate estimation of the errors.

3.5 Conclusions

In summary, we have shown a continuous measurement feedback scheme that can be

used for topological codes such as the surface code. The feedback correction preserves

the codeword fidelity over long duration against a decoherence process. Syndrome

values are estimated from measurement signals without the need for physical ancilla

qubits as well as quantum gates. This proves the feasibility of using feedback and control

methods in topological QEC. The QSDE formalism and SLH model provide us with an

3.5 Conclusions 67

efficient tool suite to tackle quantum feedback and control problems. Given the growing

interest in surface code among other alternative codes, this deems a promising method

to be used in quantum computers whereby low-level feedback control loop maintains

the fidelity of quantum memory.

We have just examined our feedback control policy on a minimised surface code

lattice under idealistic conditions. Besides, non-ideal effects, such as loop delay, limited

measurement bandwidth, measurement error, which are always present in reality will

also need to be investigated in the future to assess the robustness of our scheme. The

ultimate goal is to envision and design a coherent feedback controller for the surface

code which can make our surface code a self-correcting qubit lattice in the quantum

domain.


Chapter 4

Quantum Reservoir Engineering

The sky is filled with stars, invisible by day.

Henry Wadsworth Longfellow

The environment within which the quantum system operates typically has a contin-

uous degrading effect, i.e. decoherence, on evolution of quantum particles. Reservoir

engineering, also known as quantum dissipation control, is the term used in quantum con-

trol and information technologies to describe manipulating the environment within which

an open quantum system operates. Reservoir engineering is essential in applications

where storing quantum information is required.

When a quantum system possesses a subsystem isolated from the detrimental influ-

ence of the environment and probing fields, the quantum information associated with

dynamics of such a system is preserved and can be used for quantum computation when

needed. In a sense, decoherence-free subsystems (DFS) can play the role of memory

elements in quantum information processing. This has motivated a significant interest in

the synthesis of quantum systems with the desired DFS structure.

On the other hand, when quantum error correction is in use, dissipation control

mechanisms derived using the control-theoretical formation of Lyapunov methods can

also be applied to stabilizes the code states. This forms a “passive” error correction loop

without the need for an active controller.

70 Chapter 4. Quantum Reservoir Engineering

4.1 Introduction

Reservoir engineering refers to the process of determining and implementing coupling

operators L = [L1; ...;Ln] for an open quantum system such that desired behavior is

achieved. Examples of common objectives include quantum computation by dissi-

pation [185], entanglement [97], state preparation [171], and protection of quantum

information [26, 137]. Typically open systems have some unavoidable couplings to

the environment, and such channels may lead to loss of energy and quantum coher-

ence. However, in many systems couplings can be engineered at the fabrication stage,

providing a resource for tuning the behavior of the system.

The scheme of quantum computation by dissipation [185] suggests that the com-

putation result could be encoded into the steady states of a carefully-designed system

Hamiltonian. For instance, measurement-based quantum computation [146] can be

realized provided that we can prepare a giant entangled state often called graph state [66]

or cluster state [121]. After the coding step, the system is coupled to an engineered

environment, and the dissipative dynamics would drive the system to the ground states

which accomplish the computation. This is in contrast to the conventional quantum

computation scheme [122] in which the computation is executed by applying a sequence

of unitary gates.

The dissipative dynamics has been well studied in classical control engineering.

The so-called Lyapunov method [63] plays a vital role in establishing the stability of a

dissipative system. To be specific, a Lyapunov function is proposed as a cost function

and the control is designed such that it decreases the value of the Lyapunov function.

Efforts have been made for the purpose of applying Lyapunov techniques to quantum

systems [112, 98, 188, 76, 169, 7]. These works employ a Schrödinger-picture analysis

which facilitates the engineering of the stability of quantum states [154].

Stabilizing ground states is critical for quantum state engineering and quantum

computation [185, 139]. In this chapter, we apply the Lyapunov methods to establish the

ground-state stability of an operator-sum representation whose ground state encodes the

quantum information. The Lyapunov analysis can thus be conveniently embedded into

the framework of quantum computation by dissipation control. In particular, we focus

on a specific decomposition of the dissipation control, based on which we can derive the

4.1 Introduction 71

conditions for the Lyapunov method to be scalable. Also we will show that the Lyapunov

formalism is closely related to the stabilizer formalism for quantum error correction, in

the sense that the dissipation control can be used for passive error correction in addition

to stabilization.

In Section 4.3, an extended scalability condition which is applicable to a wider

range of applications is proposed to achieve the ground-state stability for a class of

multipartite quantum systems which may involve two-body interactions, and an explicit

procedure to construct the dissipation control is presented. Moreover, we show that

dissipation control can be used for automatic error correction in addition to stabilization.

We demonstrate the stabilization and error correction of three-qubit repetition code states

using dissipation control.

Without active error correction, from the control theory perspective, a quantum sys-

tem is capable of storing quantum information if it possesses a so-called decoherence free

subsystem (DFS). Section 4.4 explores pole placement techniques to facilitate synthesis

of decoherence free subsystems via coherent quantum feedback control. We discuss

limitations of the conventional ‘open loop’ approach and propose a constructive feedback

design methodology for decoherence free subsystem engineering. It captures a quite

general dynamic coherent feedback structure which allows systems with decoherence

free modes to be synthesized from components which do not have such modes.

The problem of DFS synthesis has been found to be nontrivial - it has been shown

in [202] that conventional measurement feedback is ineffective in producing quantum

systems having a DFS, however certain coherent controllers can overcome this limitation

of the measurement-based feedback controllers. The objective of Section 4.4 is to put

this observation on a solid systematic footing, by developing a quite general constructive

coherent synthesis procedure for generating quantum systems with a DFS of desired

dimension.

Our particular interest is in a class of quantum linear systems [129] whose dynamics

in the Heisenberg picture are described by complex quantum stochastic differential

equations expressed in terms of annihilation operators only. Such systems are known to

be passive [83]. Passivity ensures that the system does not generate energy. In addition,

in such systems the notion of system controllability by noise and that of observability


from the output field are known to be equivalent [59]. Also, one can readily identify

uncontrollable and unobservable subspaces of the passive system by analyzing the

system in the Heisenberg picture [202]. These additional features of annihilation only

passive systems facilitate the task of synthesizing decoherence free subsystems by means

of coherent feedback.

The focus on a general coherent feedback synthesis is the main distinct feature of our

work which differentiates it from other works of a similar kind, notably from [202, 126].

The paper [202] presents an analysis of quantum systems equipped with coherent

feedback for the purpose of characterizing decoherence free subsystems, quantum

nondemolished (QND) variables and measurements capable of evading backaction;

in [202] all these characteristics are expressed in geometric terms of (un)controllable

and (un)observable subspaces. In contrast, we propose constructive algebraic conditions

for the synthesis of coherent feedback to equip the system with a DFS. These conditions

are expressed in terms of linear matrix inequalities (LMIs) and reduce the DFS synthesis

problem to an algebraic pole assignment problem. We build our technique using the most

general type of dynamic linear passive coherent feedback. We show that the controller

structures from [202] are in fact special cases of our general setting. In addition, we

discuss the conventional open-loop approach to reservoir engineering and show the

shortcoming of such approach.

This chapter is based on the work presented in [137]1 and [120]2 where we derived

the condition for automatic quantum error correction by dissipation control (Theo-

rem 4.3.3) and the DFS synthesis condition (Theorem 4.4.2).

4.2 Background

4.2.1 Notations

Given an underlying Hilbert space H and an operator x : H→ H, x∗ denotes the operator

adjoint to x. In the case of a vector of operators, the vector consisting of the adjoint

components of x is denoted x#, and x† = (x#)T , where T denotes the transpose of a vector.

1the author contributed equally with the co-author2significant contributions in deriving the equations, providing examples demonstrating the synthesis

scheme

4.2 Background 73

Likewise, for a matrix A, A# is the matrix whose entries are complex conjugate of the

corresponding entries of A, and A† = (A#)T . [x,y] = xy− yx is the commutator of two

operators, and in the case where x,y are vectors of operators, [x,y†] = xy† − (y#xT )T .

A finite-level quantum system is defined on a Hilbert space H ≃ CN . Denote the

space of bounded operator on H as B(H ). A quantum state is characterized by a

density operator ρ ∈ B(H ) satisfying trace(ρ) = 1 and ρ ≥ 0. In many cases, the

interaction between the quantum system and the environment is described by a Markov

process, and the dynamical equation of the quantum state ρt can be written as

ρt = L (ρt)

= −i[H,ρt ]+J

∑j=1

L jρtL†j −

12

L jL†jρt −

12

ρtL†jL j. (4.1)

Here H ∈B(H ) is the system Hamiltonian and L j ∈B(H ), j = 1, · · ·,J are system

operators that characterize the system-environment couplings. ZV is the space spanned

by the ground states of V .

σz =

1 0

0 −1

,σx =

0 1

1 0

,σy =

0 −i

i 0

are Pauli operators acting

on a two-level system called qubit. Accordingly, σzi,σxi,σyi are the Pauli operators

defined on the i-th qubit. The vectorization of a matrix A is denoted as vec(A), which is

a column vector obtained by stacking the columns of the A on top of one another.

4.2.2 Linear Quantum Systems

Open quantum systems are systems that are coupled to an external environment or

reservoir [19]. The environment exerts an influence on the system, in the form of vectors

W (t), W †(t) consisting of quantum Wiener processes defined on a Hilbert space F

known as the Fock space. The unitary motion of the passive annihilation only system

governed by these processes is described by the stochastic differential equation

dU(t) =

((−iH− 1

2L†L)dt+dW †L−L†dW

)U(t), (4.2)

U(0) = I,


where H and L are, respectively, the system Hamiltonian and the coupling operator

through which the system couples with the environment. Then, any operator X : H→ H

generates the evolution X(t) = jt(X) =U(t)∗(X ⊗ I)U(t) in the space of operators on

the tensor product Hilbert space H⊗F,

dX = G (X)dt +dW †[X ,L]+ [L†,X ]dW, (4.3)

where

G (X) = −i[X ,H]+LL(X),

LL(X) =12

L†[X ,L]+12[L†,X ]L

are the generator and the Lindblad superoperator of the system, respectively [198]. The

field resulting from the interaction between the system and the environment constitutes

the output field of the system

dY = Ldt +dW. (4.4)

Linear annihilation only systems arise as a particular class of open quantum systems

whose operators ak, k = 1, . . . ,n, describe various modes of photon annihilation resulting

from interactions between the environment and the system. Such operators satisfy the

canonical commutation relations [a j,a∗k ] = δ jk, where δ jk is the Kronecker delta. Taking

the system Hamiltonian and the coupling operator of the system to be, respectively,

quadratic and linear functions of the vector X = a = [a1, . . .an]T ,

H = a†Ma, L =Ca, (4.5)

where M is a Hermitian n×n matrix, and C ∈ Cm×n, the dynamics and output equations

become

da = Aadt +BdW

dy = Cadt +dW, (4.6)

4.2 Background 75

where the complex matrices A ∈ Cn×n, B ∈ Cn×m, and C ∈ Cm×n satisfy

A =−iM− 12

C†C, B =−C†. (4.7)

The following fundamental identity then holds [102, 103]

A+A† +C†C = 0. (4.8)

According to [83], passivity of a quantum system P is defined as a property of the

system with respect to an output generated by an exosystem W and applied to input

channels of the given quantum system on one hand, and a performance operator Z of the

system on the other hand. To particularize the definition of [83] in relation to the specific

class of annihilation only systems, consider a class of exosystems, i.e., open quantum

systems with zero Hamiltonian, an identity scattering matrix and a coupling operator

u which couples the exosystem with its input field. The exosystem is assumed to be

independent of P in the sense that u commutes with any operator from the C∗ operator

algebra generated by X and X†. The time evolution of u is however determined by the

full interacting system P◁W, and therefore may be influenced by X , X†.

If the output of the exosystem W is fed into the input of the system P in a cascade or

series connection, the resulting system P◁W has the Hamiltonian HP◁W = H+ Im(u†L),

the identity scattering matrix and the field coupling operator LP◁W = L+u [83]. The

resulting system (P◁W) then has the generator GP◁W.

Definition 4.2.1 — [83]. A system P with a performance output Z is passive if there exists

a nonnegative observable V (called the storage observable of P) such that

GP◁W(V )≤ Z†u+u†Z +λ (4.9)

for some constant λ > 0. The operator

r(W) = Z†u+u†Z

is the supply rate which ensures passivity.


Now suppose P is a linear annihilation only system (4.5). Also, consider a perfor-

mance output for the system P◁W to be

Z =Cpa+Dpu.

Taking X = a in (4.3), the system P◁W can be written as

da = (Aa+Bu)dt +BdW, (4.10)

dY = (Ca+u)dt +dW,

Z = Cpa+Dpu.

where the complex matrices A ∈ Cn×n, B ∈ Cn×m, and C ∈ Cm×n are the coefficients of

the annihilation only system P.

Without loss of generality, we further take the storage observable V having the form

V = a†Pa, r(W) = Z†u+u†Z, then it can be shown that the system P is passive with a

storage function V and a supply rate r(W) if for some constant λ > 0,

a†(PA+A†P)a+u†BPa+a†PBu

≤ (Cpa+Dpu)†u+u†(Cpa+Dpu)+λ .

This condition is equivalent to the positive realness condition stated in Theorem 3

of [204] (letting Q = 0 in that theorem):

PA+A†P PB−C†

p

B†P−Cp −(Dp +D†p)

≤ 0. (4.11)

In the special case, where V = a†a, Dp = 0 [204] and Cp = −C, this reduces to the

condition

A+A† ≤ 0

as the condition for passivity. Clearly this condition is satisfied in the case of an

annihilation only system P in the light of the identity (4.8). Hence the annihilation only

4.2 Background 77

system (4.10) is passive with respect to performance output Z =−Ca, with the storage

function V = a†a.

4.2.3 Lyapunov Methods

The stationary states of Eq. (4.1) have been studied extensively for the purpose of state

stabilization [160, 171, 170, 154, 6]. For a multipartite quantum system, the method of

using dissipative dynamics to engineer quantum states has been generalized to the notion

of dissipatively quasi-locally stabilizable (DQLS) states [172, 173]. The theory of DQLS

states proposes a systematic approach to determine whether a given multipartite state is

asymptotically stabilizable if local dissipation controls can be engineered. Furthermore,

if the quantum state satisfies the DQLS condition, the required multipartite system

Hamiltonian and the system-environment coupling operators can be constructively

derived.

The aforementioned results are based on (4.1). Alternatively, the desired states can

be stabilized by studying the evolution of certain operators. Since the expectation of an

operator V ∈B(H ) at the state ρ is calculated by ⟨V ⟩ρ = trace(V ρ), the evolution of

the operator V (t) can be defined via the relation ⟨V (t)⟩ρ0= ⟨V ⟩ρt

. Note that V =V (0).

The generator of this Markov process is given by [83]

G (V (t)) = −i[V (t),H(t)]+L(V (t))

= −i[V (t),H(t)]+J

∑j=1

L†j(t)V (t)L j(t)

−12

L†j(t)L j(t)V (t)− 1

2V (t)L†

j(t)L j(t). (4.12)

A large class of quantum states, including graph states and cluster states (which are

DQLS states as well), can be encoded as the ground states of a multipartite operator

taking the form V = ∑Ki=1Vi [143, 185, 172]. As a result, state stabilization can be

achieved by engineering the dissipation such that the system converges to the ground

states asymptotically. The merit of this formulation is that the ground-state stability of

V can be established by Lyapunov-type operator inequalities. This scenario has been

considered before in [139], where a scalability condition is used to prove the ground-state

stability of V when each Vi is stabilized individually. However the scalability condition


proposed in [139] does not hold for certain applications, especially when Vi consist of

two-body interactions. An illustrative example can be found in Section 4.3.

Note that ground-state stability does not necessarily guarantee that a particular

ground state is stable against errors, because the erroneous state may return to a different

ground state under the dissipative dynamics. However, we can prove in Section 4.3 that if

certain types of errors occur to one of the ground states, the dissipation control can steer

the erroneous state back to the initial ground state exactly, without any measurement or

active feedback. In this regard, this result can be considered as an addition to the existing

physical literature on automatic quantum error correction (AQEC) [14, 90, 26, 88, 82].

The results of Section 4.3 are intended to deal with a specialized case, where we

apply algebraic methods to achieve the stabilization of the states by imposing scalable

Lyapunov-type conditions on the operators. If these conditions are satisfied, then the

ground states of the system are DQLS and V is frustration-free[185, 173, 139].

The multipartite quantum system is defined on H =⊗M

m=1 Hm which is a tensor

product of Hilbert spaces Hm (each Hm is associated with a subsystem). Unless

otherwise noted, we will the following assumption throughout this chapter.

Assumption 4.2.1 V can be decomposed as V = ∑Ki=1Vi, and Vi is defined on a subset of

Hm. Vi are orthogonal projections, i.e. V 2i = Vi and [Vi,Vj] = 0, i = j. Each Vi is

associated with a set of dissipation controls L j. Each L j allows the decomposition

L j = ∑iUi, jVi with Ui, j being a unitary operator.

Remark 4.2.1 Vi can be regarded as quasi-local operators [172] since they are de-

fined on a subset of Hm. Therefore, the stabilizing dynamics in this chapter can be

considered as specific realizations for the stabilization of DQLS states.

V 2i = Vi ≥ 0 is a natural assumption that holds for many applications, e.g. the

dissipation control of stabilizer states [143, 185, 139]. Vi being commutative is an

intuitive assumption which enables Vi to share common ground states. Physical

examples of the decomposition includes the dissipation control of graph states [185].

We also make the following assumption.

