Spin-Based Quantum Computing with Silicon MOS Quantum Dots by Michael Allan Fogarty

A thesis in fulfilment of the requirements for the degree of Doctor of Philosophy

School of Electrical Engineering and Telecommunications Faculty of Engineering

November 2018 PLEASE TYPE THE UNIVERSITY OF NEW SOUTH WALES Thesis/Dissertation Sheet

Surname or Family name: Fogarty

First name: Michael Other name/s: Allan

Abbreviation for degree as given in the University calendar: PhD

School: Electrical Engineering and Telecommunications Faculty: Engineering

Title: Spin-Based Quantum Computing with Silicon-MOS Quantum Dots

Abstract 350 words maximum: (PLEASE TYPE)

This thesis describes advancements in the silicon metal-oxide-semiconductor (SiMOS) quantum dot qubit platform. Recent experimental realisations of coherent single qubit and two qubit operations calls for the demonstration of a fully integrated SiMOS platform, showing a scalable implementation of all necessary elements for quantum computation operating together. Integration of qubit initialisation, control and readout via methods considered robust against scaling qubit array dimensions is the first fundamental step towards realising fully error corrected qubit registers useful for quantum computation.

This thesis commences by analysing the result of standard experiments for benchmarking performance and integrity of a singular SiMOS qubit. This experiment, known as Randomised Benchmarking, yields uncharacteristic results for the SiMOS platform when compared to experiment theory and other results in literature. This is addressed through extending the experiment to take low-frequency environmental fluctuations into account. Resulting analysis indicates that fidelities for the majority of qubit operation is considered above the threshold for fault-tolerant quantum computation.

Having demonstrated high quality qubit control using SiMOS quantum dots, the capability of preparing and measuring qubits via methods robust against scaling are detailed. Recent studies have detailed how the singlet-triplet basis can be utilised for the initialisation and readout of qubits within a scaled quantum register. These techniques are discussed and experimentally demonstrated in a SiMOS quantum dot device alongside single qubit addressability through electron spin resonance. Electrostatic control over the Heisenburg exchange coupling between two adjacent dots produces a two­ qubit SWAP operation realised in this device. These results together demonstrate, for the first time within a single silicon qubit device, the initialisation and readout of qubit pairs by scalable methods integrated with single qubit addressability and two-qubit logical operations.

Furthermore, having experimentally realised integration of fundamental building blocks of a scaled quantum register, what follows is a discussion of how this platform can be scaled into demonstrations of a fully error corrected logical qubit. Experimental state-of-the-art in SiMOS technology is discussed in the context of this logical qubit protocol, which employs quantum dots as both data qubits and singlet-triplet ancillas.

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iii Abstract

This thesis describes advancements in the silicon metal-oxide-semiconductor (SiMOS) quantum dot qubit platform. Recent experimental realisations of coherent single qubit and two qubit operations calls for the demonstration of a fully integrated SiMOS platform, showing a scalable implementation of all necessary elements for quantum computation operating together. Integration of qubit initialisation, control and readout via methods considered robust against scaling qubit array dimensions is the first fundamental step towards realising fully error corrected qubit registers useful for quantum computation. This thesis commences by analysing the result of standard experiments for benchmarking performance and integrity of a single SiMOS qubit. This experiment, known as randomised benchmarking, yields uncharacteristic re- sults for the SiMOS platform when compared to experiment theory and other results in literature. This is addressed through extending the experiment to take low-frequency environmental fluctuations into account. Resulting anal- ysis indicates that fidelities for the majority of qubit operation is considered above the threshold for fault-tolerant quantum computation. Having demonstrated high quality qubit control using SiMOS quantum dots, the capability of preparing and measuring qubits via methods robust against scaling are detailed. Recent studies have detailed how the singlet- triplet basis can be utilised for the initialisation and readout of qubits within a scaled quantum register. These techniques are discussed and experimen- tally demonstrated in a SiMOS quantum dot device alongside single qubit addressability through electron spin resonance. Electrostatic control over the Heisenburg exchange coupling between two adjacent dots produces a two-qubit SWAP operation realised in this device. These results together demonstrate, for the first time within a single silicon qubit device, the ini- tialisation and readout of qubit pairs by scalable methods integrated with single qubit addressability and two-qubit logical operations.

iv ABSTRACT

Finally, having experimentally realised integration of fundamental build- ing blocks of a scaled quantum register, what follows is a discussion of how this platform can be scaled into demonstrations of a fully error corrected log- ical qubit. Experimental state-of-the-art in SiMOS technology is discussed in the context of this logical qubit protocol, which employs constituent quantum dots as both data qubits (the carriers and propagators of quantum informa- tion) or singlet-triplet ancilla qubits for error detection.

v Acknowledgement

The work presented within this dissertation was primarily supervised by Sci- entia Professor Andrew Dzurak, with co-supervisors Dr. Menno Veldhorst and Professor Andrea Morello. Without the support I received from these three great researchers, this manuscript would not exist.

First and foremost, I would like to thank my primary supervisor Andrew Dzurak. The continual support and mentorship he has provided throughout my studies has been invaluable. The passion and enthusiasm Andrew shows for his research is truly inspiring, an attitude which continually fuels a pro- ductive and interesting workplace. He has facilitated many opportunities for me to present my research, form new and fruitful collaborations and develop as a young researcher. It has been a privilege and an honour to be a part of his research team. My co-supervisor, Professor Andrea Morello, was first person to introduce me to the concepts of quantum computing. Taking his course on quantum devices was a pivotal moment, and the advice he has pro- vided to me throughout my early career has been invaluable. The first year of my studies was co-supervised by Dr. Menno Veldhorst, who coached me through my first hands-on experience with an actual qubit device. Many of the skills I learned under his guidance during this time facilitated the success of my experiments in later years of my studies. I would like to acknowledge the past and present members of the Dzurak lab for their support, in particular the post-doctoral research team including Dr. Henry Yang, Dr. Kok Wai Chan, Dr. Tuomo Tanttu, Dr. Bas Hensen and Dr. Jason Hwang, fellow students Wister Huang, Ruichen Zhao, Yuxin Sun, Anderson West, Ross Leon. Thank you for sharing your knowledge and expertise, and for making the years a thoroughly enjoyable experience. A special mention to Bas, who went above and beyond for me throughout the past year, giving me advice on both journal articles and this dissertation.

vi ACKNOWLEDGEMENT

Also to Anderson, who was always willing to bounce ideas around or lend a helping hand. I would like to thank the CQC2T and ANFF staff and students for providing a welcoming and collaborative research environment, with a special mention to Karen Jury for her friendship and support. I would also like to acknowledge my collaborators Professor Stephen Bartlett, Associate Professor Steven Flammia, Dr. Robin Harper, Dr. Thaddeus Ladd, Dr. Cody Jones, Dr. Mark Gyure, Dr. Arne Laucht and Dr. Dimi Culcer. Their support and advice has been invaluable. Beyond the research environment, I would like to extend a huge debt of gratitude to those who kept me sane throughout the past few years. I will always cherish the memories of pub trivia and games night with my friends. Thank you to my parents, Allan and Judy, who have been ever supportive of my decisions. Thank you to my siblings, Rebecca, Jack and Emma, whom always believed in me. Without the ongoing support of my family, I would not have been able to achieve this goal. Last, but certainly not least, I would like to thank my partner, Grace. Not only was she happy to help by proof-reading, but her patience, motivation, love and support during this time saw me through to the end. Without her I surely would not have had the strength.

vii Research Contributions Peer reviewed journal articles

M. A. Fogarty, M. Veldhorst, R. Harper, C. H. Yang, S. D. Bartlett, S. T. Flammia, and A. S. Dzurak, “Non-exponential Fidelity Decay in Randomized Benchmarking with Low- Frequency Noise”, Physical Review A, 92 (2), 2015. The contribution statement for this work can be found in Appendix A. J. C. C. Hwang, C. H. Yang, M. Veldhorst, N. Hendrickx, M. A. Fogarty,W.Huang,F.E.Hudson,A.MorelloandA.S.Dzurak, “Impact of g-factors and valleys on spin qubits in a silicon double quantum dot”, Physical Review B, 96 (4), 2017. The contributions of M.A.F to this work involved preliminary characteri- sation of device behaviour, modelling via TCAD and input on the results interpretation and manuscript. C. Jones, M. A. Fogarty,A.Morello,M.F.Gyure,A.S.Dzurak,and T. D. Ladd, “A logical qubit in a linear array of semiconductor quantum dots”. Physical Review X, 8 (2), 021058. The contribution statement for this work can be found in Appendix C. M. A. Fogarty,K.W.Chan,B.Hensen,W.Huang,T.Tanttu,C.H.Yang, A.Laucht,M.Veldhorst,F.E.Hudson,K.M.Itoh,D.Culcer,T.D.Ladd, A. Morello, A. S. Dzurak, “Integrated silicon qubit platform with single-spin addressability, exchange control and robust single-shot singlet-triplet readout”. Nature communica- tions, 9 (1), 4370. The contribution statement for this work can be found in Appendix B.

viii RESEARCH CONTRIBUTIONS

Articles in submission

T. Tanttu, B. Hensen, K. W. Chan, C. H. Yang, W. Huang, M. A. Foga- rty, F. Hudson, K. Itoh, D. Culcer, A. Laucht, A. Morello, and A S. Dzurak “Controlling spin-orbit interactions in silicon quantum dots using magnetic field direction”. arXiv:1807.10415 The contributions of M.A.F to this work involved preliminary characterisa- tion of device behaviour. W. Huang, C. H. Yang, K. W. Chan, T. Tanttu, B. Hensen, R. C. C. Leon, M. A. Fogarty, J. C. C. Hwang, F. E. Hudson, K. M. Itoh, A. Morello, A. Laucht, and A. S. Dzurak “Fidelity benchmarks for two-qubit gates in silicon”. arXiv:1805.05027 The contributions of M.A.F to this work involved input on discussing device behaviour, results interpretation and manuscript preparation.

ix RESEARCH CONTRIBUTIONS

Presentations: Oral

Presenting Author: M. A. Fogarty

ˆ “High-fidelity single-shot readout and exchange control of spin qubits in silicon-MOS quantum dots” International Conference on Nanoscience and Nanotechnology, Jan 29-Feb 02, 2018, Wollongong, NSW, Aus- tralia.

ˆ “Silicon Quantum Computing” Postgraduate research symposium, Sep 26-29, 2017, UNSW Sydney, Australia.

ˆ “Integrated silicon qubit platform with single-spin addressability, ex- change control and robust single-shot singlet-triplet readout” Interna- tional workshop on silicon quantum electronics, Aug 18-21, 2017, Hills- boro, Oregon, USA.

ˆ “Nonexponential fidelity decay in randomized benchmarking with low- frequency noise”. 13th Conference on Optoelectronics and Microelec- tronic Materials and Devices, Dec 12-14, Sydney, NSW, Australia.

ˆ “Nonexponential fidelity decay in randomized benchmarking with low- frequency noise”. International Conference on Nanoscience and Nan- otechnology, Feb 07-11, Canberra, ACT, Australia.

ˆ “Nonexponential fidelity decay in randomized benchmarking with low- frequency noise”. CQC2T Internal Seminar, Nov 12, 2015, UNSW Sydney, Australia.

ˆ “Nonexponential fidelity decay in randomized benchmarking with low- frequency noise”. International workshop on silicon quantum electron- ics, Aug 03-04, Takamatsu, Japan.

Presentations: Poster

Presenting Author: M. A. Fogarty

ˆ “Integrated silicon qubit platform with single-spin addressability, ex- change control and robust single-shot singlet-triplet readout” Spin Qubit 3, Nov 06-10, 2017 Sydney, NSW, Australia.

x RESEARCH CONTRIBUTIONS

ˆ “Integrated silicon qubit platform with single-spin addressability, ex- change control and robust single-shot singlet-triplet readout” Labora- tory for Physical Science: quantum computing progress review, Aug 14-16, 2017 La Jolla, California, USA.

ˆ “Single qubit randomized benchmarking and tomography in Silicon- MOS quantum dots” Laboratory for Physical Science: quantum com- puting progress review, Jul 17-20, 2016 Alexandria, Virginia, USA.

ˆ “Randomized benchmarking in the presence of semiconductor qubit noise sources” CQC2T Annual Workshop, 2015.

ˆ “Simulation and characterization of SiMOS qubits for quantum compu- tation” UNSW honors thesis poster competition, 2014 (1st prize win- ner).

ˆ “Randomized benchmarking in the presence of semiconductor qubit noise sources” Laboratory for Physical Science: quantum computing progress review, 2014. Washington DC, USA.

ˆ “Randomized benchmarking in the presence of semiconductor qubit noise sources” International workshop on silicon quantum electronics, 2014. Albuquerque, New Mexico, USA.

Awards

ˆ Australian Postgraduate Award (2015-2018). Australian Postgraduate Awards provide a living allowance for high- quality research higher degree students. They are awarded on academic merit and research experience and/or potential.

ˆ ANFF-UNSW publication award (2016). Awarded for “Nonexponential fidelity decay in randomized benchmark- ing with low-frequency noise”, the ANFF-NSW publication award is presented in recognition of high-quality research outcomes that have been enabled by ANFF-NSW facilities.

xi Contents

Abstract iv

Acknowledgement vi

Research Contributions viii

Table of Contents xii

List of Figures xviii

List of Tables xxi

Introduction 1

1 Background theory and qubit fundamentals 12 1.1Quantumtwolevelsystemsandthequantum-bit...... 14 1.1.1 Twolevelsystems...... 14 1.1.2 Paulimatrices...... 15 1.1.3 Density matrix and the Bloch representation ...... 16 1.1.4 Anti-crossings...... 18 1.2 Postulates of quantum mechanics and measuringaquantumsystem...... 21 1.2.1 FirstPostulate...... 21 1.2.2 SecondPostulate...... 22 1.2.3 ThirdPostulate...... 22 1.2.4 FourthPostulate...... 23 1.2.5 FifthPostulate...... 24 1.2.6 SixthPostulate...... 25 1.3Anelectronwithinanexternalmagneticfield...... 26

xii CONTENTS

1.3.1 Larmor precession ...... 26 1.3.2 Therotatingframe...... 29 1.4 Operations on a single qubit ...... 30 1.4.1 TheRabiformula...... 30 1.4.2 Anti-crossingdynamics...... 35 1.5Noiseandquantumcoherence...... 41 1.5.1 Qubit noise ...... 41 1.5.2 Qubit characteristic times ...... 44 1.5.3 Noisy qubit channels ...... 51 1.6 Experiments for benchmarking qubit performance...... 56 1.6.1 Tomographic experiments ...... 57 1.6.2 RandomisedBenchmarking...... 59 1.7 Systems of two qubits ...... 64 1.7.1 Expansion from the single-spin basis to a two-spin basis 64 1.7.2 Two qubit Hamiltonian in the spin basis ...... 66 1.7.3 Systems of two coupled electrons: ...... 67 1.7.4 Two qubits in the singlet-triplet basis ...... 69 1.7.5 Coupling energies and the singlet-triplet qubit ..... 72 1.7.6 Two-qubit logical operations ...... 73 1.8 Introduction to logical qubits and fault-tolerant quantum com- putation...... 82 1.8.1 Logical qubits and encoding schemes ...... 83 1.8.2 Quantumerrordetection...... 85 1.8.3 Stabiliser codes ...... 88 1.8.4 Fault tolerant quantum computation ...... 95

2 The Silicon-MOS quantum dot platform 100 2.1 Accumulation of electrons at a semiconductor/oxide interface . 101 2.2Quantumdots...... 102 2.3Thesingleelectrontransistor...... 105 2.3.1 CoulombBlockade...... 105 2.3.2 TheSETasachargesensor...... 107 2.4 Dots as qubits ...... 108 2.4.1 Statereadoutmethods...... 110 2.4.2 State initialisation methods ...... 116 2.4.3 Qubit excited states ...... 119 2.4.4 Spin-orbit coupling ...... 122

xiii CONTENTS

2.5 SiMOS quantum dot qubit architectures ...... 128 2.5.1 Device architecture 1: Single-qubit experiments ....128 2.5.2 Device architecture 2: Singlet-triplet experiments . . . 132 2.5.3 Devicearchitecture3:Improvedconceptdevice....135

3 Randomised benchmarking for SiMOS quantum dot single- qubit fidelities 138 3.1 The Randomized Benchmarking experiment on a SiMOS qubit ...... 140 3.1.1 ExperimentalrealisationoftheCliffordgates...... 140 3.1.2 Result:Non-exponentialdecays...... 140 3.2 Randomized Benchmarking in the presence of low frequency noise ...... 142 3.2.1 Non-Markovian noise in a quantum dot qubit .....145 3.2.2 Modelling noise sources ...... 146 3.3 Extending Randomised Benchmarking ...... 148 3.3.1 Eliminating the constant for a single-qubit randomised benchmarkingmodel...... 148 3.3.2 Re-analysisofexperimentaldata...... 151 3.3.3 Interpretingthetwofidelitymodel...... 156 3.4Discussionofresults...... 159

4 Enhanced qubit readout in the singlet-triplet basis via metastable state latching 162 4.1 Physical description ...... 163 4.2Experimentsfordetection...... 165 4.3Improvementstoreadout...... 170 4.4Discussionofresults...... 173

5 Anti-crossing dynamics of SH and T− in a SiMOS quantum double-dot 175 5.1 Coupling between the singlet and polarised triplets ...... 176 5.1.1 Physicalorigins...... 176 5.2 Experimental protocols probing and utilising the SH /T− coupling...... 178 5.3 Detection and measurement of Δ(θ): Single passage Landau-Zener excitations...... 180 5.3.1 Single passage experiment details ...... 181

xiv CONTENTS

5.3.2 Single passage experiment results and discussion ....182 5.4 Further mapping of the two-qubit Hamiltonian ...... 184 5.4.1 Landau-Zener-St¨uckelbergInterferometry...... 185 5.4.2 Spin-funnel experiment ...... 187 5.5Discussionofresults...... 188

6 Integrated platform for SiMOS quantum dot qubits 190 6.1 From the singlet-triplet basis to the computational basis: Increasing applied magnetic field ...... 191 6.2 Single spin addressability via electron spin resonance techniques194 6.2.1 Spectrum...... 194 6.2.2 Experiment...... 195 6.3 Two-qubit exchange based SWAP operations...... 198 6.3.1 Experimentfordetection...... 198 6.4 Modelling exchange coupling between two quantum dots . . . 202 6.4.1 Modelling tunnel coupling ...... 202 6.4.2 Exchangefittingfromtwo-qubitdatasets...... 202 6.5 Fidelity concerns of two qubit operations ...... 205 6.5.1 Preparation/Initialisation...... 205 6.5.2 Ramping related errors ...... 205 6.5.3 Controlerrors...... 206 6.5.4 Simulations of error processes ...... 207 6.6Discussionofresults...... 212

7 Towards logical qubits and scaled quantum systems 214 7.1 An extensible logical qubit ...... 215 7.2 A linear array of SiMOS quantum dot qubits ...... 216 7.2.1 Fundamental control operations for single qubits ....217 7.2.2 Exchange control in a linear array ...... 220 7.2.3 Simulated qubit performance ...... 223 7.2.4 Tick-tockcontrolprotocol...... 230 7.3 A logical qubit in one dimension ...... 234 7.4Discussionofresults...... 238

8 Discussions, conclusions and future work 241

xv CONTENTS

Appendices 263

A Acknowledgements: Part 1 264

B Acknowledgements: Part 2 266

C Acknowledgements: Part 3 268

D Lanau-Zener Transitions 270 D.1Wavefunctioninthediabaticpicture...... 270 D.2 Non-adiabatic transitions: the solution when t →∞...... 272

E No-cloning theorem for quantum information 276

F The SiMOS fabrication process 278 F.1Microfabricationstages...... 278 F.2Nanofabricationstages...... 279

G Experimental equipment and protocols 283 G.1Devicepackaging...... 283 G.2 Cryogenic equipment and measurements ...... 285 G.2.1 Liquid helium dewar ...... 285 G.2.2 Dilution refrigerator ...... 287 G.3Electronichardware...... 288

H Configuration regimes of Architecture 3 291 H.1 Single-reservoir configuration ...... 292 H.2 Dual-reservoir configuration ...... 293

I Additional device characterisation 295 I.1 Deviceleverarms...... 295 I.2 Excitedstates...... 296

J Chronological experimental pathway for |Δ(θ)| 298

z K Fitting Bos from the spin funnel experiment 300

L Additional spin-funnel data 304

xvi CONTENTS

M Instruction set and design rules for linear nearest-neighbour error correction 306

N Encoding and concatenation schemes for linear nearest-neighbour error correction 309

O Simulated logical qubit performance of a linear array of SiMOS quantum dots 312

P Experimental pathway to a logical qubit 316 P.1ParityMeasurements...... 316 P.2Correctingonetypeoferror...... 317 P.3Correctingasingleerrorofanytype...... 319 P.4 Demonstrating a LNN logical qubit ...... 320

Bibliography 322

xvii List of Figures

1.1Blochsphereforatwolevelsystem...... 17 1.2Anti-crossingbetweentwostates...... 19 1.3 Single electron qubit Bloch sphere showing two axis control . . 35 1.4 Operation protocol and evolution Bloch spheres for dephasing andechoexperiments...... 47 1.5 Operation protocol for CP, CPMG and CDD sequences .... 50 1.6Bit-flipchannel...... 52 1.7Phase-flipandthebit-phase-flipchannels...... 53 1.8Depolarisingchannel...... 55 1.9 Operation protocol for the randomised benchmarking experi- ment...... 60 1.10 Bloch sphere representation of an error process ...... 61 1.11 Anti-crossing between the (1,1) states and the (0,2) singlet state. 70 1.12 Singlet-triplet qubit Bloch sphere...... 72 1.13 Control-based two qubit operations...... 74 1.14 Two qubit SWAP operation family circuit symbols and SWAP operationontheBlochsphere...... 76 1.15 Two qubit operations composed from the CZ gate and single qubit unitary operations...... 79 1.16 Two qubit operations composed from the SWAP-class gates and single qubit unitary operations...... 80 1.17 Encoding circuit for the logical qubit ...... 85

2.1 SiMOS quantum dot structural cross-sections and accompa- nyingschematicofelectrochemicalpotentials...... 103 2.2SETdeviceimageandoperation...... 106 2.3SETdeviceasasensor...... 107 2.4 Elzerman readout protocol for electron spin-to-charge conversion111

xviii LIST OF FIGURES

2.5Paulispinblockadefromkeyliteraturesources...... 115 2.6LiftingofPaulispinblockade...... 118 2.7 Breaking of valley degeneracy in silicon based nano-devices . . 120 2.8OrbitalstructureinaSiMOSquantumdot...... 121 2.9 Spin-orbit coupling in momentum space ...... 123 2.10 Spin-orbit coupling due to cyclotron motion of the electron . . 124 2.11SiMOSdevicearchitectureforCh.3...... 129 2.12SiMOSdevicearchitectureforCh.4-Ch.6...... 133 2.13 Concept SiMOS device architecture for improved performance 136

3.1 Non-exponential average fidelity from the randomised bench- markingexperimentduetolowfrequencynoise...... 143 3.2Modelfitstonon-exponentialfidelitydecays...... 152 3.3Two-frequencymodelfittingtobenchmarkingdata...... 158

4.1 Pictorial representation of the device cross section through the active region and the latching process...... 164 4.2 Three level pulse for spin blockade detection ...... 166 4.3 Extended study on latching ...... 168 4.4 Readout comparison between standard and latched Pauli spin- blockade...... 171

5.1 Pulse sequences for Δ(θ) coupling experiments ...... 179 5.2ResultsforsinglepassageLandau-Zenerexperiments.....183 5.3 Results for double passage Landau-Zener-St¨ukelberg experiment185 5.4 Results for the spin funnel experiment ...... 187

6.1 Energy diagram of the two-qubit charge-state anti-crossing il- lustratingthecomputationalbasis...... 192 6.2 Theory for spectral response of ESR for two electron spins under increasing exchange coupling ...... 195 6.3 Experimental observation of individual addressability of two spin qubits in a double quantum-dot ...... 196 6.4 Experiment design and results for coherent exchange driven oscillations ...... 200 6.5 Effective exchange with detuning as accumulated from multi- ple two-qubit experimental results ...... 203 6.6 Numerical simulation of exchange oscillations under realistic noise...... 208

xix LIST OF FIGURES

7.1 Proposed architectures for expansion into a logical qubit. . . . 217 7.2 Distribution and tunable range for g-factors in a linear array. . 219 7.3 Two-qubit exchange coupling via tilt control...... 221 7.4 Two-qubit exchange control via tunnel coupling modulation. . 222 7.5 Actively decoupled CZ gate...... 226 7.6 CZ simulation pulse shape characteristics...... 227 7.7ResultsofCZfidelitysimulations...... 228 7.8 Tick-tock control protocol for a linear array...... 230 7.9 Control sequence for parity-measurement experiment...... 233 7.10 Code concatenation using tiled representation...... 235 7.11 Simulated logical error rates for the two-qubit repetition code. 237 7.12 Experimental pathway to a logical qubit in quantum dots. . . 238

G.1RFenclosureusedtosupportthedevice...... 284 G.2 Liquid helium dewar device dipping stick schematic ...... 286 G.3 Hardware connection diagram for dilution refrigerator .....288

H.1 Multiple configurations of Architecture 3 ...... 292

I.1 Double dot chemical potential and valley splitting ...... 296 I.2 Energydiagramforthefirstexcitedstate...... 297

J.1 Results for single passage Landau-Zener experiments compar- ingvoltageramprates...... 299

K.1 Fitting spin funnel data ...... 302

L.1 Original spin-funnel data ...... 305

M.1 Standard-LNN instructions and their circuit-diagram symbols. 307

N.1 Encoding scheme for LNN blocks...... 310

O.1 Simulated logical error rates for the four-qubit code...... 314

P.1 Tile formalism for the three qubit repetition code...... 318

xx List of Tables

1.1TruthtableforCNOToperation...... 75 1.2 Unitaries in the stabiliser formalism ...... 91

3.1 Implementations of the single qubit Clifford set ...... 141 3.2AkaikeInformationCriterion...... 155 3.3Gatefidelityestimatesforeachofthedatasets...... 159

6.1 Key noise coupling mechanisms for singlet-triplet Hamiltonian 209

G.1 Room temperature electronics used to support dilution refrig- eratormeasurements(Table1)...... 289 G.2 Room temperature electronics used to support dilution refrig- eratormeasurements(Table2)...... 290

K.1 Fitting parameters from spin funnel data ...... 303

xxi Introduction

Quantum computers

The full realisation of error corrected quantum computation is recognised as one of today's greatest scientific and technological challenges, resulting in a rapidly growing multi-disciplinary research area which explores diverse physical systems including photons, trapped ions and nuclei, superconducting circuits and semiconductor devices. The theorised quantum computer behaves vastly different from its classi- cal counterpart through its fundamental operation. Rather than manipulat- ing a singular bit of classical information represented by either a 0 or a 1, the quantum computer encodes quantum information onto the wavefunction of a quantum two-level system, forming a quantum bit or “qubit”. Through utili- sation of a quantum mechanical property known as entanglement,or“spooky action at a distance” as it is often referred to colloquially, the quantum com- puter is capable of exploiting the full complexity of a many-particle quantum wavefunction to solve a particular set of computational problems [1].

1 INTRODUCTION

Applications of a quantum computer

Such a machine is capable of executing its own set of algorithms, with promi- nent examples outlined in Ref. [2]. These include Grover's algorithm, which is capable of performing an optimal search for specific entries in an unordered dataset [3, 4] and Simon's algorithm for determining the exclusive-or (XOR) mask over which a given black-box function is invariant [5] (corresponding to the hidden Abelian subgroup problem, which includes factoring integers into primes and calculating discrete logarithms [6]). Perhaps the most influential or iconic algorithm known today is indeed that of Shor [7], whom theorised an algorithm capable of prime factorisation of integers in near polynomial time. This example became an iconic demonstration of how a quantum com- puter would be capable of out-performing a classical computer the execution of some specific tasks, while also placing many common encryption methods at risk. The tantalising implications of being able to efficiently produce solutions to the aforementioned problems, among others, has since lead to the rapidly expanding fields of quantum computation, quantum cryptography, quantum secure networks and quantum information processing. Today, quantum com- puters are theorised to have vast and revolutionary impact on many areas in science, technology, intelligence and commerce. Specific examples stem from cryptography [8] artificial intelligence [9], and dynamic systems with a large number of input variables such as the weather or the stock market [10]. Methods of solving various logistics-based problems derived from the trav-

2 INTRODUCTION elling salesman problem [11, 12] also stand to benefit, providing efficient solutions to freight distribution, PCB drilling, wiring, X-ray crystallography sample positioning optimisation etc. Perhaps the most intriguing and impactful application of the quantum computer is, in fact, the purposes for which it was originally theorised. The original proposal by Feynman [13] was to use quantum computers for the ef- ficient simulation of quantum systems and quantum mechanics itself. While this gives the ability to simulate the very properties which allows the com- puter to function, it results in the chance to explore many other physical pro- cesses such as particle physics [14] and chemical reactions [15, 16, 17]. The latter example is of particular revolutionary importance as it would make rapid prototyping of new pharmaceutical products conceivable by adding ef- ficient simulation stages before expensive compound synthesis is required. Further, with the advantages of more efficient DNA sequencing (again, a problem related to the travelling salesman), patient-tailored treatment pro- grams can also be designed, leading to improvements in healthcare and indi- vidual quality-of-life within a community.

Approaches to quantum computation

Quantum computation has been approached using a wide range of physi- cal implementations, including chip-scale waveguide quantum circuits [18, 19], trapped ions [20, 21] and neutral atoms in an optical lattice [22, 23], liquid-state nuclear magnetic resonance [24, 25, 26, 27], superconducting res-

3 INTRODUCTION onators [28, 29, 30, 31], topological qubits [32, 33], optically addressable colour centres in diamond [34, 35], and a variety of qubit species constructed using solid-state materials based on donors and quantum-dots. Some of these species draw advantage by being optically active materials, such as self- assembled quantum dots in InGaAs/GaAs heterostructures [36] and Silicon- Carbide [37], producing qubits which naturally couple to optical resonators. Other species such electrostatically defined quantum dots [38] in GaAs [39] and Silicon (both Si/SiGe [40] heterostructures and Silicon based Metal- Oxide-Semiconductor dots [41, 42, 43]) as well as singular dopants [44, 45] or dopant clusters [46] draw upon the ability to define large arrays of densely packed qubits assembled on a solid-state host.

Dot-based architectures and scaling

For electrostatically defined quantum dots in semiconductors, the earliest proposal for a quantum computer is that of Loss and DiVincenzo [38], which details arrays of dots occupied by a singular electron whose spin state forms an individual qubit. Execution of preliminary gates forming quantum algo- rithms would be achieved by altering voltages on the lithographically defined gates to either isolate or couple adjacent qubits via the (relatively) short range electron-electron Heisenburg exchange interaction. Based on these fundamental qubit design and interaction concepts, sev- eral architecture schemes have been derived in order to address the issue of scaling these systems into quantum registers useful for quantum compu-

4 INTRODUCTION tation. Experimental proposals have been put forward in order to move such qubit architectures from the “test-bench” laboratory demonstrations of single-qubit and two-qubit control operations, to full implementations of er- ror corrected logical qubits [47]. Further, array-based quantum computer architectures are considered advantageous for achieving fault-tolerant oper- ation leading to many proposed architectures based on qubit arrays in one dimension [47, 48], while modern algorithms such as the Surface Code [49] have resulted in several 2D qubit array designs capable of achieving such a goal [50, 51, 52, 53].

CMOS industry compatibility

Modern classical semiconductor-based computational technologies are cen- tred around the CMOS industry, with silicon as the semiconductor used in these devices. As this industry is very mature in its design methods and fab- rication processes, it is apparent that compatible semiconductor-based quan- tum computing platforms stand to benefit from the available technologies of the global CMOS industry. Several material platforms used to approach constructing a quantum computer intend to exploit the state-of-the-art in semiconductor manufacturing in order to undertake rapid prototyping and to push quantum computers towards the commercial sector. These include the Silicon-Metal-Oxide-Semiconductor (SiMOS) metal-gate [41] and polysilicon- gate [43] devices, as well as the Silicon-On-Insulator (SOI) nanowire tech- nologies [54]. Recent developments have seen the SOI technology being

5 INTRODUCTION used in commercial semiconductor foundries to produce quantum dot based qubits [55, 56], and have even been integrated with a field effect transistor (FET) [57] used for switchable readout of quantum dots.