Assumption 4.2.2 The system Hamiltonian can be written as H = ∑Ki=1 Hi, where each

Hi satisfies Hi =Vi −giI. Here −gi is the smallest eigenvalue of the Hermitian operator

Hi.

4.2 Background 79

Remark 4.2.2 It is experimentally possible to engineer Hamiltonian on a multipartite

quantum system, e.g. [93].

As shown in the next section, the two assumptions allow a concise and scalable

stability analysis based on the generator (4.17). In addition, we have two definitions as

follows.

Definition 4.2.2 Vi is said to be a two-body operator if it can be decomposed as Vi =

Xm1⊗Xm2

, with Xm1,Xm2

defined on two Hilbert spaces Hm1,Hm2

, respectively.

Definition 4.2.3 The generator of the evolution of Vi that is induced by a single dissipation

control Li is defined by

G (Vi)Li=−i[Vi,H]+L†

i ViLi −12

L†i LiVi −

12

V L†i Li. (4.13)

Remark 4.2.3 Eq. (4.13) is the generator of Vi controlled by a single coupling operator

Li. If Vi is also affected by other coupling operators L j, j = i, then we have G (Vi) =

G (Vi)Li+∑ j =i L†

jViL j − 12L†

jL jVi − 12ViL

†jL j.

We also recall one theorem from [83]:

Theorem 4.2.1 If an operator X ≥ 0 satisfies the following inequality

G (X)≤−cX , c > 0, (4.14)

then the system will asymptotically converge to ZX .

Remark 4.2.4 The algebraic condition (4.21) uses X = X(0), H = H(0) and L j =

L j(0). The satisfaction of this condition implies that limt→∞⟨X(t)⟩ = 0. The other

algebraic conditions of this chapter also use the operators at the initial time. For the

details of the Heisenberg-picture stability theory, please refer to [136].

The following theorem can be derived using Theorem 4.2.1, Assumption 4.2.1 and

4.2.2.

Theorem 4.2.2 [139] If the following condition

J

∑j=1

(ViU†i, jViUi, jVi −Vi)≤−ciVi, ci > 0, (4.15)


• • •

Fig. 4.1 Each Vi has 0 as its lowest energy. The system may be stabilized to the groundstate of V1 under a local dissipation control. The local dissipation control is scalableif it maintains or decreases the energy of Vi, i = 1, i.e. the system is steered towardsthe ground states of Vi, i = 1 under the local dissipation control of V1. However forinstance, if the local dissipation control of V1 increases the energy of V2, then Li, i = 1must be able to compensate this increase in order to stabilize V2.

holds, then G (Vi)≤−ciVi and Vi is asymptotically ground-state stable under the dissipa-

tion control of L j. In particular, we say Ui, j stabilize Vi if Eq. (4.30) holds.

Now we recall the scalability condition derived in [139].

Theorem 4.2.3 Suppose for each Vi, there exists Li, j such that G (Vi)≤−ciVi,ci > 0

holds. The ground-state stability of V can be implied if the intuitive scalability condition

∑i′ =i

∑j

G (Vi)Li′ , j

≤ 0, (4.16)

can be established for all i.

Remark 4.2.5 V is asymptotically ground-state stable if the local dissipation controls

of Vi′ , i′ = i do not increase the expectation of Vi (Fig. 4.1). However, Eq. (4.23)

is easily violated if Vi involve two-body interactions. If Vi is a two-body operator,

the local control Li, j that stabilizes Vi may not act on Vi′ , i′ = i trivially. To be more

specific, if Vi = Xm1⊗Xm2

, then there exists at least one Vi′ , i′ = i which is defined on

Hm1, otherwise the resulting ground state cannot be entangled in Hm1

. As a result, the

local control Ui, j defined on Hm1⊗Hm2

may act nontrivially on this Vi′ .


The result of this chapter shows the way to construct the dissipation control for the

stabilization of V when Vi can be locally stabilized but the local dissipation controls

do not satisfy the strong scalability condition Eq. (4.23).

4.3 Quantum Error Correction by Dissipation Control

A quantum Lyapunov operator V is an observable on a Hilbert space H for which the

following properties hold [136]:

1. V ≥ 0.

2. G (V )≤ 0.

V ≥ 0 means V is positive semidefinite with its smallest eigenvalue being 0. By the

definition, V could be understood as a virtual energy operator. We define the virtual

energy as the expectation tr(V ρt) = ⟨V ⟩ρt= ⟨V (t)⟩ρ0

= tr(V (t)ρ0). ρ0 is the initial state

of the evolution. The second property G (V )≤ 0 suggests that the virtual energy will not

increase in time, i.e. the virtual energy is stable. However, V being stable is not enough

in the context of this Section. In order to stabilize the system to the ground states of V ,

we need to engineer the dissipation control such that ⟨V (t)⟩ρ0exponentially converges

to zero.

In the above definition, we have used the superoperator notation G (·) to represent

the generator of the Heisenberg-picture evolution

G (X(t)) =−i[X(t),H(t)]+L(X(t)) (4.17)

for a Heisenberg-picture operator X(t). H is the system Hamiltonian and

L(X(t)) =K

∑k=1

L†k(t)X(t)Lk(t)

− 12

L†k(t)Lk(t)X(t)− 1

2X(t)L†

k(t)Lk(t) (4.18)


is the Lindblad dissipation term. Lk,k = 1, · · ·,K are K coupling operators character-

izing the interaction between the system and the environment. Also we define

G (X(t))Lk=−i[X(t),H(t)]+L†

k(t)X(t)Lk(t)

−12

L†k(t)Lk(t)X(t)− 1

2X(t)L†

k(t)Lk(t) (4.19)

which represents a generator with regard to a specific coupling operator Lk. We use

ZX := ρ : ⟨X⟩ρ = 0 to denote the subspace spanned by the ground states of X . Then,

V is said to be asymptotically ground-state stable if

⟨V (t)⟩ρ0= ⟨V ⟩ρt

→ 0, as t → ∞. (4.20)

A sufficient condition for the asymptotic ground-state stability of V is given by the

following operator inequality [83]

G (V )≤−cV, c > 0. (4.21)

Integration of the inequality (4.21) yields ⟨V (t)⟩ρ0≤ e−ctV (0).

The inequality (4.21) can be interpreted using the ground-state spaces of the operators.

If ZA is contained in ZB for two operators A,B ≥ 0, then there must exist a constant c > 0

such that A ≥ cB. This can be seen from the decomposition of A and B into

A =

A+ 0

0 0

, B =

B 0

0 0

, (4.22)

with A+ being positive definite and B ≥ 0. Then A+− ε1I > 0 for some ε1 > 0 and

I ≥ ε2B for some ε2, which yields a solution c = ε1ε2 to A ≥ cB. As a result, the

inequality (4.21) can be established if Z−G (V ) is contained in ZV . This property will be

useful in the next section.

4.3.1 Scalability of Lyapunov Method

For multipartite quantum systems, directly solving (4.21) for Lk is computationally

hard, not to mention the physical realizability issues with the resulting multipartite


coupling operators. Indeed, the local dissipation controls designed for the subsystems

can effectively control the overall system, as long as they satisfy certain scalability

conditions. This idea was firstly conjectured in [185]. As a matter of fact, this idea is

similar to the decomposition-aggregation technique [16, 207] for the classical control

engineering of large-scale systems. We can decompose the total Lyapunov operator into

a sum of component Lyapunov operators and design environmental couplings for each

of them accordingly. If the total Lyapunov operator is asymptotically ground-state stable

under the collective action of the individually-designed couplings, we say the Lyapunov

method is scalable (Figure 4.1).

Consider a finite number of Lyapunov operators Vi, i= 1, · ··,N and the stabilization

of the system to the ground states of V = ∑iVi. We call Vi the component Lyapunov

operator. We omit the range of the summation where no confusion is raised. Suppose

for each Vi, there exists a set of local coupling operators Lik such that ∑k G (Vi)Lik≤

−ciVi, ci > 0 holds. Obviously, the ground-state stability of V = ∑iVi can be implied if

∑k

G (Vi)L jk≤ 0, j = i, (4.23)

holds for all i, j. That is, V is asymptotically ground-state stable if the dissipation controls

do not increase the energy of the other component Lyapunov operators. Eq. (4.23) is a

strong scalability condition. Generally speaking, it is possible that some of the individual

dissipation controls indeed increases the energy of the other component Lyapunov

operators but V is still asymptotically ground-state stable.

Example 4.1 (3-qubit repetition code) A logical qubit (|0L⟩ , |1L⟩) is encoded using three

physical qubits, i.e.

|0L⟩= |000⟩ , |1L⟩= |111⟩ . (4.24)

This code can be used to protect quantum information in the subspace spanned by

|0L⟩ , |1L⟩ against single bit-flip noise, with the error operators given by E = σ(i)x , i =

1,2,3. σ(i)x denotes the Pauli operator acting on qubit i, resulting in a flip of the qubit

state.

In the stabilizer formalism [53], the codewords (4.63) are characterized by stabilizers.

In particular, codewords are invariant under the action of the stabilizers. For the 3-qubit


repetition code, the group of stabilizers are given by

S = σ(1)z σ

(2)z ,σ

(2)z σ

(3)z ,σ

(1)z σ

(3)z . (4.25)

Lyapunov operator can be defined from the stabilizers as

V =12[(I −σ

(1)z σ

(2)z )+(I −σ

(2)z σ

(3)z )+(I −σ

(1)z σ

(3)z )]

= V1 +V2 +V3, (4.26)

which is a sum of component Lyapunov operators V1,V2,V3 ≥ 0. For V1 = 12(I −

σ(1)z σ

(2)z ), we can easily see that L1 =

12σ

(1)x (I−σ

(1)z σ

(2)z ) is a solution to the inequality

(4.21) due to

G (V1)L1= L†

1V1L1 −12

L†1L1V1 −

12

V1L†1L1 =−V1. (4.27)

However, L1 is not scalable. This is clear by calculating

∑j =1

G (Vj)L1= diag(0,0,1,−1,−1,1,0,0)≰ 0, (4.28)

which violates the scalability condition (4.23). Without loss of generality, we consider one coupling operator (K = 1) for each Vi. We

assume Vi are commuting and satisfying V 2i =Vi, i.e. Vi are projectors. Motivated

by [185], we use the coupling operators of the form Li = UiVi , where Ui is a unitary

operator. Also, we assume the Vi is defined by a displacement of the system Hamiltonian

Hi by Vi = Hi + gi. By this way we can ignore the unitary dynamics. Substituting

Li =UiVi into Eq. (4.17) yields

G (Vi) =ViU†i ViUiVi −Vi. (4.29)

As a result, the unitary operations Ui must satisfy

ViU†i ViUiVi ≤ (1− ci)Vi, ci > 0, (4.30)


1V1U U

6 H11 2

H1

1 2U

H1

3

H1

V V2V 3V

U 3U

Fig. 4.2 The schematic representation of three qubits governed by V1,V2,V3. Each Vi

acts non-trivially on two of the subsystems. Ui = σ(i)x is the unitary rotation on the i-th

subsystem.

in order to establish G (Vi)≤−ciVi. Fig. 4.2 depicts the systems and dissipation controls

as proposed in Example 4.1. The Li =UiVi with Ui satisfying Eq. (4.30) fails to satisfy

the general scalability condition Eq. (4.23).

Therefore, in order to derive dissipation controls that can deal with Example 4.1,

we will need weaker scalability condition. Still, we may suppose there exists a unitary

rotation Ui which stabilizes each component Lyapunov operator Vi by Eq. (4.30). Denote

V (i)n as the set of component Lyapunov operators that commute with Ui. The other

component Lyapunov operators are denoted as V (i)d ,d = i. The sufficient condition for

the asymptotic ground-state stability of V is

V (i)d U†

i V (i)d UiV

(i)d ≤V (i)

d , (4.31)

∑i(Vi ∏

dV (i)

d )≥ λ ∑i

Vi, λ > 0. (4.32)

The resulting dissipation controls are Li =UiVi ∏d V (i)d . Under these controls, ⟨V (t)⟩ρ0

will converge to 0 as t → ∞. For proof of the sufficient condition please refer to

Appendix A.2.


The dissipation control can be written as Li = UiV′i = UiVi ∏d V (i)

d . Obviously,

V′i can be regarded as the updated Lyapunov operators with the unitary rotations Ui.

As shown before, the condition (4.45) is satisfied if Z∑i V

′i

is contained in Z∑i Vi

. That

is, the common ground states of the updated Lyapunov operators are also the common

ground states of the original Vi. Particularly, if the common ground state of Vi is

unique, then V′i must share the same unique ground state.

Now we can solve for the correct dissipation control for Example 4.1 using the

weaker scalability condition (4.44)-(4.45). Obviously, the updated Lyapunov operators

V′1,V

′2,V

′3 are V ′

1 = V1V3, V ′2 = V1V2, V ′

3 = V2V3. The dissipation control for the first

Lyapunov operator is given by L1 =U1V ′1 =

12σ

(1)x (I −σ

(1)z σ

(3)z )(I −σ

(1)z σ

(2)z ). We can

derive L2 and L3 similarly. V is asymptotically ground-state stable when the three-qubit

system is coupled to a dissipative environment via L1,L2,L3.

4.3.2 Error Correction Condition

The ground-state stability concerns with driving system to the ground state of V from any

initial state. However, ground-state stability is not necessarily related to error correction.

For instance, suppose we encode the information into a steady ground state of V . If

an error Ea occurs, the perturbed state can be stabilized to ZV . However, we cannot

guarantee that the dissipation control will restore the system to the initial state since the

ground states may be degenerate. In addition, it is not clear whether the coherence of the

initial state will be preserved when subjected to the dissipation.

In this section, we will show that the controls Li =UiVi can passively correct the

errors induced by U†i if Vi satisfy a simple stabilizer-type condition.

Denote |p⟩ as basis vectors of ZV . Suppose the Kraus operators [96] describing the

error process are given by Ea. We consider a two-step passive error correction. In the

first step, the system is coupled to the noise. In the second step, the system is subjected

to the dissipation controls. The controls will return the system to the initial state if any

error occurs in the first step. If the dissipation strength is strong, it is reasonable to ignore

the effect of the noise in the second step.


The state dynamics is governed by the master equation

ρt = L (ρt), (4.33)

where L (·) is in Lindblad form

L (ρt) =−i[H,ρt ]+∑k

LkρtL†k −

12

ρtL†kLk −

12

L†kLkρt . (4.34)

An arbitrary pure state in ZV is expressed by ρ0 =∑p αp|p⟩(∑p αp|p⟩)† =∑p,q αpα∗q |p⟩⟨q|.

The erroneous states are thus given by Eaρ0E†a. The dissipative dynamics stabilizes

the erroneous states back to the initial ones if

L (|p⟩⟨q|) = 0, (4.35)

L (Ea|p⟩⟨q|E†a)

=−κpqEa|p⟩⟨q|E†a +κpq|p⟩⟨q|, κpq > 0 (4.36)

holds for all p,q. The proof of the condition is put in Appendix A.3. Eq. (4.52)-(4.53)

basically says that each of the elements of the erroneous density operator including

coherence terms can be restored to the initial value. Please note this implicitly guarantees

that the orthogonal states evolve into orthogonal erroneous states, which is a sufficient

condition for error correction [95, 14, 82].

With the controls of the form Li =UiVi, it is easy to verify that Eq. (4.52) is satisfied

since Vi|p⟩= 0 for all i, p. In addition, Eq. (4.53) can be explicitly written as


= ∑i

LiEa|p⟩⟨q|E†a L†

i

− 12

Ea|p⟩⟨q|E†a L†

i Li −12

L†i LiEa|p⟩⟨q|E†

a

= ∑i

UiViEa|p⟩⟨q|E†aViU

†i

− 12

Ea|p⟩⟨q|E†aVi −

12

ViEa|p⟩⟨q|E†a , (4.37)


if the erroneous states Ea|p⟩ are eigenstates of the Hamiltonian H. Recall that Vi is

obtained by displacing the system Hamiltonian and the nonzero eigenvalue of Vi is 1.

Hence, the requirement that Ea|p⟩ are eigenstates of H is equivalent to Ea|p⟩ are

eigenstates of Vi. Obviously, a sufficient condition for (4.62) and (4.53) to be equal is

Ea =U†i , (4.38)

ViU†i |p⟩= 1, (4.39)

VjU†i |p⟩= 0, j = i, (4.40)

for all p. Eq. (4.59)-(4.60) imply a one-to-one correspondence between each error

operator Ea =U†i to a Lyapunov operator Vi. The erroneous states Ea|p⟩=U†

i |p⟩ are

eigenstates of a unique Vi with the positive eigenvalue 1, while remaining as the ground

states of Vj for j = i. Therefore, the component Lyapunov operators are just like the

syndrome projectors proposed in the stabilizer formalism [53].

Consider Example 4.1. We can check the condition Eq. (4.59)-(4.60) by

V′i σ

(i)x |0L⟩ = V

′i σ

(i)x |1L⟩= 1,

V′jσ

(i)x |0L⟩ = V

′jσ

(i)x |1L⟩= 0, j = i. (4.41)

As a result, the dissipation controls Li proposed at the end of Section 4.3.1 can

passively correct the errors defined by the operators (σ (i)x )†= σ

(i)x , which is exactly

the single bit-flip error E .

4.3.3 Definition of AQEC

Next we will introduce the definition of AQEC [14] that is adapted to the context of this

Section.

Definition 4.3.1 The set of error operators is denoted as Ea [96]. In the error process,

an error may occur with a probability. When the error occurs, the corresponding error

operator Ea transforms the initial state ρ0 ∈ ZV to the erroneous state as Eaρ0E†a . In the

correction process that follows the error process, AQEC is defined as the dissipative

dynamics that stabilizes the system to an arbitrary initial state ρ0 ∈ ZV , when an error


modelled by an arbitrary error operator from Ea occurs, i.e.,

limt→∞

(Eaρ0E†a)t = ρ0, (4.42)

holds for arbitrary ρ0 ∈ ZV and Ea. In this case, the errors are said to be correctable by

the dissipation control.

For example, Ea can be the bit-flip error operator σx which may flip the state of the

qubit as σx|0⟩= |1⟩ with certain probability. AQEC will automatically correct the error

and return the system state to |0⟩ by dissipation control.

Note that the AQEC condition (4.42) implies that the system is ground-state stable

as every ρ0 ∈ ZV is an invariant state. As a result, if no error occurs, the ground states

will be stable under the dissipation control. However, ground-state stability does not

necessarily imply error correction capability. For instance, suppose the erroneous state is

Eaρ0E†a . The erroneous state can be automatically steered back to the invariant subspace

ZV if the system is asymptotically ground-state stable, but we cannot guarantee that

the dissipation control will restore the system to the initial state ρ0 if the ground states

are degenerate, i.e. ZV is more than one-dimensional. It is also worth mentioning that

Definition 4.3.1 corresponds to an ideal case that the error process and correction process

take place consecutively.

The following theorem is concerned with the existence of dissipation control.

Theorem 4.3.1 There always exists a set of unitary operators Ui, j, j = 1, · · ·,J that

stabilize Vi.

Proof. We provide a constructive method to prove the existence. Since Vi ≥ 0, we can

write the spectral decomposition of Vi as Vi = ∑n hn|n⟩⟨n|,hn ≥ 0, with |n⟩ being the

basis vectors of the Hilbert space. Denote one of the ground states (with eigenvalue 0)

as |0⟩⟨0|. Therefore, if for each h j > 0 we choose Ui, j = | j⟩⟨0|+ |0⟩⟨ j|+∑n= j,0 |n⟩⟨n|,then ViU

†i, jViUi, jVi = ∑n= j h3

n|n⟩⟨n|. Since hn equals either 0 or 1 for a projector Vi, we

have ∑n= j h3n|n⟩⟨n|= ∑n= j hn|n⟩⟨n|. We can verify that

J

∑j=1

(ViU†i, jViUi, jVi −Vi) =−

J

∑j=1

h j| j⟩⟨ j|=−Vi, (4.43)


where h j = 1, j = 1, · · ·,J is the set of positive eigenvalues. The condition (4.30) is

satisfied with ci = 1.

Remark 4.3.1 Theorem 4.3.1 provides a specific solution to the exponential stabilization

problem. Approaches to choose the set of unitaries have also been discussed in [185].

4.3.4 Scalability of Dissipation Control

Theorem 4.3.2 Suppose Ui stabilizes Vi. For each Ui, Vj are separated into two sets,

namely, V (i)n and V (i)

d . The definitions of the two sets are as follows

• [V (i)n ,Ui] = 0,

• [V (i)d ,Ui] = 0.

A sufficient condition for the asymptotic ground-state stability of V is given by

V (i)d U†

i V (i)d UiV

(i)d ≤V (i)

d , (4.44)

∑i(∏

dV (i)

d )≥ λV, λ > 0. (4.45)

The dissipation control is constructed as Li =Ui ∏d V (i)d .

Remark 4.3.2 By definition, V (i)d is the operator that satisfies the condition (4.44) but

does not commute with Ui. Ui acts trivially on V (i)n . In contrast, Ui acts nontrivially on

V (i)d . However, in order for the dissipation controls to be scalable V (i)

d should satisfy

the stability condition G (V (i)d )Ui

=V (i)d U†

i V (i)d UiV

(i)d −V (i)

d ≤ 0. Note that Vi ∈V (i)d since

Ui stabilizes Vi by assumption.

Proof. By assumption of Theorem 4.3.2, we have Ui satisfying Eq. (4.30) for each Vi.

Using Vi ∈V (i)d and the dissipation control Li =Ui ∏d V (i)

d we have

G (Vi)Li= ∏

dV (i)

d U†i ViUi ∏

dV (i)

d −Vi ∏d

V (i)d

= ∏d

V (i)d ViU

†i ViUiVi ∏

dV (i)

d −∏d

V (i)d

≤ (1− ci −1)∏d

V (i)d =−ci ∏

dV (i)

d , ci > 0,

(4.46)


where we have made use of the properties [Vi,Vj] = 0 and V 2i =Vi from Assumption 4.2.1.

For j = i, either [U j,Vi] = 0 which results in

G (Vi)L j= ∏

dV ( j)

d U†j ViU j ∏

dV ( j)

d −Vi ∏d

V ( j)d = 0, (4.47)

or Vi does not commute with U j and satisfies Vi ∈ V ( j)d . According to (4.44), this

implies

ViU†j ViU jVi ≤Vi. (4.48)

Consequently, we can obtain

G (Vi)L j

= ∏d

V ( j)d U†

j ViU j ∏d

V ( j)d −Vi ∏

dV ( j)

d

= ∏d

V ( j)d ViU

†j ViU jVi ∏

dV ( j)

d −Vi ∏d

V ( j)d

≤ 0, (4.49)

which further leads to

G (∑i

Vi) ≤ −∑i

ci ∏d

V (i)d ≤−cmin ∑

i∏

dV (i)

d

≤ −cminλ ∑i

Vi. (4.50)

cmin is the smallest positive number in ci. Here we have used the condition (4.45). By

Theorem 4.2.1, V = ∑iVi is asymptotically ground-state stable.

Remark 4.3.3 The strong scalability condition (4.23) is just a special case of Theo-

rem 4.3.2 with V (i)d =Ui. Furthermore, Eq. (4.44) implies

G (V (i)d )

L′i=UiV

(i)d

≤ 0. (4.51)

Therefore, ∏d V (i)d is the product of the operators in Vi whose energies are not increased

by the local controls L′i =UiV

(i)d . However, if we implement the dissipation control as

L′i =UiVi, the dissipation control may not satisfy the scalability condition Eq. (4.23).


The updated dissipation control Li =Ui ∏d V (i)d is still in the form of Li =UiV

′i

with V′i = ∏d V (i)

d . Note that V′i are still projectors. The reason we preclude the

operators V (i)n from the updated dissipation controls is that ∏d V (i)

d ∏nV (i)n = ∏ j Vj

leads to the same V′i for each dissipation control and so often results in the violation of

sufficient condition (4.45) in practical implementation.

4.3.5 Automatic Quantum Error Correction by Dissipation Control

In this section, we derive the condition such that the dissipation control Li can

automatically correct certain types of errors.

Denote |p⟩ ∈ H as the complete basis vectors of ZV . Then we can write the basis

of the bounded operators on ZV as |p⟩⟨q|, where p and q have the same range. An

arbitrary density state in ZV can thus be expanded on the basis as ρ0 = ∑p,q αpq|p⟩⟨q|.Lemma 4.3.1 The dissipation controls Li are error-correcting with respect to the error

operators Ea if

L (|p⟩⟨q|) = 0, (4.52)


=−κpqEa|p⟩⟨q|E†a +κpq|p⟩⟨q|, κpq > 0 (4.53)

hold for all p,q.

Proof. The dynamical equation Eq. (4.1) can be written as a linear system equation

˙vec(ρt) = Avec(ρt), (4.54)

using the vectorization of ρt ,H,Li and the relation vec(B1B2B3) = (BT3 ⊗B1)vec(B2).

As a result, A is determined by H and Li. The solution of the above equation is given

by

vec(ρt) = eAtvec(ρ0). (4.55)

The condition (4.52) ensures the invariance of the initial state under the dissipation

control, if no error occurs. If an error Ea occurs, the erroneous state ∑p,q αpqEa|p⟩⟨q|E†a


needs to be steered back to ρ0. Since we have

vec((Ea|p⟩⟨q|E†a)t) = eAtvec(Ea|p⟩⟨q|E†

a), (4.56)

and so

˙vec((Ea|p⟩⟨q|E†a)t)

= eAtAvec(Ea|p⟩⟨q|E†a)

= −κpqeAtvec(Ea|p⟩⟨q|E†a)+κpqvec(|p⟩⟨q|)

= −κpqvec((Ea|p⟩⟨q|E†a)t)+κpqvec(|p⟩⟨q|). (4.57)

Eq. (A.9) is an ordinary first-order differential equation which can be easily integrated

to be

vec((Ea|p⟩⟨q|E†a)t)

= e−κpqtvec(Ea|p⟩⟨q|E†a)

+ κpqvec(|p⟩⟨q|)∫ t

0e−κpq(t−r)dr

= e−κpqtvec(Ea|p⟩⟨q|E†a)+vec(|p⟩⟨q|)[1− e−κpqt ].

(4.58)

This proves (Ea|p⟩⟨q|E†a)t → |p⟩⟨q| as t → ∞. Due to the linearity of the dynamical

equation Eq. (4.54), we can deduce that (∑p,q αpqEa|p⟩⟨q|E†a)t → ρ0 for any error

operator Ea. The system is restored to the initial state exactly.

Eq. (4.53) guarantees that every element of the erroneous density state, including

the non-diagonal terms which characterize the quantum coherence, can be restored

to the initial value. In this sense, Eq. (4.52)-(4.53) are more like the definition of

quantum error correction, as compared to the sufficient conditions for the errors to be

correctable [95, 82, 14].

Based on Lemma 4.3.1, we can prove the following theorem.

Theorem 4.3.3 Suppose the dissipation control takes the form Li = UiVi, with Vibeing projectors and Ui being unitary operators. The sufficient conditions for the set


of error operators U†i to be correctable are

ViU†i |p⟩=U†

i |p⟩, (4.59)

VjU†i |p⟩= 0, j = i, (4.60)

for all |p⟩.Remark 4.3.4 The correctable errors are determined by the available dissipation controls.

For example, U†i can be the bit-flip error operator σx. Note that there exist local

dissipation controls Li = σxiVi for the stabilization of the graph states [185, 139].

Therefore, in this case the bit-flip errors caused by σ†xi= σxi are correctable if the

sufficient conditions are satisfied.

Proof. It is easy to verify that Eq. (4.52) is satisfied since ZV is an invariant subspace

under the dissipation control and |p⟩ is the basis vector of ZV . Recall that Vi is obtained

by displacing the system Hamiltonian Hi according to Assumption 4.2.2. Since the

erroneous state vector U†i |p⟩ is an eigenstate of Vi with the eigenvalue 1 and an eigenstate

of Vj, j = i with eigenvalue 0, we can conclude that U†i |p⟩ is also an eigenstate of

the system Hamiltonian H. Based on this fact, we can remove the unitary dynamics in

Eq. (4.53) due to

−i[H,U†i |p⟩⟨q|Ui] =−i[V,U†

i |p⟩⟨q|Ui]

= −i[Vi,U†i |p⟩⟨q|Ui] = 0, (4.61)


and obtain

L (U†i |p⟩⟨q|Ui)

=K

∑j=1

L jU†i |p⟩⟨q|UiL

†j

− 12

U†i |p⟩⟨q|UiL

†jL j −

12

L†jL jU

†i |p⟩⟨q|U†

i

=K

∑j=1

U jVjU†i |p⟩⟨q|UiVjU

†j

− 12

U†i |p⟩⟨q|UiVj −

12

VjU†i |p⟩⟨q|Ui

= |p⟩⟨q|−U†i |p⟩⟨q|Ui. (4.62)

Therefore, Eq. (4.53) holds with κpq = 1. By Lemma 4.3.1, the dissipation control

Li =UiVi can automatically correct errors induced by the error operators U†i .

Eq. (4.59)-(4.60) imply a one-to-one correspondence between each error operator

U†i to Vi. For each error operator U†

i , the erroneous state vector U†i |p⟩ is an eigenvector

of Vi with the positive eigenvalue 1. At the same time, U†i |p⟩ remains as the ground

state of Vj for j = i. For this reason, Vi are similar to the error syndrome projectors as

proposed in the stabilizer formalism [53].

4.3.6 Dissipation Control of 3-qubit Repetition Code States

Let us re-consider the three-qubit code introduced in Example 4.1, i.e.

|0L⟩= |000⟩, |1L⟩= |111⟩, (4.63)

where |0⟩, |1⟩ denote the basis vectors of the physical qubit and |000⟩= |0⟩⊗ |0⟩⊗ |0⟩.For this 3-qubit repetition code, the set of stabilizers is given by

S = σz1σz2,σz2σz3,σz1σz3. (4.64)


Based on the stabilizers, we can define V as

V =12[(I −σz1σz2)+(I −σz2σz3)+(I −σz1σz3)]

= V1 +V2 +V3. (4.65)

Note that V1,V2,V3 are two-body operators. |0L⟩ and |1L⟩ are two ground states of V . It

is easy to verify that L1 =12σx1(I −σz1σz2) satisfies

G (V1)L1= L†

1V1L1 −12

L†1L1V1 −

12

V1L†1L1 =−V1, (4.66)

with U1 = σx1. Similarly, we can obtain the local dissipation controls for V2,V3 as

L2 =12σx2(I −σz2σz3),L3 =

12σx3(I −σz1σz3), respectively. However, we have

∑j=2,3

G (V1)L j= σz1σz2V2 ≰ 0, (4.67)

which violates the scalability condition (4.23).

Instead, we can construct the correct dissipation control for V using Theorem 4.3.2.

Since U1 stabilizes both V1 and V3, the updated dissipation control would be designed

as L1 =U1V ′1 =

12σx1(I −σz1σz3)(I −σz1σz2) with V ′

1 =V1V3. L2 and L3 can be derived

similarly with V′2 =V1V2, U2 =σx2, V

′3 =V2V3, U3 =σx3. Furthermore, we have V

′1+V

′2+

V′3 =V1V3+V1V2+V2V3 =

12(V1+V2+V3) =

12V . By Theorem 4.3.2, V is asymptotically

ground-state stable if the 3-qubit system is coupled to a dissipative environment via

L1,L2,L3.

Next, we can verify the error correction condition Eq. (4.59)-(4.60) in Theorem 4.3.3

by

V′i σxi|0L⟩= σxi|0L⟩, V

′i σxi|1L⟩= σxi|1L⟩,

V′jσxi|0L⟩= 0, V

′jσxi|1L⟩= 0, j = i. (4.68)

As a result, the dissipation control Li can automatically correct the errors induced

by the set of error operators U†i = (σxi)

† = σxi, which is exactly the set of the

single bit-flip errors E . Fig. 4.3 is the numerical demonstration of the error correction


0.0 0.1 0.2 0.3 0.4 0.5

Time (γ−1)

0.0

0.2

0.4

0.6

0.8

1.0

Projection

Coherent evolution under Lyapunov-designed dissipation

Tr(ρtρtarget)

Tr(ρtρinitial)

Fig. 4.3 Numerical simulation of the state evolution with the coupling operators L′1,L

′2,L

′3

for 3-qubit repetition code state. Here L′1,L

′2,L

′3 are renormalized by a coupling strength√

κ , i.e. L′i =

√κLi. The initial state is an error-corrupted state ρinitial =

|001⟩+i|110⟩√2

,

which is different from the code state ρ0 = ρtarget =|000⟩+i|111⟩√

2due to the single bit-flip

error σx3. The system converges to the target state with unit probability, which meansthat no coherence is lost during this dissipative process. κ and the time scale γ are set to1 in this example.

performance of the dissipation control. An erroneous state is shown to be restored to

the code state. In particular, the coherence of the initial state is preserved under the

dissipation.

A more realistic automatic error correction scheme is applying the error correction

control in parallel with the noises [82]. In this case, the system can be modelled

as subjected to the noise and the engineered couplings simultaneously. As we have

demonstrated in Fig. 4.4, the derived system-environment couplings for the 3-qubit

repetition code states indeed can be used in parallel with the noises. The bit-flip errors

are modelled by coupling the system to the environment via the additional coupling

operators Lnoisei =

√γσxi, i = 1,2,3, with γ being the coupling strength. As a result,

the system is associated with 6 coupling channels in total. When we increase the strength

of the dissipation controls, a nearly perfect code state preservation can be achieved.


Remark 4.3.5 Coherent feedback loop in [90] is essentially implementing the environ-

mental couplings adiabatically [90, Eq. (2)]3. Physical implementation of dissipation

control has also been experimentally demonstrated for superconducting qubit systems,

such as in [156]. Recently, Cohen et al. has demonstrated a scheme which uses the

dissipative gadgets to implement automatic quantum error correction [26].

0.0 0.1 0.2 0.3 0.4 0.5

Time (γ−1)

0.75

0.80

0.85

0.90

0.95

1.00

Tr(ρtρ

0)

Ground state fidelity under errors and engineered dissipations

κ

γ= 1

κ

γ= 5

κ

γ= 10

κ

γ= 50

κ

γ= 100

Fig. 4.4 Numerical simulation of state evolution when the system is subjected to thedissipative couplings L

′1,L

′2,L

′3 and the noise operators Lnoise

i = σxi, i = 1,2,3 simul-taneously. γ = 1. The code state is defined by ρ0 = |000⟩+i|111⟩√

2. Depending on the

coupling strength κ , we achieve different levels of state preservation. In the strongcoupling regime κ/γ ≥ 50, the code state can be continuously preserved against thebit-flip noise.

4.4 Decoherence Free Subsystem Synthesis

As mentioned, a decoherence free subsystem represents a subsystem whose variables

are not affected by input fields and do not appear in the system output fields; this makes

the DFS isolated from the environment and inaccessible to measurement devices, thus

preserving the quantum information carried by the variables of the DFS. In relation to the

annihilation only system (4.6), with a = [a1, . . . ,an]T , a component a j is a decoherence-

free mode if the evolution of a j is independent of the input W and if the system output

3These couplings are obtained if we adiabatically eliminate the coherent controller dynamics.