Thesis aims and scope

This thesis undertakes various studies using the SiMOS quantum-dot qubit architecture [41]. Although these devices have not yet been produced via a commercial semiconductor foundry, the material components of this platform are compatible with the CMOS industry. The studies presented here yield strong evidence of the ability for this platform to be scaled into a large-scale quantum register, fully fledged with high single-qubit control fidelity [58], capability to individually address single qubits within an array [59, 60] and qubit readout via scalable methods [60]. Several of these requirements have been realised on a singular qubit device demonstrating the SiMOS platforms capability to fully integrate all qubit initialization, readout and control in a scalable manner [60]. Further, experimental pathways have been instigated for the study of individual requirements for quantum error correction building up to the demonstration of a fully error correctable logical qubit in a 1D array of quantum dots [47]. The extensive scope of this dissertation is broken down into the following chapters:

6 INTRODUCTION

Chapter 1: Background theory and qubit fundamentals

This chapter introduces fundamental physical concepts within the field of quantum computing. Building from the ground up, this chapter explains the basics of qubits and how the quantum information they represent can be ma- nipulated in a controlled manner. This then moves on to discuss the theory behind manipulating quantum systems through dynamic potentials and ap- plied fields within the quantum dot devices, and how this can influence the qubit states via Landau-Zener or Rabi processes. Qubit noise and effects upon qubit coherence is also discussed, leading into experiments which can assess qubit coherence and fidelity. As scaled systems often employ ancillary qubit states for initialization and/or readout of data qubit states, the ex- pansion into two-electron systems is also introduced and discussed at length. Finally, as many of the experimental results presented in this dissertation are accompanied by simulation work, a brief discussion of the simulation of qubit systems is also included.

Chapter 2: The Silicon-MOS quantum dot platform

This section introduces detailed concepts of the SiMOS quantum dot qubit platform studied. Basics of the platform are discussed including how the quantum dots are electrostatically defined, and how they can be used to pro- duce sensitive electrometers as well as qubits. For quantum dots as qubits, state initialization and readout methods are discussed, as well as qubit ex- cited states. Further, a number of fundamental concepts specific to the ma-

7 INTRODUCTION terials system used are reviewed as many are later integrated as a part of experimental processes.

Chapter 3: Randomised benchmarking for SiMOS quantum dot single-qubit fidelities

The first results chapter presented in this dissertation is that of the single- qubit randomized benchmarking (RB) results as presented in Ref. [58]. Here, the assumptions put in place for the RB experiment are challenged as en- vironmental noise sources within the SiMOS system are non-Markovian in nature. The implications of this are discussed, while the analysis of data pro- duced by the RB experiment is extended to be more tolerant of such noise sources. Using the novel analysis techniques presented, the single-qubit sys- tem analysed is shown to be capable of operating at error rates compatible with fault-tolerant operation of a scaled quantum register.

Chapter 4: Enhanced qubit readout in the singlet-triplet basis via metastable state latching

This second results chapter illustrates the first step towards demonstrating a fully integrated platform for saleable quantum computation using the SiMOS quantum dot qubit platform as presented in Ref. [60]. This chapter details how measurement signals from readout which exploits Pauli spin blockade of a system of two electrons and two quantum dots can be enhanced through a state latching protocol. This chapter presents the physical description of

8 INTRODUCTION the enhancement protocol, as well as analysing the sensitivity enhancement of the sensors to state readout. This enhancement to readout sensitivity is exploited for the remainder of the results chapters affiliated with Ref. [60].

Chapter 5: Anti-crossing dynamics of SH and T− in a SiMOS quan- tum double-dot

The third results chapter details the analysis of the interactions between the hybridising singlet state SH and polarised triplets (results presented for the

T− triplet state) following results as discussed in Ref. [60]. The origins of such an interaction are discussed in context within current studies presented in literature, as well as discussions of how such an interaction is to be measured via exploiting coherent Landau-Zener transitions over the anti-crossing. Not only is the magnitude of this interaction measured for the experiments in- volved in this dissertation, but the interaction is further utilised to probe other components of the two-electron interaction Hamiltonian (namely, the Heisenburg exchange energy J). Knowledge of the magnitude of this inter- action term and anti-crossing location in parameter Hamiltonian space are vital for the success of experiments presented in later chapters.

Chapter 6: Integrated platform for SiMOS quantum dot qubits

This fourth results chapter presents the most important advancements to the SiMOS platform presented in Ref. [60]. Utilising the results discussed in the previous two chapters, a set of experiments which demonstrate a fully inte-

9 INTRODUCTION grated platform for quantum computation using SiMOS quantum dot qubits. This chapter details how state transformations between singlet-triplet basis states, where the electrons are initialised and measured, are mapped semi- adiabatically to prepare computational basis states receptive to quantum control algorithms. These algorithms are achieved through individually ad- dressing constituent electrons within the system, as demonstrated experi- mentally through electron spin resonance (ESR) techniques. Alongside the requirement for single-qubit control is the capability to perform two-qubit entangling gates between adjacent quantum dots. This is often achieved through mediation of the Heisenburg exchange interaction between two elec- trons. This chapter details experiments which not only demonstrate the ability to measure the Heisenburg exchange interaction in parameter space, but to fully control the exchange to produce one of the first instances of a two-qubit SWAP operation in the SiMOS system. Finally, this chapter concludes with a discussion on the sources of error mechanisms within the complete exchange based protocol, including state preparation, measurement and control errors based on two prominent noise sources within the SiMOS system.

Chapter 7: Towards logical qubits and scaled quantum systems

This final results chapter is dedicated to the discussion of the experimental state of the art for the SiMOS quantum dot qubit system, and the experi- mental roadmap towards realising a fully error corrected logical qubit in a

10 INTRODUCTION linear array of quantum dots following Ref. [47]. The chapter discusses how the current SiMOS test-bench devices presented in earlier parts of this the- sis can be scaled to system sizes capable of demonstrating such intermediate experiments. This includes discussions on how elementary operations will be executed on scaled qubit systems, as well as the presentation of qubit per- formance simulations under these protocols based on realistic noise sources. This work forms a feasibility study for the use of state-of-the-art SiMOS quantum dot qubits to demonstrate elements of an extensible logical qubit. This work is extended further, describing encoding schemes and execution of stabiliser codes on a linear array, the performance of which are also simu- lated to demonstrate thresholds for error correction. These elements are the main focus of Ref. [47], and so are also summarised in this chapter.

Chapter 8: Discussion, conclusions and future work

This chapter takes a broader view of the experimental scope presented in this dissertation. The results of these experiments are discussed and compared with parallel studies produced from other qubit platforms. The impact of these results on the field to date, as well as potential avenues for future work are also highlighted.

11 Chapter 1

Background theory and qubit fundamentals

Just as a single binary bit of information is to a classical computer, the quantum-bit, or qubit is the fundamental element of a quantum computer. Any quantum two level system, or larger physical systems which can be trun- cated into a quantum two level system, can be used to form a qubit. There- fore, there are many diverse platforms upon which qubits can be studied, offering a vast array of methods to control, measure or probe qubit physics. In fact, five essential criteria have been identified [61] which any system must be able to satisfy in order to viably create a quantum computer.

12 CHAPTER 1

The five essential criteria which a qubit platform is required to satisfy are known as the DiVincenzo criteria, and are summarised as the following:

1. The platform produces a scalable physical system with well charac- terised qubits

2. The ability to reliably initialise the qubits in a known state

3. Long coherence of qubit states

4. A universal set of qubit operations

5. Ability to measure the qubit state.

Using these criteria as a framework, this chapter discusses the elements which are required form a qubit, and how qubit characteristics are used to represent and manipulate quantum information. Some specific examples will be given for electron spins in the SiMOS platform, however the core requirements to form a qubit extend to all platforms and quantum computing architectures. The chapter concludes by introducing fundamental concepts required for the expansion of qubit systems into fully scaled, error corrected quantum processors, as well as the conditions necessary for attaining fault tolerant quantum computation.

13 CHAPTER 1

1.1 Quantum two level systems and the quantum-

bit

This section introduces how a qubit is represented as a two level system, using basic concepts of linear algebra. Characteristics of a single qubit are discussed, including common concepts such as the qubit wavefunction and density matrix, visualisation on the Bloch-sphere, energy representation, cou- plings and dynamics.

1.1.1 Two level systems

A two level system is a quantum mechanical system where the system wave- function |ψ can be represented by linear combinations of two discrete and orthogonal eigenvectors |φ1 and |φ2: ⎛ ⎞ ⎜α⎟ |ψ = α |φ1 + β |φ2 = ⎝ ⎠ , (1.1) β where α, β ∈ C, the set of complex numbers, and |α2 + β2| =1asgiven by wavevector normalisation. The Hamiltonian, H, of such a system is a 2 × 2 Hermitian operator corresponding to the total energy of the system, with eigenenergies giving the energies of the two discrete basis eigenstates

{|φ1 , |φ2}. The evolution of the wavevector with respect to time can be given by the

14 CHAPTER 1

Schr¨odinger equation ∂ H |ψ(t) = i |ψ(t) , (1.2) ∂t for a time-independent Hamiltonian. This can be solved to yield the evolution of the wavefunction after time t = t0 → tf as

|ψ(tf ) = U |ψ(t0) , (1.3) where the time evolution operator

 −iHt U =exp . (1.4) 

1.1.2 Pauli matrices

Electron spin can be represented as a quantum two level system which exists in the vector space of 2 × 2 Hermitian matrices. In order to describe the dynamics of the spin as represented in this form, a set of matrices which form a basis for this vector space are the Pauli matrices:

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜10⎟ ⎜01⎟ ⎜0 −i⎟ ⎜10⎟ σ0 = ⎝ ⎠ ,σxˆ = ⎝ ⎠ ,σyˆ = ⎝ ⎠ ,σzˆ = ⎝ ⎠ , 01 10 i 0 0 −1 (1.5) where σ0 = 1 is the two dimensional identity matrix.

15 CHAPTER 1

1.1.3 Density matrix and the Bloch representation

Another convenient notation used to express the quantum two level system is by means of a density matrix given by ρ. This has the following relation to the wavevector:

ρ = Pj |φjφj| , (1.6) j

P ≥ P × where j 0and j j = 1. The result is a 2 2 Hermitian operator interpreted as describing the statistical ensemble of states in a quantum sys- tem [62], being in state |φj with probability Pj. Thus, it is clear that taking the trace over the density matrix yields

Tr(ρ)=1. (1.7)

Further, ρ2 = ρ if, and only if, the state is pure (i.e. ρ = |ψψ|). It follows that the trace over ρ2, also known as the purity,isgivenby

Tr(ρ2) ≤ 1, (1.8) which is equal to 1 only for a pure state. From what is presented in Sec. 1.1.2, this density matrix can be written as an expansion of a 3D vector over the

16 CHAPTER 1 orthonormal basis given by the Pauli matrices

1 ρ = (1 + Rxσxˆ + Ryσyˆ + Rzσzˆ) (1.9) 2 1 = 1 + R· σ , (1.10) 2

with R =(Rx, Ry, Rz), known as the Bloch vector,and σ =(σxˆ,σyˆ,σzˆ). From the purity relationship

|R| 2 +1 Tr(ρ2)= ≤ 1, (1.11) 2 which leads to important conclusions:

1. The length of the Bloch vector cannot exceed unity.

2. Any point where |R|2 = 1 corresponds to a pure state.

z

R Ɇ y ɔ x

Figure 1.1: Bloch sphere representation of the density matrix in 3D space. The state of the quantum mechanical systems is represented by Bloch vector R .

17 CHAPTER 1

The first conclusion tells us that a natural visualisation for the density matrix is as a unity sphere in spherical coordinates (1,ϑ,ϕ), known in this context as the Bloch sphere. From the second conclusion, the surface of this sphere represents a manifold of pure states. Such a sphere is visualised in Fig. 1.1.

1.1.4 Anti-crossings

In this section the system describing Hamiltonian is introduced on a more basic level. Consider a two level represented by the two states |φ1 and |φ2, which have energies given by E1 = −ε and E2 = ε respectively. The Hamiltonian for such a system can be described using the Pauli matrices introduced in Sec. 1.1.2:

H0 = εσzˆ, (1.12) which has eigenbasis representation:

H0 |φ1 = ε |φ1 , (1.13)

H0 |φ2 = −ε |φ2 ,

where eigenenergies E1,2 = ±ε.Forε = 0, these two states become degener- ate. Therefore, the term ε describes the detuning of the system away from that degeneracy point. If we now add a coupling term Δ between these two

18 CHAPTER 1 states, we result in the Hamiltonian described by:

HQ = εσzˆ +Δσxˆ. (1.14)

The associated energy diagram is pictorially illustrated in Fig. 1.2. For the

ۄɔ1| ۄɔ2|

2ȟ Energy

ۄɔ2| ۄɔ1| 0 Detuning Ԫ

Figure 1.2: Energy diagram of the two states |φ1 and |φ2. These two states form an anti-crossing due to a coupling mechanism Δ. The parameter ε represents a detuning of the system eigenenergies away from this avoided crossing point. above Hamiltonian, which is now a coupled two level system, the new eigen- basis can be represented as a combination of the eigenbasis of the decoupled

Hamiltonian H0 of Eq. (1.12):

HQ |ψe = Ee |ψe , (1.15)

HQ |ψg = Eg |ψg .

The high energy (excited) state |ψe and low energy (ground) state |ψg are

19 CHAPTER 1 represented as:   θ θ |ψ  =sin |φ  +cos |φ  , e 2 2 2 1 √ (1.16) 2 2 Ee = ε +Δ , and   θ θ |ψ  =cos |φ −sin |φ  , g 2 2 2 1 √ (1.17) 2 2 Eg = − ε +Δ ,   −1 Δ for θ =tan ε . The Hamiltonian in Eq. (1.14) is often known as the qubit Hamiltonian, and has the ability to access rotations around at least two orthogonal axes (here,x ˆ andz ˆ are involved). This demonstrates the capability to control the state of the system over the entire Hilbert space, thus producing a controllable quantum-bit, or qubit. It is observed by the commutation relation

[σ1,σ2]=2i123σ3, (1.18)

where 123 is the Levi-Civita symbol, that operations involving σyˆ can be constructed through a combination of operations from the remaining two orthogonal Pauli matrices.

20 CHAPTER 1

1.2 Postulates of quantum mechanics and

measuring a quantum system

So far, the discussion of the qubit has been limited to a basic description of how a two level system can be represented by the wavefunction and accom- panying Hamiltonian. Here, the postulates of quantum mechanics are introduced, alongside the process and effects of measuring a quantum system. The discussion below is adapted from that of Ref. [63], which should be referred to for additional details. The postulates are described below and, where relevant, linked to more in-depth discussions within the chapter. The key postulated which pertain to the discussion of the measurement of quantum systems are the Third, Fourth and Fifth, however the complete discussion of the postulates as presented in Ref. [63] is given here for additional context and completeness.

1.2.1 First Postulate

At a time t, the state of a quantum system is given by |ψ(t) which is in a suitable vector space E. This is the topic of discussion in several of the subsections of Sec. 1.1. In brief, it is stated in Eq. (1.1).

21 CHAPTER 1

1.2.2 Second Postulate

Every measurable physical quantity A is described by an observable operator A, acting within E. This is again the topic of discussion in several of the subsections of Sec. 1.1, however it does introduce the concept of system observables.The Hermitian operator A is said to be an observable if its orthonormal set of eigenvectors forms a basis in E. For example, SZ = σzˆ/2 is an observable for the basis defined in Eq. (1.12) due to the following relationship:

 S |φ  = |φ  , Z 1 2 1  (1.19) S |φ  = − |φ  . Z 2 2 2

1.2.3 Third Postulate

The only possible result of the measurement of a physical quantity A is the one of the eigenvalues of the observable A. This postulate is simply a statement of the system quantisation. The system Hamiltonian H is a description of the energy within the quantum mechanical system, therefore, the only energies possible are those described by the eigenvalues of H. Since A is Hermitian, the measurement of A must be a real value, and if the spectrum of A is discrete, results observed when measuring A are quantised.

22 CHAPTER 1

1.2.4 Fourth Postulate

When the physical quantity A is measured on a system in a normalised state

|ψ, the probability P(an) of obtaining the non-degenerate eigenvalue an of the corresponding observable A is:

2 P(an)=|un|ψ| , (1.20)

where |un is the normalised eigenvector of A associated with eigenvalue an. This introduces the probabilistic nature of measuring a quantum me- chanical system. This concept is discussed in the context of a more concrete example in Sec. 1.4, when discussing the Rabi formula. See Ref. [63] for details on systems involving degenerate energy levels. In the most generic terms, consider an arbitrary spin direction in spherical coordinates (1,ϑ,ϕ) giving the state at time t of

    ϑ −iϕ ϑ iϕ |ψ(t) =cos exp |φ  +sin exp |φ  . (1.21) 2 2 1 2 2 2

When this system is measured, the probability that the system energy will be /2 (corresponding to |φ1)isgivenby:        − 2 2  ϑ iϕ  P = |φ1|ψ| = cos exp  (1.22) 2 2 2 ϑ =cos2 . (1.23) 2

23 CHAPTER 1

It is clear that the phase angle ϕ, a value orthogonal to thez ˆ-axis from which the observable A, does not affect the projected measurement outcome. Further, the expectation value A of the measured result can be found as

A = anP(an) (1.24) n = ψ|A|ψ . (1.25)

Note: The fact that the measurement of quantum systems is probabilistic in nature requires the build-up of an adequate level of measurement statistics when determining the results of experiments performed on such systems.

1.2.5 Fifth Postulate

If the measurement of the physical property A on the system in the state |ψ gives a result an, the state of the system immediately after the measurement is the normalised projection of the state |ψ onto the eigensubspace associated with an. That is; for measurement at time t, the state immediately after the measurement is   P |ψ ψ(t+) =  n . (1.26) ψ|Pn|ψ

Here, Pn is a projector term given by

Pn = |unun| . (1.27)

24 CHAPTER 1

This implies that the state immediately after a measurement event is an eigenvector of A, with eigenvalue an.

Note: This fundamental aspect of the measurement of a quantum mechan- ical system is also utilised in the initialisation of quantum states before un- dertaking experiments. This concept is discussed further in later chapters.

1.2.6 Sixth Postulate

The time evolution of the state vector |ψ(t) is governed by the Schr¨odinger equation: ∂ i |ψ(t) = H(t) |ψ(t) , (1.28) ∂t where H(t) is the observable associated with the total energy of the system. This is discussed extensively in Sec. 1.1 and Sec. 1.4.

25 CHAPTER 1

1.3 An electron within an external magnetic

field

This section details a textbook example of a quantum mechanical two level system: the spin state of an electron. The section aims to conceptualise the spin, and draws parallels between the quantum mechanical system and an analogy based on classical mechanics. The electron as an elementary quantum two-level system is introduced, and the response of an electron to the externally applied magnetic field is discussed in the context of producing two non-degenerate eigenenergies. Im- portant implications of the of the electron mechanics in this applied field with respect to performing qubit experiments are also discussed.

1.3.1 Larmor precession

The spin of singular electron is a quantum two level system with quantized spin quantum number SZ = ±/2. Hence, the spin of the electron is often represented via the Pauli matrices given in Eq. (1.5) as

 S = σ . (1.29) Z 2 zˆ

A classical analogue of spin is a charge q with mass m with an intrinsic an- gular momentum L . Under classical electromagnetism, this system produces

26 CHAPTER 1 a magnetic dipole q M = L. (1.30) 2m

When a magnetic dipole is placed in an external magnetic field B0, the total energy of such a system can be represented as

E = −M · B0. (1.31)

For the spin of an electron in an external magnetic field, this classical analogue gives rise to the basis electron states having an energies given by

| | qe E = gS · B0. (1.32) 2me

where qe and me are the charge and mass of the electron, and g is the electron gyromagnetic ratio [63]. This energy separation forming two discrete electron states due to an externally applied magnetic field is known as the Zeeman energy, which can be instead expressed in terms of the Bohr magneton μB as

gμ E = B S · B . (1.33)  0

Treating this quantity classically once again suggests that the electron can experience a torque Γ, if S and B0 are not parallel. That is;

gμ dS Γ= B S × B = , (1.34)  0 dt

27 CHAPTER 1 which suggests a precession frequency in the time evolution of a non-parallel spin state around the vector direction of the global magnetic field B0. To treat this spin as a quantum mechanical object, we construct the Hamiltonian representing the total energy of the system:

gμ B  H = B 0 σ . (1.35)  2 zˆ

Here we have assumed B0 = |B0|·zˆ, which produces a Hamiltonian in the same form of H0 in Eq. (1.12). As an illustrative example, we prepare a state which is perpendicular to the applied field B0, one which is a superposition of the two basis states of the Hamiltonian H: 1 |ψ0 = √ (|φ1 + |φ2) , (1.36) 2 where the projection in thex ˆ direction for this state is given by

  S  = ψ |σ |ψ  = . (1.37) x 2 0 xˆ 0 2

The time evolution of this state under the Hamiltonian H is given by the Schr¨odinger equation Eq. (1.2), resulting in the evolution in time given as

  1 gμBB0 |ψ(t) = √ |φ1 exp −i t + |φ2 , (1.38) 2 

28 CHAPTER 1 whereby the projection along thex ˆ direction evolves with time as

  gμ B S  = cos B 0 t . (1.39) x 2 

Therefore, it is observed that the orthogonal component of the spin vector to an applied magnetic field B0 undergoes a precession in time with a frequency of gμ B ω = B 0 . (1.40) 0 

This physical effect is known as Lamor precession.

1.3.2 The rotating frame

As described in Sec. 1.3.1, a spin in an external magnetic field undergoes a precession at a natural frequency dictated by the applied magnetic field. As the time dynamics of off-axial components become complicated due to the time dependence of the spin, it is a common simplification to make a transformation of the system Hamiltonian into a basis which also rotates at the natural frequency of ω0. From this frame of reference, called the rotating frame, an unperturbed spin vector remains stationary. In order to transform into the rotating frame, the phase accumulation of the eigenenergies of the system described in Eq. (1.1) must be taken into

29 CHAPTER 1 account. This implies a transformation matrix

|ψr = R |ψ ⎛ ⎞   ω0 (1.41) ⎜exp i 2 t 0 ⎟ R = ⎝  ⎠ − ω0 0expi 2 t

where |ψr is the wavefunction in the rotating frame.

1.4 Operations on a single qubit

This section introduces a number of mechanisms which can be utilised in order to manipulate the qubit state. These mechanisms give a certain degree of control over the quantum information represented by the qubit, and can be used to form fundamental control operations for a quantum computer.

1.4.1 The Rabi formula

Consider once again the two level system which is described by the qubit

Hamiltonian HQ of Eq. (1.14). Choose the initial condition such that |ψ0 begins in the unperturbed ground state |φ1. This state can be instead represented in the basis states of the qubit Hamiltonian HQ as

|ψ0 = |φ1 (1.42)   θ θ =cos |ψ −sin |ψ  . (1.43) 2 g 2 e

30 CHAPTER 1

The time evolution of this state is then

    θ iE t θ iE t |ψ(t) =cos exp − g |ψ −sin exp − e |ψ  . (1.44) 2  g 2  e

The time evolution can result in the population of the |φ2 excited state. This can be observed from the state projection

     1 θ iE t iE t φ |ψ(t) = sin exp − g − exp − e , (1.45) 2 2 2   and, using the fact that

 √ θ Δ 2 2 sin = √ ,Ee = −Eg = Δ + ε , (1.46) 2 Δ2 + ε2 the probability of finding the time evolved state in the opposite basis state becomes √  iΔ Δ2 + ε2 φ2|ψ(t) = √ sin t , (1.47) Δ2 + ε2  where the measurable projection into the opposite basis state is √  Δ2 Δ2 + ε2 |φ |ψ(t)|= sin2 t . (1.48) 2 Δ2 + ε2 

This is known as the Rabi formula [63].

31 CHAPTER 1

For spins in the rotating frame: Electron spin resonance

As described in Sec. 1.3.2, an electron in an external magnetic field will undergo a Lamor precession. However, the time dependence of in-plane (ˆx andy ˆ) components of the spin vector can be mitigated through a change of basis to the rotating frame (See Sec. 1.3.2). Therefore, in order to reproduce the qubit Hamiltonian HQ within the rotating frame, one must be able to produce a coupling term Δ which is also time dependent in the same manner. Such a coupling can be produced via an in-plane, rotating magnetic field

B1 = B1(ˆx cos(ωt)+ˆy sin(ωt)), (1.49) which rotates with driving frequency ω. When applying such a driving field to the spin, the system Hamiltonian in the standard laboratory frame becomes

ω  ω  H(t)= 0 σ + 1 (cos(ωt)σ + sin(ωt)σ ), (1.50) 2 zˆ 2 xˆ yˆ where the time evolution becomes

∂ |ψ(t) i = H(t) |ψ(t) . (1.51) ∂t

32 CHAPTER 1

Under the transformation R into the rotating frame, this becomes

∂(R |ψ(t)) i = H(t)(R |ψ(t)), (1.52) ∂t ∂ |ψ (t) i r = H |ψ (t) , (1.53) ∂t r r

where Hr is the time independent Hamiltonian

 H = (Ωσ − δωσ + ω σ ), (1.54) r 2 zˆ zˆ 1 xˆ

where δω = ω − ω0 which represents the difference between the Lamor fre- quency and the frequency of the driving field B1,andΩ=ω + ω0 is a “fast” rotating term. Close to the resonance condition, where δω Ω, the ef- fect of the fast term Ω averages to almost zero, leading to the rotating wave approximation:  H ≈ (−δωσ + ω σ ), (1.55) r 2 zˆ 1 xˆ with motion dominated by the slowly rotating term. Substituting into the Rabi formula of Eq. (1.48), a driven spin system exhibits the following time dependent probability based on   ω2 ω2 + δω2 |φ |ψ(t)|= 1 sin2 1 t . (1.56) 2 2 2  ω1 + δω

33 CHAPTER 1

Driving field phase considerations

The overall phase of the driving field is important as it determines which axis in the rotating frame the electron spin rotates around. We define the following states which are aligned in the positivex ˆ andy ˆ directions

1 |ψx = √ (|φ1 + |φ2) , (1.57) 2 1 |ψy = √ (|φ1 + i |φ2) . (1.58) 2

From the derivations above, it is clear to see that under driving field

B1 = B1(ˆx cos(ωt + ϕ)+ˆy sin(ωt + ϕ)), (1.59) which includes a global phase term ϕ, the respective Rabi formulae become:   ω2 ω2 + δω2 |ψ |ψ(t)|= 1 cos2 1 t cos2(ϕ), (1.60) x 2 2  ω1 + δω   ω2 ω2 + δω2 |ψ |ψ(t)|= 1 cos2 1 t sin2(ϕ). (1.61) y 2 2  ω1 + δω

Therefore, the choice in phase ϕ is crucial when attempting coherent con- trol experiments as it represents an offset in phase with respect to the global Lamor precession from an arbitrary instance in time (such as the commence- ment of the experiment). Further, as depicted in Fig. 1.3, using phase mod- ulation with a ϕ = π/2 allows for control and manipulation of the state over two orthogonal axes in the rotating frame, and thus complete control over

34 CHAPTER 1

ۄb) |՛ ۄa)|՛

ۄ߰y| ۄ߰y|

ۄ߰x| ۄ߰x|

ۄ՝| ۄ՝|

Figure 1.3: Two axis control for a single qubit constructed from a single electron spin. Based on modulation of the phase ϕ of the driving field B1 thespincanberotateda) around the x-axis for ϕ =0,andb) around the x-axis for ϕ = π/2 the qubit state.

1.4.2 Anti-crossing dynamics

When two states are coupled in the manner presented in Section 1.1.4, via coupling energy Δ, the ground state of the system |ψg becomes dependent upon the coupling Δ and detuning energy ε. That is; as the system is detuned from one side of the anti-crossing to the other, the ground state of the system

|ψg(ε) transfers from one of the two basis states (|φ1 or |φ2 in Fig. 1.2) to the other. Consider the case when transferring from a region where detuning

ε 0and|ψg |φ1, to a region where detuning is ε 0, and |ψg |φ2. From the adiabatic theorem, the state of the system |ψ(t) tracks the ground state |ψg(θ) when the transferral ε 0 → ε 0 is performed infinitely slowly. When the adiabatic condition is not held however, the state |ψ(t)

35 CHAPTER 1 can become a mixture (or a superposition, see Sec. 1.1.1) with the excited state of the anti-crossing |ψ(t) = α1 |ψe(θ) + α2 |ψg(θ). The following sections, together with Appendix. D, detail the mechanisms under which such a superposition is produced.

Landau-Zener processes

The following is a brief overview of the Landau-Zener transition and the parameters which govern the process. A full derivation of the process is given in Appendix D which closely follows the solution in Ref. [64]. Consider again, the quantum two level system represented by the wave- function |ψ |φ1 which is initially tuned such that ε 0. The system can transfer the state from one side of the anti-crossing to the other by modulat- ing ε. The rate at which this is done is identified as the quantity ν,givenby the rate of change in the eigenenergies of the uncoupled system Hamiltonian

H0 of Eq. (1.12) d(E − E ) ν = 1 2 . (1.62) dt

Here, the interaction quantity Δ which gives rise to the qubit Hamiltonian HQ in Eq. (1.14) is considered to be the perturbation under which the interaction takes place [65, 66]. Based on the state energy velocity ν, there is a certain probability that the final state is instead the excited state |ψe, as given by the Landau-Zener

36 CHAPTER 1 excitation probability [65, 66, 67]:

 −2π|Δ|2 P =exp (1.63) LZ ν

By controlling ν during the passage through the anti-crossing, the adiabatic- ity of the passage is modulated to produce a certain transition probability into the excited state. For the case where ν Δ, the passage is adiabatic with respect to the perturbation Δ, and it can be seen that PLZ → 0. Conversely, the diabatic limit is the case where ν Δ, where PLZ → 1. For experiments where the coupling energy Δ is a fixed quantity, control over ν can not only be used to probe the magnitude of this coupling energy (as explored in Chapter. 5), but can also be utilised to coherently produce a superposition state of an arbitrary two level system of the form HQ. The creation of such a superposition state can be described in the tran- sition matrix formalism [68, 69, 70, 71], where the transition at the level crossing is given by matrix ⎛ ⎞ √ √ ⎜ PLZ 1 − PLZ exp(iϕS)⎟ M = ⎝ √ √ ⎠ (1.64) − 1 − PLZ exp(−iϕS) PLZ

for the case where |φ1 crosses |φ2 from the low energy side. For a passage through the other direction, the equivalent transition matrix is simply the transpose of M. The phase factor ϕS from Eq. (1.64) is termed the Stokes

37 CHAPTER 1 phase,givenby

   π |Δ|2 |Δ|2 |Δ|2 ϕ = +argΓ 1 − i + ln − 1 . S 4 |ν| |ν| |ν|

Here, Γ(...) is the gamma function, which has asymptotic limits [71] ⎧  ⎪ | |2 2 ⎨ Δ | |2 | | |Δ| C | | , Δ ν arg Γ 1 − i ν ⎪ | |2 | |2 |ν| ⎩ − π − Δ Δ − | |2 | | 4 |ν| ln |ν| 1 , Δ ν where C is the Euler constant.

Landau-Zener-St¨ukelberg processes

The transition matrix given in Eq. (1.64) is dependent upon the direction of passage through the anti-crossing, and it can be observed that for multiple passages back and forth through this anti-crossing, the Stokes phase factor ϕS is accumulative. It is therefore more convenient to describe each transition by a diabatic unitary evolution matrix N,givenby ⎛ ⎞ √ √ ⎜ 1 − PLZ exp(−iϕS) − PLZ ⎟ N = ⎝ √ √ ⎠ (1.65) PLZ 1 − PLZ exp(iϕS) as derived in detail in Ref. [71]. We have a modification to the Stokes phase parameter ϕS = ϕS − π/2 due to the evolution. Landau-Zener-St¨ukelberg (LZS) processes describe the passage through the anti-crossing followed by a dwelling time τD and second, identical passage

38 CHAPTER 1 back through the anti-crossing. The time evolution during the dwell time simply results in the unitary ⎛ ⎞ ⎜ 10⎟ U = ⎝ ⎠ , (1.66) − − δE 0 exp( i  τD) representing the accumulation of phase based on the energy separation of the two eigenenergies at the dwelling position in ε detuning (recall Fig. 1.2). Therefore, the evolution of this system under this regime becomes

|ψf  = N · U · N |ψi , (1.67)

for the case where the initial state is the low energy state |ψi = |ψg |φ1 (i.e. the case for a negative detuning, where |ε| |Δ| in Fig. 1.2). After passing through the anti-crossing to/from a location |ε| |Δ|, which is positive in detuning ε, the probability of measuring the excited state |ψe =

|φ2 is given by

2 2 Pe = |φ2|ψf | =4PLZ (1 − PLZ )sin (ΦSt), (1.68)

where ΦSt is the St¨uckelberg phase [65, 72]

δE · τ ϕ + ϕ Φ = D + S1 S2 , (1.69) St 2 2

th where ϕSn is the Stokes phase for the n passage.