Y is independent of a j. The collection of decoherence-free modes forms a subspace,

called the decoherence-free subspace4. According to this, decoherence free subspaces

are uncontrollable and unobservable subspaces of a quantum system.

An important fact about the existence of a decoherence-free subsystem for linear

annihilation only systems follows from the results established in [59]:

Proposition 4.4.1 The linear annihilation only system (4.6) has a decoherence-free sub-

system if and only if the matrix A has some of its poles on the imaginary axis, with the

remaining poles residing in the open left half-plane of the complex plane.

Proof. According to [59, Lemma 2], for the system (4.6), the properties of controllability,

observability and Hurwitz stability are equivalent. The statement of the proposition then

follows by contraposition, after noting that being passive, the system (4.6) cannot have

eigenvalues in the open right hand-side of the complex plane due to (4.8).

4.4.1 Open-loop Reservoir Engineering for DFS Generation

With reference to Fig. 4.5, we investigate conditions to enable the synthesis of a quantum

coherent controller-system network to generate a DFS in the interconnected system

through interactions between the principal quantum system and the controller.

The quantum linear passive system in Fig. 4.5 is the system of the form (4.6), and its

input fields are further partitioned as W = [wT ,uT , f T ]T . Here, w represents a ‘natural’

environment for the system, and f and u represent an open-loop and feedback engineered

fields, respectively. According to this partitioning, the system evolution is described as

dap = Apapdt +B1dw+B2du+B3d f , (4.69a)

dy =Capdt +dw. (4.69b)

Accordingly, the matrices of the system have dimensions as follows: Ap ∈ Cn×n,

B1 ∈ Cn×nw , B2 ∈ Cn×nu B3 ∈ Cn×n f , and C ∈ Cnw×n (n,nw,nu,n f ∈ N). We also use

the notation ap for the vector ap (t) =[ap1

(t) , . . .apn(t)]T of the system annihilation

operators defined on its underlying Hilbert space Hp.

4A decoherence-free subspace exists if we can partition the Hilbert space into two subspaces HS =HDF ⊕HN , where the subspace of the Hilbert space not affected by decoherence is denoted by HDF andHN denotes the decoherence-affected, “noisy” subspace.


w

Quantumsystemf s

y

Quantumcontroller

SW

y’u’

z

v

z’u1’

ru

u1

Fig. 4.5 Coherent feedback network for DFS generation.

In terms of the Hamiltonian and coupling operators, the system has the Hamiltonian

Hp = a†pMap where M is an n×n complex Hermitian matrix, and is linearly coupled to

the input fields via the coupling operators Lp1= α1ap, Lp2

= α2ap, Lp3= α3ap where

α1 ∈ Cnw×n, α2 ∈ Cnu×n, α3 ∈ Cn f×n are complex matrices. Then the relations (4.7)

specialize as follows:

Ap =−(

iM+12

α†1 α1 +

12

α†2 α2 +

12

α†3 α3

),

B1 =−α†1 ,

B2 =−α†2 ,

B3 =−α†3 ,

C = α1.

The starting point of the discussion that follows is the assumption that under the

influence of its natural environment w alone, (i.e., in the absence of the engineered

fields f and u), the system does not possess a DFS. Mathematically, this assumption

corresponds to the assumption that(Ap,B1

)is controllable and

(Ap,C

)is observable,


since these properties rule out the existence of a DFS in the plant (4.69) when B2 = 0,

B3 = 0; see Proposition 4.4.1 and [202, 59].

In many cases, system couplings can be engineered at a fabrication stage to reduce

unavoidable loss of energy due to decoherence [145, 185]. The process of tuning the

system at the fabrication stage does not involve feedback, and we let Lp2= 0, which

corresponds to α2 = 0 and B2 = 0 in (4.69); see Fig. 4.6.

w

Quantumsystemf s

y

Fig. 4.6 Open loop setup for DFS generation.

Then the system (4.69) reduces to that of the form

dap = Apapdt +B1dw+B3d f , (4.70a)

dy =Capdt +dw (4.70b)

Here, w and f symbolize the natural environment and the fabricated open-loop field,

respectively. Accordingly, the coupling operator Lp1corresponds to a fixed coupling

with the natural environment, while the coupling Lp3corresponds to the engineered

coupling. The physical realizability requirement imposes the constraint that

Ap +A†p +B1B†

1 +B3B†3 = 0, (4.71)

cf. (4.8). Recall [102] that a quantum stochastic differential equation of the form (4.70)

is said to be (canonically) physically realizable if it preserves the canonical commutation

relations, [ap,a†p] = apa†

p − (a∗paTp )

T = I, and is a representation of an open harmonic

oscillator, i.e., it possesses a Hamiltonian and a coupling operator. The satisfaction of

the identity (4.71) is a necessary and sufficient condition for physical realizability [102,

Theorem 5.1].


Theorem 4.4.1 Suppose (−iM,B1) is controllable. Then a DFS cannot be created by

coupling the system to an engineered environment.

Proof. To prove the theorem we will show that the matrix Ap has all its eigenvalues

in the open left half-plane of the complex plane, and therefore it cannot have a DFS,

according to Proposition 4.4.1; see [59, Lemma 2].

First consider the system with a fixed coupling with the environment, i.e., Lp3= 0.

For this system, the physical realizability properties dictate that

Ap1 +A†p1 +B1B†

1 = 0, (4.72)

with Ap1 =−iM− 12B1B†

1; see (4.7).

Recall that for an arbitrary n×n matrix Φ and an n×m matrix B, the pair (Φ,B) is

controllable if and only if (Φ+ 12BB†,B) is controllable. Applying this fact to the pair

(−iM,B1) which is controllable by the assumption of the theorem, we conclude that

(Ap1,B1) is controllable. Thus, equation (4.72) can be regarded as a Lyapunov equation

Ap1P+PA†p1 +B1B†

1 = 0

with controllable (Ap1,B1), which has a positive definite solution P = I. Since B1B†1 ≥ 0,

according to the inertia theorem [22, Theorem 3], the above observation about the

existence of a positive definite solution to the Lyapunov equation implies that Ap1 must

have all its eigenvalues in the open left half-plane of the complex plane. As a result, if

(−iM,B1) is controllable, the corresponding passive quantum system with fixed coupling

cannot have a DFS, according to Proposition 4.4.1.

Next consider this system when it is coupled to an engineered environment, i.e.,

Lp3= 0 and B3 = 0. Since Ap1 has been shown to have all eigenvalues in the open

left half-plane of the complex plane, there exists a positive definite Hermitian matrix

P = P† > 0 such that

A†p1P+PAp1 < 0.

On the other hand, according to Corollary 4 of [134], the matrix 12B3B†

3P cannot have

eigenvalues in the open left half-plane of the complex plane, and therefore −12B3B†

3P−


12PB3B†

3 ≤ 0. This implies that

(Ap1 −12

B3B†3)

†P+P(Ap1 −12

B3B†3)< 0

and therefore Ap = Ap1 − 12B3B†

3 must have all its eigenvalues in the open left half-plane

of the complex plane. According to Proposition 4.4.1, this rules out the possibility for

the system with engineered coupling to have a DFS.

Next, suppose that (−iM, [B1 B3]) is not controllable5, therefore (−iM,B1) is not

controllable either. Theorem 4.4.1 does not rule out a possibility for a DFS to exist in

this case. It is easy to show that

ker(C T ) = ker(C Tw )∩ker(C T

f ),

where Cw, C f are the controllability matrices with respect to the inputs w and f , respec-

tively. From this observation, it follows that the dimension of the DFS of system (4.70)

is less or equal to the dimension of each of the decoherence free subsystems arising

when the quantum plant is coupled with the fixed and engineered fields only. This leads

to the conclusion that coupling the system with additional engineered fields can only

reduce the dimension of the DFS. In the remainder of the paper, we will show that using

coherent feedback, on the other hand, does allow to create or increase dimension of a

DFS.

4.4.2 Coherent Feedback Reservoir Engineering

In this section we consider a system of the form (4.69). To simplify the notation we

will combine two static channels w and f into a single channel, which will again be

denoted as w. More precisely, we combine the coupling operators Lp1and Lp3

into a

single operator Lp1. Then the system (4.69) reduces to a system of the form

dap = Apapdt +B1dw+B2du, (4.73a)

dy =Capdt +dw, (4.73b)

5Here, [B1 B3] is the matrix obtained by concatenating the rows of B1 and B3.


where the new matrix B1 is composed of the previous matrices B1 and B3, so that using

the new notation we have

Ap =−(

iM+12

α†1 α1 +

12

α†2 α2

),

B1 =−α†1 ,

B2 =−α†2 ,

C = α1. (4.74)

For a coherent quantum controller for the quantum plant (4.69), we will consider

another open quantum linear annihilation only system. Such a system will be assumed

to be coupled with three environment noise channels, y′, z′ and v. The fields y′, z′

are to produce output fields which will be used to form the feedback, and the channel

v will be used to ensure that the constructed observer is physically realizable. As is

known [102], once physical realizability of the observer is ensured, one can readily

construct a scattering matrix, a Hamiltonian and a collection of coupling operators

describing the quantum evolution of the controller in the form of a quantum stochastic

differential equation (4.3). Alternatively, a physically realizable coherent controller can

be represented in the form of the quantum stochastic differential equation (4.6) [102],

i.e., in the form

dac =Acacdt +G1dy′+G2dz′+G3dv, (4.75a)

du′ =Kacdt +dy′, (4.75b)

du′ =Kacdt +dz′, (4.75c)

where for physical realizability, the following constraints must be satisfied [102, Theo-

rem 5.1]:

Ac +A†c +G1G†

1 +G2G†2 +G3G†

3 = 0, (4.76)

K =−G†1, (4.77)

K =−G†2. (4.78)


Interconnection between the controller and the plant are through scattering equations

relating the output fields of the plant with the input channels of the controller and vice

versa. Specifically, the scattering equation

y′

z′

= S

y

z

, (4.79)

links the output field of the plant y and the controller environment z with the input

controller channels y′, z′. Here, S is a unitary matrix partitioned as

S =

S11 S12

S21 S22

. (4.80)

Likewise, feedback from the controller (4.75) is via a unitary matrix W ,

u

u

=W

u′

u′

, W =

W11 W12

W21 W22

. (4.81)

The matrices Ac, G1 = −K†, G2 = −K†, G3, and the scattering matrices S, W are

regarded as the controller design parameters. Our objective in this paper is to find a

procedure for selecting those parameters so that the resulting coherently interconnected

quantum system in Fig. 4.5 possesses a decoherence free subsystem.

To devise the DFS synthesis procedure, we first note that the control system governed

by y,z,v and output u can be represented as

dac = (Acac − (G1S11 +G2S21)B†1ap)dt

+(G1S11 +G2S21)dw

+(G1S12 +G2S22)dz+G3dv, (4.82)

du = (−(W11G†1 +W12G†

2)ac

−(W11S11 +W12S21)B†1ap)dt

+(W11S11 +W12S21)dw

+(W11S12 +W12S22)dz


Also, the closed loop system is described by the quantum stochastic differential equation

d

ap

ac

= Acl

ap

ac

dt +Bcl

dw

dz

dv

, (4.83)

with block matrices Acl , Bcl partitioned as follows

Acl =

Ap −B2(W11S11 +W12S21)B

†1 −B2(W11G†

1 +W12G†2)

−(G1S11 +G2S21)B†1 Ac

,

Bcl =

B1 +B2(W11S11 +W12S21) B2(W11S12 +W12S22) 0

G1S11 +G2S21 G1S12 +G2S22 G3

.

Let

Ac = Ap − B2(W11S11 +W12S21)B†1

+(G1S11 +G2S21)B†1

−B2(W11G†1 +W12G†

2) (4.84)

Then for Acl to have all eigenvalues on the imaginary axis or in the left half-plane of the

complex plane it is necessary and sufficient that the following matrices

A = Ap −B2(W11S11 +W12S21)B†1

−B2(W11G†1 +W12G†

2), (4.85)

A = Ap −B2(W11S11 +W12S21)B†1

+(G1S11 +G2S21)B†1 (4.86)

have all eigenvalues on the imaginary axis or in the left half-plane of the complex plane.


Proof. The matrix Acl has the same eigenvalues as the matrix

I 0

I −I

Acl

I 0

I −I

=

Ap −B2W11(S11B†1 +G†

1)

−B2W12(S21B†1 +G†

2)B2(W11G†

1 +W12G†2)

Ap −Ac

−B2(W11S11 +W12S21)B†1

+(G1S11 +G2S21)B†1

−B2(W11G†1 +W12G†

2)

B2(W11G†1 +W12G†

2)+Ac

.

Hence the lemma follows, due to the definition of Ac in (4.84).

Theorem 4.4.2 Suppose matrices S, W are given. Let G1, G2 be such that

1. The following linear matrix inequality (LMI) in G1, G2 is satisfied

R G1 G2

G†1 −I 0

G†2 0 −I

≤ 0, (4.87)

where

R =−B1B†1 −B2B†

2 −B2(W11S11 +W12S21)B†1

−B1(S†21W †

12 +S†11W †

11)B†2 +(G1S11 +G2S21)B

†1

+B1(S†11G†

1 +S†21G†

2)−B2(W11G†1 +W12G†

2)

−(G1W †11 +G2W †

12)B†2; (4.88)

2. The matrices A and A, defined in equations (4.85) and (4.86) respectively, have

all their eigenvalues in the closed left half-plane, with at least one of them having

eigenvalues on the imaginary axis.

Then a matrix G3 can be found such that the closed loop system (4.83) admits a DFS.

Proof. Via the Schur complement, (4.87) is equivalent to

R+G1G†1 +G2G†

2 ≤ 0.


Therefore one can find G3 such that

R+G1G†1 +G2G†

2 +G3G†3 = 0.

From this identity and the expression (4.84), the identity (4.76) follows. This shows

that the feasibility of the LMI (4.87) ensures that the controller system (4.75) can be

made physically realizable by appropriately choosing G3. As a result, the closed loop

system, being a feedback interconnection of physically realizable systems, is a physically

realizable annihilation only system. Also, condition (b) and Lemma 4.4.2 ensure that

Acl has eigenvalues on the imaginary axis. Then it follows from Proposition 4.4.1 that

the closed loop system (4.83) has a DFS.

Note that matrices A and A can be rewritten as

A = Ap −B2(W11S11 +W12S21)B†1

−B2[W11 W12]

G†

1

G†2

, (4.89)

A = Ap −B2(W11S11 +W12S21)B†1

+[G1 G2]

S11

S21

B†

1 (4.90)

A necessary condition to ensure that an eigenvalue assignment can be carried out

for these matrices by selecting G1, G2, is that the pair (Ap,B2[W11 W12]) is control-

lable and the pair (Ap,

S11

S21

B†

1) is observable; the latter condition is equivalent to

the controllability of the pair (A†p,B1

[S†

11 S†21

]). Indeed, these controllability and

observability conditions imply that (Ap −B2(W11S11 +W12S21)B†1,

S11

S21

B†

1) is ob-

servable and (Ap −B2(W11S11 +W12S21)B†1,B2[W11 W12]) is controllable. Therefore, if

(Ap,B2[W11 W12]) and (A†p,B1

[S†

11 S†21

]) are controllable, one can always select G1

and G2 so that the matrices A, A have a required eigenvalue distribution. Thus the condi-


tions of Theorem 4.4.2 boil down to solving a simultaneous pole assignment problem

under an LMI constraint.

We next demonstrate that our pole assignment problem captured quantum plant-

controller DFS architectures considered in [138, 202].

4.4.3 Special Case 1: DFS synthesis using a coherent observer [138]

In [138], the DFS synthesis was carried out using a quantum analog of the Luenberger

observer for a class of linear annihilation only systems as described by (4.5); see Fig. 4.7.

This controller structure is a special case of the architecture in Fig. 4.5, when

S =

I 0

0 I

, W =

0 I

I 0

With this choice of S and W , we have from (4.84)

Ac = Ap +G1B†1 −B2G†

2 (4.91)

w

Quantumsystem

y

Quantumcontroller

u

z

v

Fig. 4.7 Coherent plant-observer network considered in [138].


Corollary 4.4.3 Suppose the pair (Ap,C) is observable and the pair (Ap,B2) is control-

lable. Let G1, G2 be such that

1. The following linear matrix inequality (LMI) is satisfied

R G1 G2

G†1 −I 0

G†2 0 −I

≤ 0, (4.92)

where

R =−B1B†1 −B2B†

2

+G1B†1 +B1G†

1 −B2G†2 −G2B†

2 (4.93)

2. The matrices

A = Ap −B2G†2, (4.94)

A = Ap +G1B†1 (4.95)

have all eigenvalues on the imaginary axis or in the left half-plane of the complex

plane, with at least one of them having eigenvalues on the imaginary axis.

Then the closed loop system admits a DFS.

Proof. Via the Schur complement, condition (4.92) is equivalent to the condition

−B1B†1 −B2B†

2 +G1B†1 +B1G†

1

−B2G†2 −G2B†

2 +G1G†1 +G2G†

2 ≤ 0.

This ensures that

Ac +A†c +G1G†

1 +G2G†2 ≤ 0.

Therefore, one can find G3 such that the controller is physically realizable. The claim

then follows from Theorem 4.4.2.


4.4.4 Special Case 2: Coherent feedback DFS generation model from [202]

Consider a system of Fig. 4.5 in which S = I, W = I, and let G2 = 0, G3 = 0. This

corresponds to the system shown in Fig. 4.8; which was considered in [202]. In this

case, the controller matrix becomes

Ac = Ap −B2B†1 +G1B†

1 −B2G†1. (4.96)

w

Quantumsystem

y

Quantumcontroller

r

u

Fig. 4.8 Special Case 2: Coherent feedback network for DFS generation consideredin [202].

Corollary 4.4.4 Suppose the pair (Ap,C) is observable and the pair (Ap,B2) is control-

lable. Let G1 be such that

1. The following equation is satisfied

−(B1 +B2)(B1 +B2)† +G1(B1 −B2)

†

+(B1 −B2)G†1 +G1G†

1 = 0; (4.97)


2. The matrices

A = Ap −B2B†1 −B2G†

1, (4.98)

A = Ap −B2B†1 +G1B†

1 (4.99)

have all eigenvalues on the imaginary axis or in the left half-plane of the complex

plane, with at least one of them having eigenvalues on the imaginary axis.

Then the closed loop system admits a DFS.

Proof. Condition (4.97) ensures that

Ac +A†c +G1G†

1 = 0.

Next, A and A have eigenvalues on the imaginary axis or in the open left half-plane,

hence the statement of the corollary follows from Theorem 4.4.2.

4.4.5 Examples

Example 1

To illustrate the DFS synthesis procedure developed in the previous section, consider

a system consisting of two optical cavities interconnected as shown in Fig. 4.7. The

system is similar to those considered in [126].

The cavity to be controlled is described by equation (4.73), with all matrices becom-

ing complex numbers

Ap = −iM− κ1 +κ22

, B1 =−√κ1, B2 =−√

κ2,

C = −B∗1 =

√κ1. (4.100)

Here, κ1, κ2 are real non-negative numbers, characterizing the strength of the couplings

between the cavity and the input fields w and u, respectively, and M characterizes the

Hamiltonian of the cavity.

Clearly, the pair (Ap,C) is observable and the pair (Ap,B2) is controllable, therefore

the optical cavity cannot have a DFS unless the cavity is lossless. To synthesize a DFS,

let us connect this cavity to another optical cavity with the same Hamiltonian, as shown


in Fig. 4.7. This corresponds to letting the controller have the coefficients

Ac = −iM− κ3 +κ42

, G1 =−√κ3, G2 =−√

κ4,

K =√

κ4, G3 = 0. (4.101)

and letting the scattering matrices S and W be

S =

1 0

0 1

, W =

0 1

1 0

. (4.102)

We now apply Corollary 4.4.3 to show that the parameters κ3,κ4 for the controller cavity

can be chosen so that the two-cavity system has a DFS. It is readily verified that the

matrices A and A in (4.94), (4.95) reduce to

A = −iM− κ1 +κ22

−√κ2κ4, (4.103)

A = −iM− κ1 +κ22

+√

κ1κ3. (4.104)

From Corollary 4.4.3, we need either A or A to have poles on the imaginary axis in

order to create a DFS within the closed-loop system. Clearly, for the two-cavity system

under consideration this can only be achieved by placing the pole of A at the origin. For

this, the coupling rate κ3 of the controller must be set to

κ3 =(κ1 +κ2)

2

4κ1. (4.105)

Also we must satisfy the LMI condition (4.92). The matrix R in this example reduces to

R =−κ1 −κ2 +2√

κ1κ3 −2√

κ2κ4. (4.106)

Hence, using (4.105) and (4.106), the LMI condition (4.92) reduces to the two following

inequalities:

Re[−κ1 −2κ1

√κ2κ4 ±

√D]≤ 0, (4.107)


where

D = κ21 +κ

31 +2κ

21 κ2 +κ1κ

22 +4κ

21 κ4

+4κ21 κ2κ4 −4κ

21√

κ2κ4.

The inequality (4.107) is the only constraint for the remaining coupling parameter κ4

to be determined. Notice that there is an obvious solution to this inequality in the case

where κ1 = κ2 = κ . The solution is κ3 = κ4 = κ which satisfies both (4.105) and (4.107).

The above calculations demonstrate that by placing the pole of the controller on the

imaginary axis, one can effectively create a DF mode which did not exist in the original

system. This fact has been established previously in [126] by calculating the system

poles, whereas we have arrived at this conclusion from a more general Corollary 4.4.3,

as a special case.

Example 2

We now present an example in which, the DFS is created which is shared between

the controlled system and the controller. The controlled system in this example consists

of two cavities as shown in Fig. 4.9.

Denote the matrices associated of the Hamiltonians corresponding to the each cavity

internal dynamics as M1, M2. Also for the convenience of notation, define the complex

numbers

γ j =√

κ j, j = 1, . . . ,4,

associated with the coupling strengths within the cavities. All four constants are assumed

to be nonzero.


w

u

y

r

k1

k2

k4

Cavity 1

k3

Cavity 2

Fig. 4.9 The two-cavity system for Example 2.

Then the equations governing the dynamics of the two-cavity system have the form

of (4.73) with

Ap =

−

(iM1 +

|γ1|2+|γ2|22 + γ∗1 γ2

)−γ2γ∗3

−γ∗1 γ3 −(

iM2 +|γ3|2

2

) ,

B1 =

−(γ1 + γ2)

−γ3

, B2 =

−γ4

0

,

C = −B†1 =

[γ1 + γ2 γ3

]. (4.108)


To verify observability of the pair (Ap,C), we observe that

det

C

CAp

=

12

γ∗3(|γ3|2(γ∗1 − γ

∗2 )

+(γ∗1 + γ∗2 )(|γ1|2 −|γ2|2)

+2i(γ∗1 + γ∗2 )(M1 −M2)) .

Suppose γ1 = −γ2, then det

C

CAp

= γ∗3 γ∗1 |γ3|2, and we conclude that the matrix

C

CAp

is full rank. This implies that in the case γ1 = −γ2, the pair (Ap,C) is

observable. Also, the pair (Ap,B2) is controllable, since

det[B2 ApB2

]=−γ

24 γ3γ

∗1 = 0.

These observations allow us to apply Corollary 4.4.3 to construct a DFS by interconnect-

ing the two-cavity system with a coherent quantum observer, which we now construct.

For simplicity, choose

G1 =

g1

0

, G2 =

0

g2

.

With this choice of G1 and G2 and under the condition γ1 = −γ2, the matrices A =

Ap −B2G†2 and A = Ap +G1B†

1 take the form

A =

−iM1 −γ2γ∗3 + γ4g∗2

−γ∗1 γ3 −(

iM2 +|γ3|2

2

) ,

A =

−iM1 −γ2γ∗3 −g1γ∗3

−γ∗1 γ3 −(

iM2 +|γ3|2

2

) .

Letting g2 =γ∗2 γ3γ∗4

, g1 =−γ2 allows us to conclude that each of the matrices A and A have

one imaginary eigenvalue and one eigenvalue with negative real part, −(

iM2 +|γ3|2

2

).

4.5 Concluding Remarks 117

It remains to show that the LMI condition (4.92) is satisfied in this example. Noting

that with the above choice of g1, g2,

R =

−|γ4|2 2γ2γ∗3

2γ∗2 γ3 −|γ3|2

,

R < 0 holds provided |γ4|2 > 4|γ2|2. Next, the LMI (4.92) in this example requires that

−|γ4|2 2γ2γ∗3 −γ2 0

2γ∗2 γ3 −|γ3|2 0 γ∗2 γ3γ∗4

−γ∗2 0 −1 0

0 γ2γ∗3γ4

0 −1

< 0 (4.109)

Using the Schur complement, this requirement is equivalent to

−|γ4|2 2γ2γ∗3

2γ∗2 γ3 −|γ3|2

+

|γ2|2 0

0 |γ2|2|γ3|2|γ4|2

< 0.

The latter condition holds when |γ4|2 > (3+√

10)|γ2|2.

4.5 Concluding Remarks

In this chapter, we have applied methods from quantum reservoir engineering and

coherent control to address one of the most challenging aspects of quantum computing,

namely decoherence.

Firstly, we derive an algebraic procedure based on Lyapunov formalism for the

engineering of dissipation control. The scalability and error correction capacity of

the dissipation control of the form L = UV has been investigated in detail. Hence,

the Lyapunov formalism may facilitate the application of the classical decomposition-

aggregation technique to a large scale quantum system.

Secondly, we have proposed a general coherent quantum controller synthesis proce-

dure for generating decoherence-free subspaces in quantum systems. Decoherence-free


components capable of storing quantum information are regarded to be essential for

quantum computation and communication, as quantum memory elements [126].

Chapter 5

Quantum Programming and

Simulation

Let the computer itself be built of quantum mechanical elements which obey

quantum mechanical laws.

Richard Feynman

The first publicly-available quantum computer was announced in 2016 by IBM [20],

which provides a prototype platform for researchers around the world to test their

quantum computing programs on a real quantum computer. Despite offering an excellent

experience for quantum programmers, the platform is still only a proof-of-concept and

could not provide the scale and reliability needed for algorithm discovery or development.

Indeed, from the evolution of conventional computer science, we have learned that

software is as important as hardware. Software innovations can even lead and guide

hardware development. For quantum computing, the current set of applications is very

limited. Hence, computer simulation and emulation of quantum computing algorithms

on classical computers will still be the most accessible and reliable way that we can rely

on to explore quantum software and applications in the foreseeable future. Thanks to

the constant innovation of the microelectronic industry, we now have high-performance

computing platforms which are capable of providing simulation and emulation of sizable

quantum computers. Furthermore, innovative mathematical approaches could be used to

120 Chapter 5. Quantum Programming and Simulation

extend the capability of those simulators in specific use cases beyond the memory-bound

limit of the Hilbert space.

The work presented in this chapter involves the definition of a quantum assembly

language; the complete tool-chain which supports compilation from high-level procedu-

ral wrapper language to the assembly instructions and the serialization of the assembly

instructions into a binary data structure which could be sent to a simulator or a real

quantum computers; and lastly a simulator engine which is based on Feynman path

integral that provides unique performance advantages under various test cases as well

as could be integrated with a conventional linear algebra simulators to achieve optimal

time-to-result for all cases. These were done as part of my research internship at Atos

Quantum, a research lab of the Big Data and Security division of Atos SE.1

5.1 Quantum Programming Language

A quantum computer, as described by the quantum gate model, can execute arbitrary

quantum programs (algorithms) by applying a specific sequence of gate operations.

This computing model is essentially similar to classical computing devices where those

low-level operations are described by machine code instructions.

These instructions are directly mapped to a human-readable form which is often

called assembly languages. In principle, every computer programs (e.g. games, simula-

tion, web browsers, etc.) can and in fact must be programmed by the assembly language

specific to the hardware that they are running on. The tedious process of writing com-

puter programs instruction-by-instruction is thus not very practical and is best reserved

for very specific use cases (e.g. low-level firmware). Most computer programmers

learned to code in the so-called high-level programming languages (e.g. C/C++, Python,

Java, Javascript, etc.) which are both hardware agnostic and algorithmically descriptive

(as supposed to be operationally descriptive).

Hence, a quantum programming paradigm could mimic that of its classical coun-

terpart, i.e. providing multi levels of abstraction from the low-level gate operations to

the high-level algorithmic descriptions. This includes an assembly-language definition

which closely matches (modulo trivial gate transformations) and directly executable on

1This work does not reflect the views or opinions of Atos SE or any of its employees.


quantum computer hardware. On top of that, a higher level language and the correspond-

ing compiler is ultimately necessary for practical quantum programming purposes. A

comprehensive solution for quantum programming language also needs to be flexible

and agnostic in the sense that it can interface with both quantum hardware as well as

simulators/emulators.

5.1.1 Quantum Assembly Language

The quantum assembly (QASM) language is an abstract description of operations of

a general-purpose quantum computer. This includes a set of quantum instructions as

well as the syntax and semantics rules. In designing our own QASM language, the

Atos Quantum Assembly Language (AQASM), we take into consideration the following

criteria:

• Fully quantum: Although the first generation of AQASM will only interact with a

classical simulator/emulator back-end, the long-term goal is to be able to use it on

a real quantum computer.

• Low-level: AQASM is designed around the quantum gate model. The basic

instructions correspond to elementary quantum gates.

• Extendable: AQASM also supports user-defined quantum gates and arbitrary gate

extensions by adding control or gate parameterisation.

• Standard: AQASM is the universal language for all future ATOS-developed

quantum simulators to maintain interoperability.

• Complete toolchain: We also plan to develop high-level wrappers together with

proper compilation programs in order to provide a high-level abstraction to devel-

opers. Also, it is essential to provide parsing/translating support for other existing

quantum programming languages to AQASM.

The AQASM has native support for arbitrary quantum gates (including user-defined

unitary gates and gate parameterisation), expanding quantum gates (e.g. complex

conjugate and adding control), and basic quantum operations (e.g. quantum Fourier

transform). An example of AQASM programs is given in Fig. 5.1.

The compilation of AQASM into quantum circuit data structures is implemented in

Coco/R [113], which is a widely-used framework for syntax validation and compiler


DEFINE GATE qft2 = QFT[2]DEFINE GATE qft10 = QFT[5]

BEGINqubits 3

QFT q[0],q[1],q[2]H q[0]S q[1]T q[2]CNOT q[0],q[1]GATE ex q[1]X q[2]CNOT q[1],q[2]IQFT q[0],q[1],q[2]DAG(CTRL(RX[PI/2])) q[2],q[0]

END

Fig. 5.1 An example of arbitrary quantum circuit described by the Atos quantum assem-bly language.

generation. By defining the compiler descriptions, which includes the list of keywords

(tokens) as well as all the syntax and semantics rules, we can use Coco/R to generate the

corresponding scanner and parser to process the AQASM inputs.

5.1.2 High-level Programming and Data Processing

The advantages of assembly languages are the strong correspondence between the

language, i.e. the instructions, and the physical operations, i.e. quantum gates, on

quantum hardware. Experimentalists can import such a program as in Fig. 5.1 and

readily generate the control sequence for lab instruments to implement the AQASM

program. Or as in the case of our simulation back-end, it can take those AQASM

programs and perform the simulation accordingly.

However, the main drawback of low-level languages, e.g. AQASM, is its verbosity,

which often hides the algorithmic structure and hierarchy of the program. For example,

let’s take a look at an 18-qubit quantum Fourier transform program written in AQASM2

as depicted in Fig. 5.2. There are approximately 170 AQASM instructions in that

program which has been abbreviated for brevity.2This is an explicit AQASM representation of the QFT algorithm. The language does have native

support for QFT and inverse QFT (IQFT) as demonstrated in Fig. 5.1.


BEGINqubits 18H q[0]CTRL(PH[PI/2]) q[1], q[0]H q[1]CTRL(PH[PI/4]) q[2], q[0]CTRL(PH[PI/2]) q[2], q[1]H q[2]CTRL(PH[PI/8]) q[3], q[0]CTRL(PH[PI/4]) q[3], q[1]CTRL(PH[PI/2]) q[3], q[2]H q[3]CTRL(PH[PI/16]) q[4], q[0]// Skipping...//H q[17]END

Fig. 5.2 An example of 18-qubit QFT program written in AQASM. The body of theprogram has been abbreviated for clarity.

Most people will find it difficult to comprehend a program as in Fig. 5.2. More

importantly, it is extremely tedious and error-prone for developers/programmers to write

quantum programs in that manner. Hence, upon the complete definition of AQASM

in terms of language features, semantics and syntax rules, we embarked on creating

high-level wrappers for AQASM in commonly-used classical programming languages.

For instance, for Python, we developed PyAQASM package 3 which supports looping,

subroutine declarations, function calls, classical processing of intermediate data, etc. For

comparison, the same 18-qubit QFT program in Fig. 5.2 can be expressed in PyAQASM

using the program listed in Fig. 5.3.

The looping and parametrisation supported by PyAQASM not only shorted the

program significantly but also allow novice readers to comprehend the overall structure

of the algorithm. In addition, by having Python support with PyAQASM, we can

integrate the programming workflow more seamlessly. For example, a web interface

can be used to input the PyAQASM program; the user’s computer can then generate

3The author and other researchers of the ATOS Quantum Lab collaborated on defining the AQASMlanguage itself. The implementation of the Coco/R-based compilation unit was developed by anotherresearcher. The author supported the testing of the compiler and implemented the first high-level Python-based wrapper as described in this section.


from AQASM import *

nbqubits = 18

AQASM_prog = AQASM("QFT.aqasm", nbqubits)

for i in range(nbqubits - 1):AQASM_prog.apply(qGate("H"), qubit(i))for j in range(i+1):

phase = "PI/" + str(2**(i-j+1))AQASM_prog.apply(

qGate("PH", CTRL = ’true’, PHASE = phase),[qubit(i+1), qubit(j)])

AQASM_prog.apply(qGate("H"), qubit(nbqubits - 1))

AQASM_prog.finish()

Fig. 5.3 An example of 18-qubit QFT program written in PyAQASM. PyAQASM isa Python-based wrapper for AQASM which can export the corresponding AQASMprogram for simulation and experiment purposes.

the AQASM and serialise it via the Internet to the simulator (e.g. a supercomputer in a

remote server room) or real quantum hardware in a remote laboratory; the result data of

that algorithm is returned to the user for further data analysis and visualisation.

In the next session, we will discuss the ATOS quantum simulator which is a High-

Performance Computing (HPC) software program designed specifically to simulate

quantum programs written in AQASM.

5.2 Simulation Engine

The exponential size of the Hilbert space which describes the computation space of

quantum computers makes the quantum simulation on classical computers impossible

in general. Indeed, a conventional linear-algebra based approach will be limited firstly

by the computer memory to store the quantum state. In this section, we present an

alternative approach to the classical simulation which employs the space-time trade-

off 4 to extend the ability to simulate a larger number of qubits and achieve shorter

time-to-result in specific use cases.

4memory footprint vs. runtime


The program code was written by the author in C++ programming language (includ-

ing OpenMP [130] extension), with the primary goal is to reach high performance on

High-Performance Computing (HPC) systems.