39 CHAPTER 1

Due to the dependence of the excited state probability on the energy separation δE within the time evolution, the LZS process is a useful inter- ferometric tool for Hamiltonian characterisation [73, 60] and benchmarking the effect of decoherence due to the environment [74, 75].

Landau-Zener-St¨ukelberg-Majorana interferometry

The theory presented in the previous section can be extended for strongly driven systems, giving Landau-Zener-St¨ucelberg-Majorana (LZSM) interfer- ometry. An extensive discussion of these experiments is given in Ref. [71] and the citations therein. A brief introduction is as follows: The LZSM experi- ments are performed on systems that are continuously driven in time using input signal ε(t)=A cos(2πft). This is a simple extension to the above theory which studies the dynamics of n passages through the anti-crossing, where the state is simply modelled as

n |ψf  =(UR · N · UL · N) |ψi . (1.70)

Here, |ψi is the ground state on the right side of the anti-crossing and

UL/UR represent the phase evolution on the left/right side of the anti-crossing respectively.

40 CHAPTER 1

1.5 Noise and quantum coherence

This section discusses the concept of a qubit which is coupled to a noisy environment, and how these noise sources can affect the overall coherence of a qubit and the integrity of the quantum information it represents.

1.5.1 Qubit noise

An ideal qubit Hamiltonian HQ is expressed in Eq. (1.14). For this sim- ple, time independent Hamiltonian, the dynamics are well understood and described by detuning ε and coupling Δ terms. Noise sources in the qubit en- vironment are effects which result in time variance of these two parameters, such that the Hamiltonian is instead described as:

HQ(t)=ε(t)σzˆ +Δ(t)σxˆ, (1.71) where the time dependent components of ε(t) and Δ(t) are the variation terms δε(t)andδΔ(t) respectively, such that

ε(t)=ε + δε(t), (1.72)

Δ(t)=Δ+δΔ(t). (1.73)

The postulates of quantum mechanics as presented in Sec. 1.2 have fun- damental implications of how environmental noise fluctuations impact qubit performance. From the Schr¨odinger equation, it is clear that qubit evolution

41 CHAPTER 1 in a noisy environment leads to the temporal integration over those noise sources. One of the implications of the fourth postulate (Sec. 1.2.4) is that experimental results will often require measurement statistics to probabilisti- cally determine the final state. The integration over noise spectra will result in this final state being subject to a probability envelope when comparing shot-to-shot experiments. This often leads to concepts such as the visibility of qubit readout. These effects must be taken into account when determining an adequate level of measurement statistics. What follows is a brief discus- sion of common examples for each noise type in the context of semiconductor qubits. This discussion highlights the fact that the time dependence in the qubit Hamiltonian is intrinsically tied to fluctuations of environmental elec- tric and magnetic fields.

Environmental sources of δε(t)

As discussed in Sec. 1.4.1 for single electrons driven by a resonant magnetic field, the dynamics are described by the Rabi formula. For this system, the qubit energy separation ε is represented as the difference between the driv- ing frequency ω and Larmor frequency ω0. This fundamental frequency of the spin is defined by the externally applied magnetic field B0,andthere- fore fluctuations in this field can result in δε(t) during electron resonance. Fluctuations of this variety can, for example, be produced by the ensemble effect of a nuclear spin bath in a wide range of semiconductor materials. The detuning parameter ε is also often given in the context of charge based anti-

42 CHAPTER 1 crossings produced by tunnel coupling between two quantum dots [76]. Here the detuning is a function of the electrostatic potentials used to define the quantum wells, often manipulated in semiconductor quantum dots through voltages applied to gate electrodes. Variation in these potentials can arise due to phenomena such as volatile charge or spin fluctuations due to mate- rial defects in oxides and at interfaces (often giving the classic 1/f α noise spectra) [77], or as voltage noise capacitively coupled from surrounding gate electrodes.

Environmental sources of δΔ(t)

For electrons driven by a resonant magnetic field, as described by the Rabi formula, the qubit energy coupling term Δ is described by

gμ |B | Δ=ω = B 1 (1.74) 1 

and therefore any fluctuations in magnitude |B1(t)| of the driving field will result in a δΔ(t) term. This effect has been discussed in Ref. [78] with the potential source being thermal Johnson-Nyquist radiation, coupling to an electron spin qubit through the on-chip microwave antenna [79]. For the Landau-Zener crossings discussed in Sec. 1.4.2, the coupling term Δ is instead described by a number of different phenomena. Common phys- ical phenomena such as the tunnel coupling, t0, between two dots can give rise to such effects [80] as well as coupling between the two-electron singlet

43 CHAPTER 1

|S and polarised triplets |T± due to spin-orbit coupling [81, 82, 83, 84, 85]. Both of these physical phenomena are also intrinsically tied to the variation in electrostatic potentials which define the qubit, similar to δε(t). Another means of creating noise of this type is the Overhauser effect between the qubit and a fluctuating nuclear spin environment [86, 87, 88].

1.5.2 Qubit characteristic times

The lifetime and integrity of quantum states intrinsic to the qubit can be represented through two characteristic time-frames known as T1 and T2. These characteristics are adopted from the field of nuclear magnetic reso- nance (NMR) and are a method through which the qubit interaction with the noisy environment can be assessed.

T1 processes

Noisy fluctuations can perturb the qubit operation, adding and subtracting energy from the system. Eventually, any quantum system is brought to an equilibrium with the environment, and the timescale over which the qubit can persist out of equilibrium is referred to as the T1 time. In nuclear physics, this timescale is known as spin-lattice relaxation time, and is a measure of the mechanisms under which the spin vector within a static magnetic field reaches thermodynamic equilibrium with its surroundings (i.e. the “lattice”). For a singular electron, this is the relaxation from an excited state into the ground eigenstate as determined by an externally applied magnetic field, and results

44 CHAPTER 1 in a change in the energy of the qubit (thus, termed a “relaxation” process). Experimentally, the time can be gauged by preparing some superposition state |ψ = α |↑ + β |↓, and monitoring relaxation of this state into the spin down ground state |↓. The characteristic T1 time is extracted from fitting the excitation probability to the exponential form:

2 P↑(t)=α exp(−t/T1). (1.75)

T2 processes and related measures

A qubit interacting with the environment can also undergo an interaction where energy is absorbed or lost in the form of a phase shift. The result is the loss in coherence in the plane orthogonal to the magnetisation axis. This process is given by a timescale called T2. From NMR theory, the T2 time is the mechanism by which the transverse component of the magnetisation vector exponentially decays towards an equilibrium value. Therefore, it is often associated as a measure of the NMR signal strength. The characteristic time is observed due to the interactions of an ensemble of spins dephasing from each other through sampling local magnetic field inhomogeneities. This results in the respective accumulated phases of each spin in the ensemble to deviate from the expected value.

In the context of quantum computers, the value of T2 is used as an initial characterisation of a qubit, as the elements of a quantum computer must operate on timescales much faster than T2 in order to execute algorithms

45 CHAPTER 1 in a fault-tolerant manner. This necessity for fast drive with respect to

T2 is particularly evident in the context of the work presented in Ref. [89], which describes the power-law decay of the Rabi formula in the presence of decoherence as

P A ↑(tMW)= α cos (2πfRabitMW + ϕ) (1.76) tMW

where tMW is the microwave burst duration. As microwave power (i.e. |B1|) decreases, the susceptibility to the influences of dephasing increases. This is due to slower Rabi oscillations resulting in a longer sample over the environ- ment for each rotation period.

The actual value of T2 for a singular qubit is a difficult value to attain experimentally, as the act of taking an ensemble of measurement statistics tends to obscure the information gathered. As a result, there are several experiments which are employed to probe qubit coherence, resulting in a family of T2 measurements as described below.

∗ Pure dephasing T2 ∗ The pure dephasing time T2 represents trial-to-trial variations in qubit res- onant frequency, leading to an apparent damping of wave interference in an ensemble of measurements [1]. Fitting this damping rate yields the time

∗ constant T2 , which also is also representative of the coherence of the Lamor oscillations of a spin [90]. The standard experiment used to gauge this de-

46 CHAPTER 1

a) X Dwell Time ɒ X 2 2

ۄ՛| ۄ՛| ۄ՛| ۄ՛|

y y y y x x x x

ۄ՝| ۄ՝| ۄ՝| ۄ՝|

b) X X Dwell Time ɒȀʹ X Dwell Time ɒȀʹ 2 2

ۄ՛| ۄ՛| ۄ՛| ۄ՛| ۄ՛| ۄ՛|

y y y y y y x x x x x x

ۄ՝| ۄ՝| ۄ՝| ۄ՝| ۄ՝| ۄ՝|

∗ Figure 1.4: a) Ramsey experimental protocol yielding T2 . For each stage of the protocol an ensemble of qubit measurements is shown on Bloch spheres in the rotating frame. As a function of τ, a distribution of dephased states is projected onto thez ˆ-axis for probabilistic quantum measurement. b) H Hahn echo experimental protocol yielding T2 as a function of decaying spin- up probability P↑(τ). As illustrated by accompanying Bloch spheres, the X refocusing pulse applied at τ/2 reverses the dephasing due to fluctuations with timescales greater than τ. cay timeframe is the Ramsey interference experiment, which is described in Fig. 1.4a. For a qubit initial state of |↓, the first X /2 pulse (π/2 rotation around thex ˆ-axis) rotates the spin vector to they ˆ-axis. The qubit then pre- cesses about thez ˆ-axis for a time τ. Finally, the second X /2 pulse rotates they ˆ component of the state back to thez ˆ-axis. If the driving field is ex- actly on-resonance with the Larmor frequency, the qubit is stationary in the

47 CHAPTER 1 rotating frame. However, off resonant driving is detuned in frequency by δω, and bothx ˆ andz ˆ components of the final state beat as a function of δω and τ. This beating produces an oscillatory pattern which is characteristically known as Ramsey fringes. The decay time of these oscillations in turn yields

∗ the T2 dephasing time.

H Hahn echo: T2 The pure dephasing time is subject to all noise sources coupling into the qubit system, however, a single experimental trial may retain its phase co-

∗ herence for much longer than T2 . Certain environmental fluctuations which cause the pure dephasing to be shorter than T2 can be removed by applying additional refocusing pulses such as those shown in Fig 1.4b and Fig 1.5. The simplest of such a sequence is the Hahn echo [91] depicted in Fig 1.4b. The additional X (full π) rotation in the centre of the sequence decouples the ef- fect of scalar spin-spin couplings and spatial variations of the static magnetic field along thez ˆ-axis [91, 90], provided the variations are constant through- out the experiment. This results in the spin being in-phase by the end of the second time interval τ/2, and the resulting echo signal decays as a func-

H tion of τ. The decay time constant is given by T2 , and is a measure of the intrinsic T2 time.

48 CHAPTER 1

CP CP decoupling: T2 The Hahn echo can only decouple fluctuations which occur on a timescale greater than τ. However, this limit can be reduced to decouple fluctuations which occur on the order of τ/n. This is achieved through applying a train of n refocusing X pulses as shown in Fig 1.5a. This is known as the Carr-

CP Purcell sequence [92] and produces the characteristic decay time T2 , which

H extends further towards T2 when compared to T2 for increasing refocusing pulses n. However, this pulse train is limited through the susceptibility to accumulate fixed over- or under-rotation errors in the control pulse.

CPMG CPMG decoupling: T2 This limitation of the CP experiment was addressed by shifting the phase of the refocusing pulses to an orthogonal axis [93] (i.e. Y pulses rather than X ) as shown in Fig. 1.5b. Thus, pulse length errors are compensated on even-numbered echoes. Due to this additional robustness against control

CPMG errors, the T2 time is perhaps one of the best estimates for the intrinsic

T2 coherence time of the qubit.

Dynamical Decoupling A core assumption for the previously presented noise decoupling algorithms is the application of periodic, zero-width π-pulses. These decoupling tech- niques have been extended to preservation of an arbitrary quantum state

49 CHAPTER 1

a) ɒ ɒ X x”‡’‡ƒ–• X ʹ X ɒȀ ʹ 2 2

b) ɒ ɒ X x”‡’‡ƒ–• X ʹ Y ɒȀ ʹ 2 2

c) Y ɒȀX ɒȀY ɒȀX ɒȀ

Y p1 X p1 Y p1 X p1

Y p2 X p2 Y p2 X p2

ɒ ɒ X X ʹ Y p X p Y p X p ʹ 2 Ǧͳ Ǧͳ Ǧͳ Ǧͳ 2

CP Figure 1.5: a) CP experimental protocol yielding T2 , refocussing fluctua- tions which occur down to timescales of τ/n. b) CPMG experimental proto- CPMG col yielding T2 . Refocussing down to τ/n is maintained, however the se- quence is robust against control errors in the form of over- or under-rotations c) Protocol for Concatenated Dynamical Decoupling illustrating the tempo- ral recursive structure which aims to correct for errors at different levels of resolution.

50 CHAPTER 1

(i.e. a quantum memory) through processes known as Dynamical Decoupling (DD). One example of particular note in literature is that of Concatenated Dynamical Decoupling (CDD). It has been shown in Ref. [94] that (CDD) sequences can be made to correct for errors at different levels of resolution through temporal recursive structure as shown in Fig. 1.5c. The sequence type can prevent error build-up often observed in standard decoupling ex- periments. Such CDD pulses are often discussed in the context of quantum memory [94] or as a means of preserving the qubit state during idling tasks for times as close to the limitation of the qubit T2 coherence time as possi- ble [95].

1.5.3 Noisy qubit channels

It is often convenient to represent the application of a noisy operation on a quantum bit as the passage of that qubit though a noisy quantum channel. The channel is often represented by a state transformation given by a linear map E, with operational elements {Ej} such that

E ≡ † (ρ) EjρEj . (1.77) j

Summarised below are common channels used to represent noisy quantum processes or errors.

51 CHAPTER 1

Bit-flip channel

The bit-flip channel represents operations ρ →EX (ρ), which makes the trans- formation between |↓ ↔ |↑ states with an error probability p as represented in Fig. 1.6a.

a) b) z

1-p ۄͳ| ۄͳ| p y p x ۄp |Ͳ-1 ۄͲ|

Figure 1.6: a) Representation of the bit-flip channel EX where bit-flip errors occur with probability p. b) Deformed surface illustrating the passage of all pure states through the bit-flip channel. Thex ˆ components of each state are preserved, while they ˆ andz ˆ components are reduced by 1 − 2p

The operational elements of this channel are given by

√  E0 = p 1,E1 = 1 − pσxˆ. (1.78)

52 CHAPTER 1

Phase-flip channel

The phase flip channel is represented by the operation ρ →EZ (ρ) where the operational elements of EZ are given by

√  E0 = p 1,E1 = 1 − pσzˆ. (1.79)

The effect of this channel is illustrated in Fig. 1.7a where thez ˆ components a) b) z z

y y x x

Figure 1.7: Surfaces on the Bloch sphere representing all pure states passing through the a) EZ phase-flip channel and b) EY bit-phase-flip channel. of each state are preserved, while thex ˆ andy ˆ components are reduced by a factor of 1 − 2p.

53 CHAPTER 1

Bit-phase-flip channel

The bit-phase-flip channel EY is illustrated in Fig. 1.7c and is the combination of both a bit-flip and phase-flip error occurring on the system simultaneously. The operational elements are

√  E0 = p 1,E1 = 1 − pσyˆ. (1.80)

Depolarising channel

The depolarising channel ρ →E(ρ) represents a channel which replaces the input state ρ with a completely mixed (or depolarised) state 1/2 with prob- ability p. The channel is described by

p1 E(ρ)= +(1− p)ρ. (1.81) 2

The effect on the set of pure states is illustrated in Fig. 1.8. Using the fact that 1 1 = (ρ + σ ρσ + σ ρσ + σ ρσ ) , (1.82) 2 4 xˆ xˆ yˆ yˆ zˆ zˆ the depolarising channel can be re-written as

 3p p E(ρ)= 1 − ρ + (σ ρσ + σ ρσ + σ ρσ ) . (1.83) 4 4 xˆ xˆ yˆ yˆ zˆ zˆ

54 CHAPTER 1

z

y x

Figure 1.8: Surfaces on the Bloch sphere representing all pure states passing through the E depolarising channel.

Therefore, the operational elements of the channel are given by

 E0 = 1 − 3p/4 1, (1.84) √ E1 = pσxˆ/2, (1.85) √ E2 = pσyˆ/2, (1.86) √ E3 = pσzˆ/2, (1.87) demonstrating that the depolarising channel combines the action of all three previous channels equally together with a combined error probability of p. This depolarising channel is an important tool used to model qubit behaviour in the presence of a noisy environment, and will be re-visited again in the discussion of the Randomised Benchmarking experiment.

55 CHAPTER 1

1.6 Experiments for benchmarking qubit

performance

In this section two different families of experiments which aim to bench- mark qubit performance are introduced. The tomographic family (including state and process tomography) are considered very useful for analysing the performance of qubit systems with the capacity to assist in error diagno- sis. However, these tomographic experiments do not scale efficiently with increasing numbers of qubits, requiring more and more experimental over- head in order to extract all required data. These experiments are also unable to isolate qubit control errors from for state-preparation-and-measurement (SPAM) errors. On the other hand, the Randomised Benchmarking family of experiments is robust against SPAM errors and the experiment has been shown to scale efficiently with increasing numbers of qubits. The experiment is useful in the regard that the extracted information and experimental output is expressly the qubit control fidelity as a singular unit of merit. However, the experiment often suffers from strict assumptions being made on input noise processes and from having arguably less capacity to diagnose system error processes for self-improvement of qubit control fidelity.

56 CHAPTER 1

1.6.1 Tomographic experiments

The two varieties of tomographic experiments attempt to characterise the quantum system by determining elements of the density matrix ρ or an arbi- trary quantum process E. The description of these two experiments presented below follows the summarised discussion of Ref. [90].

State Tomography

In order to determine the overall state of a prepared quantum system, mul- tiple measurements of the same prepared state are performed to determine the individual elements of ρ [96, 97, 98] ⎛ ⎞

⎜ρ11 ρ01⎟ ρ = ⎝ ⎠ . (1.88) ρ10 ρ00

The repeated measurements of the same state are performed with respect to various measurement bases until all elements of ρ can be extracted from a set of linear equations. Measurement in the standard qubit basis of {|↓ , |↑} yields ρ00,withρ11 =1− ρ00. For the off-diagonal elements of ρ, it is often more convenient to make a basis transformation to rotate ρ with respect to the standard measurement basis, rather than performing the measurements in another basis. Therefore, transforming ρ using a 90◦ rotation around thex ˆ-axis and measuring in the standard basis yields Im(ρ10)=−Im(ρ01). Further, a similar measurement after the same rotation around they ˆ-axis

57 CHAPTER 1

will yield Re(ρ10)=Re(ρ01), allowing for the full density matrix ρ to be determined. The expansion into state tomography of multiple qubits results in the production of linear equations from which the density matrix can still be determined, the details of which are summarised in Ref. [90].

Process Tomography

Process tomography is an experiment which builds upon state tomography, but is instead used to determine some process E which acts upon an arbitrary quantum state ρ [99, 100, 101, 102]. The overall procedure is to determine the output state of E for a set of input states which form a basis for the system Hilbert space, and then compute the entire transfer function from the finite set of input-output pairs. The transformation of the arbitrary quantum state ρ by E is the linear map E(ρ) ρ → , (1.89) Tr(E(ρ)) where E(ρ) can be expressed in operator-sum representation [103, 15]

E † (ρ)= AjρAj. (1.90) j

This expansion is not unique, but a fixed set of operators Aj whichforma basis for the set of operators on the state space can always be found such

58 CHAPTER 1 that

E † (ρ)= χpqApρAq. (1.91) p,q

Because Aj are fixed, the Hermitian operator χ completely describes E. Determining χ for a system of n qubits requires a basis of 4n linearly independent density matrices ρk which span the Hilbert space. A set of linear equations of the form presented in Eq. (1.91) can then be constructed from measuring E(ρk) for all k, allowing χpq to be determined.

1.6.2 Randomised Benchmarking

Acknowledgement: This introduction to the Randomised Benchmarking (RB) experiment was largely influenced by the introduction presented in Ref. [58]. The discussions and results presented therein were produced in collaboration with Dr. M. Veld- horst, Dr. R. Harper, Dr. C. H. Yang, Dr. S. D. Bartlett, Dr. S. T. Flammia, and Dr. A. S. Dzurak. The full acknowledgements for this manuscript can be found in Appendix A.

This experiment will be elaborated upon in Chapter. 3, but will be dis- cussed briefly for comparison with the tomographic experiments. The RB experiments have the useful advantage of being robust against these SPAM errors, but pay the price of being subject to a near over-constraining set of

59 CHAPTER 1 assumptions to be able to extract meaningful results. The standard random-

Figure 1.9: Randomised benchmarking consists of applying multiple se- quences of random Clifford gates, a final recovery Clifford gate to ensure that each sequence ends with the qubit in an eigenstate, and reading out the qubit state. In interleaved randomized benchmarking, an additional test Clif- ford gate is inserted in between the random Clifford gates. Figure is sourced from Ref. [58]. ized benchmarking procedure involves subjecting a quantum system to long sequences of randomly sampled Clifford gates followed by an inversion step and a measurement, as depicted in Fig. 1.9. The unitary operations of the Clifford group G are those that map the set of Pauli operators to itself under conjugation. They are a discrete set of gates that exactly reproduce the uni- form average gate fidelity, averaged over the set of all input pure states [104]. An alternate version known as interleaved benchmarking [105] inserts a sys- tematic application of a given gate, such as the H gate shown in Fig. 1.9. The difference from the reference sequence gives information about the spe- cific average gate fidelity of the given gate, rather than the average fidelity additionally averaged over the ensemble of gates.

60 CHAPTER 1

z

y D

CN x CI

Figure 1.10: Bloch sphere representation for the breakdown of a general noisy operation CN into an ideal CI rotation followed by a noise operation D. Figure is sourced from Ref. [58].

Consider a general noise process D, depicted in Fig. 1.10, which repre- sents the deviation of a noisy Clifford gate CN from an ideal unitary Clifford operation CI :

CN = DCI . (1.92)

The above equation uses the formalism of completely positive maps [15], and the multiplication corresponds to composition of maps. The standard ap- proach to randomised benchmarking makes the assumption that D does not depend on the choice of CI or other details such as time, but our simulations and of course real experiments will include such a dependence. The fundamental result of randomised benchmarking [106] is that for suf- ficiently well-behaved noise the observed fidelities only depend on the average

61 CHAPTER 1

error operation ED averaged over the Clifford group G given by

1 −1 ED = C DC , (1.93) |G| I I CI ∈G as well as any SPAM errors present in the system. Furthermore, standard tools from representation theory reduce this average error operation to one that is nearly independent of D, and is characterised by just a single param- eter p. In particular, it is a depolarising channel E with p = p(D) being the polarisation parameter (i.e., the probability of the information remaining un- corrupted as it passes through the channel). For a d-dimensional quantum system, the action of the depolarising channel is given by

1 E(ρ)=pρ +(1− p) , (1.94) d and the polarisation parameter is related to the noisy deviation D by the ¯ average gate fidelity Favg(D) according to [106]

1 − p F¯ (D)= dψψ|D(|ψψ|)|ψ = p + , (1.95) avg d where the integral is a uniform average over all pure states. For a randomised benchmarking sequence comprised of m+1 total Clifford gates (including the +1 for the recovery operation), the average sequence fidelity is given by [106] ¯ m Fm = Ap + B. (1.96)

62 CHAPTER 1

Here the parameters A and B quantify the SPAM errors and are given by [106] A =Tr ED(ρ − 1/d) ,B=Tr ED(1/d) , (1.97) and ρ and E are the noisy state preparations and measurements implemented instead of the ideal desired states and measurements. ¯ A typical benchmarking experiment proceeds by estimating Fm for several values of m and fitting to the model in Eq. (1.96) to extract the p, A,andB fit parameters, and then using Eq. (1.95) to report an ensemble average of ¯ the average gate fidelities Favg of the gates. This derivation of Eq. (1.96) assumes certain features about the noise, namely that it has negligible time and gate dependence, and that non- Markovian effects are not present at time-scales on the order of the gate time. The limits to the validity of this assumption have been probed be- fore [107, 108, 109], and in particular it was noted via numerical simulations by Epstein et al. [107] that the exponential model of fidelity decay no longer holds in the presence of 1/f noise, resulting in a noise floor to the accuracy of the benchmarking experiment.

63 CHAPTER 1

1.7 Systems of two qubits

This section moves from the description of a single qubit system, to the complete Hamiltonian description of two coupled qubits. This presents the first stage of moving towards larger coupled quantum systems such as those which form quantum processors. Here, expanding the basis to describe two qubits is discussed, as well as systems of two coupled electrons.

1.7.1 Expansion from the single-spin basis to a two-

spin basis

The single qubit basis is presented in Sec. 1.1. In contrast, this section details how the computational basis can be expanded to include two electrons. As each electron is a two-level system, we require 2n basis states to represent the n different electrons. For two electrons in the computational basis, these states are: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜1⎟ ⎜0⎟ ⎜0⎟ ⎜0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜0⎟ ⎜1⎟ ⎜0⎟ ⎜0⎟ |↑↑ ⎜ ⎟ |↑↓ ⎜ ⎟ |↓↑ ⎜ ⎟ |↓↓ ⎜ ⎟ = ⎜ ⎟ , = ⎜ ⎟ , = ⎜ ⎟ , = ⎜ ⎟ . ⎜0⎟ ⎜0⎟ ⎜1⎟ ⎜0⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 0 0 0 1 (1.98) It can be observed that these states can be produced via the tensor prod-

64 CHAPTER 1 uct between two individual spins:

|ψφ = |ψ⊗|φ . (1.99)

It follows that the Pauli spin operators can be expanded in a similar manner to form a two qubit set:

⊗ 1 Q1: σmˆ 1 = σmˆ (1.100) 1 ⊗ Q2: σmˆ 2 = σmˆ

where σmˆ are the single spin Pauli spin matrices as outlines in Eq. (1.5) for Cartesian elementsm ˆ ∈{x,ˆ y,ˆ zˆ}. Further, vectorial representation as presented in Sec. 1.1.3 holds for the two qubit case:

σn = σmˆ n . (1.101) mˆ

65 CHAPTER 1

1.7.2 Two qubit Hamiltonian in the spin basis

When considering a stable picture of two electrons in separated quantum dots, the system expansion described above allows us to asses the effects of a magnetic field vector B = Bx + By + Bz on the electrons in two dots. An

mˆ n electron on dot n experiences a Zeeman energy EZ in vector directionm ˆ , such that the system Hamiltonian for two electrons under magnetic field B , is expressed as:

mˆ n · H2e,B = EZ σmˆ n . (1.102) n,mˆ

Here, σn,mˆ is now a two spin Pauli matrix as outlined in Sec. 1.7.1. In expanded form this appears as the following: ⎛ ⎞ zˆ1 zˆ2 xˆ2 − yˆ2 xˆ1 − yˆ1 ⎜ EZ + EZ EZ iEZ EZ iEZ 0 ⎟ ⎜ ⎟ ⎜ xˆ yˆ zˆ zˆ xˆ yˆ ⎟ ⎜ E 2 + iE 2 E 1 − E 2 0 E 1 − iE 1 ⎟ |  ⎜ Z Z Z Z Z Z ⎟ |  H2e,B ψ = ⎜ ⎟ ψ , ⎜ Exˆ1 + iEyˆ1 0 −Ezˆ1 + Ezˆ2 Exˆ2 − iEyˆ2 ⎟ ⎝ Z Z Z Z Z Z ⎠ xˆ1 yˆ1 xˆ2 yˆ2 − zˆ1 − zˆ2 0 EZ + iEZ EZ + iEZ EZ EZ (1.103)

|  |↑↑ |↑↓ |↓↑ |↓↓ 2 where ψ = a1 + a2 + a3 + a4 ,and j aj = 1 It is conve- mˆ mˆ 1 mˆ 2 mˆ mˆ 1 − mˆ 2 nient to make the simplification of EZ = EZ + EZ and δEZ = EZ EZ wherever possible. These represent the average and difference between the Zeeman energies respectively, which often become the relevant field depen- dent energy scales within the system of two spins as investigated in Ch. 6.

66 CHAPTER 1

1.7.3 Systems of two coupled electrons:

This section discusses the unique physical effects which are introduced through allowing coupling between the two qubits. This coupling energy takes a sim- ilar form of Δ for the qubit Hamiltonian HQ.

Exchange energy and the singlet-triplet basis

The previous sections discuss the expansion of the computational basis for a qubit system from one electron on one dot, to a basis which is capable of representing two qubits on two individual and isolated quantum dots. If, in- stead, those two individual electrons are configured such that the two dots allow a degree of coupling between the electrons, the system is modified by that coupling. One such effect is known as Heisenburg exchange coupling, denoted by J, and arises due to the exchange symmetry of the two indistin- guishable electrons [63, 110]:

H2e,J = J( σ1 · σ2). (1.104)

This is a system of three degenerate eigenenergies (triplet states) of ET =

J/4, and a (singlet state) eigenenergy of ES = −3J/4. The eigenstates cor- responding to these are represented in the computational basis of Eq. (1.98) as

1 Triplet States: |T+ = |↑↑ , |T0 = √ (|↑↓ + |↓↑), |T− = |↓↓ 2 1 Singlet States: |S = √ (|↑↓ − |↓↑) . 2 (1.105)

67 CHAPTER 1

Tunnel coupling, system detuning and exchange

The ability for an electron to tunnel from one dot to another is governed by a coupling energy tc known as the tunnel coupling. This tunnel coupling sets up an anti-crossing between certain quantum states in a similar fashion to that which is observed in Fig. 1.2. This allows for a system of two electrons on two individual dots (such as the one illustrated in section. 1.7.2), to be transferred into a system of two electrons on a single dot based on a detuning ε via the same principles discussed in Sec. 1.4. However, for a system of two electrons on a single quantum dot, Pauli exclusion principle dictates that the lowest energy of the two electron system is a singlet state. Therefore, the triplets are only accessible via higher energy states which offer a new degree of freedom other than the spin of the two electrons. For silicon, these are the valley and orbital states (introduced in more detail in Ch. 2). Therefore, to describe the new system Hamiltonian which includes the effects of exchange, the isolated system H2e,B of Eq. (1.103) needs to be aug- mented with an additional tunnel coupled state, the lowest state representa- tive of two electrons on the same dot, |(0, 2)S. For this state, both electrons are placed on a singular dot, and the combined spin state is a singlet. The resulting Hamiltonian is as follows:

68 CHAPTER 1

⎛ ⎞ zˆ − xˆ2 − yˆ2 xˆ1 − yˆ1 ⎜ EZ ε/2 EZ iEZ EZ iEZ 0ΔSOC ⎟ ⎜ √ ⎟ ⎜ xˆ2 yˆ2 zˆ − xˆ1 − yˆ1 ⎟ ⎜ EZ + iEZ δEZ ε/20EZ iEZ tc(ε)/ 2 ⎟ ⎜ √ ⎟ ⎜ xˆ1 yˆ1 zˆ xˆ2 yˆ2 ⎟ H2e |ψ = ⎜ E + iE 0 −δE − ε/2 E − iE −t (ε)/ 2⎟ |ψ , ⎜ Z Z Z Z Z c ⎟ ⎜ zˆ ⎟ ⎜ 0 Exˆ1 + iEyˆ1 Exˆ2 + iEyˆ2 −E − ε/2 −Δ ⎟ ⎝ Z Z Z Z Z SOC ⎠ √ √ ΔSOC tc(ε)/ 2 −tc(ε)/ 2 −ΔSOC ε/2 (1.106)

where |ψ = a1 |↑↑ + a2 |↑↓ + a3 |↓↑ + a4 |↓↓ + a5 |(0, 2)S. This represen- tation is illustrated in Fig. 1.11, and demonstrates an anti-crossing between the two singlets |(0, 2)S and |(1, 1)S, in charge configuration (0,2) and (1,1) respectively. The distance from the anti-crossing is given by detuning ε,and represents the shifting of chemical potential from one dot to the other, with- out changing the net energy of the system [39]. The system also introduces

a coupling term ΔSOC which will be discussed later.

1.7.4 Two qubits in the singlet-triplet basis

The system Hamiltonian presented in Eq. (1.106) can be transformed from the computational basis to that of a basis solely described by singlet and triplet states of Eq. (1.105).

69 CHAPTER 1

|T ሺɂሻ ۄൌȁ՛՛ۄ + |T_ ʹο ۄൌȁ՝՝ۄ

2tc Energy T| ۄ ۄ 0(1,1)| |(0,2)S ۄ (0,2) (1,1) 0 Detuning Ԫ

Figure 1.11: Anti-crossing between the (1,1) states and the (0,2) singlet state. As the two dots are coupled together, a singlet-hybridizing exchange energy J exists.