5.2.1 Overall Architecture

The modular representation of our simulation suite is depicted in Fig. 5.4, in which

the top right-hand side is the core simulation engine running on high-performance

computer clusters. The left-hand side represents user modules (i.e. running on a personal

computer/laptop) which handle user interfaces for code editing or importing (from other

programming languages) as well as other post-simulation data analysis and visualisation.

The ATOS compiler, which also runs on the user side, compiles and serialise AQASM

Fig. 5.4 The functional diagram of ATOS quantum simulator.

code into a binary file for remote execution.

The technical specifications of the computing hardware and software packages are

listed in Table 5.1.

On the hardware side, the ATOS Bullion server is built with an extremely large

amount of high-bandwidth memory (per CPU socket as well as in total) which is to


Table 5.1 Technical details of the ATOS path-integral-based quantum simulator.

Programming Language C++Compiler gcc (6.2.1)Multithreading OpenMP

Bullion (8 sockets, 6TB of main memory)Platform Intel Xeon CPU E7-8890 v4 (2.20GHz)

24 cores/socketLibraries Thrift (serialise/deserialise)

Coco/R (AQASM compiler)

address the memory-bound nature of quantum mechanics simulation. The memory

specifications of the specific machine that is used for this project is listed in Table 5.2.

It is worth noting the drop in bandwidth (by more than an order of magnitude) when a

CPU wants to access memory from another node. This is one of the reasons why we

want to explore the path integral approach since it does not require access to the global

state vector during the computation. On the other hand, a linear algebra simulator, which

constructs a global state vector and distributes it among multi-node clusters, will need

node-to-node data exchange in order to perform the matrix-vector multiplication since

each node keeps a portion the state vector of the size (2m) locally and performs state

update on its part (which requires inter-process communication from time to time).

Table 5.2 Computer memory capacity and bandwidth specifications of the simulationhardware platform.

Memory per socket 768GBMemory per node 1.5TBNumber of nodes 4Memory bandwidth 58 GB/sInter-socket BW 30 GB/sInter-node BW 4.9 GB/s

5.2.2 Technical Implementation

In the Feynman path integral approach, the simulation of a quantum operation is divided

into individual paths whose results are then combined as shown in equation (5.1). The

term “path” here refers to the fact that we have an intermediate node (z) which can


take any random values. Each assigned value of z represents a path from x to y and

by summing the contributions of all possible paths, we can compute the propagation

amplitude from x to y through the quantum operation U .

⟨x|U |y⟩= ∑z⟨x|U1 |z⟩⟨z|U2 |y⟩ , U =U1U2 (5.1)

In the context of quantum computing simulation, Fig. 5.5 demonstrates a simple

example where we have a three-qubit register with the initial state of |010⟩. The paths

generated by applying quantum gates on the qubit register are explicitly visualised which

highlights the fact that in this path integral simulation, each path “walker” does not

need access to the global state vector. In an HPC multi-threading environment, each

computing thread can handle each path independently hence can be beneficial when we

have a massively parallel computer.

Fig. 5.5 Example of a path integral diagram on a 3-qubit register starting at the state

|010⟩ and applying the a sequence of two gates[

α γ

γ∗ β

]and

[a bb∗ c

]on the second

qubit.

In terms of resources, since each path computation is simply a traversal from a

classical input and classical output (n-bit string), the memory requirement scales linearly

with the number of qubits (O(n)). On the other hand, the upper-bound for run-time is

O(2nx2depth), where the depth is the number of quantum gates. Therefore, there are only

a few scenarios where path integral is the preferred method of simulation, e.g. classical


input state (initial state is classical), or fat-short quantum circuits (many qubits but only

a few gates), etc. More importantly, by analyzing the input circuit, we can devise the

optimal simulation strategy in terms of runtime or memory consumption.

At the high level, the simulation program is designed using the object-oriented model.

Some key objects classes and their relative relationship is depicted in Fig. 5.6.

Fig. 5.6 Object-oriented design: Quantum circuit simulation objects

The serialised byte-code from the front-end will be translated into the internal data

structures from which the simulator further derives an execution model in terms of paths.

Those paths are distributed among computing threads and recollected at the end to form

the final simulation result.

5.2.3 Benchmark Results

In order to validate the simulator, we performed some initial tests using basic quantum

circuits such as all Hadamard gates (on each qubit) and the Quantum Fourier Transform

(QFT). The main purpose of those tests is to confirm the threading scalability of the

path integral approach, i.e. we get a linear reduction in runtime when more computing

resources (CPU cores) are added. This is the main difference between the path integral

and the linear-algebra methods where the latter is often constrained by memory traffic.


First, we ran a scalability test in which we gradually increased the number of

computing threads available for the simulator. The data for a test suite consists of all

Hadamard gates on a 30-qubit register is shown in Fig. 5.7.

Fig. 5.7 Run-time vs. number of computing thread benchmark of the simulator runningall Hadamard gates on a 30-qubit register.

Fig. 5.7 clearly demonstrates the linear scaling that we expect from a path-integral

simulator, especially when the thread count is in the range from 16 to 256 threads.

Outside that range of core count, there are a few hardware-related effects kick in which

changes the expected scaling behaviours. In the low core count range (8 vs. 16), since the

number of cores per CPU socket is 24, running only a few cores, e.g. 8 cores, will allow

the CPU to throttle the clock frequency of those executing cores hence the difference in

runtime is minimal when we double the core count in this case. On the other hand, the

total number of physical hardware cores that are available is 384. Hence, using more

threads than that will not result in any significant performance improvement as we can

see in the tail-end of Fig. 5.7.

Next, we compare the ATOS path-integral simulator against the publicly available

Microsoft LIQUi |⟩ (Fig. 5.8) and a generic linear algebra simulator such as Intel qHiP-

STER (Fig. 5.9). In both cases, we use Quantum Fourier Transform as the test case.


Fig. 5.8 Run-time comparison between the ATOS simulator and LIQUi |⟩ running theQuantum Fourier Transform using the same number of threads (128). The initial state is|00...0⟩.

Fig. 5.9 Run-time comparison between the simulator and linear algebra (LA) basedsimulator. We implemented a generic LA quantum simulator using the same computingmodel as qHiPSTER so that we can run on the local Bullion server. The runtime data ofthe LA simulator matches the public data of qHiPSTER (after compensating the memorybandwidth differences) as shown by the red triangles in the cases of 29 and 30 qubits.The ATOS path-integral simulator provides faster time-to-result in this case as comparedto the LA-based simulator.

When comparing to Microsoft LIQUi |⟩, Fig. 5.8 shows a significant performance

gap in favour of the ATOS simulator. This can be explained partly because the ATOS

simulator is heavily optimised for this particular hardware configuration while the

LIQUi |⟩ simulator is designed mainly for general PC-based setups. Secondly, the path-


integral based simulator can take advantage of the fact that the input state is classical

hence eliminating a lot of configuration paths.

Fig. 5.9, on the other hand, shows a much closer gap between an optimised LA-based

simulator and a path integral based counterpart. The path-integral simulator is still faster

in this case stemming from the fact that the input state is singular. One thing that we

can observe in Fig. 5.9 is the almost doubling of run time for the path integral simulator

when a qubit is added. This is due to the fact that the number of paths which need to be

explored doubles.

Preliminary data from the ATOS quantum simulator has been promising thanks

to its innovative approach to maximising low-level parallelism to achieve speed-up

in various use cases. The performance advantage is also derived from software and

hardware co-design for the high-end Bull Bullion HPC platforms. A proposal to study

the feasibility of using FPGA as an accelerator for this type of simulation is summarized

in Appendix B. The simulator is further enhanced and maintained by the ATOS Quantum

Lab after the author finished the research internship there.


Chapter 6

Simulating Input-Output Quantum

Systems with LIQUi|⟩

Never try to solve all the problems at once — make them line up for you

one-by-one.

Richard Sloma

Quantum computation is usually described by unitary operations which is based

on closed quantum system dynamics. In contrast, open quantum dynamical systems

are often modeled by the master equation which takes into account the environmental

interaction. Using the complementary Heisenberg-picture approach, we can derive the

quantum stochastic differential equations that describe the dynamics of system operators

of an open quantum plant. This is commonly referred to as the quantum input-output

formalism, which underpins the burgeoning field of quantum feedback control.

The inclusion of dissipation processes, allows us to passively and autonomously

perform a wide variety of quantum operations, such as enhancing adiabatic quantum

computation, and state preparation/stabilization. For instance, Chapter 3 and 4 have

demonstrated the application of open quantum system models in developing protocols

to improve the stability of qubit systems. More importantly, fully coherent feedback

loops mediated by field coupling without measurement have been shown to have major

advantages such as high operating speed for quantum optics applications and the non-

demolition nature of coherent operations.

134 Chapter 6. Simulating Input-Output Quantum Systems with LIQUi|⟩

In this chapter, we describe the use of Microsoft Language-Integrated Quantum

Operations framework (LIQUi|⟩[194]) in simulating complex engineered open quantum

systems by discretizing the physical system composition, and introducing a dissipation

emulator gadget. The Hamiltonian simulator module of LIQUi|⟩ is used for time

evolution simulation of the Trotter decomposed unitary.

In particular, we will apply this method to compound gradient echo quantum memory

which is capable of efficiently capturing and storing a quantum state of light. In these

experiments, memory hardware consists of a continuous solid-state material (rare-earth-

ion-doped crystal) which will be represented in simulation by a network of qubits. This

not only provides analysis results for such a discrete quantum memory model, but can

also potentially be used as a blueprint for a quantum digital signal processor capable of

processing incoming continuous variable quantum information.

6.1 Background

As introduced in Chapter 1, quantum network theory has been developed based on

the quantum stochastic differential equation model. In particular, the modular and

system-theoretic SLH parametrization scheme [56] provides the mathematical frame-

work for quantum network description and analysis. The SLH-based Quantum Hardware

Description Language (QHDL)[168], which is illustrated in Figure 6.1, has been suc-

cessfully deployed to design a number of quantum optics devices and applications such

as quantum NAND logic gate and latch[107], optical Set-Reset flip-flop[105], and a

coherent error correction loop[91]. Backend differential equation solver based tools such

as QuTiP[85] and XMDS[31] are often used to simulate the system dynamics described

by its SLH parameters.

The Microsoft Language-Integrated Quantum Operations framework (LIQUi|⟩[194]),

which is based on quantum gate formalism, is not specifically designed for this type

of low level dynamical simulation. However, since LIQUi|⟩ is capable of faithfully

simulating a general purpose quantum computer, porting the dynamical SLH model onto

LIQUi|⟩ will pave the way for the implementation of discrete/digital quantum emulators

of open quantum systems.

6.1 Background 135

Functional Spec.(e.g. gate/latch/memory/control)

Quantum OpticsCircuit Editor

Component Netlist

QHDL Definition

DynamicalSimulation

ComponentLibrary

Symbolic AlgebraNumerical Simulation

Fig. 6.1 QHDL design flow which incorporates quantum optics schematic editor, netlist-ing tool, Hardware Description Language (HDL) parser, and a full quantum dynamicalsimulation backend.

One recent development of the SLH modeling approach is the effort to extend its

application to a wide variety of input states besides the conventional vacuum inputs,

such as thermal field, single-photon and two-photon states. Next, we will introduce the

SLH model for a system driven by a continuous-mode single-photon wave packet.

6.1.1 Single-photon State

Depending on its source, a photon can be single or multi-modal. In this work, continuous-

mode single photon state is concerned. We can decomposed such state in the time and

frequency domains as followings,

∣∣∣1ξ

⟩=∫ +∞

0ξ (t)dB†(t) |0⟩ ,

∣∣∣1ξ

⟩=∫ +∞

−∞

dωξ (ω)a†(ω) |0⟩ . (6.1)


Plant(SLH)

λ(t)σ−

Fig. 6.2 Single-photon generating filter cascading realization.

The function ξ (t) is the time-domain wave packet shape of the photon, which can be

interpreted as the probability amplitude of detecting the photon at time t. ξ (ω) is the

Fourier transform of ξ (t).

In the following simulation, we will consider incoming single-photon wave packets

which have Gaussian shape parametrized by:

ξ (t) =(

Ω2

2π

)1/4

exp[−Ω2

4(t − tc)

2], (6.2)

where tc specifies the peak arrival time and Ω is the frequency bandwidth of the pulse.

Using a pre-“signal-generating filter” which is driven by vacuum, we can derive

the SLH model for the augmented system. Equations (6.3) and (6.4) show the SLH

parameters of the pre-filter ancilla and the overall network, respectively.

Ganc = (I,λ (t)σ−,0), λ (t) =ξ (t)√∫

∞

t |ξ (s)|2ds, (6.3)

GT =(

S, L+λ (t)Sσ−, H +λ (t)Im(L†Sσ−)), (6.4)

in which Gs = (S,L,H) is the original system parameters.

The signal-generating filter (Ganc) is a vacuum-driven two-level atom initially pre-

pared in the excited state which will decay in time to the ground state via spontaneous

emission of a photon into the light field. That output field is then channeled to the system

of interest as shown in Figure 6.2.

6.1 Background 137

Fig. 6.3 Gradient settings for GEM operations

6.1.2 Compound Gradient Echo Memory

In the conventional quantum computation paradigm, there is little or no use of a “memory”

device since in most cases the problems are formulated as isolated computing tasks

with classical information inputs, e.g. factoring, optimization, etc. Nevertheless, if

networking between distant quantum computers are required, photons are the best

candidate for quantum information carriers. In that scenario, a quantum memory is

needed to capture and release photons on-demand.

One of the most promising schemes for quantum memory is the so-called Controlled

Reversible Inhomogeneous Broadening (CRIB). CRIB can be implemented on any

“controllable” absorbing physical system including NV centers, quantum dots, atomic

vapors and rare-earth-metal-ion-doped solids. Gradient echo memories (GEM) fall

under the CRIB category, in which an electric field induced Stark shift is deployed for

broadening. Two different linear gradients are applied for writing and reading phases

that causes the rephasing effect as shown in Figure 6.3.

While semi-classical approaches [41] can be used to simulate the memory operations,

no direct access to individual quantum state of each ion can be attained that way. On

the other hand, using the SLH formalism, Hush et al. [80] modeled the memory as a

series of optical cavities. Each cavity represents a collection of atomic absorbers at the

corresponding location in the memory, which all absorb light of a single frequency. The

number of excitations within a given cavity may be thought of as the number of excited

atoms in the ensemble at the corresponding location. In this framework, the SLH model


of a gradient echo memory is given by

GGEM = G1 ▷G2...▷Gn (6.5)

=

(SGEM = I, LGEM = ∑

k

√βak,

HGEM = ∑k

ξka†kak +

β

2i

N

∑j=2

j−1

∑k

(a†

jak −h.c))

.

The above model is a vacuum input model which has not taken into account the source

term. To simulate non-classical input such as the single photon state, the generating

filter approach mentioned previously becomes handy and can be apply directly to get the

complete model as followings,

GT = Ga ▷GGEM (6.6)

=

(I,LGEM +λ (t)σ−,HGEM +

12i

(λ∗(t)σ+LGEM −h.c

)),

in which λ (t) term dictates the photon wave packet shape as given by Eq. (6.3).

The cascading cavity approach is extremely accurate in the case that there are high

concentrations of light absorbers (ions). What we want to investigate is the performance

of such memory protocols in the discrete settings where only a countable number of

two-level qubits involved as absorbers. More importantly, the dynamics is simulated on

a discrete-time basis which is executable on a general-purpose quantum computer. The

analogy could be using a Analog-to-Digital converter (ADC) with limited resolution and

sampling rate to digitize continuous signals for further processing. Not only will this

provide new insights into such discrete memory operations, with a synthesis flow to a

target quantum hardware, we can implement such protocol on a real quantum computer.

The last piece of this simulation/emulation approach is a method of implementing

engineering dissipation which is given by the L operator in the SLH model. This will be

introduced in the next section.

6.1 Background 139

SystemEnv.

AncillaHL

√γσ−

HL = ω(σ−L† + σ+L)

EquivalentModel

L

Fig. 6.4 Engineered dissipation by direct coupling to an amplitude damping ancillarysystem.

6.1.3 Engineering Dissipation Control

At first glance, the input-output open quantum system approach can be easily imple-

mented on LIQUi|⟩ by a set of suitably designed Kraus super-operators. While being

ideal for simulation purposes, hardware synthesis to real physical systems will be

extremely challenging. An alternative approach could be implementing a universal

dissipative gadget which is just an amplitude-damped qubit as proposed by Verstraete

et al.[185] The damped ancillary qubit is directly coupled to the other qubits (such

as the memory qubits) to provide the desired dissipation. The engineering dissipation

configuration is illustrated in Figure 6.4.

The coupling Hamiltonian between the ancilla and the system has the form Hc =

ω(σ−L† +σ+L). In the limit that the damping rate, γ , of this ancilla is much greater

than the strength ω of the coupling Hamiltonian, one can adiabatically eliminate the

excited level of the ancilla, thus reconstructing the desired dissipation L for the system.

The upper bound for error of this adiabatic approximation is ∥ε∥∼ 1γ2 .

The coupling Hamiltonian can be easily transformed into a sequence of discrete

quantum gates executable on a quantum computer while there are many physical systems

that may act as a dissipative gadget, such as low-Q optical or superconducting cavities.

Therefore, a LIQUi|⟩ program written in this way is more likely to be ready for hardware

synthesis.


6.1.4 Trotter Decomposition

One key aspect of simulating quantum dynamics is the time evolution simulation, i.e.

applying exp(−iHt) to the quantum state. Unlike differential equation solvers which

directly solve for this time evolution, LIQUi|⟩ uses the quantum gate model where the

complex exponential term can be approximated by[195]:

exp(−iHt) =

(∏

kexp(−ihk∆t)

)t/∆t

+O(∆t), (6.7)

in which, the original Hamiltonian has the summation form H = ∑k hk and ∆t is the

Trotter step size.

The term exp(−ihk∆t) is in turn implemented by a sequence of quantum gates. This

is known as first-order Trotter decomposition which is often used in quantum chemistry

simulation[193].

For open quantum systems with engineered dissipation coupling, we can use the

“dissipative gadget” that was introduced in the previous section to adiabatically emulate

the environmental coupling. As shown in Eq. (6.6), the GEM Hamiltonian consists

of two classes of terms. The first one is the number operator term (a†a) which can

be implemented directly by a T (θ) gate. The coupling Hamiltonian Hc required for

engineering dissipation, Hc = ω(σ−L† +σ+L), has the form of the excitation operator

(hpq(a†paq +a†

qap)) which can be realized by the sequence of gates as shown in FIG. 5

(ii) in Wecker et al.[193]

Last but not least, the cascading term, i.e. β

2i ∑Nj=2 ∑

j−1k

(a†

jak −h.c)

, which is not

commonly seen in the quantum chemistry model, can also be decomposed in the same

manner, i.e.

a†jak −h.c. ≡ σ

( j)x − iσ ( j)

y

2σ(k)x + iσ (k)

y

2−h.c.

= iσ(k)y σ

( j)x −σ

(k)x σ

( j)y

2. (6.8)

The resulting sequence of quantum gates (rendered by LIQUi|⟩) which can be used to

implement complex exponential of the terms in Eq. (6.8) is shown in Figure 6.5 which

6.1 Background 141

H H Y Y †

Y θZ/2 Y † H −θZ/2 H

Fig. 6.5 Quantum circuit that represents the unitary generated by the cascading Hamilto-nian term.

involves change-of-basis gates (H and Y ) and the standard unitary exp[−i(θ/2)(σz⊗σz)].

The LIQUi|⟩ code snippet that is used to run this Trotter decomposed term is shown as

below.

//Trotterization of Cascading

Hamiltonian

let HamiltonianCasOp(theta:float, p:

int, q:int) (qs:Qubits) =

let a_qubit = qs.[p..]

let s_qubit = qs.[q..]

Ybasis s_qubit

H a_qubit

CNOT !! (qs,q,p)

Rz(-theta) 1.0 ’’’’ a_qubit

CNOT !! (qs,q,p)

YbasisAdj s_qubit

H a_qubit

H s_qubit

Ybasis a_qubit

CNOT !! (qs,q,p)

Rz(theta) 1.0 ’’’’ a_qubit

CNOT !! (qs,q,p)

H s_qubit

YbasisAdj a_qubit


Qubit 1Single-photonAncilla

Qubit N

Dissipation Ancilla

√γσ−

Fig. 6.6 Discretized GEM model driven by single-photon source used for LIQUi|⟩simulation.

6.2 Models and Results

6.2.1 Amplitude Damping Set-up

Using the dissipation gadget, the discretized gradient echo memory model used for

LIQUi|⟩ simulation is shown in Figure 6.6 below. The dissipation ancilla at the bottom

is a qubit which is subjected to general amplitude damping. This is done by running idle

gate at each Trotter step with the error probability of (1− exp(−γ∆t)). A LIQUi|⟩ code

snapshot for the amplitude damping set-up is also shown in Figure 6.7.

The LIQUi|⟩ simulation is performed on a Windows 7 desktop machine using Intel

Core i7-4770 3.4 GHz (Haswell) with 16 GB of main memory. In the case of running

4-qubit gradient echo memory model (6 qubits in total), the runtime for 1,000 trajectories

each with 10,000 Trotter steps is 3 hours.

6.2.2 Toy Example: Single-Atom Memory

Before delving into the discretized GEM simulation, we will test the LIQUi|⟩ set-up

in a much simpler situation, namely a single-atom memory. Here we consider a single

two-level atom driven by a Gaussian continuous-mode single photon as given by Eq.

(6.2). This problem has been investigated thoroughly by both conventional and quantum

network approaches.

The coupling operator of the atom is L =√

Γσ−. The atom is assumed to be free of

internal Hamiltonian dynamics and field scattering for simplicity, i.e., H = 0 and S = I .

Here Γ > 0 is the normalized coupling rate (Γ = 1).


//Probability of amplitude dampinglet probDamp_E = 1.0 - Math.Exp(-

gamma*dt)let noise = Noise(circ,ket,models)ket.TraceRun <- 0noise.LogGates <- falsenoise.TraceWrap <- falsenoise.TraceNoise <- falsenoise.DampProb(0) <- 0.0 //single-

photon ancillanoise.DampProb(1) <- 0.0 //qubit 1noise.DampProb(2) <- 0.0 //qubit 2...noise.DampProb(Nq+1) <- probDamp_E

//dissipation ancilla

Fig. 6.7 LIQUi|⟩ code snapshot for the amplitude damping set-up.

By using Trotter decomposition technique and LIQUi|⟩, we were able to simulate

the dynamics of this open quantum system. In our simulation, dissipation is induced by

an ancilla which undergoes amplitude damping. In Figure 6.8, we plot the “detection

time” histogram and the excited-state population for the case of Ω = 1.5Γ, which is

known to be the optimal Gaussian wave packet for atomic excitation. The detection time

is triggered by the state-collapse event of amplitude damping noise and histogram is

plotted from 10,000 LIQUi|⟩ runs.

This is consistent with theoretical analysis which shows that that the maximum

excitation probability of an atom driven by a Gaussian-shaped single photon is about

80%[189]. Since we are essentially running a stochastic simulation on the input-output

model, we have the benefit of being able to simulate the conditional dynamics and output

observable statistics (photon detection in this case) if we choose to observe the output.

By sweeping the Gaussian bandwidth (larger Ω means broader spectrum in frequency

domain and vice versa), we can get the maximum excitation (absorption) rate of our

single atom model, as shown in Figure 6.9 (black curve). There is a very narrow range

of bandwidth where the absorption peaks. The optical “depth” created by cascading


0 5 10 15 20

Time

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Fre

qu

ency

Single-photon pulse shape

0 5 10 15 20

Time

0.0

0.2

0.4

0.6

0.8

1.0

PE

Fig. 6.8 Histogram of photon detection events from a two-level atom driven by aGaussian-shape single photon state. Inset: the excited state population of the atom. Thephoton wave-packet is included for reference (red curve in main and dotted curve ininset). Trotter step is chosen to be 2π×10−4, and the number of trajectories (realizations)is 10,000.

absorbers of gradient echo memory can enhance the photon absorption as shown in the

below section.

6.2.3 Discretized Gradient Echo Memory

Using the full GEM model with signal generating filter as in Eq. (6.7) and Figure 6.6, we

can investigate the performance of the discrete memory in terms of photon absorption

(efficiency). Due to computing resource and runtime constraints, only models of two-

and four-atom GEM are considered in this report (4 and 6 qubits in total, respectively).

Despite minimal system size, the performance enhancement is significant as shown in

Figure 6.9.

Almost perfect absorption (100% excitation) can be achieved for some Gaussian

wave packet shapes by using just 4 atoms in series. More importantly, a monotonic

improvement over the full range of bandwidth is achieved by introducing more qubit

resources. Notice that in the case of the multi-atom memory, the single excitation from


0 2 4 6 8 10

Ω/Γ

0.0

0.2

0.4

0.6

0.8

1.0

Pmax

E

Single Ion

Two-ion GEM

Four-ion GEM

Fig. 6.9 Single photon absorption probability as a function of photon bandwidth. Thebaseline (black) is the single atom model which has the max absorption of 80% for avery narrow range of bandwidth. The more atoms involve, the better absorption and alsowider range of bandwidth can be achieved. Each data point is a LIQUi|⟩ run of 1,000trajectories. Solid line is polynomial fit.

the photon is distributed among all memory qubits in a Fourier-transformed (frequency)

basis. Therefore, the choice of gradient is of importance and can be optimized for a given

target input wave packet. Since we use a linear symmetric gradient and the number of

atoms are even (thus missing a zero point), the absorption rate has bi-modal behavior, as

is clearly seen in the four-atom case. If, as for continuous solid-state GEM, the gradient

is a smooth continuous function, the memory can effectively capture the full Fourier

spectrum of incoming photons.

Besides the fact that this discrete memory model can be implemented on a general

purpose quantum computer, one major advantage is that we can potentially extend the

range of gradient. Inhomogeneous broadening (by the electric field induced Stark shift,

for instance) has difficulty in achieving the bandwidth required for telecommunication

wavelength photon absorption. If implemented on a digital quantum computer, the

gradient term is controlled by the Trotter decomposed unitary thus can be changed freely

provided that the time step can be reduced for sufficient error bound.


This discretized memory model has close analogy to classical digital signal pro-

cessing technology. The gradient quantum memory program running on the a quantum

computer is equivalent to an Analog-to-Digital Converter (ADC), where (i) the Trotter

step serves as the sampling rate; (ii) the resolution is dictated by the number of qubits;

and (iii) the sampling method is frequency gradient mapping (classical ADC’s have a

wide variety of sampling methods such as integrating and sigma-delta.)

6.3 Conclusions and Future Work

6.3.1 Summary

In this chapter, we have adopted LIQUi|⟩ to simulate quantum open systems using the

input-output formalism. A combination of theoretical (Trotter decomposition, dissipation

gadgets) and practical (Hamiltonian simulator and the noise models available in LIQUi|⟩)solutions can provide accurate and consistent simulation results with other tool suites as

demonstrated in the “single atom driven by single photon” case.

Simulation-wise, the use of LIQUi|⟩ for this sort of numerical simulation struggles

to compete with more specialized direct differential solver based tools such as XMDS

or QuTiP due to (i) extra efforts needed to perform the Trotter decomposition and set-

up the dissipation ancilla; (ii) simulation errors associated with these approximations

(could be trade-off by additional runtime). However, if we look at the grand scheme of

quantum computation with reference to the classical computing paradigm, this approach

is essentially bridging the analog and digital worlds of the quantum framework. The

fact that LIQUi|⟩ is a quantum simulator which guarantees a direct “plug-and-play” on a

future digital quantum computer. Being able to simulate this system on LIQUi|⟩ shows

that we can readily emulate this sort of quantum dynamics on a general purpose quantum

computer. This is what we have shown in the discrete quantum memory example where

a small collection of qubits running synchronously to perform photon quantum state

storage. An initial dry run of this hypothetical memory demonstrates great improvement

in terms of the absorption rate even with an unoptimized gradient scheme.

Using this discretized method to demonstrate open quantum system approach, we

can get new insights about hardware requirements and potential errors associated with

emulating continuous “analog” quantum functions and inputs. This is somewhat missing

6.3 Conclusions and Future Work 147

in the current quantum computing picture where computing problems are often formu-

lated as an algorithm started on a given initial state without the need for external quantum

inputs. The SLH formalism provides a modular and systematic approach toward this

modeling problem thus worthwhile integrating to the LIQUi|⟩ framework, especially

when quantum hardware synthesis is taken into account since the SLH framework has

native models for a lot of quantum optics, opto-mechanical and cavity QED devices.

6.3.2 The Path Forward

In the discretized GEM simulation, we have introduced an analogy between a GEM

with a finite number of qubits and a classical A/D converter. There are some immediate

considerations that warrant further investigation such as error bound estimation regarding

the system size and Trotter step size; whether aliasing effect exists in this context; some

optimization techniques that we can do to improve the performance.

As this approach can be used to treat a large variety of quantum input-output dynam-

ical models besides the GEM, some future research avenues which can be considered

are listed below.

• Non-Markovian quantum noise model: Recent work[201] has shown that the

amplitude damping dissipation ancilla, when treated non-adiabatically, is equiva-

lent to a non-Markovian (colored) Lorentzian noise source. By combining multiple

ancillas, we can construct an arbitrary noise spectrum. We can thus use this model

to evaluate the performance of a quantum computer under influence of colored

noise, especially regarding the performance of quantum error correction codes.

The advantage of this dynamical approach is that an experimentally measured

noise spectrum can be fed directly into the model by adjusting the parameters

of the dissipation ancillas. For instance, quantum dot - cavity QED is a good

candidate, since the noise in this platform is intrinsically colored.

• Coherent digital quantum controller: Another direction is to look at the use of

a quantum computer in quantum control settings which usually involve continuous

input-output models. Especially in the case of coherent feedback control whereby

control objectives are achieved by field coupling quantum systems together. This

removes the measurement step and can perform control actions at the intrinsic


system speed (e.g. for quantum optics systems). A coherent digital controller can

perform equally well in this setting if equipped with appropriate quantum coherent

A/D and D/A converters at its input and output ports. By allowing quantum

programing, a coherent digital quantum controller can potentially achieve much

better performance than just wiring up quantum components (e.g. cavities, beam-

splitters, Kerr non-linear media.)[104]

• Quantum synthesis: As LIQUi|⟩ is part of a larger effort to create a full quantum

design flow from software to target hardware implementation, it is worthwhile

considering the integration of the SLH network model at the low-level physical

description. It is one of the most accurate and systematic dynamical descriptions

for quantum optics devices and circuits.

In comparison, a classical VLSI design flow always has a so-called “parasitic

extraction and analysis” step after “place and route” during which the full RLC

model of the entire circuit plus routing is compiled and simulated. Besides

simulations at the logic and transistor levels, it is deemed necessary to perform such

low level verification especially for complex and high speed designs. The approach

taken by classical integrated circuit design is by no mean the absolute solution

for quantum computer design. However, by incorporating this “analog” model

of quantum devices, we can definitely improve the versatility of the synthesis

process especially in cases where continuous time is the most natural description

(e.g. continuous variable quantum computation).

Given the “Big Hairy Audacious Goal” of creating a functional quantum computer in

the next one or two decades, a solid foundation for quantum system design is imperative.

And LIQUi|⟩ created by Microsoft is the first bold step in that direction. Augmenting

the quantum gate model of LIQUi|⟩ with the SLH dynamical model either by emulating

or intrinsic integration will open up new applications as well as cater to a larger research

community.

Chapter 7

Concluding Remarks

Any sufficiently advanced technology is indistinguishable from magic

Arthur C. Clark

Thanks to the first-wave of quantum innovation in the 20th century, we have achieved

unimaginable technologies, especially in the electronics industry. The second wave of

quantum revolution that we are witnessing is set to experience exponential growth in

a foreseeable future. Building a functional quantum computer which can outperform

any of our current computing devices and systems is one of the grand challenges of the

quantum technology sector.

Hence, the primary goal of this thesis is to study the scalability and engineering

aspects of quantum computing architectures. That includes the estimation of the overhead

incurred in quantum error correction, the potential use of advanced quantum control

techniques in migrating quantum errors as well as enhancing existing quantum error

correction protocols, and the development of quantum software simulation tools to aid

early quantum programmers.

7.1 Where We Stand

• In Chapter 2, we applied the classical electronics know-how regarding interconnect

routing to provide a concrete procedure for fan-out scalability analysis. The

150 Chapter 7. Concluding Remarks

results not only indicate the potential scalability bottleneck but, more importantly,

highlight the need for innovative solutions w.r.t. the scale-up of quantum circuits.

• In Chapter 3, we derived a continuous measurement feedback scheme that can be

used for topological quantum error codes. Quantum control modelling techniques,

namely the Quantum Stochastic Differential Equation (QSDE) and the (S, L,

H) model are demonstrated in designing the controller to tackle the problem of

correcting surface code qubit lattice automatically.

• In Chapter 4, we take a step back and look at the problem of quantum decoherence

at a fundamental/theoretical level. In particular, we demonstrated a pathway to

the synthesis of the so-called decoherence-free subsystem (DFS) via quantum

coherent feedback. Despite not being universal, the technique is nonetheless

applicable to a wide variety of passive quantum systems. Furthermore, we adopted

the Lyapunov formalism to the design and synthesis of passive quantum error

correction.

• In Chapter 5, we summarised the quantum programming suite which includes the

definition of a gate-based quantum assembly language, the design of a quantum

programming work-flow, and most importantly the implementation of a high-

performance quantum simulator. The framework that we derived bridges the gap

between computer science and quantum physics hence assists the development

and testing of quantum algorithms and application.

• In Chapter 6, we demonstrated that a digital quantum simulator, such as LIQUi|⟩is capable of simulating quantum open systems using the input-output formalism.

This bridges the gap between gate-based quantum computation and open quantum

system approaches.

7.2 The Way Forward

Quantum computing architecture is a rapidly-evolving field which takes advantages of

both the academic and industry research. For example, the surface-code quantum error

correction, which is currently state-of-the-art, could be surpassed by other codes. Also,

all the assumptions about the qubit devices in terms of geometry, operating conditions,

7.2 The Way Forward 151

and characteristics could dramatically change in the near future given the substantial

investment of technology companies around the world.

Hence, the study of quantum computing architecture needs to keep up with the rate of

development in the field. In particular, the fan-out scalability study in Chapter 2 has laid

the foundation for other types of scalability analysis where we could predict in advance

potential scaling issues and hence adjust the design to circumvent such limitation.

Control and coherence of large quantum systems remain one of the most challenging

aspects of quantum technologies. Quantum control engineering is still an emerging field,

yet very promising. Indeed, we can translate many quantum computing problems into

control problems as what we demonstrated in this thesis with the quantum error cor-

rection. Quantum control could provide a scalable approach, given its autonomous and

passive nature, toward the construction of large quantum networks which is crucial for

quantum computation. The problem of quantum decoherence suppression or elimination

is particularly well-suited to quantum control engineering.

Last but not least, the software stack will play a vital role in the roll out of any future

quantum computing platforms. At this stage, the lack of real quantum hardware is the

limiting factor in quantum application development. However, as the field progresses, the

need for an end-to-end quantum software development environment will become more

prevalent. This area, in particular, requires a strong collaboration between academic and

industry. The ultimate goal is to provide quantum software engineers with the tools they

need to explore and fully-utilise the capability of quantum computers.