⎛ √ ⎞ zˆ − − xˆ yˆ ⎜ EZ ε/20(δEZ + iδEZ)/ 20 ΔSOC ⎟ ⎜ ⎟ ⎜ − zˆ ⎟ ⎜ 0 ε/2 δEZ 00⎟ ⎜ √ √ ⎟ ⎜ xˆ yˆ zˆ xˆ yˆ ⎟ HST |ψ = ⎜ (−δE − iδE )/ 2 δE −ε/2(δE − iδE )/ 2 t ⎟ |ψ , ⎜ Z Z Z Z Z c ⎟ ⎜ √ yˆ ⎟ ⎜ 00(δExˆ + iδEyˆ)/ 2 −E − ε/2 −Δ ⎟ ⎝ Z Z Z SOC⎠

ΔSOC 0 t(ε) −ΔSOC ε/2 (1.107) |  |   |   |  |  |  where ψ = a1 T+ + a2 T0 + a3 (1, 1)S + a4 T− + a5 (0, 2)S is the same wavevector from Eq. (1.106), corresponding to the system illustrated in Fig. 1.11, but transformed into the singlet and triplet basis. We can take the liberty of choosingz ˆ such that the average Zeeman energy in thex ˆ andy ˆ

mˆ directions are zero, leaving only the difference δEZ as a relevant parameter.

70 CHAPTER 1

The above Eq. (1.107) is transformed again to represent the Hybridised sin- glet ground state |SH  and excited state |GH . By diagonalising the singlet terms [87] we form this change of basis by introducing tan(θ)= √2t(ε) : −ε− 4t(ε)2+ε2

⎛ ⎞ − ⎜ EZ ε/20 0 Δ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 −ε/20δEZ cos(θ)⎟  |  ⎜ ⎟ |  HST ψ = ⎜ ⎟ ψ , ⎜ 00−E − ε/2 −Δ ⎟ ⎝ Z ⎠ ∗ − ∗ Δ δEZ cos(θ) Δ ESH (1.108) where |ψ = b1 |T++b2 |T0+b3 |T−+b4 |SH , and the higher energy excited state |GH  has been dropped. The overall coupling term between the hybrid singlet state |SH  and polarised triplets |T± is given by the term   xˆ yˆ δEZ + iδEZ Δ=− √ cos(θ)+ΔSOC sin(θ) . (1.109) 2

This overall coupling is a combination of spin-orbit effects and magnetic field gradients perpendicular to applied field Bz (such as those attributed to the Overhauser field of a spin bath).

It is clear that when ε →∞,θ → 0andtheδEZ coupling terms dom- inate in the (1, 1), the ΔSOC coupling contributions introduced via (2S, 0) − − 1 are suppressed. Further, the exchange energy J(ε)=ET0 ESH = 2 ε + √ 1 2 2 → 2 ε +4t 0 and is dependent upon the Hybridised singlet energy ESH = √ − 1 2 2 → 2 ε +4t ET0 .

71 CHAPTER 1

1.7.5 Coupling energies and the singlet-triplet qubit

ۄ|

J

ۄ՛՝| ۄ՝՛|

ɁEz

ۄT0|

Figure 1.12: Two axis control for a single qubit constructed from a two- electron system.

It has been observed that this two electron system can, itself, be used to form a singular qubit [111] with two axis control given by the combina-

zˆ tion of exchange J(ε) and electron Zeeman energy splitting δEZ as shown in Fig. 1.12. This qubit was first demonstrated in the GaAs heterostruc- ture qubit platform (see Ref. [112]) but has been widely adapted to many spin based qubit architectures. This qubit is capable of being controlled completely via electrical means [112]. Further, the coupling energy J(ε)is capable of driving the |↓↑ ↔ |↑↓, which can be interpreted in the compu- tational basis as a two-qubit SWAP operation.

72 CHAPTER 1

1.7.6 Two-qubit logical operations

The Bloch sphere shown in Fig. 1.12 illustrates the capability of producing logical operations (or entangling operations) between two qubits. Some of these two-qubit gates, together with single-qubit gates, provide all of the nec- essary operations required for universal quantum computation. Collectively, they compose elements of what is known as a universal gate set for quantum computation. In this section, the two main classes of two-qubit gates seen in qubit systems are introduced. The first class are control-based operations, which take the form of the unitary ⎛ ⎞ ⎜ 10 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 01 0 0 ⎟ ⎜ ⎟ UB = ⎜ ⎟ (1.110) ⎜ 00B B ⎟ ⎝ 11 12 ⎠

00B21 B22 which will apply operation B to Q2 based on the logical state of Q1 (an opposite class of UB with Q2 as the control bit can also be found). The sec- ond class is based on performing swapping (or partial swapping) operations between two states. Below, some of the most commonly discussed gates are presented, as well as a brief description of how these gates can be transformed into other two-qubit operations.

73 CHAPTER 1

CZ or CPhase

This elementary two-qubit interaction is the manifestation of an accumulated phase between the |↑↓ and |↓↑ computational basis states. The circuit symbol for this operation is shown in Fig. 1.13a. This operation has been

(c ۄ| (a) b

ۄ՛՝| ۄ՝՛|

ۄT0|

Figure 1.13: a) Circuit symbol for the two qubit CZ gate. b) This operation is achieved via accumulating a phase of ϕ = π between the |↑↓ and |↓↑ states. c) Circuit symbol for the two qubit CNOT operation demonstrated on spin based SiMOS quantum dot qubits in Ref. [113]. As shown in Fig. 1.13b, the operation is based on a time evolution under energy splitting δEZ (to a global phase). The unitary representing this operation in the spin basis is given by ⎛ ⎞ ⎜ 100 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 010 0 ⎟ ⎜ ⎟ UCZ = ⎜ ⎟ . (1.111) ⎜ 001 0 ⎟ ⎝ ⎠ 000−1

It will be shown in a later section how the CZ operation can be combined with single qubit unitary operations to produce many other two qubit gates.

74 CHAPTER 1

CNOT

This gate simply performs a bit-flip operation on Q2 based on the state of Q1. The circuit symbol is shown in Fig. 1.13c, and the unitary associated with this operation is given by ⎛ ⎞ ⎜ 1000⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0100⎟ ⎜ ⎟ UCZ = ⎜ ⎟ . (1.112) ⎜ 0001⎟ ⎝ ⎠ 0010

A truth table which describes this operation is shown in Table. 1.1

Table 1.1: Truth table for CNOT operation. Input |Q1Q2 Output |Q1Q2 |↓↓ |↓↓ |↓↑ |↓↑ |↑↓ |↑↑ |↑↑ |↑↓

This CNOT operation is one such gate which can be used to form a universal gate set for quantum computation [15].

75 CHAPTER 1

SWAP family

As the name suggests, the two qubit SWAP operation directly swaps two computational basis states.

ۄ| (a) b

ۄ՛՝| ۄ՝՛|

ۄT0| c) d) e)

ξ ξ

Figure 1.14: a) Circuit symbol for the two qubit SWAP gate. b) This operation is achieved via exchanging√ between the |↑↓ and |↓↑ states. c) Circuit symbol for the two qubit SWAP gate. d) Circuit√ symbol for the two qubit iSWAP gate. e) Circuit symbol for the two qubit iSWAP gate.

The circuit symbol is shown in Fig. 1.14a, while the execution of this gate is shown in Fig. 1.14b. The unitary operation associated with this gate is

76 CHAPTER 1 given by ⎛ ⎞ ⎜ 1000⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0010⎟ ⎜ ⎟ USWAP = ⎜ ⎟ . (1.113) ⎜ 0100⎟ ⎝ ⎠ 0001

This variety of operation becomes especially useful when implementing quan- tum algorithms using qubits which have nearest neighbour coupling [62].

√ SWAP operations: In order to introduce a degree of entanglement between the two qubits, often √ √ the SWAP (such that ( SWAP)2 = SWAP) is implemented. The circuit element is illustrated in Fig. 1.14c, and is described by the unitary ⎛ ⎞ ⎜ 1+i 00 0 ⎟ ⎜ ⎟ ⎜ ⎟ 1 ⎜ 01i 0 ⎟ U√ = ⎜ ⎟ . (1.114) SWAP ⎜ ⎟ 1+i ⎜ 0 i 10⎟ ⎝ ⎠ 0001+i

√ As shown in a later section, the SWAP with single qubit unitaries can be √ shown to produce a CNOT operation, thus the SWAP is also an operation which yields a universal gate set for quantum computation. iSWAP operations: This operation is a natural operation for several systems as discussed in

77 CHAPTER 1

Ref. [114]. The effect is the combination of the SWAP with the CNOT (up to a phase factor). The circuit symbol is seen in Fig. 1.14d and unitary operation given by ⎛ ⎞ ⎜ i 000⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 00i 0 ⎟ ⎜ ⎟ UiSWAP = ⎜ ⎟ . (1.115) ⎜ 0 i 00⎟ ⎝ ⎠ 000i

This operation is also shown to be necessary for accessing the full set of two qubit Clifford gates [115, 116].

√ iSWAP operations: √ √ Similar to the the SWAP gate, the iSWAP can be used for a universal gate set. The circuit symbol is seen in Fig. 1.14e and the associated unitary is ⎛ ⎞ ⎜ 1+i 00 0 ⎟ ⎜ ⎟ ⎜ ⎟ 1 ⎜ 01i 0 ⎟ U√ = √ ⎜ ⎟ . (1.116) iSWAP ⎜ ⎟ 2 ⎜ 0 i 10⎟ ⎝ ⎠ 0001+i

Compositions of two qubit gates

As mentioned earlier (and in part, rigorously studied in Refs [117, 115, 116]), many of these two qubit gates can be transformed into another with the assistance of single qubit unitary operations.

78 CHAPTER 1

a)

-Y Y 2 2

b) -Y Y Y 2 2 2

-X -X X 2 2 2

c) -Y Y 2 2

-Y Y -Y Y 2 2 2 2

Figure 1.15: Two qubit operations composed from the CZ gate and single qubit unitary operations. These include formation of an equivalent a) CNOT, b) iSWAP, and c) SWAP operation. These combinations are reappropriated from Ref [116].

Figure. 1.15a demonstrates how another control-based two qubit opera- tion, the CNOT, can be constructed using the CZ gates. Further, The CZ can also be used to construct elements of the SWAP class of two qubit oper- ations as shown in Figure. 1.15b&c. This characteristic of the CZ operation makes it incredibly useful as it can be used as the sole two-qubit operation for constructing large scale qubit algorithms [116, 113, 47]. This topic will

79 CHAPTER 1 be discussed in later sections, and is indeed the focus of discussion in Ch. 7.

a) X -Y 2 2 X -Y

ξ ξ

X Y 2 2

b) -Y X Y 2 2 X 2

ξ ξ

-X 2

c)

Figure 1.16: Two qubit operations composed from the SWAP-class gates and single qubit unitary√ operations. These√ include formation of an equivalent CNOT from a) SWAP [38] and b) iSWAP [115]. c) TheCNOTcanbe linked back to a full SWAP operation through the use of three back-to-back CNOT operations, where control qubit alternates from Q2, then Q1, and then Q2 again.

The ability to construct many forms of two qubit operations with the assistance of single qubit unitary operations is also shown in Fig. 1.16a&b, where elements of the SWAP class are capable of creating the all-important CNOT operation. Further, Fig. 1.16c shows the capability to wrap back

80 CHAPTER 1 around to elements in the SWAP class from the CNOT. It is noted that the operations presented could be made more efficient through access to Z or ±Z/2 [38, 115], however, this example is presented in the context of two- axis control via ESR in a SiMOS electron spin qubit double-dot system.

81 CHAPTER 1

1.8 Introduction to logical qubits and fault-

tolerant quantum computation

This section introduces the basic concepts of logical qubits and the funda- mental error-detection and correction processes surrounding the concept of fault tolerant quantum computation. This section acts as the theoretical background for the concepts and goals presented in Ch. 7 (or Ref. [47]), and follows much of the discussion presented in Ref. [15]. The pinnacle goal of quantum computational hardware is to produce a 2D array of qubits which acts as a processor for the quantum computer. The constituent qubits of these processors will each be subject to environmental noise as discussed in Sec. 1.5. Similar to classical information theory, in order become more robust against the corrupting influence of these noise sources, logical qubits can be encoded upon the states of multiple constituent qubits within the processor. In order to encode a logical qubit for the further development of quantum error correction codes, there are a number of physical differences between classical and quantum information which need to be taken into account. These are presented in Ref. [15], and summarised below:

1. No cloning: Quantum information cannot be duplicated or copied as information can in classical systems. This is outlined in the no cloning theorem presented in Appendix E.

82 CHAPTER 1

2. Continuous errors: A quantum two level system operating within a noisy environment is subject to a continuum of environmental states. Therefore, there are an infinite number of error processes which can occur upon the qubit. For these error processes to be characterised and subsequently corrected, an infinite amount of precision is needed, leading to infinite resources.

3. Measurement destroys quantum information: The output of a classical channel can be directly observed to determine what error process has occurred. Postulate 5, presented in Sec. 1.2.5, states that the act of measuring the quantum system projects that system to a basis state of the measured observable. This irreparably alters the information carried by that channel/qubit.

Each of these characteristic differences between classical information and quantum information is addressed when constructing quantum error correct- ing codes.

1.8.1 Logical qubits and encoding schemes

The constituent qubits within a quantum processor are represented by quan- tum two level systems with two discrete eigenstates. For an electron spin in silicon, the computational basis is represented by the |↑ and |↓ eigenstates. These states can be mapped directly to logical states of the processor in the same way the current through a transistor is mapped for classical logic: The

83 CHAPTER 1 low-energy eigenstate |↓ ≡ |0 and the high-energy eigenstate |↑ ≡ |1. The inability to clone information fundamentally affects how a logical qubit is encoded upon the constituent qubits of a quantum processor. Sup- pose we have an arbitrary pure quantum state |ψ = α |1 + β |0 on a single qubit within the quantum processor. The state of this qubit can be encoded onto the state of three constituent qubits (including the original) within the quantum processor, such that

|0→|0L = |000 (1.117)

|1→|1L = |111 . (1.118)

Here, the state of the logical qubit is |ψL = α |1L + β |0L and represents the states of three encoded qubits |Q1,Q2,Q3 (with original qubit as Q1). This demonstrates that a superposition of the basis states in the original con- stituent qubit lead to an equivalent superposition of the logical states |1L and |0L for the logical qubit. The circuit executed on the quantum proces- sor which leads to the encoding of this logical qubit is shown in Fig. 1.17.

Note. The logical qubit presented above can be represented as a [3,1] logical qubit, with [n, k] system notation describing k logical qubits encoded form n constituent qubits. This notation can be extended to describe error correction codes as [n, k, d]. Here, d is the code “distance” metric, with d−1 representing the maximum number errors detectable by the code.

84 CHAPTER 1

ۄ߰|

ۄ߰L| ۄͲ| { ۄͲ|

Figure 1.17: Quantum processor circuit which encodes the state of a single qubit |ψ onto the states of three constituent qubits within the quantum processor (including the original). The final state is the encoded logical state |ψL.

1.8.2 Quantum error detection

Similar to classical information theory, the encoding of a single qubit state onto a logical qubit adds redundant information which allows for the detec- tion of error processes. Consider the example where the logical qubit |ψL undergoes some arbitrary operation within a noisy environment such that |   the state becomes ψL . For the sake of discussion, the effect of the en- vironment will be modelled by a passage through the bit-flipping channel

EX introduced in Sec. 1.5.3. There are direct measurements which can be |   performedonthestate ψL in order to determine what kind of error has occurred. The results of these specific measurements are termed error syn- dromes. These syndromes can be determined in a number of ways, with two common examples for the bit-flip channel detailed below.

85 CHAPTER 1

Error detection via projective measurement

For a [3,1] logical qubit, detection of bit-flip errors can be achieved by mea- suring four projection-based error syndromes. These projectors are:

P0 ≡|000000| + |111111| , No error, (1.119)

P1 ≡|100100| + |011011| , Bit-flip on Q1, (1.120)

P2 ≡|010010| + |101101| , Bit-flip on Q2, (1.121)

P3 ≡|001001| + |110110| , Bit-flip on Q3. (1.122)

E |  −E→|X   As an example, consider the effect of a passage through X : ψL ψL |   |  |  such that ψL = α 100 + β 011 . All error syndromes will produce a result   |P |   of 0, except ψL 1 ψL = 1, which indicates a bit-flip error had occurred on Q1. Note. An important feature of the syndrome measurements is that they do |   not cause any change in the state of ψL . As a result, the syndrome only contains information of the error which has occurred, and is independent of α and β. This characteristic of the syndrome measurement circumvents Postulate 5 (the projection of the quantum state through measurement) from limiting error detection.

86 CHAPTER 1

Error detection via parity measurement

A more compact method which performs the same analysis as the previous section is to simply compare the parity between all pairs of adjacent qubits forming the logical qubit. For the [3,1] qubit, this is performed via the two measurements Z1Z2 = σzˆ ⊗ σzˆ ⊗ 1 and Z2Z3 = 1 ⊗ σzˆ ⊗ σzˆ. The action of

Z1Z2 is to compare thez ˆ component of Q1 qubit to thez ˆ component of Q2, producing a +1 if they are the same and a −1 if they oppose each other. Thus, the syndromes can be expressed as:

Z1Z2 =(|0000| + |1111|) ⊗ 1 − (|0101| + |1010|) ⊗ 1, (1.123)

Z2Z3 = 1 ⊗ (|0000| + |1111|) − 1 ⊗ (|0101| + |1010|) . (1.124)

Combining the two measurement results allows the error process to be de- |   |  |  duced. For the same example of ψL = α 100 + β 011 ,theZ1Z2 error syndrome is −1, while the Z2Z3 error syndrome is +1, indicating that the most likely event is that an error occurred on Q1.

87 CHAPTER 1

1.8.3 Stabiliser codes

The following section is a brief introduction into the stabiliser formalism and the production of stabiliser codes for quantum error correction.

The stabiliser formalism

The stabiliser formalism is built around elements of group-theory, particu- larly associated with the Pauli group. For the case of the single qubit, this

Pauli group G1 is defined by

G1 ≡{±1, ±i1, ±σxˆ, ±iσxˆ, ±σyˆ, ±iσyˆ, ±σzˆ, ±iσzˆ} (1.125)

which is closed under multiplication. For n qubits, the Pauli group is Gn, and the stabiliser [118] is defined by the set of operations M ∈ Gn under which M |ψ = |ψ , (1.126) for all |ψ within the code. That is to say, given an Abelian subgroup S of the Pauli group Gn, the quantum code T (S) can be defined as the set of states |ψ given by Eq. (1.126), for all M ∈ S [118]. Therefore, if M,N ∈ S, then

MN |ψ = M |ψ = |ψ , (1.127)

NM |ψ = N |ψ = |ψ , (1.128)

⇒ [M,N] |ψ = MN |ψ−NM |ψ = 0 (1.129)

88 CHAPTER 1 as elements of the Pauli group either commute or anticommute, [M,N]=0. The description of the set S can be simplified through writing S in the form of its generators. A set of elements g1, ..., gl in some group S is said to generate S if every element of S can be written in the form of some product of the elements g1, ..., gl. This introduces the notation S = g1, ..., gl and, using this notation, and in order to check that a particular vector is stabilised by S, it simply needs to be stabilised by at least one of the generators. A simple example which highlights the characteristics of the stabiliser S is given in Ref. [15], and is summarised below:

Consider once again the logical qubit |ψL from Sec. 1.8.2, where n =3.

The set S can be chosen such that S ≡{1,Z1Z2,Z2Z3,Z1Z3}.Itcanbede- duced that the states stabilised by Z1Z2 is spanned by |000 , |001 , |110 , |111, and likewise |000 , |010 , |101 , |111 for Z1Z3. The common elements are

2 T (S)=|000 , |111. Further, as Z1Z3 =(Z1Z2)(Z2Z3)and1 =(Z1Z2) , S can be written in terms of the generators S = Z1Z2,Z2Z3.

Unitary gates in the stabiliser formalism

Suppose unitary operation U is applied to the vector space T (S) which is stabilised by S.For|ψ∈T (S)andg ∈ S:

U |ψ = Ug|ψ = UgU† |ψ , (1.130)

89 CHAPTER 1 indicating that the state U |ψ is stabilised by the element UgU†. This pro- duces the deduction that UT(S) is stabilised by the group USU† ≡{UgU†|g ∈ U}.  † † † From the description S = g1, ..., gl is it clear that USU = Ug1U , ..., UglU . This states that in order to compute change in the stabiliser, we only need to know how the change affects the generators. The implications of this yields a simplistic method of describing the effect of unitary operations on the codespace [15]. For example, the Hadamard operation H is described by

HXH† = Z, HZH† = X. (1.131)

This indicates that quantum states which are originally stabilised by S = Z

† (i.e. |0 and |1), will instead be stabilised by HSH = X (i.e. |ψx = √ √ (|1+|0)/ 2and|ψ−x =(|1−|0)/ 2). This simple example demonstrates the ease at which the codespace can be transformed from one which stabilise errors generated by the bit-flip channel EX , to another which stabilises phase-

flip errors from EZ . It is this characteristic which leads to the Hadamard operation being heavily utilised within even the earliest of codes which aim for robustness against both bit- and phase-errors [119, 120, 121, 122]. The transformation properties of other common elements from the Pauli group are illustrated in Table 1.2 [15]. The CNOT operation was introduced in Sec. 1.7.6 and the input/output parameters discussed in Table. 1.1 can be compared to the stabiliser formalism here. As discussed in Ref. [15], any

† unitary operation U for which UGnU ∈ Gn is termed the normaliser N(Gn),

90 CHAPTER 1

Table 1.2: Transformation properties of common elements of G1 in the form of UAU† = B. For the CNOT gate, Q1 is the control bit with Q2 as the target. This table is adapted from Ref. [15] Operation U Input A Output B XXX Z −Z YX−X Z −Z ZX−X ZZ Hadamard XZ (H) ZX Phase XY (P ) ZZ

CNOT X1 X1X2

(UCNOT) X2 X2

Z1 Z1

Z2 Z1Z2

and is generated by (i.e. U can be composed of) the gates H, UCNOT,and P . This concept is identical to that of a universal gate set as discussed in Sec. 1.7.6.

Measurement in the stabiliser formalism

Consider an element g ∈ Gn.Asg is Hermitian it can be considered an observable, and therefore a measure of g can be taken. If the system is in a state |ψ which is stabilised by S = g1, ..., gl, it follows from Postulate 5 that the measurement of the observable g can also transform the state. Two

91 CHAPTER 1 cases must be considered:

1. g commutes with all elements of the generator g1, ..., gl.

2. g anticommutes with at least one generator.

For the first case, it follows from the stabiliser formalism that the commuting elements yield gjg |ψ = ggj |ψ = g |ψ = |ψ, thus the measurement of g yields +1 and leaves the stabiliser invariant.

For the second case, say g anticommutes with gj.Asg has eigenvalues of ±1 it yields a projected measurement outcome of ±1givenby(1 ± g)/2. The measurement probabilities are given by

 1 + g P+1 =Tr |ψψ| (1.132)  2 1 − g P− =Tr |ψψ| . (1.133) 1 2

− † However, exploiting the anticomutativity relation ggj = gjg, gj +gj and gj |ψ = |ψ from the stabiliser formalism it can shown that  1 + g P+1 =Tr gj |ψψ| (1.134)  2 g (1 − g) =Tr j |ψψ| (1.135)  2 1 − g |  | † =Tr ψ ψ gj (1.136)  2 1 − g =Tr |ψψ| = P− . (1.137) 2 1

92 CHAPTER 1

As P±1 = 1, the probability of the outcome P+1 = P−1 =1/2. Therefore, if the measurement outcome is +1, the new state is |ψ+≡(1 + g) |ψ /2

+ with S = g1, ..., g, ...gl,whereg replaces the element gj from S. Similarly, for the result of −1, the state becomes |ψ−≡(1 − g) |ψ /2 and is stabilised

+ by S = g1, ..., −g, ...gl. These two examples play an essential role in the determination of the errors which are identifiable through syndrome measurement of a stabiliser code. In the next section these are integrated with the rest of the stabiliser formalism to define a stabiliser code for quantum computation.

Constructing stabiliser codes for error correction

This section details how a codespace described by the stabiliser formalism, combined with measurement, can produce the detection of errors within a stabiliser code. This section follows closely the discussions from Refs. [15, 118]. An [n, k] stabiliser code C(S) is defined to be the vector space T (S)sta- bilised by subgroup S ∈ Gn such that −1 ∈/ S (to avoid trivial solutions).

S has n − k independent and commuting generators S = g1, ..., gn−k.The logical computational basis states of C(S) can be chosen to be any 2k or- thonormal vectors in codespace C(S).

Consider an error process E ∈ Gn which corrupts the stabiliser code. The effect of E follows one of three possibilities:

1. E ∈ S, and the error does not corrupt the codespace.

93 CHAPTER 1

2. E anticommutes with an element gj, mapping C(S) to an orthonormal space resulting in an error. This error can then be determined by

performing syndrome measurements of the generators g1, ...gl.

3. Eg = gE for all g ∈ S. The set of E which satisfy this condition are

known as the centraliser of S in Gn, which is identical to the normaliser N(S). In this case the error E is identical to a logical operation and will remain undetected by the stabiliser code causing an irreversible corruption of the encoded information.

Hence, the theorem for quantum error correction in terms of stabiliser codes is given by the following: { } † ∈ − Suppose Ej is a set of operators in Gn such that Ej Ek / N(S) S fo all j and k. Then {Ej} is a correctable set of errors for the code C(S). This code can detect the presence of q arbitrary errors occurring provided d ≥ 2q − 1whered, the code distance, is defined as the minimum weight of an element N(S) − S. The error detection for such a code C(S) is performed by measuring each of the stabilisers gj ∈{g1, ..., gn−k} to obtain error syndromes β1, ..., βn−k.

† Consider an error E ∈{Ej} for which the error syndrome is EgjE = βjgj. For the case where E is unique, an error correction is performed by applying

† E to the system. Consider now F ∈{Ej} which is distinct from E, but gives

† rise to the same error syndrome FgjF = βjgj. It follows for the projector P onto the codespace, E†PE = F †PF, and therefore FE†PEF† = P ,where

94 CHAPTER 1

EF† ∈ S. Hence, applying E† to the system will also correct for error process F . Therefore, for each error syndrome, only knowledge of a single error process E which flags that syndrome measurement is required for the correction operation E† to be known and applied.

1.8.4 Fault tolerant quantum computation

Quantum error correction as presented in the previous sections not only protects stored or transmitted information from corruption, it is also capable of being integrated into the quantum computation such that information is dynamically protected throughout the execution of a quantum algorithm. The processing of the quantum information can be made completely tolerant of any error or faults within the system provided the error probability is below a certain threshold. The basic concept of fault tolerant quantum computation is to execute a computation directly upon encoded systems without the requirement of any decoding of logical states. As environmental noise is inherent to all processes within a quantum system, the constituent qubits within a quantum processor are encoded into logical qubits. Error correction can then be executed upon each logical qubit in between each logical operation performed during the computation.

95 CHAPTER 1

Fault tolerant logical operations

The logical operations performed upon the encoded logical qubits can also be carefully designed in order to minimise error propagation within or between logical qubits. This is designed such that the error correction stages between each operation is enough to clean up the residual errors. These specifically designed operations are known as fault tolerant operations. An operation on one or more logical qubits falls into this category if, upon the event of a fault in one component of the procedure, at most one error in each encoded output is produced from that operation [15]. The “component”, in this instance, can represent a noisy element intrinsic to the overall procedure such as a single gate, measurement, state perturbation etc. Another issue which must be taken into account is the fact that the error correction process itself is subject to the same environmental noise sources, resulting in the need to develop error correction procedures which are also fault tolerant.

As a high level example, consider a circuit C1 which is comprised of w in- ternal processes upon the logical qubit (including syndrome measurements). The individual components which make up this overall circuit have a fail rate of p. If the total number of failure modes for the syndrome measurements is given by c0, the injected error into the circuit C1 is c0p, as generated by the recovery stages of the previous measurement of the logical qubit. Due

2 2 to the chance of error propagation, the error at the output is at most c0p .

For the condition of fault tolerance, the circuit C1 is designed such that it

96 CHAPTER 1 does not introduce two or more errors at the output. Therefore, the maxi-

2 mum error bound for the total circuit C1 is the output error rate cp ,where

2 w c = c0 + x=1 cx, which depends on each internal process x executed with error probability cx.

2 Depending upon the value of c for the process C1, the condition cp

Code concatenation

The effective error rates produced by the quantum error corrected computa- tion can be suppressed even further through concatenation of the code struc- ture. This concept involves applying the same encoding operation utilised to produce (tier-1) logical qubits from the processor constituent qubits, to produce an additional layer of encoding upon these logical qubits (forming a tier-2 logical qubit).

As an example, consider once again the fault tolerant process C1,with individual component failure rate p, and maximum output error rate cp2.

Now consider C2, which executes the same operation of C1, but upon tier- 2 logical qubits which are encoded from a set of tier-1 logical qubits. The addition of each layer of concatenation adds a recursive factor to the output

2 2 error rate of the logical process, such that C2 has output error rate c(cp ) .

This is achieved by constructing C2 using the exact form of C1, but employing fault tolerant elements from the tier-1 encoding level interleaved with error correction. This concatenation procedure can be executed any number of

97 CHAPTER 1

k times, resulting in an output error rate of (cp)2 /c at the kth concatenation level. The size of the algorithm executed increases as dk,whered represents the total number of operations present in the fault tolerant procedure of an encoded gate (including error correction) at the first level of encoding.

Threshold theorem

Consider a quantum computer tasked with solving a certain problem. If the problem being investigated has some intrinsic “size” metric described by n, the algorithm contains a total of P (n) gates, where P (n) is a polynomial function in n. If the accuracy of the computation output is chosen to be , each gate in the algorithm must be accurate to within a factor of /P (n). This simple formula can give an estimate for the required number of code concatenations based on the known error probability c as

k (cp)2  ≤ . (1.138) c P (n)

The condition for k to be extracted is that p

 log(P (n)/c) log(d) dk = (1.139) log(1/pc) = O(poly(log(P (n)/))). (1.140)

98 CHAPTER 1

This shows that the growth rate O of the algorithm with respect to the “size” of the problem n is a function of some polynomial poly(x) of fixed degree. This result leads to the statement of the threshold theorem: For a noisy quantum computer using hardware with intrinsic component failure rate p, an ideal quantum circuit containing P (n) gates may be simu- lated with probability of error  using

O(poly(log(P (n)/))) (1.141)

gates, provided p is below some threshold pth given from noise assumptions within hardware. It is seen that for successful application of the threshold theorem, a good understanding of the error rate p of the qubit system must be known. This is one of the key motivations for many of the qubit characterisation experiments outlined in Sec. 1.5. Further, a key design consideration for quantum com- putation algorithms is the minimisation of c, such that the error threshold can be as low as possible.

99 Chapter 2

The Silicon-MOS quantum dot platform

This chapter introduces the SiMOS quantum dot platform studied within this dissertation. It presents in more detail core concepts which are more specific to the SiMOS materials system, including how the quantum dots are electrostatically defined, and how they are employed to produce both sensitive electrometers as well as qubits. For the description of quantum dots as qubits, the DiVincenzo criteria (see Ch. 1) requires the platform to be capable of qubit state initialisation and readout, and the methods for the SiMOS platform are discussed here. In addition, a number of other fundamental concepts specific to the mate- rials system used are reviewed and many are later integrated as an element of experimental processes. These include the common excited states seen in

100 CHAPTER 2

SiMOS quantum dot qubits and the spin-orbit coupling. The chapter con- cludes by discussing the SiMOS device designs used in this thesis. Details concerning the SiMOS device fabrication process can be found in Appendix F, and the discussion of experimental methods and equipment used to measure these devices are presented in Appendix G.

2.1 Accumulation of electrons at a semicon-

ductor/oxide interface

A two-dimensional electron gas (2DEG) can be created in MOSFET struc- tures [123] at the Si/SiOx interface, when a potential above the material threshold voltage VTH is applied to an electrode above the oxide layer. Here, electrons are unconfined in the x-y plane, but have confinement perpendicu- lar to the semiconductor/oxide interface. In the SiMOS devices, metallic electrodes (or gates) are lithographically patterned on the SiOX layer and, just as in the channel of a conventional MOSFET, the 2DEG can extend below this gate towards a highly doped region. These doped regions can acts as an Ohmic contact, supplying carriers to the induced 2DEG layer. These 2DEG regions often act as conductive electron reservoirs (such as the source or drain of a single electron transistor).