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Appendix A

Mathematical Proofs

A.1 General form of the QSDE for continuous error correction in Chap-

ter 3

Conditional master equation for continuous error correction monitor:

dρc =γ

3

n

∑m=1

∑j

D [σ(m)j ]ρcdt +κ

k

∑i=1

D [gi]ρcdt +√

κ

k

∑i=1

H [gi]ρcdWi

In which,

• σ(m)j represent Pauli operators (error channels), i.e. σ

(1)x = X1, etc.

• gi is set of stabilizer operators: for our distance-2 surface code, they are

Z1Z2Z3Z4, X1X2, X1X3, X2X4, and X3X4

We want to derive a set of stochastic differential equations for π(gi) = gi, which is

the expectation of stabilizer operator (syndrome value)

gi = π(gi) = Tr[giρc];dgi = Tr[gidρc]

dgl =γ

3

n

∑m=1

∑j

Tr[glD [σ(m)j ]ρc]dt +κ

k

∑i=1

Tr[glD [gi]ρc]dt +√

κ

k

∑i=1

Tr[glH [gi]ρc]dWi

170 Chapter A. Mathematical Proofs

• First term

Tr[glD [σ(m)j ]ρc] = Tr[gl(σ

(m)j ρcσ

(m)j −ρc)]

= Tr[glσ(m)j ρcσ

(m)j ]−Tr[glρc]

= Tr[σ (m)j glσ

(m)j ρc]−Tr[glρc]

=

0 if gl(m) ∈ I,σ j

−2gl if gl(m) /∈ I,σ j

The gl(m) in the last equation denotes the position m Pauli operator in the stabilizer

gl . In deed, if the stabilizer gl contains σ(m)j or has an identity operator at that

position, σ(m)j glσ

(m)j = gl . Otherwise, σ

(m)j glσ

(m)j =−gl , eg. X1(Z1Z2Z3Z4)X1 =

−Z1Z2Z3Z4 or Y1(X1X2)Y1 =−X1X2

For a depolarizing channel, we can generalize the results as following:

γ

3

n

∑m=1

∑j

Tr[glD [σ(m)j ]ρc]dt =−(4w)γ

3gl,w = Pauli weight of gl

• Second term

κ

k

∑i=1

Tr[glD [gi]ρc] = κ

k

∑i=1

(Tr[gl(giρcgi)]−Tr[glρc])

= κ

k

∑i=1

(Tr[(giglgi)ρc]−Tr[glρc]) = 0

This is the basis property of stabilizer operators: commuting and unitary.

• Last term

A.1 General form of the QSDE for continuous error correction in Chapter 3 171

√κ

k

∑i=1

Tr[glH [gi]ρc]dWi =√

κ

k

∑i=1

Tr[gl(giρc +ρcgi −ρcTr[giρc +ρcgi])]dWi

=√

κ

k

∑i=1

Tr[gl(giρc +ρcgi)]−Tr[glρc]Tr[giρc +ρcgi]dWi

= 2√

κ

k

∑i=1

Tr[glgiρc]−Tr[glρc]Tr[giρc]dWi

= 2√

κ

k

∑i=1

glgi − gl gidWi

Notice in this equation that:

– The product glgi can become a new operator outside the set of initial stabilizer

set. However, stabilizer code properties tell us that glgi is also a stabilizer of

the code and they will be closed.

– The resulting QSDE is non-linear as it involves second-order terms like gl gi

In conclusion, we have derive the set of QSDE’s tracking the parity syndrome of the

surface code which have the following form:

dgl =−(4w)γ3

gl +2√

κ

k

∑i=1

glgi − gl gidWi

with w is the weight if gl and l is the index of all members of the set generated by giki=1

For example, the distance-2 surface code has 5 stabilizer operators: Z1Z2Z3Z4,

X1X2, X1X3, X2X4, and X3X4. With that, we can form a close set of all stabilizer opera-

tors: Z1Z2Z3Z4, X1X2, X1X3, X2X4, X3X4, X2X3, X1X4, X1X2X3X4, Y1Y2Z3Z4, Y1Z2Y3Z4,

Z1Y2Z3Y4, Z1Z2Y3Y4, Z1Y2Y3Z4, Y1Z2Z3Y4, Y1Y2Y3Y4.

There are a total number of 15 equations, which is a great reduction from propagating

the full density matrix having 4n elements (256 for the distance-2 four qubit code).

However, there is also a caveat associated with this QSDE approach. We have not

yet included the feedback in our conditional master equation. Adding feedback will then

break the closeness of our stabilizer QSDE’s.


Let’s consider a specific type of feedback which is Hamiltonian (unitary) feedback,

i.e. adding the commutation term −i[F,ρc]dt into the conditional master equation, where

F is the feedback Hamiltonian.

Using a similar approach, we will have an additional term: Tr(−igl[F,ρc]) which

can be rewritten as Tr(−i[gl,F ]ρc) using the cyclic property of the trace. At this point,

we can see that additional operators will appear namely the commutator [gl,F ]. If we

choose F = σ(m)j to correct Pauli errors, the commutator will certainly be outside the

stabilizer set. Thus, a reduced QSDE model as before cannot be realized. Approximation

methods are required to reduce the dimension of the syndrome filter in order to tackle

real surface codes as they contain a big number of qubits.

A.2 Proof of the scalability condition (4.44)-(4.45)

Proof : By assumption, we have Ui satisfying Eq. (4.30). Let Li =UiVi ∏d V (i)d and we

have

G (Vi)Li= Vi ∏

dV (i)

d U†i ViUiVi ∏

dV (i)

d −Vi ∏d

V (i)d

≤ (1− ci −1)Vi ∏d

V (i)d =−ciVi ∏

dV (i)

d , ci > 0.

(A.1)

For j = i, either [U j,Vi] = 0 which results in

G (Vi)L j= Vj ∏

dV ( j)

d U†j ViU jVj ∏

dV ( j)

d −ViVj ∏d

V ( j)d

= 0, (A.2)

or U j satisfies (4.44) as

ViU†j ViU jVi ≤Vi. (A.3)

A.3 Proof of the condition (4.52)-(4.53) 173

Note that Eq. (A.3) implies Vi ∈ V ( j)d according to the assumption. Based on this fact,

we obtain

G (Vi)L j

= Vj ∏d

V ( j)d U†

j ViU jVj ∏d

V ( j)d −ViVj ∏

dV ( j)

d

= Vj ∏d

V ( j)d ViU

†j ViU jViVj ∏

dV ( j)

d −ViVj ∏d

V ( j)d

≤ 0, (A.4)

which further leads to

G (∑i

Vi) ≤ −∑i

ciVi ∏d

V (i)d ≤−cmin ∑

iVi ∏

dV (i)

d

≤ −cminλ ∑i

Vi. (A.5)

cmin is the smallest number in ci. Here we have used the condition (4.45).

A.3 Proof of the condition (4.52)-(4.53)

Proof. The Liouville-space representation of the master equation (4.33) is

vec(ρt) = Avec(ρt), (A.6)

whose solution is

vec(ρt) = eAtvec(ρ0), (A.7)

where vec(ρt) is a column vector obtained by piling up the column vectors of ρt , and A

is matrix determined by H,Lk.

The condition (4.52) ensures the invariance of the state if no error occurs in the first

step. If the error Ea occurs, the erroneous state Ea|p⟩⟨q|E†a needs to be steered back to

|p⟩⟨q|. Since we have

vec((Ea|p⟩⟨q|E†a)t) = eAtvec(Ea|p⟩⟨q|E†

a), (A.8)


and so

˙vec((Ea|p⟩⟨q|E†a)t)

= eAtAvec(Ea|p⟩⟨q|E†a)

= eAt [−κpqvec(Ea|p⟩⟨q|E†a)+κpqvec(|p⟩⟨q|)]

= −κpqeAtvec(Ea|p⟩⟨q|E†a)+κpqvec(|p⟩⟨q|)

= −κpqvec((Ea|p⟩⟨q|E†a)t)+κpqvec(|p⟩⟨q|). (A.9)

Eq. (A.9) is an ordinary first-order differential equation which can be easily integrated

to be

vec((Ea|p⟩⟨q|E†a)t)

= e−κpqtvec(Ea|p⟩⟨q|E†a)+κpqvec(|p⟩⟨q|)

∫ t

0e−κpq(t−r)dr

= e−κpqtvec(Ea|p⟩⟨q|E†a)+ vec(|p⟩⟨q|)[1− e−κpqt ]. (A.10)

This proves (Ea|p⟩⟨q|E†a)t → |p⟩⟨q| as t → ∞. Due to the linearity of the Liouville

equation, finally we have (∑p,q αpα∗q Ea|p⟩⟨q|E†

a)t → ρ0.

Appendix B

Feynman Path Integral Simulation

With FPGA

B.1 Method Overview

The Feynman path integral method can be used to calculate the propagation amplitude

from the input state (classical) to a particular classical output state. To illustrate this,

considering a quantum algorithm f operating on N qubits:

f : |i1i2...iN⟩ 7→2N−1

∑k=0

αk |k⟩ , (B.1)

In the Feynman path integral framework, we theoretically need to compute the

coefficients αk’s iteratively for all possible outcomes (2N). Then, this is completely

equivalent to the linear algebra approach which represents the output as a tall column

vector of length 2N . An unique advantage of the path integral method is its memory

efficiency and distributed-computing compatibility. Firstly, since we do not store the

whole state vector at any given time, we do not have the memory “hard” limit (space-time

trade-off). Secondly, path calculation is an independent workload, thus can be distributed

to a large number of processing units without worrying about data dependency. For

example, each unit compute the αk amplitude for a specific value of k.

176 Chapter B. Feynman Path Integral Simulation With FPGA

Let’s look at the method in more details. Consider a quantum circuit with d layers

of quantum gate and N qubits. Assuming only 1- and 2-qubit gates are used (satisfying

quantum computing universality).

G1

G2

G3

G4

Gd

The input is a classical state (|x⟩) that we load to the qubits. We calculate the

amplitude αk to get a readout of |y⟩= |k⟩, k ∈ [0,2N −1] as

⟨x| f |y⟩= ∑internal nodes

⟨x1|G1 |a1⟩⟨x2x3|G2 |b1c1⟩ ..Gd |y4y5⟩ , (B.2)

in which we used the notations a to h to represent the 8 qubit lines and the subscript

numbers to denote the sequence of gates on that line. We shall call a connection inside the

circuit between gates as “wire”. The idea is to sum over all possible binary combinations

of internal wires.

In order to explain this concept better, let’s look at a simple example which is a

3-qubit quantum Fourier transform (QFT) circuit.

We have labeled all the internal wires which need to be summed over. Since we have 4

internal wires, each path calculation will require a sum of 24 = 16 terms. Each term is

computed from a product of 6 complex numbers (equal to the number of gate layers).

B.2 FPGA Implementation 177

And to get the complete final state vector, we need to repeat this task 23 = 8 times to

cover all possible outcomes of |y1y2y3⟩. Explicitly, we have:

⟨x1x2x3|QFT3 |y1y2y3⟩ = ∑c1c2b1c3

(⟨x1|H |c1⟩⟨x2c1|CR(π/2) |x2c2⟩⟨x2|H |b1⟩ (B.3)

× ⟨x3c2|CR(π/4) |x3c3⟩⟨x3b1|CR(π/2) |x3y2⟩⟨x3|H |y1⟩).

For each layer of gate, we only deal with the local gate which is applied at that layer.

The operands for the multiplier are elements of the corresponding gate matrix indexed

by the current classical bra and ket states.

B.2 FPGA Implementation

As shown in the previous example, the calculation of Feynman path integral can be

efficiently simulated on the FPGA by:

1. Streaming the complex matrix elements, i.e. ⟨a1b1|G |a2b2⟩, into the FPGA. The

host CPU will serve as a parser for the input quantum circuit and dispatch the

coefficients onto a continuous stream sending to the FPGA.

// Define complex number operation functions

//multiplyfloat2 comp_mult(float2 a, float2 b)

float2 res;res.x = a.x * b.x - a.y * b.y;res.y = a.x * b.y + a.y * b.x;return res;

//addfloat2 comp_add(float2 a, float2 b)

float2 res;res.x = a.x + b.x;res.y = a.y + b.y;return res;

Fig. B.1 OpenCL implementation of complex arithmetic.


2. Implement a pipeline of sum operations with enough depth to hide the latency in-

volved in the data loading and multiplication (multiple copies of the multiplication

unit). The high-level structure can be seen as following:

+ + + +

m-way

Complex

Multiplier

m-way

Complex

Multiplier

m-way

Complex

Multiplier

m-way

Complex

Multiplier

RESULT STORE

LOAD LOAD LOAD LOADCOMPUTE UNIT

3. Multiple compute units can be mapped onto the FPGA so long as the hardware

complexity is permitted. Many path can be calculated independently on the same

FPGA or even across multiple FPGA’s. For a very large quantum circuit, we can

choose between fast single amplitude calculation (latency optimized) or parallel

multiple amplitudes (throughput optimized). Data access can be organized in

SIMD fashion for more efficient bandwidth.

4. Perform quantum circuit optimization before the simulation, e.g. combining gates

together for less memory transfer between FPGA and CPU, reducing internal

nodes, etc. Innovative data structure for gate coefficient storage to quickly identify

multiplying sequences which contain zero element(s), thus don’t send to the

FPGA.


B.3 Software Components

B.3.1 Host Program

Task: Manage the overall execution of the simulator, including: reading the input

quantum gate sequence and configurations; assembling the required data to send to the

FPGA; collecting the results returned by all the compute units of the FPGA; performing

any circuit optimization if required.

Components:

• Quantum circuit parser: parsing the quantum assembly language. For example,

the sequence for the above 3-qubit QFT circuit is shown below.

num_operations: 6

// Quantum Fourier Transform on 3-qubit register x

H x[1]

cR_Piover2 x[2], x[1]

H x[2]



H x[3]

• FPGA control and management for initialization, send and receive data, etc. This

part is closely related to the kernel implementation, e.g. the assembled data stream

needs to match the work-group size of the kernel and the returned amplitude data

needs to be mapped to the corresponding state, etc.

• Quantum circuit optimization: reorganizing and combining gates to reduce internal

nodes; eliminating zero or insignificant paths; optimizing data transfer between

host and kernel; etc.

B.3.2 FPGA Kernel

Task: Fetching data in main memory sent by the host, computing the complex mul-

tiplication, adding the amplitudes of multiple paths. Depending on the problem size

(number of gates/circuit depth), multiple compute units can be instantiated in one FPGA

for computing multiple output states or segmenting the sum into chunks to optimize for

low latency in case only one amplitude is concerned.


// Define the work group size__attribute__((reqd_work_group_size(NUM_WORKGROUPS, 1, 1)))

// Multiplication kernel__kernel void Feynman_Mult (global const float2 * restrict src)

//get global IDuint gid = get_global_id(0);

//temp var for running productfloat2 running_prod;running_prod.x = 1.0f;running_prod.y = 0.0f;

// Compute the multiplication of corresponding elements// in the data stream.#pragma unrollfor (unsigned a = 0; a < MUL_POINTS; a ++)

running_prod =comp_mult(running_prod, src[gid*MUL_POINTS + a]);

// Write the result to the FIFO channel// for the next kernel (sum) to processwrite_channel_altera (chanin, running_prod);

// Summation kernel__kernel void Feynman_Sum(global float2 * restrict result)

//temp var for running sumfloat2 running_sum;running_sum.x = 0.0f;running_sum.y = 0.0f;

float2 tempVar;for (unsigned i = 0; i < NUM_WORKGROUPS; i++)

// read data from the channeltempVar = read_channel_altera(chanin);

// cumulative sumrunning_sum = comp_add(running_sum, tempVar);

//get global IDuint gid = get_global_id(0);//send back the sum result with respect to the assigned IDresult[gid] = running_sum;

Fig. B.2 OpenCL implementation of multiplication and summation kernels.


Internally, the compute unit has a couple of kernels passing data through FPGA FIFO

channel as shown in the figure below. This will be more efficient and flexible in terms of

memory access (potential bottleneck). Data input and output to and from the kernel are

fetched and assembled before accessing the memory.

Components:

• Fetcher: fetch data stream from memory. Optimize for efficient data access.

• Multiplier: need to be sufficiently wide and balance for fast execution.

• Sum: deeply pipelined to mask the latency of the fetcher and multiplier. For real

circuits with a sufficient number of internal nodes (many terms in the sum), this

strategy is feasible since after the initial delay to do the first fetch and multiplica-

tion, sum operation can be done in every cycle.


Index

automatic quantum error correction, 78

Bloch sphere, 5

coherence time, 11

coherent feedback, 25

decoherence-free subsystem, 69

dissipatively quasi-locally stabilizable, 77

DiVincenzo’s criteria, 3

donor-based qubits, 6, 30

entanglement, 1, 2

fault-tolerance threshold, 12

Fock space, 73

Hilbert space, 2

Lindblad superoperator, 74

logical qubits, 12

master equation, 18

measurement feedback, 24

open quantum system, 73

open quantum systems, 18

quantum assembly language, 121

quantum circuits, 6

quantum computer architecture, 30

quantum computer micro-architectures, 30

quantum computers, 3

quantum computing, 2

quantum dissipation control, 69

quantum dots, 30

quantum error correction, 11, 12

quantum feedback control, 24

quantum filtering, 19

quantum gates, 6

quantum input-output model, 20

quantum noise, 19

quantum programming language, 16

quantum stochastic differential equation,

20

quantum Wiener process, 73

qubit, 3

spin qubits, 30

superposition, 1, 2

surface code, 12

tensor product, 2, 5

universal quantum gates, 9

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