101 CHAPTER 2

2.2 Quantum dots

For a 2DEG-based quantum dot, lateral confinement is achieved by control- ling voltages on lithographically defined metallic gate structures as shown in Fig. 2.1. These top-gate electrodes produce relatively smooth walls of the potential wells and enabling tunnel coupling to source and drain 2DEG elec- tron reservoirs. The resulting potential landscape is highly tunable, as the top-gate electrodes not only allow for accumulation of electrons to form dots or reservoirs, but are also employed to deplete local regions forming tunnel barriers between quantum dots and electron reservoir 2DEGs. Due to the additional lateral electrostatic confinement forming the quan- tum dot, energy levels of an electron in this potential well are quantised according to basic quantum mechanics (see, for example, Refs. [63, 110]). The energy required to add an electron to the dot is known as the addition energy:

EAE = EC + δE, (2.1) where δE is the energy level spacing for two discrete quantum levels, which is zero when two consecutive electrons are added to the same spin-degenerate level (discussed further in Sec. 2.4.3 and Fig. 2.8). Energy EC is the charging energy that is required to overcome the electrostatic repulsion between the new electron and the electrons already on the dot. The charging energy is

102 CHAPTER 2

Figure 2.1: SiMOS quantum dot structural cross-sections and accompanying schematic of electrochemical potentials. For device cross sections, different metallic layers are indicated in light grey, with isolating AlOX colour-coded to indicate deposition layers (Purple, Blue then Orange). a) Device cross section of a single electron transistor (SET) where 2DEG source and drain electron reservoirs are accumulated by applying positive VG. The barrier volt- ages VB1 and VB2 electrostatically define a central dot known as the “island” by depleting tunnel barriers beneath each barrier electrode. b) Electrochem- ical potential across the SET device. c) Device cross section of a double-dot qubit device. Here, gates are used to accumulate a 2DEG to form an elec- tron reservoir (VRes), while other electrodes are used for confinement of side potentials (VC) and the accumulation of singular electron occupancies of elec- trostatically defined dots (VG1 and VG2) d) Electrochemical potential across the double dot device, indicating that there is a sufficient electrostatic barrier present at the Al/AlOX /Al interfaces to produce a tunnel barrier yielding discrete dots under adjacent gates (this is further discussed in Ref. [42]).

103 CHAPTER 2 given by the geometry and materials platform of the quantum dot:

2π2 e2 EC = ∗ = . (2.2) gm A 2CΣ

Here, g and m∗ are the material dependent electron g-factor and effective mass respectively. The geometry of the quantum dot is also a key factor, with A being the dot area due to lateral electrostatic confinement. This energy can also be determined from variation in the surrounding electrostatics of the quantum dot, as the charging energy is related to the total capacitance of the dot, the surrounding environment CΣ and e, the electron charge. This fundamental aspect demonstrates the capability for the electron occupancy of the dot (or in broader terms, the chemical potential μ with respect to the environment) to be indirectly controlled by the voltage potential of the gates implemented to define the quantum dot through lever arms

C α = G , (2.3) CΣ

where CG represents the capacitive coupling between the dot and some gate electrode “G”. Common quantum-dot device cross-sections and their accompanying elec- trochemical potentials are illustrated in Fig. 2.1. These two specific geome- tries are discussed in greater detail in the next two sections.

104 CHAPTER 2

2.3 The single electron transistor

The most basic quantum dot device is the Single Electron Transistor (SET), and a simple cartoon cross-section of an SET device can be seen in Fig. 2.1 a&b. Here, a single quantum dot has been defined in series with a 2DEG connec- tion between the source and drain electrodes. After the 2DEG channel is induced using VG, the dot is formed by depleting the 2DEG under barrier gates. The low voltage settings on VB1 and VB2 to counteract the effects of

VG for these regions.

2.3.1 Coulomb Blockade

The electrochemical potential μ of the dot is controlled by all surrounding voltages VG,VB1 and VB2, but is usually varied dynamically by VG.When addition energy EAE lies within the source drain bias window (produced by

VSD in Fig. 2.1 b) an electron can tunnel from the source to the dot and then from dot to the drain. This tunnelling from source to drain via the quantum dot produces a transport current ISET. As gate voltages are altered, the energy levels shift up or down via the capacitive lever arms α. At the point where there is no available level sitting within the bias window, no electron tunnelling occurs and ISET is blocked, resulting in a phenomena known as Coulomb Blockade. A SiMOS SET device (published in Ref. [41]) is shown in Fig. 2.2 a, where the top-down view of the device is imaged via a scanning electron microscope

105 CHAPTER 2

Figure 2.2: a) Scanning electron microscopy image of the SET device. b) Transconductance through the SET based on VG and VSD, showing clear coulomb blockade behaviour. Figure is adapted from Ref. [41]

(SEM). For this device, the Coulomb Blockade as a function of key tunable device parameters VG and VSD are shown in Fig. 2.2 b, where the FWHM of the Coulomb peaks δVG = VFWHM are broadened by increasing the bias window VSD. The broadening of the coulomb peak VFWHM can be dependent upon several other parameters, including lifetime broadening due the opacity of the tunnel barriers: Γ V = , (2.4) FWHM e where Γ is the tunnel rate through the barriers defined by both the barrier width and height. The other key mechanism is the thermal broadening of the Fermi Energy [124] k T V = B , (2.5) FWHM e where kB is Boltzmanns constant, and T is the electron temperature in the 2DEG. It describes how the temperature T dictates a fundamental limit for

106 CHAPTER 2 the observation of distinguishable quantised energy levels, an issue which is overcome through the use of cryogenic measurement apparatus (see Ap- pendix G). Together, the thermal and lifetime broadening effects yield the minimum broadening (when VSD = 0) which is a fundamental element within these quantum devices.

2.3.2 The SET as a charge sensor

The SET device can be used as a sensor for changes in the surrounding electrostatic environment. As the chemical potential μ is capacitively coupled to the surrounding environment, any electrostatic changes (for example; the changing of the charge configuration of a nearby quantum dot) will result in a shift in μ, detectable as a change in ISET as illustrated in Fig. 2.3. This shift in μ can be compensated via the top-gate electrode shown in Fig. 2.1, or by voltage change δVG asshowninFig.2.3.

Figure 2.3: Shift in Coulomb peaks by gate referred voltage δVG due to changes in the electrostatic environment

Consider a quantum dot which is coupled to the SET island by a mutual

107 CHAPTER 2

capacitance CSET↔Dot. If that quantum dot undergoes a charge transition, the SET device experiences a gate referred voltage shift [125]

δq δVG = , (2.6) CSET↔Dot where δq is the fraction of charge induced on the SET island due to the mutual capacitance between the island and the dot. Hence, if the SET is tuned to a VG where the conductance δISET/δVG is large, the SET current will be the most sensitive to changes in the electrostatic environment given by δVG.

2.4 Dots as qubits

The previous section details how an electrostatically defined quantum dot can be employed as an electrometer, sensitive to capacitive changes in the environment. Typically such devices operate with a large number of elec- tron occupancy N (typically 10-100 electrons). These same electrostatically defined quantum dots can be used to implement qubits, with typical elec- tron occupancy N around 1-4 for each dot. A schematic device cross-section showing how quantum dots can be implemented as a linear array of qubits is illustrated in Fig. 2.1c, with the accompanying electrochemical poten- tial cross section in Fig. 2.1d. Not only are the lithographically defined gates used as a top-gate for the individual quantum dots (VG1 and VG2 in

108 CHAPTER 2

Fig. 2.1c) and for 2DEG reservoirs to supply electrons (VRES in Fig. 2.1c), but other gates surrounding the dot regions are also used for electrostatic con-

finement (VRES in Fig. 2.1c). Further details of how this is achieved in the SiMOS system, and the contemporary architectures involved can be found in Refs. [126, 42, 59, 60], as well as in Sec. 2.5. Consider a quantum dot with occupancy of only a singular electron (N=1). In order to realise a qubit device, there are a number of requirements in line with the DiVincenzo criteria, which must be experimentally demonstrated:

1. The two quantised states of the electron, namely spin-up and spin- down, must be made distinguishable through a quantum measurement protocol.

2. The system must have a protocol set in place such that the state of the electron can be robustly prepared into the chosen initial state.

3. The state of the electron must be controllable over the Bloch sphere. This is experimentally implemented in many cases through electron spin resonance (ESR), giving two axis control of the qubit as discussed in Sec. 1.4.1.

The first two points, regarding initialisation and readout of a quantum system, are discussed in theory Sec. 1.2, but are explained in context with the SiMOS quantum dot qubit platform in the sections below. Further, the first experimental demonstration of an actual SiMOS quantum dot qubit (i.e. an

109 CHAPTER 2 experimental demonstration of the above three requirements combined in a single test-bench device) is reported in Ref. [42].

2.4.1 State readout methods

Readout via an electron reservoir

Readout of the state of an electron in electrostatically defined quantum dots was first pioneered in GaAs/AlGaAs heterostructure devices [127]. The test- bench device is designed to have a dot which is tunnel coupled to a 2DEG electron reservoir and capacitively coupled to the sensor device (in this case, a quantum point contact - see Ref. [127] for details). The readout scheme was adapted for use with an SET in SiMOS to sense the ionisation states of a phosphorous donor in Ref. [125]. For SiMOS quantum dot qubit devices, the readout mechanism was first shown in Ref. [126]. For a system where the Zeeman energy is much greater than the thermal population around the electron reservoir (discussed below), the readout pro- cess for an N = 1 quantum dot is illustrated in Fig. 2.4. The electrode with the highest capacitive coupling to the dot (VG in this example) is commonly employed to alter the dot electrochemical potential with respect to the Fermi energy of the 2DEG electron reservoir. The voltage protocol is illustrated in Fig. 2.4a, and is accompanied by the response of the sensor current in Fig. 2.4b. For each execution of the protocol there are two options depend- ing upon the spin states as illustrated in Fig. 2.4 c where only one option

110 CHAPTER 2

Figure 2.4: Elzerman protocol for reading out electron spin states for quan- tum dots or donors tunnel coupled to electron reservoirs. a) Shape of the voltage pulse applied to quantum dot top-gate. b) Measured current through the capacitively coupled sensor device highlighting key features indicative of electron tunnelling events. c) Schematic energy diagrams for spin-up (en- ergy level E↑) and spin-down (energy level E↓) during the different stages of the pulse. Detailed explanation of each stage, numbered 1-4, is given in the main text. This figure is adapted from Ref. [127]

111 CHAPTER 2 results in a detectable response in the readout window. The overall protocol is described as follows:

1. Begin with the dot completely empty of electrons, both of the energy states are well above the Fermi energy as defined by the electrochemical

potential influenced by the top-gate voltage VG.

2. Rapidly (with respect to dot-reservoir tunnel rate Γ) plunge the volt-

age VG into a region where the electrochemical potential dictates there should be one electron on the dot. As there is an electrostatic change in the system, there will be an equivalent shift in the sensor current due to mutual capacitance. A random electron state will tunnel into the quantum dot with characteristic tunnel rate Γ, resulting in a shift in the sensor current due to the addition of charge in the quantum dot system.

3. The top-gate VG is used to align the chemical potential such that the Fermi energy of the reservoir bisects the two discrete spin-up and spin-down eigenenergies. Following the fourth and fifth postulates in Sec. 1.2.4 and Sec. 1.2.5, upon measurement of the state, the quan- tum system projects onto either the spin-up or spin-down eigenstates. For the case where the state projects to spin-down, the accompanying

eigenenergy E↓ is sufficiently less than the Fermi energy of the reser- voir that the electron on the quantum dot remains in the low energy state. Otherwise the electron projects to the spin-up state, where the

112 CHAPTER 2

eigenenergy E↑ is sufficiently greater than the Fermi energy such that it tunnels off the dot and onto the reservoir 2DEG with characteris- tic time dictated by Γ. This process is followed by an electron with the spin-down orientation supplied from the reservoir and tunnelling onto the quantum dot with rate Γ once again. Both of these electron tunnelling events cause a change in the eletrostatic environment of the sensor and is thus detected as a “blip” in the sensor current. Such an event is often denoted spin-to-charge conversion as this event is due to a charge state being created on the quantum dot, conditional upon the spin state of the electron. After this cycle is completed the quantum dot contains a low-energy spin-down electron state.

4. The final element of the Elzerman scheme [127] is to once again empty the quantum dot. Here, the top-gate potential pulls the energy levels of the quantum dot above the Fermi energy, where the electron (no matter what state) will tunnel out of the dot. Once again, there is a

response in the sensor due to change in VG and detection of the electron leaving the system.

It was discussed in Sec. 2.3.1 that there is a thermal broadening of the Fermi energy. This broadening plays a crucial role in the ability to distinguish the discrete electron states within this measurement system. In order for the two electron states to be made distinguishable with respect to this electron temperature, the Zeeman energy as presented in Eq. (1.32) must exceed this

113 CHAPTER 2 broadening of the Fermi energy. The electron temperature is measured to be of the order 100 mK, equivalent to 8.6 eV which is achieved with an applied

z magnetic field of B0 150 mT. This is of course equivalent to the FWHM of the Fermi energy broadening, therefore in order to suppress readout error attributed to thermal population of electrons, an external magnetic field much greater than this value must be applied (often seen in literature is an

z external field of B0 1.4T).

Readout via Pauli spin blockade

Readout of quantum dot states can also be conducted using the state of two electrons spread over two tunnel coupled quantum dots. Spin-dependent tun- nelling events attributed to the Pauli exclusion principle, known as Pauli spin blockade (PSB) were first observed in AlGaAs/InGaAs quantum dot struc- tures [128], and extensively studied in GaAs/AlGaAs quantum dots [129, 130, 112, 131], Si/SiGe quantum dots [132, 133, 134] and SiMOS quantum dots [135, 136, 85, 60]. Several of these were achieved using single-shot read- out of the charge state [131, 132, 133, 134, 85, 60]. PSB can be made ob- servable on the charge stability diagram through the methods outlined in Ref. [130], and key results from multiple materials systems utilising PSB are illustrated in Fig. 2.5. The Hamiltonian describing a system of two electrons was introduced in Sec. 1.7, including the singlet-triplet basis states in Sec. 1.7.3. As observed in

z Fig. 1.11, for low externally applied fields B0 and gate voltage configurations

114 CHAPTER 2

Figure 2.5: Pauli spin blockade as presented by key literature sources. a) Pauli spin blockade is observed in this GaAs/AlGaAs heterostructure dou- ble quantum dot as a metastable triangular area located in the (0,2) charge region (See Ref. [130] for further details). Here the cyclic pulse sequence (E-R-M) which allows the mapping of the double dot stability diagram with the observation of the metastable charge region. b) Using the same puls- ing sequence as Ref. [130], the metastable charge region is observed to be truncated to a trapezoidal area for the Si/SiGe heterostructure double quan- tum dot device studied in Ref. [133]. This truncation is due to accessing an additional degree of freedom which lifts the blockade (such as encountering the first valley excited state). c) Pauli spin blockade in a SiMOS quantum double dot device (Ref. [85]) which also observes truncation due to access- ing excited states. Other features within this stability diagram include state latching (see Refs. [137, 136] and Ch. 4 for further details). The sections in this figure are adopted from a) Ref. [130], b) Ref. [133], and c) Ref. [85].

115 CHAPTER 2 leading to positive detuning ε into the (1,1) charge region, the triplet states and the hybridised-singlet ground state are relatively close together in energy. Conversely, for negative detuning ε into the (2,0) or (0,2) charge region, the triplet states and the hybridised-singlet ground state become further separated in energy. This indicates that for the pair of electrons configured as any one of the triplet states, the charge configuration will remain blockaded in the (1,1) while the singlet configuration can transition between (1,1) to (2,0) or (1,1) to (0,2). This blockaded state, which is dependent upon the spin state of the two-electron system, remains configured as a (1,1) state against the ground state dictated by the electrochemical potential. This absence of a tunnelling event for blockaded states can be detected through a capacitively coupled sensor resulting once again in a detectable spin-to-charge conversion.

2.4.2 State initialisation methods

As dictated by Postulate 5 in Sec. 1.2.5, the measurement of the quantum system results in the projection onto a system eigenstate. This process can be intrinsically tied to the initialisation of the quantum mechanical state.

Initialisation via an electron reservoir

Sec. 2.4.1 describes the Elzerman readout process via a 2DEG electron reser- voir. As is shown in Fig. 2.4 c, after stage 3 has been executed successfully, the spin state of the electron is prepared in the spin-down ground state. This effectively ties together the readout of one instance of an experiment with

116 CHAPTER 2 the initialisation protocol of the next instance [45]. For the first instance of the experiment, spin dependent loading [125, 45] is achieved by executing a sequence defined by 1 → 3 → 2 from Fig. 2.4. Thus, the empty dot (stage-1) is brought to an electrochemical potential such that E↓

Initialisation in the singlet-triplet basis

The initialisation of the two electron singlet-triplet states can be more com- plicated than the initialisation of a single electron system. Similarly for the reservoir based approach, the (0,2) or (2,0) singlet ground state is signifi- cantly split from the (1,1) triplet eigenstates (as observed in energy diagram Fig. 1.11). This means it is also common to initialise the system in the sin- glet ground state and then transfer into the (1,1) configuration [112]. For a double dot system with electrons in a triplet state, a metastable state is cre- ated for certain regions of (0,2) and (2,0) as shown in Fig. 2.5. When the chemical potential is configured in this metastable region, the relaxation pro-

117 CHAPTER 2

Figure 2.6: Lifting of Pauli spin blockade by accessing a) charge states with additional electrons supplied by the 2DEG reservoir. This figure accompanies the coloured dots seen in Fig. 2.5 a). b) Lifting Pauli spin blockade by accessing an excited state. For the case represented by the purple dot, the electron spin states are not relaxed to the |SH  ground state but remain in a |T0,± triplet state, but for the charge ground state as disctated by the electrochemical potential. This figure is adopted from Ref. [130].

cess between |T0,± states and the hybridised singlet |SH  is a T1 relaxation process (see Sec. 1.5.2 for details). This relaxation process can be enhanced by approaching the reservoir in voltage space, resulting in the lifting of Pauli spin blockade through swapping an electron with the reservoir or accessing other charge states [130, 129, 133]. Relaxation via charge states is observed as lifting of metastable states in Fig 2.5a at locations marked by the orange and red circles, corresponding to relaxation paths depicted in Fig. 2.6a. One potential pitfall to this method is the lifting of blockade via accessing qubit excited states. These excited states have access to additional degrees of freedom allowing for a triplet to exist in the (0,2) or (2,0) charge configu- ration. A process such as this is illustrated in Fig. 2.6b, corresponds to the

118 CHAPTER 2 purple descriptors. This process can be described in the SiMOS devices as    the metastable (1, 1)Tv−− state of the lowest energy v−− valley configura-    tion (see Sec. 2.4.3), relaxing to a (0, 2)Tv−+ triplet of the higher energy v−+ valley configuration. In this configuration, the triplet will relax based on the T1 processes of the excited state (for the SiMOS platform, these are stud- ied in Ref. [140]). Otherwise, the electrochemical potential can be brought into proximity with the reservoir to enhance rapid initialisation in a similar fashion to stage-3 in the Elzerman scheme.

2.4.3 Qubit excited states

As identified in the previous section, the excited states of a qubit system can play a detrimental role in the successful operation of a qubit. In the following sections, the key excited states for an electron in a SiMOS quantum dot are introduced.

Valley states

Silicon is an indirect band-gap semiconductor with conduction band minima at momentum k0 > 0. Cubic symmetry gives six energy minima, known as valleys, which are degenerate in bulk silicon. For silicon nano-devices, the degeneracy of these valleys can be lifted by various means including material strain and the application of electric fields [141]. Figure 2.7 illustrates the breaking of valley degeneracy in SiMOS quan- tum dots. Surface strain at the Si/SiOx interface results in splitting off

119 CHAPTER 2

Figure 2.7: For a quantum well, at the Si/SiOx interface, the valley degen- eracy of bulk silicon is broken by the in-plane strain at the interface. The remaining twofold degeneracy is broken by additional electrostatic confine- ment in the quantum well and by electric fields. This figure is adapted from Ref. [141] into two-fold degenerate Γ-valley ground states and four-fold degenerate Δ- valley excited states. This results from vertical confinement of electrons with different effective mass along the longitudinal and transverse directions re- spectively [142]. The Γ-valley degeneracy can be further lifted by a sharp perpendicular potential [143]. For silicon-based quantum dots the energy sep- aration between valleys can be measured experimentally via pulsed-voltage spectroscopy [140], and was later demonstrated to be tunable via modulat- ing the surrounding electrostatic environment [126]. From these works, the valley splitting energy EVS is expected to fall within a range of 0.4 − 0.8eV.

120 CHAPTER 2

Orbital levels

When adding multiple electrons to a singular dot, the addition energy demon- strates shell-structure-like behaviour, where the addition energy is defined by either just the changing energy EC of the quantum dot, or the charging en- ergy plus the difference between orbital energies δE. For the SiMOS quantum dot platform, the shell filling can be represented by 2D Fock-Darwin states in the presence of two valley states [126, 144, 145] as shown in Fig. 2.8.

Figure 2.8: a) Charging energy as a function of electron number N. Increases when transitioning from N ↔ N + 1 correspond to completed 2D shells, whereby the addition energy increases by the difference in energy between the two orbitals. b) 2D FockDarwin states with two valleys. Each state can hold two electrons of anti-parallel spin and is identified by a pair of quantum numbers (n, l) and valley occupancy (v). This figure is adapted from Ref. [126]

121 CHAPTER 2

2.4.4 Spin-orbit coupling

The inversion asymmetry of silicon crystal at a material interface (such as the oxide) leads to a spin-orbit (SO) interaction as outlined in Refs. [81, 82, 83, 84, 85]. Thus, the coupling is intrinsically tied to the the kinetic momentum k of the electron ∂ k = −i + eA ( r), (2.7) ∂ r for vector potential A ( r). The interactions come in two key varieties. The Rashba SO effect derives from asymmetry of electric fields at the material interface due to structural inversion asymmetry. For the electron confined to this interface this is rep- resented as the Hamiltonian

HR = αR(kyσx − kxσy), (2.8)

where αR represents the interaction magnitude. The second form of the in- teraction is the Dresselhaus SO coupling which is due to microscopic interface inversion asymmetry (also known as “surface roughness”). The representa- tive Hamiltonian is given by

HD = αD(kxσx − kyσy), (2.9)

for αD representing the interaction magnitude of the Dresselhaus SO com- ponent. The two varieties of SO components are illustrated in Fig. 2.9.

122 CHAPTER 2

Figure 2.9: The two varieties of spin-orbit coupling as given in momentum space. High-symmetry directions x  [110] and y  [110]¯ are shown. This figure is sourced from Ref. [84]

Ref.[85] presents the effects of the spin-orbit terms within the SiMOS system represented as the following Hamiltonian

μ H = B · B · g · σ, (2.10) SO 2 SO with a tensor form of the electron g-factor given by the Rashba and Dressel- hause terms ⎛ ⎞ ⎜−α β 0⎟ ⎜ R D ⎟ ⎜ ⎟ 2 ⎜ ⎟ gSO = ⎜ β −α 0⎟ . (2.11) μ ⎜ D R ⎟ B ⎝ ⎠ 000

This results in a modified g-tensor for quantum dots as ⎛ ⎞

2αR 2βD ⎜g⊥ − 0 ⎟ ⎜ μB μB ⎟ ⎜ ⎟ ⎜ 2βD 2αR ⎟ g = g0 + gSO = ⎜ g⊥ − 0 ⎟ . (2.12) ⎜ μB μB ⎟ ⎝ ⎠ 00g

123 CHAPTER 2

The physical origins of the SO effects are discussed in Ref. [85], and observed in Fig. 2.10. Here, the authors link the cyclotron motion experienced by an electron under an applied magnetic field parallel to the interface (thus generating appreciable kx and ky) to the SO interaction observed in an SiMOS device.

Figure 2.10: Spin-orbit coupling due to cyclotron motion of the electron. a) Schematic cross section of the SiMOS quantum dots with polysilicon gates. b) Kinetic momentum density and cyclotron motion of the quantum dot ground state with B = By. Blue colour scale shows the electron probability density. c) Interface cyclotron motion combined with details of the interface at the inter-atomic bond level govern elements of the spin-orbit interaction. This figure is sourced from Ref. [85].

The analysis above describes how the SO coupling can lead to the ob- servation of a g-tensor. Other studies, including Refs. [146, 88], present an alternate theory with the same theoretical origins (i.e. the Rashba and Dres- selhause SO couplings) which results in SO coupling between charge states of two coupled quantum dots.

124 CHAPTER 2

The discussion presented in Ref. [146] refers to an electron which tunnels between to dots. The dots are separated by some distance d along unit vector direction eζ with respect to the crystallographic axis. This tunnelling gives † † the electron momentum kζ . Using second quantisation notation where cL (cR) creates an electron in the Wannier state [147] ΦL (ΦR), HSO = HR + HD can be expressed as

iΩ H = c† σ c − h.c. , (2.13) SO 2 La ab Rb a,b=↑↓ where σ is the vector of Pauli spin matrices and Ωisthe spin-orbit field

 | |  iΩ= ΦL kζ ΦR aΩ . (2.14)

This spin-orbit field highlights the mixing between charge states of two quan- tum dots (L and R), where the vector aΩ yields dependence of the dot ori- entation with respect to the crystallographic axes. For a 2DEG in the (001) plane, aΩ is given by

− aΩ =(αD αR)cos(θ) e[110]¯ +(αD + αR) sin(θ) e[110], (2.15)

where θ is the angle between the [110] crystallographic axis and the eζ tun- nelling direction.

125 CHAPTER 2

Effects of spin-orbit coupling in silicon devices

The interpretation presented in Ref. [85] accounts for a number of different observations connected to the SO components in SiMOS quantum dot qubits. Due to the cyclotron motion set up by the applied magnetic field, the SO component of the g-tensor presented in Eq. (2.12) is observed to vary with orientation to the crystal axis [148, 85]. Further, the Rashba term is derived from electric field asymmetry at the interface. This gives rise to the observation of a Stark shift in the g-factor via modulating potential landscape of the quantum dot [113]. The spin orbit coupling term is also intrinsically tied to the valley state and phase of the quantum dot energy levels [84, 148], resulting in a distri- bution in the observed g-factor as seen between dots, as well as opposing direction of the Stark shift for different electron energy levels on the same quantum dot [149]. The surface roughness intrinsic to each individual quantum dot results in the modification of each g-factor [42, 148]. Not only does this give each quantum dot an individual resonance frequency for ESR, the SO interaction has implications on the singlet-triplet experiments via contributions to the

Δ coupling between the |SH  hybridising singlet and |T± polarised triplets as seen in Eq. (1.109). As such, the SO effects must be accounted for during the semi-adiabatic state transformation from the hybridising singlet state of the singlet-triplet basis to either the |↑, ↓ or |↓, ↑ state of the two-qubit computational basis [85, 60] (details are presented in Ch. 6).

126 CHAPTER 2

With regards to the origin of the coupling Δ, it is observed in Eq. (1.109) that the coupling is broken into two parts. The first part mixes |T± and |(1, 1)S states, attributed to the difference in g-tensors between quantum dots [85] and acts in a similar fashion to a gradient in the external magnetic

field [112, 87]. The other term is attributed to mixing |T± and |(0, 2)S states [146, 88]. The coupling term is presented in Ref. [146] as

  √  ΔST =2 2(δB z sin ϕ − δB x cos ϕ + iδB y)cosϑ − iΩsinϕ sin ϑ , (2.16) which is related to several geometric quantities including the angle ϕ between applied field B and spin-orbit field Ω, and a mixing angle ϑ describing the relative size of mixing and splitting of σzˆ eigenstates [146].

127 CHAPTER 2

2.5 SiMOS quantum dot qubit architectures

This section details the two key SiMOS architectures for quantum dot qubits upon which the experiments presented in this dissertation were conducted. The two architectures are shown in Fig. 2.11 and Fig. 2.12.

2.5.1 Device architecture 1: Single-qubit experiments

The architecture shown in Fig. 2.11a was first introduced in Ref. [126], but was further employed to produce the first SiMOS quantum dot qubit [42], benchmark qubit performance [58] and study the effects of mulit-valley qubits and spin-orbit coupling [149]. This device was further utilised in a double- dot configuration to study interaction effects between two electrons in the context of a two-qubit logic gate [113]. The device electrodes are colour coded into the five major components of the test-bench qubit device:

Sensor electrodes

Shown in grey/brown, these include ST, LB and RB electrodes, used to form the SET sensor (see Sec. 2.3 for discussion).

Confinement electrode

Patterned on the first layer of the multi-gated aluminium stack, the purpose of the confinement barrier in this device is two-fold:

128 CHAPTER 2

Figure 2.11: a) False-coloured scanning electron microscopy (SEM) image of the qubit device architecture which produced the results presented in Ch. 3 b) Vertical cross-section through the quantum dot locations illustrating a single qubit configuration. The quantum dot is tunnel coupled to an extended 2DEG reservoir with tunnel rate controlled via G1-G3. The SET island is capacitively coupled to the dot and remains isolated through the confinement barrier CB. c) Horizontal cross section through gate electrode G2 illustrating 2DEG from an extended reservoir and electron accumulation under fan-out leads. d) Horizontal cross section through the SET showing the same features as Fig 2.1 a&b. Subfigure a) of this figure is sourced from Ref. [42].

129 CHAPTER 2

1. Providing the lateral confinement of quantum dots, the confinement barrier has two parallel arms between which the quantum dots are induced.

2. Isolating the sensor ohmics (on the SET-side of the device) and the electron reservoir ohmic region (dot-side). Such a conduction path between these ohmics would result in a leakage current, known to cause excess heating and noise in devices.

The confinement barrier achieves these two distinct goals by intentionally reverse-biasing the regions beneath the electrodes. This region is not con- ducive to the existence of a 2DEG, and thus it creates an isolation barrier along the entire gate fan-out for the thin-oxide region.

Electron reservoir electrode

This electrode supplies electrons to the qubit side of the device. For this architecture, the reservoir is also on the first patterned layer, as one of the two prongs is situated below the qubit gates. The fan-out of the electrode runs across the n-type ohmic contact regions where electrons are readily available. When the electrode potential surpasses the threshold voltage VTh a 2DEG will form at the Si/SiOx interface directly beneath the gate, with electrons fed from the n-type region. For this design the reservoir also has a two-prong shape, one is patterned to align with the qubit window created by the confinement barrier, thus being capable of supplying electrons to the

130 CHAPTER 2 quantum dots. The other prong sits beneath the qubit gate electrode feed- out region, where the purpose is to directly supply electrons to regions which can source or sink large amounts of electrons as 2DEGs are charged and discharged due to the voltage pulsing undertaken during qubit experiments (visible in Fig. 2.11 d). The separation of the reservoir into the dot reservoir 2DEG, and the largely dynamic electrode fan-out 2DEG, was estimated to be far enough from the dots that the dot 2DEG would be relatively static.

Qubit top-gate electrodes

The qubit top-gate electrodes are formed using two metal layers such that each adjacent gate is overlapping at the edges (≈ 4nm overlap), yet remains electrically isolated. The role of the gates in these devices is three-fold:

1. Accumulate electrons on a quantum dot directly beneath the electrode. This is discussed in Sec. 2.4.

2. Control the electron tunnel rate of an adjacent dot and the electron reservoir. Such a purpose is discussed in Ref. [126]. This can see gates being tuned to a “high accumulation” mode, where there is a very large number of electrons on the dot (N>10). Under these conditions a quantum dot adjacent to the reservoir 2DEG begins to merge with the reservoir itself. Such a configuration could be interpreted as extend- ing the electron reservoir into the confinement window as shown in Fig. 2.11b.

131 CHAPTER 2

3. As the dot is capacitively coupled to all the surrounding gates, the electrodes adjacent to a quantum dot have a degree of control over the electrochemical potential. For certain configurations (such as in Ref. [113]), an adjacent electrode can be used Stark shift the ESR frequency or even cross the dot energy level and incur a tunnelling event.

ESR transmission line

Often the last gate patterned, the transmission lines are produced via an on- chip balun waveguide configuration. The design, as well as the transmission spectrum for both electric and magnetic fields are discussed in Ref. [150]. The transmission line is aligned parallel to the axis of collinear dots such that each dot experiences a similar ac magnetic field amplitude.

2.5.2 Device architecture 2: Singlet-triplet experiments

The elements of the device design discussed in this section function in a very similar fashion to that of the previous design. The architecture is inspired by the discussions and theoretical designs found in Ref. [47]. The key elements of comparison between the two architectures are summarised the following:

SET orientation

The orientation of the SET is such that the sensor axis (direction of current flow) has been rotated to be parallel with the quantum dot axis. While this

132 CHAPTER 2

Figure 2.12: a) False-coloured SEM image of the qubit device architecture which produced the results presented in Ch. 4-Ch. 6. b) Horizontal cross- section illustrating dots under G1 and G2 are tunnel-coupled to an electron reservoir under gate ST, with fast and slow tunnel rates ΓFast and ΓSlow c) Vertical cross-section through dot top gate G2, illustrating the lateral confinement of the dot due to gate CB. The dot is capacitively coupled to the SET island. Subfigure a)&b) of this figure is sourced from Ref. [60]. has the advantages of being able to determine tunnelling events from the reservoir to dots beneath G1-G3 with relatively similar accuracy, it has the disadvantage of being insensitive inter-dot tunnelling events as discussed in Chapter 4.

Electron reservoir

A singular electron 2DEG reservoir feeds the SET and both the dots and the gate fan-out. This is performed via a branch which extends from the ST electrode to the confinement window. This branch extends an electron reservoir from the drain side of the sensor 2DEG to the dots in a similar fashion to electrode R in the previous architecture.

133 CHAPTER 2

As a result of feeding all electrons from a singular 2DEG, substantial charging/discharging events associated with electrode fan-out are observable in the sensor current path. This dynamic response result in large capacitive transients observed in ISET due to gate pulsing. These transients would also exist in the previous device design, but are not directly detected via the SET in this fashion due to the separation of the dot and sensor ohmics, as well as the isolating fan-out of the confinement barrier.

Gate fan-out and confinement

In order to have both the ESR transmission line and the sensor axis parallel to the dot axis, the fan-out of the dot top-gate electrodes must “dog-leg” and fan out parallel to the ESR transmission line. Such a fan-out strategy results in larger amounts of metal placed in between the transmission line and the qubits, which could result in a screening effect and requires more distance between the qubits and the transmission line. These can result in higher ESR operating powers to produce the same field strength at the qubit. The ST, G2 and G3 electrodes all fan-out to the same side of the chip, however the confinement barrier fan-out in this design isolates the source ohmic from the drain by separating the source 2DEG from the fan-out 2DEG of G2 and G3. Mitigation of the fan-out 2DEG under these two gate electrodes was attempted by extending a large confinement plane below the gates in the device region. In practice, this was attributed to a much slower transient observed for the G2 and G3 gates than compared to G1 and B electrodes.

134 CHAPTER 2

2.5.3 Device architecture 3: Improved concept device

Acknowledgement: The device architecture presented in Fig. 2.13 and further discussed in this section was theorised and designed in collaboration with Mr. A. West.The device was fabricated and imaged by Mr. A. West at ANFF-NSW, via the standard processes as presented in Appendix F.

This device architecture was designed for the proposed use in singlet- triplet based experiments. The architecture is designed to have the following benefits and improvements:

Multi-regime tunability for increased experimental capacity

This design can be configured to perform standard single-reservoir qubit ex- periments through using CB1, CB2 and B1 in depletion-mode. The combined effect of these gates results in the confined qubit window seen in the previous architectures. Conversely, as B1 can be independently tuned, it can also be used as a tunnel barrier to the dot region. Changing the SET barrier gates such that SLB is transparent and SRB is a hard barrier to the 2DEG, a conduction path can be joined between the S and R ohmics. This results in the ability to operate the device in a transport regime which is useful for exploring and benchmarking Pauli spin blockade in the SiMOS device, similar to the

135 CHAPTER 2

Figure 2.13: Improved device (in production) which is capable of operating in both single-reservoir and dual-reservoir regimes. A key design difference is the isolation of the reservoir 2DEG and ohmic contact region from all other device components. SEM image supplied as courtesy of Mr. A. West. experiments conducted in Ref. [135]. The electrostatic configurations of these two regimes are further discussed in Appendix H.

Split confinement gates

The confinement gates have been split into two or three components (de- pending on the regime configured), resulting in greater flexibility in qubit configuration. Electrode B1 can be independently tuned as either a con-

136 CHAPTER 2

finement barrier or as an opaque barrier with tunable electron tunnelling rate. Separation of CB1 and CB2 results in the independent tuning of lat- eral confinement, useful to counteract device electrostatic asymmetry and strain-induced effects as discussed in Refs. [151, 152]. Electrode CB2 would generally need greater confinement levels that CB1 as it is required to iso- late the R and SG ohmics, as well as contending with the electrostatics of gate fan-out from dot top-gates and barrier gates.

Dot-reservoir 2DEG isolation

As observed in Fig. 2.13, the fan-out of CB1 and CB2 run along either side of the electron reservoir gate R. This intentionally isolates the qubit electron reservoir 2DEG from having to supply any other regions such as the gate fan-out as seen in the previous two architectures. The additional isolation of the reservoir intends to present the most stable Fermi energy possible for single-reservoir qubit experiments. In order to supply the fan-out of gates with access to a 2DEG, a screening gate SG is patterned beneath the fan- out. This gate acts in the same way as the additional prong attached to the reservoir in the device architecture of Fig. 2.11a, but with the electrons being supplied from a dedicated ohmic. For the two-reservoir configuration, the SG gate is held low and the fan-out is instead supplied by the drain ohmic. Here, the S and D ohmics are isolated via SRB.

137 Chapter 3

Randomised benchmarking for SiMOS quantum dot single-qubit fidelities

Chapter Acknowledgement: This chapter concerns the findings presented in part within M. A. Fogarty et. al., Non-exponential Fidelity Decay in Randomized Benchmarking with Low-Frequency Noise, Physical Review A 92, 022326 (2015) (also referred to as Ref. [58]). The full acknowledgements for this manuscript can be found in Appendix A. The SiMOS architecture used to generate these results is shown in Fig. 2.11 with details discussed in Ch. 2.5.1. The measurement protocols used to generate the results are discussed in Ref. [42] and Appendix G.

138 CHAPTER 3

The results in this chapter follow those presented in Ref. [58], regarding the study of non-exponential decays produced by the Randomised Benchmarking (RB) experiment in the presence of non-Markovian noise sources. Randomised benchmarking experiments [153, 106] quantify the accuracy of quantum gates by estimating the average decay in control fidelity as a func- tion of the number of operations applied to a qubit. Benchmarking enjoys several advantages over the traditional methods of characterising gate fidelity that involve quantum process tomography [99, 100], namely that it is insen- sitive to state preparation and measurement (SPAM) errors, and scales effi- ciently with the system size. As such, benchmarking protocols (see Fig. 1.9) have become a standard against which different qubit technologies and archi- tectures are compared. Benchmarking experiments have been performed in many different technologies, including trapped ions [153, 154, 155], supercon- ducting qubits [156, 105, 116], nuclear magnetic resonance architectures [157], nitrogen-vacancy centers in diamond [158], semiconductor quantum dots in silicon [42], and phosphorous atoms in silicon [78]. Most experiments are fitted using an exponential fidelity decay, which is in line with original the- oretical predictions [153, 159], and consistent with the assumption of weak correlation between noise on the gates that is important for accurate fidelity estimates.

139 CHAPTER 3

3.1 The Randomized Benchmarking

experiment on a SiMOS qubit

In the experiment, the qubit is defined by the spin state of a single electron.

A magnetic field B0 =1.4 T is applied to create a Zeeman splitting and the qubit is operated using electron spin resonance (ESR) techniques by applying

gμB B0 an AC magnetic field with frequency ω0 =  . A Rabi π-pulse is realised µ ∗ µ in τop =1.6 s and using a Ramsey sequence the dephasing time T2 = 120 s has been obtained with state-preparation and measurement fidelities of 95% and 92%, respectively [42]. In between consecutive pulses, a waiting time

τw =0.5 µs has to be incorporated, due to the operation of the analogue microwave source.

3.1.1 Experimental realisation of the Clifford gates

The set of Clifford gates is generated using the set [±X , ±X /2, ±Y, ±Y/2] that are realised using Rabi pulses, and the identity simulated with a waiting time equal to a π-pulse. The 24 single qubit Clifford gates are produced by combining elements of the generating set together as illustrated in Table. 3.1.

3.1.2 Result: Non-exponential decays

A clear deviation from the fidelity decay predicted by Eq. (1.96) has been observed in a silicon quantum dot qubit [42]. A non-exponential fidelity decay

140 CHAPTER 3

Table 3.1: Implementations of the single qubit Clifford set. Gate Set Implementation[107, 42] Pauli Set I X Y X , Y 2π/3 Rotations Y/2,X /2 −Y/2, X /2 Y/2,−X /2 −Y/2,−X /2 X /2,Y/2 −X /2,Y/2 X /2,−Y/2 −X /2,−Y/2 π/2 Rotations X /2 −X /2 Y/2 −Y/2 X /2, Y/2, −X /2 X /2, −Y/2, −X /2 Hadamard-like Rotations Y/2, X −Y/2, X X /2, Y −X /2, Y X /2, Y/2, X /2 X /2, −Y/2, X /2

141 CHAPTER 3 can be caused by leakage, where population in the two qubit states is lost to other levels [107, 108, 160]. For example, in Ref. [107] it has been shown that in the presence of a third level, the sequence fidelity for large m could approach 1/3, instead of 1/2 (although this benchmarking protocol used a different gate-set than the standard one). While leakage is an important aspect in multi-dot qubits, such as singlet-triplet qubits or exchange-only qubits that possess accessible non-qubit spin states, a qubit encoded in a spin-1/2 system is inherently two-dimensional. Higher energy levels of the quantum dot, or valley degeneracies, represent different degrees of freedom, rather than leakage channels. Leakage can occur through loss of the electron, but it is noted that the qubit system experiences a T1 time on the order of seconds and the absence of tunnelling during a sequence was measured. As further evidence of negligible leakage, it is noted that the spin-up and down fractions are symmetric around the half-visibility-plus-offset, as observed in Fig. 3.1b&c.

3.2 Randomized Benchmarking in the

presence of low frequency noise

In this section a study is outlined which shows that non-exponential fidelity decays in randomised benchmarking experiments on quantum dot qubits are consistent with numerical simulations that incorporate low-frequency noise, corresponding to a control fidelity that varies slowly with time.

142 CHAPTER 3

Figure 3.1: a) Sequence fidelity as a function of sequence length m,with the ∗ qubit subject to Gaussian distributed T2 associated noise. Each black line represents a fidelity decay for one particular value of detuning δω (only 10% of traces shown). The linear decay on the logarithmic scale illustrates that these individual traces are indeed exponential, while the ensemble average (thick blue line) is non-exponential. The thick red line is the ensemble average for a Lorentzian distributed noise. b) Gaussian distributed detuning frequencies and c) Lorentzian distributed detuning frequencies associated with individual traces. Figure is sourced from Ref. [58].

143 CHAPTER 3

Non-exponential fidelity decay in this semiconductor qubit[42] is indica- tive of a dephasing-limited decay caused by non-Markovian noise. A numerical simulation method is proposed that incorporates time-dependent effects, primarily a drift in frequency detuning. This detuning drift and other time-dependent low-frequency noise sources lead to decay curves that are effectively integrated over an ensemble of experimental results, each with slightly different “instantaneous” average fidelities, i.e., fidelities that are ap- proximately stable over the course of a single benchmarking run, but that drift over the course of the entire sequence of experiments. These simulations show good qualitative agreement with the observed non-exponential decay from the experiments on isotopically-purified silicon quantum dot qubits [42]. Quantitative analysis which compares two very simple models that both give good fits to the data are studied: the first is a simplified version of the drift model that postulates that each experimental run has one of only two possible average fidelities; the second model attributes the non-exponential decay to fluctuating SPAM errors. Both of these models have only one ad- ditional parameter over the standard benchmarking model, but the quanti- tative likelihood analysis shows that the simplified drift model is much more probable. The conclusion of this analysis for the SiMOS quantum-dot qubit is that, while the total average fidelity over a long series of benchmarking runs is 99.6% [42], the instantaneous fidelity can be as high as 99.9% or more when naturally fluctuating environmental noise sources and system calibrations are

144 CHAPTER 3 most favourable. Consistent high fidelities such as these may be within reach: further improvements in the system calibration, such as more frequent and accurate estimates of the qubit detuning, could allow these high fidelities di- rectly by exploiting the low-frequency character of the noise. Achieving such high fidelities for single-qubit gate operations gives optimism for exceeding the demanding error thresholds for fault-tolerant quantum computation.

3.2.1 Non-Markovian noise in a quantum dot qubit

Within the experiment, Ramsey sequences have been performed in between benchmarking sequences to recalibrate the resonance frequency of the qubit and to compensate drifts due to, for example, the superconducting mag- net. These drifts, in combination with errors in setting the resonance fre- quency, cumulate in a time dependency within the system and result in an

∗ apparent T2 for the randomised benchmarking experiment. This decoher- ence time is also dependent upon the duration of the data acquisition. The

∗ non-Markovian noise processes that are expected to determine T2 can be modelled as a random walk of the detuning δω away from the ideal oper- ation frequency ω0, over time-scales greater than a single run of a random Clifford sequence.

145 CHAPTER 3

3.2.2 Modelling noise sources

In order to simulate an ensemble of results, the δω term is selected randomly from a Gaussian distribution of normalised variance:

 τop σop = ∗ . (3.1) 2π 2 ln(2)T2

Using this distribution, benchmarking experiments have been numerically simulated and the results are shown in Fig. 3.1. In this simulation the time- scale of the low frequency noise is approximated to be on the order of a single run, and as such the detuning is constant over a single trace. However, be- tween each trace the detuning is sampled randomly from a distribution as shown in Fig. 3.1b&c. The individual traces correspond to a given detuning δω and result in the “instantaneous” fidelity of the qubit. While the indi- vidual traces are decaying exponentially, the averaged fidelity (bold blue) obtained from the Gaussian ensemble is clearly non-exponential. The case of a Lorentzian distribution of detunings is included (red), resulting in a non-exponential decay as well. In the simulation, the only error source is dephasing, whereas in the experiment, other errors might be present such as pulse-errors. Inclusion of such errors will still result in non-exponential decays, provided dephasing is a significant source of error. As low-frequency drift of the qubit resonance frequency can lead to non- exponential fidelity decay, it is hypothesised that some ensemble of exper- iments with varying decay rates is the correct explanation for the non-

146 CHAPTER 3 exponential behaviour of the experimental benchmarking data [42]. To sup- port this hypothesis, the Akaike information criterion is employed to show that a simple model allowing for differing fidelity rates better explains the data than an alternative explanation that assumes fluctuating SPAM errors in the standard (zeroth order) model.

147 CHAPTER 3

3.3 Extending Randomised Benchmarking

This section details how the randomised benchmarking analysis was extended to take non-Markovian noise in the form of low-frequency drifts in qubit con- trol parameters into account. The theoretical ideas related to the experimen- tal design in this section were developed in group discussions, with support from collaborators outlined in Appendix A.

3.3.1 Eliminating the constant for a single-qubit ran-

domised benchmarking model

Acknowledgement: While the theoretical ideas related to the experimental design were devel- oped in group discussions, the theory for eliminating the constant from this section was made rigorous with the support of R.H, S.T.F and S.D.B.

The parameters A and B in Eq. (1.96) are nuisance parameters that do not convey information about the desired control fidelity. Eliminating one of these parameters, in this case B, will further constrain the zero order model and allows deviations to be more clearly identifiable. A further advantage of removing the parameter B is to allow fitting of a linear function on a log-linear plot where deviation from standard assumptions of randomised benchmark- ing will show clearly as a non-exponential decay trace. In Ref. [42] the ran- domised benchmarking protocol was modified to eliminate B from the zero

148 CHAPTER 3 order model. First, a theoretical justification for this approach that conforms with the standard assumptions of randomised benchmarking is provided. It is noted that this approach applies only to qubits (d = 2), and demonstrate the deviation of the measured data from the expected exponential is high- lighted via this method. Recall that the zero-order model fits the average fidelity of a gate sequence to a simple formula as follows [106]:

¯↑ ↑ m ↑ Fm = A p + B , (3.2) where the qubit is initialised as |↑↑|, the final gate in the random bench- |↑↑| ¯↑ marking sequence is chosen to return the state to ,andFm is the survival probability of this state. To eliminate the constant B↑ from this sequence, a second set of similar randomised sequences is executed, with the difference being that the final (m +1)th gate is set to change the state to |↓↓|.For these runs, the survival probability for yielding the measurement outcome E↓, where in the ideal case the final state ρ = E↓ = |↓↓| are considered. ¯↓ This is the survival probability for each run Fm. Under the same assumptions we have ¯↓ ↓ m ↓ Fm = A p + B . (3.3)

˜ ≡ ¯↑ − − ¯↓ Combining these two equations by defining Fm Fm (1 Fm), we have:

˜ ˜ m ↑ ↓ Fm = Ap +(B + B ) − 1 , (3.4)

149 CHAPTER 3 where A˜ = A↑ + A↓. Recall that B↑ =TrE↑D(1/d) ,whereD is the average noise operator. ¯↓ For the Fm runs, the derivation is identical, apart from the final change to the |↓↓| state using a π pulse X ,sowehaveB↓ =Tr E↓D(X (1/d)) .One expects that X is close to unital, i.e., X (1/d) 1/d. Under the assumption that this is true (an assumption that will be respected up to violations no larger than the gate infidelity), and noting that E↑ + E↓ = 1 for qubits (d =2)andthatD is trace-preserving, B↓ can be re-expressed as follows:

B↓ =Tr E↓D(1/2) =Tr (1 − E↑)D(1/2) =Tr D(1/2) − Tr E↑D(1/2) =1− B↑ . (3.5)

− ¯↓ Therefore by subtracting the average results of the data-set (1 Fm)fromthe ¯↑ average results of the data-set Fm, a data set that is distributed according to the model

˜ ˜ m Fm = Ap (3.6) is obtained under the standard benchmarking assumptions on the noise. The data from Ref. [42] consist of 8 data sets (one reference set and 7 interleaved sets). The experiment, in order of operation, comprised of ¯↑ measuring 50 single shots which were randomly distributed over Fm and ¯↓ Fm. The sequence randomisation protocol was carried out 10 times, first for

150 CHAPTER 3 the reference and then all 7 interleaved sets. For each data set, the gate sequences of the lengths m ∈{2, 3, 5, 8, 13, 21, 30, 40, 50, 70, 100, 150} were measured. This entire process was then repeated 50 times for a total of 2,400,000 single shot measurements. The fast Ramsey recalibrations were performed at approximately 10 minute intervals. This amount of randomisation is at least an order of magnitude more than in previous experiments [153, 154, 105, 156, 116, 157, 158, 78]. Each randomised protocol was performed 50 times in order to estimate the survival probability.

3.3.2 Re-analysis of experimental data

Acknowledgement: While the theoretical ideas related to the experimental design were devel- oped in group discussions, the statistical analysis summarised in this next section was performed by R.H, with the aid of S.D.B and S.T.F. This in- cludes the introduction of alternative (residual SPAM) models, calculation and write up of the Akaike bounds.

To quantify the quality of experimental data fits, estimates of the variance in the data of Ref. [42] are required. The observed variance of the data matched to within 5-40% of the theoretical upper bounds derived in Ref. [161] when ¯ the gate length was shorter than 20 (so that the m(1−Favg) 1 assumption discussed in that reference was satisfied). Accordingly, the observed experi-

151 CHAPTER 3 mental variance was used as a reliable estimate of the actual variance of the distribution. It should be noted that the observed variance actually decreased for gate lengths of 100 or greater. One explanation for this unexpected be- haviour is that some of the sequences become saturated to something close to a completely mixed state before reaching those sequence lengths. Figure 3.2 shows the data from the reference dataset plotted on a semi-log plot. The confidence bounds are 95% and the data is clearly non-linear (i.e. the decay is not a simple exponential). Similar deviation from the linear fit was noted in each of the data sets, with the best-fit linear model consistently ¯ underestimating Fm for m ≥ 100. Ref.[106] outlines a higher (first) order

¯↑ − − ¯↓ Figure 3.2: Semilog plot of Fm (1 Fm) for the reference sequence of randomised benchmarking on a silicon quantum-dot qubit [42]. Both the two-fidelity model and a single-fidelity model including residual SPAM can fit the data, but for the single-fidelity model an unreasonably large SPAM has to be included. Figure is sourced from Ref. [58].

152 CHAPTER 3

fitting for the fidelity decay which includes gate dependent noise. With the elimination of the parameter B as outlined above, the inclusion of higher

˜1 ˜ m ˜ − − 2 m−2 orders results in Fm = Ap +C(m 1)(q p )p . The first order equation did not fit the data significantly better than the zero order equation. Two other possible explanations are considered. First, it may not be pos- sible to entirely eliminate the constant term (B) due to a violation of one of the assumptions in the above derivation. A second explanation is that low-frequency noise leads to detuning, and hence time-dependent errors on the gates in some of the experiments. The first, which is denoted the resid- ual SPAM model, can be modelled by reverting to a formula of the form

˜ ˜ m ˜ ˜ Fm = Ap + B, where now B represents residual SPAM errors that were not eliminated under the assumptions that led to the derivation of Eq. (3.6). Considering the simplest possible model for the second explanation – the two fidelity model – by fitting the fidelity decay to a formula of the form

˜ ˜ m ˜ m Fm = Ap + Aq . This represents an attempt to model the data by sim- plifying the ensemble of experiments by reducing them to just two different equally weighted sequence behaviours: one with a high-fidelity rate (related by the usual measure to p) and one with a lower fidelity rate (similarly related to q). This model has fewer parameters than the Gaussian or Lorentzian drift models, and is much easier to fit. In this interpretation, the B parameter as per Eq. (3.6) is successfully eliminated, but time variation gives the two dif- ferent polarisation parameters p and q, with the decay rate for each sequence sampled randomly with equal probability. As can be seen in Fig. 3.2, both

153 CHAPTER 3 models fit the data substantially better than the simple exponential of the zero order model. Although the residual SPAM model produces a good fit to the experi- mental data, it does so with the equivalent of an unusually large SPAM pa- rameter B˜ of around 0.14 corresponding to an individual fitting of B↑ =0.56 and B↓ =0.58. This represents in the theoretical model a very large bias in the expectation value of the spin-up measurement on the asymptotic value of the sequence fidelity away from the theoretical value of 0.5, which is not observed in the experiment. To compare the residual SPAM and two-fidelity models quantitatively, it is possible to calculate the log likelihood and Akaike information crite- rion [162] for the two models. Because the actual distribution of the test statistic is not accessible, an appropriate assumption that the samples con- tained in the underlying data are independent and the Gaussian distributed limit is made. This assumption is well-justified as a large number of in- ˜ dependent data sets are used here. The distribution Fm can therefore be approximated by a Gaussian distribution with a variance estimated by the observed variance at each gate length. The log likelihood of the observed data, given each of the two models, can then be calculated using standard methods, as follows.

154 CHAPTER 3

Table 3.2: Akaike information criterion for standard and interleaved ran- domised benchmarking. The comparison column specifies how many times as probable is the Ap˜ m + Aq˜ m model to minimise information loss as com- pared to the Ap˜ m + B˜ model.

Akaike Information Criteria Dataset Ap˜ m + B˜ Ap˜ m + Aq˜ m Comparison

Ref -16.93 -25.29 65.44 I -46.19 -57.12 238.10 X -54.52 -59.99 15.43 X /2 -62.89 -63.79 1.56 -X /2 -57.77 -64.34 26.69 Y -36.06 -50.43 1317.00 Y/2 -36.04 -46.39 172.00 -Y/2 -46.37 -63.32 4815.00

The Akaike information criteria used was 2 ∗ (Number of parameters − loglikehood). The log likelihood is calculated as:    

1 1 2 ln  − [μs − xs] (3.7) 2 2 sequence 2πσs 2σs s= lengths

2 where σs , is the variance for sequence s, μs is the measured average at that sequence and xs is the predicted average. Table 3.2 shows the calculated Akaike information criterion for each of the experimental datasets. As can be seen, the two fidelity model better

155 CHAPTER 3 explains the data, significantly so on all but one of the datasets. Although such a model is a simplified version of the drift model, the fact that it fits the data well and is physically motivated supports its adoption as the most likely explanation of the non-exponential curve seen in the data.

3.3.3 Interpreting the two fidelity model

Since the two fidelity model is the quantitatively preferred model, a natu- ral question arises: how should the model parameters be interpreted? The obvious interpretation of the two parameters p and q is as presented in ta- ble 3.3; that their difference represents the characteristic spread of the actual underlying ensemble of fidelities from which the benchmarking data are sam- pled. Such an interpretation is natural and compelling, however it remains an open problem to quantify such a connection more carefully. In particu- lar, it would be interesting to give a direct connection to a more general drift model, since these are easier to interpret physically, but much harder to fit and analyse statistically. By considering the non-exponential decay manifesting as the average over an ensemble of results, the fidelity can be considered to be operating under two regimes as depicted in Fig. 3.3a. Firstly, dominating the observed fidelity decayatlowm, there is a rapid decay rate dominated by traces of large detuning δω. Secondly, for large m, these traces of large detuning will fast approach the constant B and so will not influence the decay slope in the fidelity; the fidelity decay rate for large m is then governed by long-lived

156 CHAPTER 3 traces of smaller detuning. In Fig. 3.3a, each of the data points are an average over 25,000 experimen- tal repetitions as presented by the two accompanying histograms (Fig. 3.3b for m =2andFig.3.3cform = 150). Each histogram separately shows the measured probability, averaged over 50 repetitions, for the spin-up and spin-down observables as expected at the end of a noiseless version of the ap- plied random sequence. From Fig. 3.3 it is inferred that the instantaneous fidelity approaches a peak fidelity of 99.9% when the microwave frequency is on resonance and that the fidelity can drop to 98.9% in the presence of large detunings. The time-average performance will lead to a fidelity in between, consistent with the 99.6% fidelity quoted in Ref. [42]. A similar analysis can also be applied to the interleaved gate sequences, to extract average fidelities for the individual gates. See Table 3.3 for p and q values for each set. For each of the interleaved gates, a high and a low gate fidelity comparable with the two values quoted for the standard benchmarking scheme above are found, although the numerical instability in calculating the gate fidelities for interleaved benchmarking leads to larger uncertainties in these values, of the order of 1% for 95% confidence margins. It is noted that a naive application of the method to derive fidelities for interleaved gate sequences as outlined by Ref. [105] will, for certain gates ¯ such as the Y gate, yield a fidelity Favg > 100%. Such results may be an indication of correlated low frequency noise, where some Clifford gates can echo out noise; such an effect would result in decays which are slower than

157 CHAPTER 3

Figure 3.3: a) Reference sequence of randomised benchmarking on a silicon quantum-dot qubit [42]. The separate fidelities from the two-fidelity model have been plotted to show how the initial decay is dominated by the low q value, whereas the higher value of p is indicative of the average decay in the longer-lived high-fidelity regime. Histogram of spin up |↑ and spin down |↓ corresponding to data point b) m =2andc) m = 150. Results with expected spin-up outcome are shown in red, while blue represents data with expected spin-down result. The grey regions illustrate the overlapping areas. Figure is sourced from Ref. [58].

158 CHAPTER 3 that of the reference set, and break the assumptions of interleaved randomised benchmarking. However, the large uncertainty in the estimated fidelities for interleaved gates due to statistical noise does not allow a definitive test for such an effect with the current dataset.

3.4 Discussion of results

This chapter have analysed the non-exponential decay in randomised bench- marking experiments on SiMOS quantum dot qubits, and found that the

Table 3.3: Calculated p and q values for the two fidelity model. The gate ¯ fidelity estimates (Favg) reported for the reference run are the high (p) gate fidelity estimate and low (q) gate fidelity with the 95% confidence margins are ±0.06% and ±0.5% respectively. It is further noted that calculated p and q values will result in an inaccurate interleaved gate fidelity as given by the process outlined by Ref. [105] due to the low-frequency noise.

F¯p F¯q Dataset pq Dataset avg avg Ref 0.995 0.959 Ref 99.9% 98.9% I 0.993 0.946 X 0.993 0.952 X /2 0.993 0.947 -X /2 0.991 0.947 Y 0.993 0.964 Y/2 0.991 0.952 -Y/2 0.990 0.911

159 CHAPTER 3 most plausible explanation of this decay is drift in detuning frequencies. These simulation of temporal integration over a spectrum of time-dependent detuning frequencies qualitatively reproduces the observed fidelity decay of previously conducted experiments [42]. In addition, a competing model has been quantitatively ruled out by showing agreement of a simplified ensemble (the two fidelities model) that is much more probable. This yields confi- dence that detuning drift is the correct explanation for the origin of such a non-exponential fidelity decay. Low frequency noise leads to a time-varying fidelity that is relatively con- stant over a given sequence but can vary between sequences. Consequently, an “instantaneous fidelity” is defined to characterise the performance of a gate during a single sequence, and we can consider how this instantaneous fidelity varies in time from sequence to sequence. Fitting the randomised benchmarking data with a two-fidelity model demonstrates that silicon MOS quantum dot qubits can already exhibit an “instantaneous” control fidelity of 99.9%. This is achieved when the system is correctly calibrated and the microwave frequency is on resonance. However when the noise causes large detuning the fidelity drops to 98.9%. It is anticipated that the higher fidelity can be achieved consistently (i.e., made time-independent) as improvements in the readout fidelity appear feasible and better calibration could be ob- tained by performing optimised Ramsey protocols to calibrate the resonance frequency for each experiment [163]. These results raise several intriguing questions. The first is to quantita-

160 CHAPTER 3 tively link the simple and easy to analyse two fidelity model to the Gaussian or Lorentzian drift models. Alternatively, directly fitting a drift ensemble to the data would give a better picture of the source of the non-exponential fi- delity decay, but this approach risks over-fitting, and is already difficult for the simple case of Gaussian-distributed detuning. Secondly, there are sev- eral different models discussed here, each of which are capable of analysing different forms of breaches in the standard assumptions of RB. The reduced parameter representation of fidelity (F˜) not only allows for higher accuracy in fitting, but it is capable of immediately identifying a deviation from the expected result under standard assumptions. Further it is capable of identi- fying the presence of certain types of noise if F˜ is to asymptote to a non-zero offset. Finally, there is at least one other natural competing explanation for the non-exponential decay. It might be the case that long benchmarking se- quences saturate the exponential decay rates and have slower decay on very long time-scales. If this were the case, then fitting to sequences that were “too long” would bias toward non-exponential decays and reporting fidelities that were higher than warranted by the analysis. Therefore, deriving stop- ping criteria for the maximum sequence length, and tests that rule out this alternate explanation is an important open question for future work.

161 Chapter 4

Enhanced qubit readout in the singlet-triplet basis via metastable state latching

Chapter Acknowledgement: This chapter concerns in part the findings presented in M. A. Fogarty et. al. arXiv preprint arXiv:1708.03445v2 (also referred to as Ref. [60]). The full acknowledgements for this manuscript can be found in Appendix B. The SiMOS architecture used to generate these results is shown in Fig. 2.12 with details discussed in Ch. 2.5.2. The measurement protocols used to generate the results are discussed in Ref. [60] and Appendix G.

162 CHAPTER 4

The role of excited states for measured signal enhancement has been studied in a number of semiconductor quantum dot systems [164, 165, 166, 136, 80, 60]. In the SiMOS system, state latching is often observed when tunnel rates to an electron reservoir is lower than inter-dot co-tunnelling rates [167]. In the presence of low tunnelling to a reservoir, a dot charge state can become latched in a certain configuration. This condition can be alleviated by a co- tunnelling event via an intermediate state which possesses stronger coupling to the reservoir [167]. Recent studies of reservoir-charge-state latching in semiconductor quantum dot devices have led to novel methods to reduce readout error by almost an order of magnitude [136]. The latched condition can arise via a number of different mechanisms, and is therefore dependent upon a range of factors. The example detailed in the following sections involves employing charge state hysteresis [137] produced by weak coupling of one dot to a common electron spin reservoir [85, 60].

4.1 Physical description

The latching is produced via asymmetric couplings of the two dots to the common electron reservoir [137, 85, 60], where a (1,1)-(1,2) dot-reservoir metastable charge state is produced via a combination of the low tunnel rate between QD2 and the reservoir (shown as ΓSlow in Fig. 4.1a) and co- tunnelling between QD1, QD2 and the reservoir (ΓFast in Fig. 4.1a). The latching mechanism produced a metastable spin-reservoir-charge state upon

163 CHAPTER 4

a) CB G3 G2 G1 B ST SiO2 QD2 QD1 Reservoir 28Si ȽFast ȽSlow Si (Natural)

b) ȽFast (0,2)T0 ȽSlow

(0,2)S (1,1)T0 (1,2T0) PSB (1,1)S (1,2S)

Figure 4.1: a) Pictorial representation of the device cross section through the active region. This illustrates dots under G1 and G2 electrodes are tunnel- coupled to an electron reservoir under gate ST. State latching requires tunnel rates to be asymmetric, where fast and slow tunnel rates are given by ΓFast and ΓSlow respectively. b) Energy level representation of the latching process; in order to rapidly populate the (1,2) ground state, an existing (1,1) state must co-tunnel via (0,2) where the standard Pauli spin blockade exists. If the state is not blocked (i.e. a |S state) then an electron is free to tunnel from the reservoir to fill the strongly coupled dot. Otherwise, the tunnelling from the reservoir is blocked, resulting in a spin-to-reservoir charge state conversion. Image reproduced from Ref. [60] attempting to configure a (1,2) charge configuration. As demonstrated in Fig. 4.1b, in order to rapidly populate the (1,2) ground state, an existing (1,1) state must be able to co-tunnel via (0,2) where the standard Pauli spin blockade exists (see Sec. 2.4.1). A singlet |S state does not experience blockade, and therefore the electron under the G1 electrode can co-tunnel

164 CHAPTER 4

at rate ΓFast to the dot under G2 as a reservoir electron is supplied. Any triplet |T0,± state experiences spin blockade where the path to the (1,2) state involves a slow tunnelling process via ΓSlow.

4.2 Experiments for detection

As demonstrated in the previous section, the goal of these latched readout mechanisms is to enhance Pauli spin blockade (PSB) signals. Thus, pulsing schemes designed to demonstrate or detect PSB[168, 133] can often lead to visualisation of these excited states[60]. Figure. 4.2 illustrates the stability diagram given by such a scheme. For standard PSB readout, pulsing from (1,1)→(0,2) charge configuration results in the QD1 electron tunnelling to QD2 only when the two spatially separated electrons form a singlet spin configuration. The triplet states are blockaded from tunnelling due to the large exchange interaction in (0,2). After first flushing the system of a QD1 electron to create the (0,1) state at A, a (1,1) state at B loads a randomly configured mixture of singlet and triplet states (solid arrow in Fig. 4.2a. The current through the nearby single electron tran- sistor (SET) is recorded at this position, tuned to be at the half-maximum point of a Coulomb peak. The system is then ramped rapidly to a variable measurement point (dashed arrows in Fig. 4.2) where the SET current is measured again. A map of the comparison current ΔISET between these two points is created, where the derivative in sweep direction d(ΔISET)/d(ΔVG1)

165 CHAPTER 4

a) 30

(N1=0,N2=2) ((1,2)1,2) 20 C

PSB Latched PSB 10 (0,1) (1,1) (mV)

G2 B V

ǻ 0 A b) (0,2) (1,2) B -10

A C (0,1) (1,1) -20 0 20406080100 ǻVG1 (mV) Figure 4.2: a) Cyclic pulsing [168, 133] (arrows) through sequence A(0,1)- B(1,1)-C, where the location of point C is rastered to form the image, reveals latched spin blockade features (orange dot & top zoom-in). Shown is the dif- ferential transconductance d(ΔISET)/d(ΔVG1), where ΔISET is the difference in SET current recorded at points B and C. b) When point B lies in the (0,2) charge region, no blockade is observed, as expected for an initial singlet state. Image reproduced from Ref.[60]

(as shown in Fig. 4.2a&b) decorrelates the capacitive coupling of the con- trol gates to the SET island. A change in the charge configuration marks a shift in the SET current, clearly observed as bright/dark bands. The bright

166 CHAPTER 4 band in the centre of the (1,1)-(0,2) anti-crossing of Fig. 4.2a is consistent with PSB, where the blockade triangle is restricted to a narrow trapezoidal area, bounded by state co-tunnelling via the reservoir and the first available excited triplet state[133]. For this type of measurement, a dot-reservoir metastable charge state can be produced via a combination of the low tunnel rate between QD2 and the reservoir (shown as ΓSlow in Fig. 4.1a) and co-tunnelling between QD1, QD2 and the reservoir (ΓFast in Fig. 4.1a). The latching results in a prominent metastable feature observed at the (1,1)-(1,2) transition in Fig. 4.2a. In contrast, when the system is initialised in the (0,2) charge configuration (Fig. 4.2b) the singlet state is prepared robustly due to large energy splitting, and the resulting map in Fig. 4.2b manifests no metastable regions, as would be expected. As demonstrated in Ref. [137], state latching can be lifted by co-tunnelling via an intermediate state energy which has strong coupling to both the latched dot and the electron reservoir. For this readout protocol, the en- ergy level which lifts the latching condition must also produce a relaxation path for the triplet states as well, indicating a valley or orbital degree of free- dom. The energy splitting between the (0,2) singlet ground state and the first available state capable of producing a triplet relaxation path is measured to be 1.72%±0.2% of the charging energy EC (see Appendix I for further de- tails). Typically EC ∼ 10meV [140], indicating that this splitting exceeds electron thermal energies by two orders of magnitude. This first available

167 CHAPTER 4 state observed here is likely to be the first excited valley state [169, 170, 140] for the (0,2) charge configuration. The conditions favourable for the observation of enhanced readout via state latching can be observed in the full stability diagram Fig. 4.3 (left). This map is produced by the methods detailed in Ref. [167], and will not be discussed in this thesis. Within this diagram, strong charge state hystere-

12345 (0,3) (1,3) -12 ǻISET x10 1.50

(0,2) (1,2) (2,2) (V) G2 V 1.40 (0,1) (1,1) (2,1)

(0,0) (1,0) (2,0)

1.30 (N1,N2)

1.00 1.10 1.20 1.30 VG1 (V)

Figure 4.3: Extension to latching: Charge stability diagram (left) of the double-dot system studied [60]. Mapping at each anti-crossing (0,N2+1)− (1,N2) is shown (right) for random loading of (1,N2) state. The latched readout is observed only at every second anti-crossing, consistent with map- ping of the PSB charge state to a dot-reservoir charge state. Image repro- duced from Ref. [60]

168 CHAPTER 4 sis [137] under the G2 dot is observed after the unloading of the last electron under G1 (i.e. N1 = 0). Often, this can be attributed to the inability for any co-tunnelling of the G2dotviaG1 to the reservoir due to the G1 dot being empty of all states. The presence of charge hysteresis is also observable in the accompanying maps of Fig. 4.3 (right) produced via the three level pulse sequence described above. These four maps detail state latching at the anti-crossings attributed to the N1=1,N2=1− 4 charge configurations. Here, it is observed that latching occurs only at every second anti-crossing, where an even number of electrons exist on QD2. This reinforces the theory that the enhanced readout via latching is indeed the projective mapping of (0, 2)S − (1, 1)T → (1, 2)S − (1, 1)T .

169 CHAPTER 4

4.3 Improvements to readout

In order to estimate the increase in visibility due to the latched readout, the characteristic relaxation time of the latched state is measured by initialising a random (1, 1) state, pulsing rapidly to the Standard-PSB location in (0, 2) before pulsing to the a latched readout location close to the anti-crossing (Or- ange dot in Fig. 4.2a). The state is taken via (0, 2) in order to emulate the measurement protocols used in spin readout experiments, and therefore the latched readout histograms should contain the same measurement errors as these protocols (to be detailed in Ch. 5 & 6). Relaxation of the latched state

τL is shown in Fig. 4.4d with a characteristic time of 2.8 ms. Using an op- timal measurement integration time of τM =0.2 ms a model for single-shot statistics which accounts for relaxation under τL can be implemented [131], yielding a maximum visibility of 98%, by optimising the threshold current as shown in Fig. 4.4c. The blue and red dashed lines are singlet state mea- surement fidelity FS and triplet state FT measurement fidelity as given by equations in Ref. [131]:

∞ IT FS =1− nS(ΔISET)dI, FT =1− nT (ΔISET)dI. (4.1) IT ∞

These two fidelities are dependent upon the threshold current ISET chosen in the experiment, and are the integral over the probability density of mea- surement outcomes of nS(ΔISET) for singlets, and nT (ΔISET) for triplets. These two values are modelled as noise broadened Gaussian distributions

170 CHAPTER 4

a120 b 120 PSB Latched-PSB, Initialized: (1,1) 80 80 (0,2)

40 40 S T S T Counts (arb.) Counts (arb.) 0 0 -0.05 0 0.05 -0.05 0 0.05 ǻISET (pA) ǻISET (pA) c d 1 0.6 Relaxation time Latched-PSB Standard-PSB IJ =2.831ms Max Visibility Max Visibility L 98.04% 69.50% 0.4

Visibility 0.2

0 0.0 -3 -2 -1 0 1 2 -0.050 0.05 Blockade Probability 10 10 10 10 10 10 ǻIT (pA) Time at latch point (ms)

Figure 4.4: a) Histogram of detected current ΔISET for the standard Pauli spin-blockade measurement detection scheme. This location coincides with the blue dot in Fig. 4.2a. b) Histogram of detected current ΔISET for the latched Pauli spin-blockade measurement detection scheme. This location coincides with the orange dot in Fig. 4.2a. A clear increase in signal detection (histogram separation) is observed due to conversion into a reservoir charge state. Fidelity of conversion between a inter-dot charge-state to a reservoir charge-state is also observable from the red histogram, given by red marker in Fig. 4.2b. c) State measurement fidelities and measurement visibility as a function of the SET threshold current level ΔIT as analysed for signal detection histograms. d) Relaxation time of the latched state indicating the range of measurement integration time available under the latched protocol. Image reproduced from Ref. [60]

171 CHAPTER 4 around peak amplitude currents ΔIS and ΔIT as detected by the SET. For the singlet state, the probability density is given by (Ref. [131]):

  S 2 (ΔISET − ΔI ) 1 nS(ΔISET)=(1−PT )exp − √ (4.2) 2σ2 2πσ

where PT  is the triplet population. As described in Ref. [131], due to metastability of the triplet population, the formula for nT (ΔISET) is more complicated. The probability density for triplets also takes into account the proportion of triplets which are expected to relax over the measurement integration time τM , including the relaxation value TL extracted from Fig. 4.4d for the latched system.     τ (ΔI − ΔIT )2 1 n (ΔI )=P  exp − M exp − SET √ T SET T 2 TL 2σ 2πσ T   ΔI τ P  (ΔI − ΔIS)2 τ + M T exp − SET M × (4.3) T − S T − S ΔIS TL ΔI ΔI  (ΔI ΔI ) TL (ΔI − I)2 dI exp − SET √ 2σ2 2πσ

By fitting the histogram data shown in Fig. 4.4a&b using Eq. 4.2 and Eq. 4.3, the fidelity of identifying the correct state can be calculated as a function of the threshold current ΔIT using Eq. 4.1 as shown in Fig. 4.4c. From these measurement fidelities, the readout visibility (defined by Ref. [131]) is defined by:

V = FS + FT − 1. (4.4)

172 CHAPTER 4

This is calculated to be 69.50% for the standard PSB process, while the latched PSB process returns a maximum readout visibility of 98.04%. This improved fidelity due to the latched readout mechanism produces a reduction in misidentification error of 93.57% when compared to the standard-PSB.

4.4 Discussion of results

As outlined in Ch. 2, the device architecture used for these experiments was not designed for PSB, and has both dots near symmetrically coupled to the SET island. As a result, the metastable region in the (0,2) charge configu- ration attributed to PSB is almost indistinguishable. In order to counteract this effect, the enhancement mechanism described here is employed to in- crease the visibility of PSB from 60.50% to 98.04%. Other enhancement mechanisms seen in the literature demonstrate similarly high readout visibil- ities for enhanced-PSB, with Ref. [136] reporting a factor of 8 improvement in readout error for an improved visibility of 99.86% in a dot-donor system in silicon. Similarly, Ref. [171] reports 98.4% in a double-donor device in silicon. Advantages in visibility are also reported in GaAs/AlGaAs heterostructure double-dot systems, with Ref. [80] demonstrating improved readout visibility from 83.8% to 99.5% using similar enhancement methods. The use of the reservoir for this signal enhancement implies that the current form of this protocol is considered non-scalable, however, this en- hancement to readout visibility via dot-reservoir change states is an enabling

173 CHAPTER 4 protocol for further exploration of experiments focussing on the single-triplet basis. Further work is required to increase readout visibility of PSB ob- served in the (0,2) charge region in order for this mechanism to be employed within densely-packed arrays of quantum dots such as those proposed in Refs [172, 47, 50].

174 Chapter 5

Anti-crossing dynamics of SH and T− in a SiMOS quantum double-dot

Chapter Acknowledgement: This chapter concerns in part the findings presented in M. A. Fogarty et. al. arXiv preprint arXiv:1708.03445v2 (also referred to as Ref. [60]). The full acknowledgements for this manuscript can be found in Appendix B. The SiMOS architecture used to generate these results is shown in Fig. 2.12 with details discussed in Ch. 2.5.2. The measurement protocols used to generate the results are discussed in Ref. [60] and Appendix G.

175 CHAPTER 5

5.1 Coupling between the singlet and polarised

triplets

As described in Sec. 1.7, the singlet-triplet basis is commonly used to define interactions between multiple electrons, such as the two-qubit SWAP [112] and controlled phase (CPhase) [113] logical operations. Both of these two operations employ energy splitting between the |↑↓ and |↓↑ states (CPhase) or the |(1, 1)S and |T0 states (SWAP) as shown on the Bloch sphere in Fig. 1.12. These experiments are detailed in Sec. 6.3. The presence of a bit-flipping term, identified in Refs. [129, 73, 173, 174, 88], needs to be taken into account when designing these experiments. This term, introduced in Eq. (1.109) of Ch. 1, is represented as Δ(θ). In this chapter, the physical origins of Δ(θ) are discussed in the context of the SiMOS quantum dot architecture, building upon preliminary discussions on the spin-orbit coupling from Ch. 2. A number of experiments are presented which either attempt to measure this Δ(θ) coupling, or utilise the interaction to benchmark other characteristics of the singlet-triplet Hamiltonian.

5.1.1 Physical origins

The original context of this bit-flipping term Δ(θ) was in the GaAs quantum dot qubit platform, where it was identified in Refs. [112, 87] that, due to strong hyperfine field contributions of the nuclear spin bath [129], a Zeeman field gradient between the two dots in thex ˆ andy ˆ directions gives rise to a

176 CHAPTER 5

coupling between the |(1, 1)S and polarised triplet states |T± [87]. Within the context of the SiMOS system, as described in Appendix F, the devices are fabricated upon an epilayer of isotopically purified 28Si which has zero nuclear spin [175]. Within this epilayer exists a residual 800 ppm 29Si, which is a naturally occurring nuclear-spin-1/2 isotope. This level of residual isotope is predicted give rise to a Δ(θ)-coupling contribution on the order of 50 kHz [134]. Another possible contribution to the Δ(θ) coupling arises due to magnetic field gradients between the two dots is produced by the Meissner effect. The SiMOS devices are defined and addressed by Aluminium gates which are a type 1 superconductor. This means the device leads can be superconducting when operating below the critical external magnetic field required to quench this effect. It has been identified in the Si/SiGe heterostructure quantum dot platform, that the Meissner effect, produced by superconducting aluminium leads, can give rise to magnetic field gradients δBz between dots, effects contributing up to a few MHz have been reported [176]. Beyond the magnetic field effects of the Overhauser field and Meissner screening, there are a number of other effects intrinsic to silicon devices which result in contributions to Δ(θ). As outlined in Sec. 2.4.4 the Rashba and Dresselhause spin-orbit-coupling effects can result in Stark shift of the electron g-factor [149, 148]. In the presence of a g-tensor, the difference in off diagonal elements for each dot can produce the same Zeeman energies as the hyperfine field contributions [85].

177 CHAPTER 5

A separate contribution to Δ(θ) can be produced by inter-dot tunnel coupling in the presence of the interface spin-orbit interaction. This effect results in mixture between the |(0, 2)S state and the polarised triplets |T±, producing the ΔSOC term identified in Eq. (1.109).

5.2 Experimental protocols probing and

utilising the SH/T− coupling.

The energy diagram around the charge region presented in Fig. 4.2 is shown in Fig. 1.11. In order to initialise and manipulate these states, the five-level pulsing sequence shown in Fig. 5.1a is utilised, where each level within the sequence is described as the following:

F : Location on the (0, 2) side of the (1, 2)−(0, 2) charge transition. Dwelling

close to this transition (τF 50us) results in rapid relaxation into the singlet state [133] via resonantly tunnelling to the reservoir. Here, the location is deep in the (0, 2) with respect to the ε to ensure minimal singlet hybridisation.

I: Strong coupling between the (0, 2) charge state to the reservoir is re- moved by ramping to this position.

P1: First instance of ramping to location P in Fig. 5.1a. Here the ground state |ψ |(0, 2)S as P should not be so close to the anti-crossing that a charge superposition state is the ground state.

178 CHAPTER 5

a) b) (0,2) (1,2) P Initialization Measure F 0 Ramp Rate Ramp Rate (in) (out) I M Ԫ P

c) Initialization Measure Ԫ P 0 Ramp Rate Ramp Rate (in) (out) Ԫ Dwell Time ɒD

d) Initialization Measure P 0 Ramp Rate Ramp Rate (in) (out) Ԫ (0,1) (1,1) Dwell Time ɒD

Figure 5.1: a) Five-level pulse sequence used for the experiments. A |(0, 2)S state is initialised by moving from M, through point F where rapid tunnelling occurs with the reservoir, to point I.FrompointP , we plunge into the (1,1) region to probe the anti-crossing via sequences illustrated in panels b,c and d. After returning via P to the latched spin blockade measurement point at M. b-e) Cartoons of the pulse sequence against time for the probe sequence P −E−P for b), Single passage Landau-Zener experiments, c) Landau- Zener-St¨ukelberg interferometry and d) spin-funnel experiments.

E: Location in detuning ε where the experiment is performed (see Fig. 5.1b- d). The location of this point in voltage space can often be a variable within the experiment (represented as the dashed line extending from

P ). As described below, the dwell time τD at this point as well as the voltage ramping rates to/from this location are also experimental variables.

179 CHAPTER 5

P2: Second instance of ramping to location P in Fig. 5.1a. Careful cal- ibration is performed to keep P within the PSB window while also minimising dwell time so not to incur relaxation to the (0, 2) singlet ground state.

M: Measurement point located within the latched region. Parameters for readout at this location are discussed in Chapter. 4, Section. 4.3. After measurement the sequence returns to F to rapidly reinitialise the singlet state.

5.3 Detection and measurement of Δ(θ):

Single passage Landau-Zener excitations.

In this section, the final results for the detection and measurement of Δ(θ) are presented. The experimental pathway to achieve the fittings for these measurements presented here was convoluted, with a chronological discus- sion of Δ presented in the Appendix J for completeness. At this stage, estimations of |Δ(θ)| with respect to voltage ramp rates could be fed for- ward into pulse sequence designs for experiments discussed in later sections and chapters. These experiments measure independent Hamiltonian param- eters, ultimately producing the Hamiltonian model which feeds back into more accurate measures of |Δ(θ)|. For example, the final model for energy level velocity ν (see Sec. 1.4.2 and Appendix D) is extracted from the spin

180 CHAPTER 5 funnel via data processing outlined in Appendix K. As discussed in Ch. 1.7.4 and illustrated in Fig. 1.11, the coupling Δ(θ) results in an anti-crossing between the |SH  and |T− states, where the system detuning ε can modulate these energy levels as described in Eq. (1.108). Further, from the discussion in Ch. 1.4.2, it is known that pulsing through this anti-crossing at a given energy level velocity ν results in a Landau-Zener excitation of the ground state based on the Landau-Zener formula described in Eq. (1.63).

5.3.1 Single passage experiment details

The single passage Landau-Zener excitation experiment,is executed using the pulsing protocol detailed in Fig. 5.1a&b. Here, the voltage ramp rate dV/dt through the anti-crossing is swept as the independent variable. To transform from voltage to detuning ε, the lever arm relationship (see Appendix I) is utilised, resulting in dε/dt. From here, the calculation to extract the energy level velocity is given by:

d(E − E ) dε ν = T− SH · , (5.1) dε dt which relies on the Hamiltonian modelling discussed in Ch. 6.4 and Appen- dices J&K. − ¯ − Further, it is noted that as the triplet energy ET− = EZ ε/2 is directly √ z − 2 2 proportional to the applied field B0 , and singlet energy ESH = ε +4t /2,

181 CHAPTER 5 the same ramp rates in the voltage space will yield vastly different ν at

z differing B0 settings. This can be deduced from Fig. 5.2a where increas- ing magnetic field will move the location of the anti-crossing along a near- exponentially increasing energy slope.

5.3.2 Single passage experiment results and discussion

The results from this experiment are illustrated in Fig. 5.2b, as produced for two distinct magnetic field settings. By fitting this data to the Landau- Zener formula, using appropriate ν, produces an estimate for |Δ(θ)| at the

± z location of the minimum energy gap as 196 6.3kHzatB0 = 0. Further | | ± z Δ(θ) =16.72 1.64 MHz at B0 = 155 mT, where the uncertainty here (and elsewhere) corresponds to 95% confidence intervals.

These results indicate that the dominant element of Δ(θ)istheΔSOC term from Eq. (1.109). As discussed in Sec. 5.1.1, this term is theorised to be derived solely from the spin orbit coupling between the |(0, 2)S and |T± states (outlined in Refs. [146, 88]). Due to the dependency upon θ, these terms are suppressed when the SH /T− anti-crossing is far from the SH /T0, z deep in the (1,1) region. This is the case for low field (B0 0), where the

Δ(θ) term is dominated by the mixture of |(1, 1)S and |T± states. These

z terms are also often dependent on B0 , increasing with the applied magnetic field, but are in turn suppressed by θ as the anti-crossing moves closer to the (0,2) charge region.

z Finally, it is indicated in Fig. 5.2b that data was taken at B0 applied field,

182 CHAPTER 5

a) b) 10 z z B0= 0mT B0= 155mT ۄ+T|

5 T 0.8 P ۄS| 2 ۄT0| ɋinǿο /h 0 2 ۄ_T| ɋinǾο /h 0.6 -5 -10 0.4 -15 2 ɋoutǿο /h Probability Triplet

Normalized Energy (MHz) -20 0.2 0.0 0.1 0.2 10-4 10-2 100 102 104

Detuning ፴ (meV) Energy level velocity ɋin (eV/s)

Figure 5.2: a) Normalized energy level diagram highlighting the anti-crossing of the two states involved in the single passage Landau-Zener experiment. The effect of varying ramp rate νin with respect to Δ, while keeping νout diabatic is illustrated. b) Results of the single passage experiment. Curve fittings illustrate fits to the Landau-Zener formula and yield measures for |Δ(θ)| at two different applied magnetic fields. Fits include a 95% confidence interval on all fitting parameters. when the Zeeman energy is theoretically zero and the triplets are degenerate.

Here, the ESH energy level only approaches ET− asymptotically, but never crosses. To explain this, any magnetic field offsets which occur within the sample must be taken into account. These field offsets can come from a number of different sources:

z 1. Offset magnetic field BOS produced by the superconducting coil. This z − ± data does, in fact, possess an offset field of BOS = 1.04 0.06 mT as estimated from spin funnel asymmetry (see Appendix K for details).

183 CHAPTER 5

2. Below a critical field BC the aluminium gates in the device are super-

z conducting. This results in screening of smaller applied B0 fields via the Meissner effect [176].

3. The Overhauser field produced by the background nuclear spins of residual 29Si. This is expected to produce a minimum energy split- ting of 50 kHz for 800 ppm residual concentrations [134].

Further discussion on the single passage Landau-Zener experiment is given in Appendix J.

5.4 Further mapping of the two-qubit

Hamiltonian

The interaction under Δ(θ) can be utilised as a probe for the other cou- pling energies within the two-qubit Hamiltonian shown in Eq. (1.108). Due

zˆ to the fact that the applied magnetic field B0 is low, the remaining energy coupling to be probed is the tunnel coupling tc which leads to the hybridi- sation of the singlet state. This tunnel coupling is a leading contributor to the exchange coupling J(ε), which can be observed through a number of ex- perimental methods [112, 73]. In particular, this probing of the magnitude of the exchange coupling can be performed by the two experiments detailed below, both of which employ the coupling Δ(θ) between the |SH  and |T− states.

184 CHAPTER 5

5.4.1 Landau-Zener-St¨uckelberg Interferometry

We can further characterise the Hamiltonian in Eq. (1.108) at very large de- tuning ε by performing a Landau-Zener-St¨uckelberg (LZS) interference ex- periment [71, 73] (Fig. 5.3a) as described by the pulse sequence in Fig. 5.1a&c. a b c 0.8 0.8 10 ۄ+T| 5 0.9 0.9 z ۄS| ۄT0| BOS 0  ۄ_meV) 1.0 (meV) 1.0 |T) Ԫ Ԫ -5 1.2 1.2 -10 z

Detuning B ൌͲ Detuning 1.3 0 1.3 -15 0.00 0.15 0.30 PT 1.4 1.4 Normalized Energy (MHz) -20 0.0 0.5 1.0 1.5 2.0 0510 0.0 0.1 0.2 Dwell Time ɒD (μs) Frequency (MHz) Detuning ፴ (meV)

Figure 5.3: a) Landau-Zener-St¨uckelberg interference pattern produced by semi-diabatic double-passage through the S/T− anti-crossing under zero-field z BOS offset b) Fourier transform of the time series data illustrating the en- ergy separation of the |SH  and |T− states with respect to detuning ε be- yond the anti-crossing. c) Normalised energy level diagram highlighting the anti-crossing of the two states involved in the double passage Landau-Zener experiment. The frequency fits to the LZS equation (shown as red circles) allow for a low-field approximation to the exchange coupling J(ε)canbe found, producing the resulting energy level diagram depicted.

z This is performed at B0 = 0 applied field, however the residual magnetic field present which may include some nuclear polarisation [177] is sufficient to split the |T0 and |T± states. By setting the ramp rate across the |SH /|T− anti-crossing to 2πν ≈|Δ|2/, an approximately equal superposition of both states is created. Dwelling for varying times τD and detunings ε results

185 CHAPTER 5

−  in a St¨uckelberg phase accumulation φ = (ESH (ε[t]) ET− (ε[t]))dt/ ,with |  |  ESH (ET− ) the energy of the SH ( T− ) state. Depending on the accumulated phase, the returning passage through the anti-crossing either constructively interferes, resulting in the blockaded |T−, or destructively interferes, bringing the system back to |SH . By keeping ν constant throughout the experiment the Fourier transform of the interference pattern (Fig. 5.3b) directly extracts | − | the energy separation ESH (ε) ET− (ε) as a function of detuning. The Hamiltonian for modelling these experiments is simplified to include only the hybridised singlet state and |T− polarised triplet state, with system described as ⎛ ⎞ ⎛ ⎞

⎜ EZ − J(ε)Δ(θ)⎟ ⎜|SH ⎟ |  HS/T− ψ = ⎝ ⎠ ⎝ ⎠ , (5.2) ∗ Δ(θ) 0 |T− which is a truncated version of Eq. (1.108) with normalisation to the triplet energy ET− . For these preliminary results, a simple exponential model for the exchange is fit, which results in an asymptotic approach to a Zeeman en-

z ergy of EZ =4.832 MHz, corresponding to residual magnetic field BOS = 0.164 mT. It is noted that this residual field is of the correct order of mag- nitude to result from complete nuclear polarisation [177] but it may also be due to a residual field from the superconducting solenoid apparatus.

186 CHAPTER 5

5.4.2 Spin-funnel experiment

The spin funnel experiments were first performed in GaAs quantum dots [112] and utilise the coupling Δ(θ) as a means to rapidly mix the |SH  and |T− states. By performing the pulse sequence as outlined by Fig. 5.1a&d, two rapid diabatic passages through the anti-crossing partitioned by a dwell time

τD produces a final state which maintains the initialised singlet population, so long as τD is substantially less than the T1 relaxation time between the |  |  − SH and T− states when ESH ET− Δ(θ).

PT 0.0 0.0 0.5 1.0

(meV) 0.2 Ԫ

0.4

Detuning 0.6

-20 -10 0 10 z Applied Field B0 (mT)

Figure 5.4: Spin funnel experiment illustrating the location of the SH /T− z anti-crossing with respect to applied magnetic field B0 and detuning ε.The polarised triplet energies are Zeeman split with field, while the hybirdized singlet energy bends with exchange coupling J(ε). The energy degeneracy produces a conversion between the applied field and exchange energy

As the depth of the plunge into the (1,1) charge region is varied, a re- laxation “hot-spot” is produced when ESH ET− as the equivalent T1 time becomes shortened. This yields an increase in the returned triplet probabil-

187 CHAPTER 5

ity at the location of the SH /T− anti-crossing with respect to independent variable ε. − As the triplet energy splitting ET0 ET− is given by the average Zeeman ¯ energy EZ , this hotspot can be traced along ESH by sweeping the applied z − magnetic field B0 . Thus, the splitting ET0 ESH = J(ε) can be deduced. The offset in the applied field of the solenoid can further be deduced from this spin funnel experiment, however, as this process requires more sophisticated models for exchange as presented in later chapters, this data processing stage is instead presented in Appendix K.

5.5 Discussion of results

For singlets prepared in the (0,2) state, the first accessible element of the sys- tem Hamiltonian is the bit-flipping term Δ introduced in Ch. 1. This term has been extensively discussed with respect to nuclear-rich materials such as GaAs [168, 112, 87, 73, 165, 88], and likewise there are a number of pos- sible processes that can contribute to Δ in the silicon-MOS platform. For 800 ppm nuclear-spin-1/2 29Si in the isotopically enriched 28Si epilayer [175], the expected magnitude of the hyperfine field is of root-mean-square order 50 kHz [134] for unpolarised nuclei. However, other effects may have a com- parable contributions, including Meissner effects from superconducting alu-

z minium gates for experiments below critical applied field B0 ,aswellastwo independent spin-orbit contributions. In order to separate and explore each

188 CHAPTER 5 of these individual effects, detailed studies on magnetic field magnitude and angle dependence, such as those performed to isolate hyperfine from spin- orbit contributions in nuclear-rich materials such as GaAs [178, 174], are required. This type of experiment requires the ability to sweep the magnetic field in thex, ˆ yˆ andz ˆ directions, which was not available in the experiments presented here. The single passage LZ excitation experiment allows for the required rate of passage with respect to control voltages to be known, such that the state diabatically passes through this anti-crossing with little mixing. This is a re- quired characterisation element for further experiments [112, 133] focussing on the |SH  and |T0 states, however does not need explicit knowledge of |Δ| to execute. The interaction term Δ provides a means for further character- isation experiments which probe the exchange coupling J via the |SH  and

|T− states, including the spin-funnel [112] and coherent LZS oscillations [73] shown here. In order to explicitly extract the magnitude of Δ, more sophisticated − models of the energy splitting ESH ET− are required, and are extracted from the full characterisation of the Hamiltonian as presented in Ch. 6 and Appendix K.

189 Chapter 6

Integrated platform for SiMOS quantum dot qubits

Chapter Acknowledgement: This chapter concerns in part the findings presented in M. A. Fogarty et. al. arXiv preprint arXiv:1708.03445v2 (also referred to as Ref. [60]). The full acknowledgements for this manuscript can be found in Appendix B. The SiMOS architecture used to generate these results is shown in Fig. 2.12 with details discussed in Ch. 2.5.2. The measurement protocols used to generate the results are discussed in Ref. [60] and Appendix G.

190 CHAPTER 6

In this chapter the readout methods of Ch. 4 as well as the knowledge of Δ from Ch. 5 (and more specifically, how to avoid the effects of this anti-crossing) are utilised for the demonstration of the combination of Loss- DiVincenzo [38] (one-electron) and singlet-triplet [111, 112] (two-electron) qubits for the first time, showing how one can perform singlet-triplet oper- ations and readout together with resonant single spin addressing in a single integrated platform.

6.1 From the singlet-triplet basis to

the computational basis:

Increasing applied magnetic field

The energy diagram of the system analysed in Ch. 4&5 is shown in Fig. (1.11) and illustrates a system which can be readily described using the singlet- triplet basis states in Eq. (1.105). As highlighted in several contemporary articles [47, 50] which focus on the scaling of SiMOS architectures, PSB can serve as a useful tool for the readout of spins in a large array of qubits. However, this indicates that the energy levels attributed the computational basis states introduced in Eq. (1.98) must be distinct oas shown in Fig. 6.1, and easily accessible when operating in the (1,1) charge region (where there is one electron in each quantum dot [38]).

z |↑ ↓ Under the condition that B0 0, the computational basis states ,

191 CHAPTER 6

ۄ՛ǡ՛| ۄ՛ǡ՝|

ۄ՝ǡ՛| ۄ՝ǡ՝| 2tc J

Energy 2ο

œɁ ۄȁሺʹǡͲሻ

ሺʹǡͲሻ ሺͳǡͳሻ 0 Detuning Ԫ Figure 6.1: Energy diagram of the two-qubit charge-state anti-crossing illus- trating the computational (spin-projection) basis. and |↓, ↑ are near-degenerate, indicating that the individual addressability of either one electron is near impossible via conventional ESR methods. That is, the driving frequency of both electron are indistinguishably close. This has been rectified in many different solid state systems through the understanding that single-spin distinguishability arises from the vary- ing Zeeman energy between each dot. This results in an energy difference

z − z |↑ ↓ |↓ ↑ δEZ = g2μBB2 g1μBB1 between the , and , states, as described in Eq. (1.108). This non-degeneracy marks the ability for two-axis control given by the Bloch sphere in Fig. 1.12, and the advent of a two-electron, qubits based on singlet-triplet spin states. These were first demonstrated in GaAs heterostructures [112, 73] and have now been operated in a variety of

192 CHAPTER 6 silicon-based structures [133, 134, 179, 136, 85, 60]. Many materials systems such as GaAs possess background nuclear spins which give rise to a non-degeneracy based on local Overhauser field gradi- ents [112], where this field can be quenched [178] to produce longer qubit coherence times. For SiMOS quantum dots, this state non-degeneracy is instead interpreted as a site-specific effective g-factor [85, 60] and yields the energy diagram il- lustrated in Fig. 6.1. For high in-plane magnetic field, the varying effective g-factors result from a combination of interface spin-orbit terms, which de- pend on local strain, electric fields, and the atomistic details of the oxide interface [140, 149, 148]. Previous devices have produced g-factor differences between QDs as large as 0.5% [113]; at high-field, Overhauser contributions to δEZ are negligible in isotopically purified samples. At lower magnetic fields, magnetic screening from the superconducting aluminium gates may also contribute significantly to δEZ [176].

193 CHAPTER 6

6.2 Single spin addressability via electron spin

resonance techniques

A requirement for scalable quantum computing is the ability to indepen- dently address the constituent qubits within a quantum register. In this section an experiment is detailed which demonstrates how this is achieved in the SiMOS double quantum-dot. The four distinct Eigenstates of the com- putational basis for the two-qubit system are made non-degenerate through the methods given in the previous section. With the ability to coherently transform between the singlet-triplet basis where the qubits are jointly ini- tialised/measured, to the computational basis where quantum algorithms are to be executed, a certain set of selected transitions are addressable which is consistent with manipulating the qubits in each individual quantum dot.

6.2.1 Spectrum

Non-degenerate energy levels in the computational basis are present at large, positive ε detuning in the (1,1) region in Fig. 6.1. The energy transitions are addressable via electron spin resonance (see Ch. 1.4.1) which incur a singular bit-flipping π rotation around thex ˆ axis for any one spin in the system as illustrated in Fig. 6.2a. Accompanying this image is the spectral response of these transitions in Fig. 6.2b, colour-coded with the same level transitions indicated in Fig. 6.2a. As the detuning ε is decreased, the energies within the system hybridise with the singlet-triplet due to increasing exchange energy.

194 CHAPTER 6

a) b) ESR

|՛ ۄ՛, Energy E = hf

|՛ ߜ ۄ՝, |՝ ۄ՛, |՝ ۄ՝, (1,1) 0 0 Detuning Ԫ Detuning Ԫ Figure 6.2: a) Energy spectrum in the (1,1) region illustrating the transi- tions between computational basis states as addressable via electron spin resonance. b) Theoretical frequency spectrum observable via ESR illustrat- ing the funnelling behaviour attributed to increasing exchange coupling.

6.2.2 Experiment

The compatibility of spin blockade readout with individual QD (i.e. single spin) addressability via electron spin resonance (ESR) [42] is investigated. This combination is desirable for scalable spin qubit architectures incor- porating error correction [47, 50]. Using the pulse sequence illustrated in Fig. 6.3a&b, a semi-adiabatic ramp prepares the large-ε ground state |↓↑.

The sequence P1 −E−P2 as illustrated in Fig. 6.3b can be described as the following:

P1: First instance of ramping to location P in Fig. 6.3a&b. Here the ground state |ψ |(0, 2)S as P is calibrated to be not so close to the anti-crossing that a charge superposition state is the ground state.

195 CHAPTER 6

a) c) (0,2)F (1,2) 0.0 I M P 0.5 Ԫ (0,1) (1,1) 1.0

b) (meV) 1.5 Ԫ

MW 2.0 P Initialization Measure 0 Detuning 2.5 Plunge Depth Ԫ z B0=150mT 3.0 0.0 0.2 0.4 0.6 PT Dwell Time ɒD -5 0 5 Frequency fESR-f0 (MHz)

Figure 6.3: a) Five-level pulse similar to that of Fig. 5.1, including measure- ment and state preparation in the singlet-triplet basis with the key differ- ence being the semi-adiabatic preparation of the hybridising ground state. b) Pulsing sequence P1 −E−P2 including the application of a microwave pulse c) Triplet probability as a function of detuning ε and applied ESR frequency with f0 =4.205 GHz. ESR spin rotations of the spin in the left dot (upper branch) and right dot (lower branch), using an on-chip microwave ESR line. |↓↑ is prepared similar to b), a 25 s ESR pulse of varying frequency is ap- |↓↑ → |↑↑ z |↓↑ → |↓↓ plied rotating when g2μBB0 /h = fESR,and ,when z |↓↑ |  g1μBB0 /h = fESR; is again mapped back (0, 2)S .

A semi-adiabatic ramp, diabatic through the S/T− crossing, but adia-

batic through with respect to tc, prepares |↓↑ (assuming g2 >g1). This process coherently maps the initial state which is in the singlet-triplet basis into a state which is in the computational basis.

E: Location in detuning ε where the experiment is performed. The loca- tion of this point in voltage space is a variable within the experiment, represented as the dashed line extending from P − D. At this loca-

196 CHAPTER 6

tion an a.c. magnetic field is applied to perform ESR. The final state then undergoes a reversal of the semi-adiabatic ramp, whereby not only

are the |↓↑ mapped to singlets while |↑↓ is mapped via the |T0 to blockaded states, but the reverse mapping also unwinds any phase ac- cumulation during the initial semi-adiabatic passage [133].

P2: Second instance of ramping to location P in Fig. 6.3a. Careful cal- ibration is performed to keep P within the PSB window while also minimising dwell time so not to incur relaxation of blockaded states to the (0, 2) singlet ground state.

The a.c. magnetic field is applied to perform ESR with pulse duration 25 µs, supplied by the on-chip microwave transmission line [150], to drive transitions that correspond to |↓↑ ↔ |↓↓ and |↓↑ ↔ |↑↑ at large detuning, when exchange is small (see Fig. 6.1). Any excitation from the ground state will now map to the blockaded triplet state population. Figure 6.3c shows the measured ESR spectrum as a function of detuning ε. The higher frequency fESR2 branch corresponds to a coherent rotation of the electron spin in QD2, while the lower frequency fESR1 rotates the QD1 spin. At large detuning ∼ z fESR1 4.2GHz, consistent with the applied magnetic field B0 = 150mT for this experiment. As ε decreases (and J(ε) increases), the ground state is better described as |SH , so the transitions become |SH  ↔ |↓↓ and

|SH  ↔ |↑↑ and exchange now competes with ESR, resulting in a lower visibility.

197 CHAPTER 6

6.3 Two-qubit exchange based SWAP

operations

The theory for this experiment is largely covered in Ch. 1.7, particularly in sections 1.7.4 and 1.7.5. The energy splitting between the hybridising singlet and unpolarised triplet is given by the tunnel coupling induced exchange energy J(ε). As illustrated in the Bloch sphere for a singlet-triplet qubit (see Fig. 1.12), the exchange energy can be utilised as a two-qubit SWAP operation between the computational basis states |↑↓ and |↓↑. The first experimental realisation of this was in GaAs, and is presented in Ref. [112]. The experimental methods presented below were adapted from Refs. [112, 133].

In order to investigate exchange between the hybridised singlet |SH  and |  z unpolarized triplet T0 , an external magnetic field B0 = 200 mT is applied to strongly split away the T± triplet states. At these fields the Zeeman energy difference δEZ dominates exchange J(ε) deep in the (1,1) region, and the eigenstates there become |↓↑ and |↑↓, as depicted in Fig. 6.1.

6.3.1 Experiment for detection

The pulse sequence for the experiment is illustrated in Fig. 6.4a which shows the experiment is executed by employing eight pulsing stages. The prepara- tion and single-shot latched-PSB readout sequence P2 − M − F − I − P1 is identical to that which is presented and discussed in Ch. 4 and Ch. 5.2.

198 CHAPTER 6

From P1, a ramp rate ν which is fast enough to be diabatic with respect to Δ, but slow enough to be adiabatic with respect to tc(ε), ensures adiabatic preparation of a ground state |↓↑ or |↑↓, depending upon the sign of δEZ = z − z z g2μBB2 g1μBB1 .AtB0 = 200 mT the Meissner effect is expected to be to be quenched, so that δEZ is dominated by the effective g-factor difference between the dots.

For simplicity it is henceforth assumed δEZ > 0, so that |↓↑ is adia- batically prepared for large ε. Following the pulse sequence illustrated in Fig. 6.4a&b, coherent-exchange-driven oscillations can then be observed be- tween |↓↑ and |↑↓ by rapidly plunging the prepared state |↓↑ back towards the (1,1)-(0,2) anti-crossing where J(ε) is no longer negligible. Variable dwell time τD results in coherent exchange oscillations, and the reversal of the rapid plunge leaves the state in a superposition of |↓↑ and |↑↓. The semi-adiabatic ramp back to (0,2) maps the final state |↓↑ to the |(0, 2)S singlet, while |↑↓ is mapped to a blockaded state via the T0 triplet [112, 133]. The resulting data is shown in Fig. 6.4c&d. To achieve the exchange driven oscillations within the (1, 1) region, the following sequence P1 − D1 −E−D2 − P2 is executed [112, 133]. Broken down into the components as represented in Fig. 6.4a&b, each step within this sequence is given in detail as the following:

P1: First instance of ramping to location P in Fig. 6.4a&b. Here the ground state |ψ |(0, 2)S as P should not be so close to the anti-crossing that a charge superposition state is the ground state.

199 CHAPTER 6

a) c) 0.1 d) 0.1 (0,2)F (1,2) I M P Ԫ 0.4 0.4 (0,1)D (1,1) 0.7 0.7 b)

Initialization (meV) (meV) P Measure Ԫ 1.0 Ԫ 1.0 0 Plunge Depth Ԫ 1.3 1.3 Dwell Time ɒD

Detuning z Detuning B0 ൌʹͲͲ 1.6  1.6 0.0 0.2 0.4 0.6 0.8 1.0 D PT 1.9 1.9 0.0 0.6 1.2 1.8 01020 Dwell Time ɒD (Ɋs) Frequency MHz

Figure 6.4: Experiment design and results for coherent exchange driven oscil- lations. a) Eight-level pulse sequence used for the experiments. A |(0, 2)S state is initialised by moving from M, through point F where rapid tun- nelling occurs with the reservoir, to point I.FrompointP , we plunge into the (1,1) region to probe the anti-crossing via sequences illustrated in panel b. After returning via P , the state is ramped to the latched spin blockade measurement point at M. b) Cartoons of the pulse sequence against time for the probe sequence P − D −E−D − P (see text for details). c) Coher- ent Rabi oscillations between |↓↑ and |↑↓ states, driven by exchange J(ε). d) Fourier transform of time series which shows the frequency of exchange driven oscillations between the |↓↑ and |↑↓ states.

D1: First instance of ramping to location D in Fig. 6.4a&b. A semi-

adiabatic ramp, diabatic through the S/T− crossing, but adiabatic

through with respect to tc, prepares |↓↑ (assuming g2 >g1). This process coherently maps the initial state which is in the singlet-triplet basis into a state which is in the computational basis.

E: Location in detuning ε where the experiment is performed. The loca- tion of this point in voltage space can often be a variable within the experiment (represented as the dashed line extending from P − D). A

200 CHAPTER 6

diabatic pulse from D1 −E preserves the prepared |↓↑ state while turn-

ing the exchange energy on. Time evolution under τD results in SWAP operations between |↓↑ ↔ |↑↓ (see Fig. 1.12).

D2: Second instance of ramping to location D in Fig. 6.4a&b. A diabatic

pulse from E−D2 preserves the state after time evolution while turning the exchange energy off. This final state then undergoes a reversal of the semi-adiabatic ramp, whereby not only are the |↓↑ mapped to

singlets while |↑↓ is mapped via the |T0 to blockaded states, but the reverse mapping also unwinds any phase accumulation during the initial semi-adiabatic passage [133].

P2: Second instance of ramping to location P in Fig. 5.1a. Careful cal- ibration is performed to keep P within the PSB window while also minimising dwell time so not to incur relaxation of blockaded states to the (0, 2) singlet ground state.

201 CHAPTER 6

6.4 Modelling exchange coupling between two

quantum dots

6.4.1 Modelling tunnel coupling

It has been documented in Si/SiGe based double-dots such as those in Refs. [134, 180] that the tunnel coupling can be dependent upon detuning, and can be modelled using the Wentzel-Kramers-Brillouin (WKB) approximation [134]. While full device simulations were not performed here, the assumption is made that a one-dimensional approximation holds [181]. The model used is of a similar form to Ref. [180], giving:

 tc(ε)=Λ exp(2ϕ(ε)) + 1 − exp(ϕ(ε)) (6.1) where ϕ is an integral over electron momentum between classical turning points of the 1D potential φ(ε) [181]. Following the findings in Refs. [134, 180], a phenomenological model for the integral ϕ is simply ϕ = ε/ζ

6.4.2 Exchange fitting from two-qubit datasets

Each of the experiments described above probes the Hamiltonian in Eq. (1.107) for different ranges of detuning. Figure 6.5 collates the results of all experi- ments and plots the energy splitting between the hybridised singlet |SH  and unpolarised triplet |T0 across all detuning values. Close to the (0,2)-(1,1)

202 CHAPTER 6

| − | Figure 6.5: Exchange energy splitting ESH (ε) ET0 (ε) as a function of detuning ε, as extracted from the spin-funnel (see Appendix L), Landau- Zener-St¨uckelberg interferometry (Fig. 5.3a), coherent exchange oscillations (Fig. 6.4c) and ESR funnel data (Fig. 6.3c). Each include 95% confidence intervals based on data fits uncertainties or measurement resolution. The solid/dashed lines represent fits to the data based on the model Hamiltonian in Eq. (1.108), for which tc(ε =0)=1.86 ± 0.03 GHz is found. This figure was prepared with the assistance of Dr. B. Hensen, and is adopted from Ref. [60].

203 CHAPTER 6 anti-crossing, for low ε, the splitting is dominated by exchange coupling

J, while for large ε, δEZ dominates. As expected, the energy differences z ≈ obtained from the LZS interferometry (for B0 0) diverge from those ob- z z ≈ tained via ESR (where B0 = 150 mT), since when B0 0 there remains only a small residual δEZ due to combined Meissner screening and weak Overhauser fields. Figure. 6.5 also shows a fit to the data employing the

Hamiltonian of Eq. (1.107) as documented elsewhere. A constant tc fits poorly; instead a model for a dependence of the tunnel coupling on ε is em- ployed. At the anti-crossing (ε = 0), the curve fit to this model indicates

−3 tc(ε =0)=1.864±0.033 GHz and δg =(0.43±0.02)×10 . It is noted that this tunnel coupling is comparable to that observed for a separate two-qubit √ device [113] for which tc(ε)=900 2 MHz. In order to adequately fit the exchange, a phenomenological model for tunnel coupling, as described in Eq. (6.1), is employed. This model yields ζ numerically fit to 0.59 ± 0.02 meV and Λ = 4.36 ± 0.07 GHz. Magnetic field offsets in low-field experiments as presented in the main text are incorporated in Fig. 6.5, and are able to tie together the spin-funnel and LZS interferometry data. Remaining variables in this dataset are lim- ited to small shifts in ε for spin-funnel, coherent-exchange and ESR fittings attributed to the separation of preparation point P and ε = 0 which can vary between device calibrations.

204 CHAPTER 6

6.5 Fidelity concerns of two qubit operations

This section follows the discussions presented in Ref. [60], specifically from the supplementary section which directly addresses fidelity concerns when operating in the singlet-triplet basis. Using the example of the driven exchange oscillations |↑↓ ↔ |↓↑ illus- trated in Fig. 6.4c, the sources of errors are discussed including how they might be mitigated in future experiments.

6.5.1 Preparation/Initialisation

The first source of error ePrep manifests during the preparation of the |(0, 2)S state, while eMeas refers to the error in conversion from (0, 2) − (1, 2). The combined error e(0,2)SPAM is directly observed in the histogram of Fig. 4.4b, measured to be 0.8%.

6.5.2 Ramping related errors

High fidelity operation of this device requires consideration of the voltage ramp rates. In general, a ramp rate νX , for any process X, is the deriva- tive of some energy E with respect to time as a particular anti-crossing is traversed. These are then related to an experimentally controlled voltage ramp-rate through the associated function E(V ) given by the Hamiltonian model Eq. (1.108). The results presented all use linear ramps, enabling sim- pler analysis of bounds of constant ramp-rates; a future, more sophisticated

205 CHAPTER 6 approach might employ pulse-shaping to accelerate and decelerate voltage ramps to avoid errors.

The first such ramp-rate considered is νin, which impacts etransfer, the error of adiabatically transforming |(0, 2)S into |↑↓. To avoid mixing with |T−, as well as to maintain adiabaticity in the transfer to |SH , νin must satisfy 2  2  2  Δ / νin 4tc ()/ . Further, νin δEZ/ is required as detuning ε increases so not as to populate a mixture of |↓↑ and |↑↓. Failure to maintain these conditions may lead to leakage of the desired |SH  or |↑↓ states into undesired states. Likewise, the error in conversion from the final state back to the |(0, 2)S stateisgivenbyeMap. The resulting state is required to transfer back to the

2 (0, 2) charge state, where it can initially mix again based on νOut (δEZ) /.

2 2 The |↓↑ state is mapped back to |(0, 2)S,whereΔ/ νOut 4t (ε)/.

2 For |↑↓, the state is blocked by Pauli-exclusion, and νOut 2t (ε)/ is required near the anti-crossing.

2 2 Therefore, constant ramp rates require Δ / νIn/Out (δEZ) / to be satisfied to maintain low transfer errors eTransfer and eMap.

6.5.3 Control errors

The final error mechanism tested is eControl, the error associated with control- ling the exchange operation between the ramp in/out stages. After the (1, 1) state is prepared, this state is pulsed rapidly back to a point of non-zero ex-

2 change coupling at rate νPulse. This rate must abide by νPulse J (ε)/,oth-

206 CHAPTER 6 erwise a phase error can be introduced based on the integration of exchange during the non-negligible ramping times. Ramping close to the (1, 1) − (0, 2) transition can also result in leakage to a (0, 2) charge state. This is achieved

2 via νPulse t (ε)/.

6.5.4 Simulations of error processes

The above sections present a strict set of bounds which could be achieved via shaped pulses. Falling short of this, a trade-off between eTransfer/eMap and eControl can be produced by shallower preparation in the (1, 1). For the experimental conditions presented here, the measurement errors discussed above can be observed in the simulation of state preparation and measurement in Fig. 6.6a. Here, state leakage can be observed as both a population of the |(0, 2)S state in the (1, 1) region and population of the T± triplet states, due to passage through (or in the vicinity of) ε =0. The populations illustrated are produced via Monte-Carlo integration over realistic noise parameters for SiMOS [78, 134], during the pulse sequence shown in Fig. 6.6b. Key noise sources include charge noise modelled as gate-referred voltage noise, and magnetic gradient noise. Each spectra can

2 α−1 α be given by S(f)=A f0 /f with f0 = 1 Hz [134]. Magnetic gradient noise is reconstructed from measured parameters in literature [134, 78], with

Am =0.2neV,αm =1.5. Electric field noise (charge and voltage fluc- tuations) is likewise reconstructed [78], with Ae =4.5 µeV, αc =1,and Johnson-Nyquist noise attributed to 100Ω at room temperature.

207 CHAPTER 6

a) 1.0 c) 1.0 Data (Fig.3d) 0.8 Combined Noise 0.6 Detuning Noise 0.8 Magnetic Gradient œƸ 0.4 Noiseless Simulation Visibility ۄS(1,1)| 0.6 0.2 0.0 ۄ՝՛| d) 1.0 ۄ՛՝| Probability 0.4 0.8 0.6 ۄT0|

0.4 ۄൌȁ՛՛ۄ+T| 0.2 ۄS(0,2)| 0.2 ۄൌȁ՝՝ۄ_T|

0.0 Saturation Value 0.0 b) e) -3 -0.4 10 Semi-adiabatic rate (0,2) 0.0 94.1x10-3eV/s Ԫ (1,1) (meV) -6 0.4 Dwell time 10

0.8 Exchange pulse rate Decay Time (s) Time Decay Detuning 0.188x106eV/s -9 1.2 10 Simulation time progression 0.0 0.5 1.0 1.5 2.0 (Non-linear time scale) Exchange point detuning Ԫ(meV)

Figure 6.6: a) Simulation example illustrating the combination of all prepa- ration and measurement errors including state leakages. The process involves transfer of a |(0, 2)S state to |(1, 1)S, followed by semi-adiabatic transfer to |↓, ↑, pulse to the exchange point for a dwell time, followed by reversal of the exchange pulse and semi-adiabatic mapping. b) Shows the detuning pulse sequence for a) against simulation time steps (non-linear time progression for illustrative purposes). Simulation time-steps are normalised to ramping speeds for each individual pulse region. c-e) Shows fitting parameters from Eq. 6.2 after time evolution at the exchange point ε. Comparisons are drawn between the separate contributions of the most detrimental noise couplings from Table 6.1 and a cumulative noise of all mechanisms. c) Upper bound visibility metric VSig from Eq. 6.2 for the experimental data compared against the same visibility metric produced from simulations involving electric and magnetic field fluctuations. d) Illustrates the blockade saturation value VSat, which can indicate eTransfer and eMap error processes. e) Fitting parameter ρ for driven oscillation decay time T2 produced from frequency variation σF (see text) for both the experimental data and the simulations.

208 CHAPTER 6

Table 6.1: Key noise coupling mechanisms as studied in the accompanying simulations presented in Fig. 6.6. These parameters are defined in Hamilto- nian equation Eq. (1.108) and include components attributed to electric field such as exchange noise as well as single- and double-dot spin-orbit (SO) ef- fects. Also included are Zeeman energy terms dependent upon magnetic field noise. Noise Type Hamiltonian Parameter Governing noise source Exchange noise J(ε) Electric field noise

z Stark (single dot SO) δEZ Electric field noise

Stark (double dot SO) Δ (via ΔSOC) Electric field noise z Magnetic gradientzδE ˆ Z Magnetic field noise x,y Magnetic gradientx, ˆ yˆ Δ (via δEZ ) Magnetic field noise

These two noise sources can couple into the experimental system in var- ious ways, and the key mechanisms studied here are shown in Table. 6.1. Under these noise mechanisms, a simulation which involves a zero-dwell time exchange pulse towards the anti-crossing is shown in Fig. 6.6a, with time pre- cision normalised to the associated ramping rates νX (as shown by Fig. 6.6b). When this preparation, semi-adiabatic transfer, exchange pulse in/out, semi- adiabatic map and readout mechanism are also combined with a dwell time at the exchange point ε, the resulting oscillation characteristics can be fit and compared to the experimental data to Fig. 6.4. The resulting visibility

ρ VSig, saturation value VSat and oscillation decay times T2 can be extracted by fitting with the following |(0, 2)S return probability function:  2 VSig − t PS(t)= cos(2πFt + ϑ)exp ρ + VSat, (6.2) 2 T2

209 CHAPTER 6 where t is the dwell time at the exchange point ε. The oscillation frequency indicated in Eq. 6.2 is given by the energy separation of the hybridised singlet  2 2 and the T0 triplet state F = J + δEZ/h. The variation in this frequency due to the noise sources within the system is given by σF and relates to the

ρ decay time T2 =1/(πσF ). The phase term ϑ is close to zero, and is accumulated as an error during

ρ state transfer and mapping processes. The key parameters VSig, VSat and T2 are shown in Fig. 6.6c-e for the dominant noise coupling mechanisms from in Table 6.1. The first key noise coupling mechanism is attributed to electric field fluctuations which couples directly into the exchange term J via ε.A second term is attributed to the magnetic field noise gradients which couple

z directly into δEZ. The individual simulations for these two key coupling mechanisms are also presented alongside a simulation showing the cumulative noise from all of the mechanisms from Table 6.1 acting together. Comparing the cumulative noise simulation to the individual exchange noise andz ˆ magnetic gradients shows that the remaining coupling mechanisms from Table 6.1 are suppressed due to coupling via weaker physical effects such as the Stark shift. These simulated evolutions demonstrate that the two different noise sources have drastically different effects on the experiment. It is observed that the electric field noise components which couple into the system via the exchange

ρ J(ε), lead to a shortening of decay time T2 as shown for shallow ε in Fig. 6.6e, while the transfer and mapping processes remain largely unaffected. Under

210 CHAPTER 6 high exchange, the visibility of the oscillations is also degraded due to expo- nential dependence upon ε. The other major error source is due to magnetic gradient noise in thez ˆ

z direction. Due to direct modulation of the δEZ term, this noise type directly affects the ability to consistently transfer (map) the initial (final) state via the semi-adiabatic ramp. This is evident from the lifting of VSat for regions z |↓↑ where J(ε) δEZ in Fig. 6.6d. Due to the error in preparation of the state, the resulting visibility of oscillations also suffers. When these noise mechanisms are brought together, the overall simulation resembles that of the data collected for the exchange driven oscillations. It is noted, however, that reasonable trade-off between noise amplitudes Ae and

Am can tweak the overall shape of the visibility curve observed within the cumulative noise simulations.

211 CHAPTER 6

6.6 Discussion of results

For the results observed in this chapter, the preparation of a computational basis state, which is an equal superposition of the |(1, 1)S and |T0 states, provides an avenue for measuring the exchange energy J as a function of detuning within a region where J δEZ. Together with the spin-funnel and zero-field Landau-Zener-St¨ukelberg oscillations, a consistent picture of the J(ε) is extracted. The energy gap fit from the ESR funnel is also a measure of not only this exchange energy, but also δg for these qubits. Collectively, the results yield full Hammiltonian characterisation for this system, yielding similar results to a previous SiMOS device [113]. Previous studies on SiMOS quantum dots [113] were interpreted with

fixed coupling tc between dots. Modelling the exchange function J(ε), and thus extracting tc, did not yield a good fit for constant tunnel coupling with respect to ε, and a similar result to that which was found in a Si/SiGe device [180] was extracted. The overall visibility observed after readout of an experiment can be at- tributed to combined state-preparation and measurement processes which arise independently of the enhanced readout process. The characterisation and calibration of these processes are crucial, yet remains poorly represented experimentally in SiMOS devices in current literature. For the device pre- sented here, any errors collected in the process of mapping from the standard PSB region to the enhancement region are difficult to disentangle experimen-

212 CHAPTER 6 tally due to poor readout visibility of the standard (0,2) PSB region. Re- gardless, having access to full characterisation of all Hamiltonian parameters with respect to the detuning ε facilitates realistic simulations of state trans- fer dynamics and spin control in the singlet-triplet basis. It was shown that state-transfer or mapping errors could benefit from further detuning pulse optimisation which was not explored within this system. For the simulation results presented in this chapter, there is also a discrep- ancy between simulation visibility compared to data visibility in Fig. 6.6c. A probable explanation for this is the time resolution limitations within the experiment prevents an accurate estimate for the visibility at large exchange due to under-sampling. This issue is avoided within the simulations by nor- malising the evolution time-steps at each detuning point ε by the exchange energy J. This approach maintains adequate sampling of the exchange os- cillations, facilitating good fits for Eq. 6.2 return probability envelope. Aside from exploring two-qubit system characterisation, the results from this chapter demonstrate for the first time in a silicon device, experimentally combined single spin qubit addressability using ESR with high-fidelity single- shot readout in the singlet-triplet basis, as well as two qubit control via SWAP operations integrated together on a single SiMOS test-bench device. This is an important milestone for SiMOS quantum dot qubits as it is a preliminary demonstration of all required elements for operating a large scale quantum system integrated together in a single test-bench system.

213 Chapter 7

Towards logical qubits and scaled quantum systems

Chapter Acknowledgement: This chapter concerns in part the findings presented in C. Jones et. al. arXiv preprint arXiv:1608.06335v2 (also referred to as Ref. [47]). The full acknowl- edgements for this manuscript can be found in Appendix C.

Of the many approaches to constructing hardware for a quantum com- puter, and the quantum error correction schemes developed for those hard- ware systems, state-of-the-art experiments now involve operations on two to nine coherently controllable qubits [182, 183, 184, 185, 186, 113, 187, 188, 189]. Despite this rapidly accelerating progress, an extensible logical qubit

214 CHAPTER 7 has not yet been demonstrated. The works presented in Ref. [47], and discussed in this chapter, presents an experimentally realisable logical qubit in a linear array of SiMOS quan- tum dots. As scaling of the SiMOS architectures is discussed elsewhere [50], this chapter presents a study of a single logical qubit produced under the constraints of nearest-neighbour couplings in a linear array [172].

7.1 An extensible logical qubit

Acknowledgement: The criteria for an extensible logical qubit was not directly developed by the dissertation author. They are presented here as a means of motivation for the remainder of the chapter. See Ref. [47] for additional detail.

As introduced in Ref. [47], an extensible logical qubit must demonstrate all of the four criteria described below.

1. Error threshold: Error-detection procedures that introduce errors at a low enough prob- ability must be achievable to allow for the threshold theorem. For stabiliser codes, this is the stabiliser measurement. These concepts are outlined in the earlier discussion of fault tolerant quantum computation in Ch. 1 (Sec. 1.8).

215 CHAPTER 7

2. Fault tolerance: Any single fault generated in the system is detectable.

3. Parallel measurement: The logical qubit must be designed for the capacity to perform error- detection measurements at multiple locations simultaneously. This im- plies multiple measurement apparatuses within the system. For the system to be considered extensible, the number of measurement appa- ratuses must be proportional to the number of physical qubits [190].

4. Extensible encoding

The encoding strategy for the logical qubit must be capable of extend- ing to correct any number of errors. For stabiliser codes, this means code distance d can increase without compromising any of the preced- ing criteria.

7.2 A linear array of SiMOS quantum dot

qubits

Much of the hardware platform for the proposed logical qubit has been de- scribed in introductory chapters, or within the details of results chapters. Here, a brief summary of the requirements for the initialisation, control and measurement protocols are discussed. Further, potential modifications to ex- isting device architectures for the extension into a linear array of quantum

216 CHAPTER 7

Figure 7.1: Device schematic for a linear array of quantum dots. a) Top view of the device where dispersive readout is implemented through the gate electrodes [191, 192] This design is similar to Ref [172] and architectures seen in Fig. 2.11 & 2.13. b) Alternative top view where readout is implemented using SETs located near the dots [42, 113]. This design is a direct extension of the architecture seen in Fig. 2.12. This figure is appropriated from Ref. [47]. dots is shown in Fig. 7.1. These architectures are similar to those illustrated in Fig. 2.11, 2.12 & 2.13, and draws several advantages from another similar architecture presented in Ref [172].

7.2.1 Fundamental control operations for single qubits

Acknowledgement: The work presented in this section was originally drafted by the dissertation author and was further developed jointly between the researchers referenced in Appendix C.

The set of spin-control operations must be as simple as possible to support a logical qubit, and all the control operations needed for the logical qubit have been demonstrated in silicon quantum dots. These include ESR [42, 193, 78, 134], electrically controllable exchange

217 CHAPTER 7 interaction between spins in neighbouring dots [112, 194, 133, 113, 134, 60], and electrically controllable preparation and measurement of two spins in the singlet/triplet basis using Pauli spin-blockade [129, 195, 196, 133, 134, 60]. The use of selective ESR, which addresses a single spin, can become sub- ject to frequency crowding for an increasing number of qubits. Rather, to favour extensibility, the logical qubit control implements global ESR which addresses all spins as in an ensemble experiment. Simultaneously addressing all spins with broadband ESR avoids frequency crowding and the associated cross-talk errors. However, this approach limits what overall control is possi- ble. Global ESR is also beneficial for dynamical decoupling (Sec. 1.5.2) which is to be integrated within the control sequence for intrinsic error robustness of operations (Sec. 1.8.4). High-fidelity control of spin ensembles has been demonstrated using mag- netic resonance [197], where composite pulse sequences are used to correct for systematic over-rotation errors [198, 199]. Global ESR on all spins in this proposal will require broadband pulsing as the expected spread in values for electron g-factor will lead to a spread in Zeeman energies of tens of MHz at

B0 = 1 T [148]. The ability to electrically Stark shift the electron g-factor (Sec. 2.4.4) provides another resource for maintaining high-fidelity control using only global ESR pulses.

As discussed in Sec. 2.4.4, disorder perturbations at the Si/SiO2 interface lead to a stochastic and bias-dependent variation in g-factors. A randomly generated distribution of electron g-factors for a linear chain of 20 qubits is

218 CHAPTER 7

Figure 7.2: Randomly generated sample of spread in g-factors for a lin- ear chain of 20 quantum dots. The underlying distribution is based on the measured variance in g-factors observed in devices like that in Ref. [42]. Each data point shows the range of g-factor tuning possible with Stark shift- z ing [140, 42, 193], corresponding to 10 MHz at B0 =1.5T.Dots5and6 are coloured red to indicate that they have small Δg splitting and require the g-factors for those dots to be tuned apart for the CZ gate. This figure is appropriated from Ref. [47]. illustrated in Fig. 7.2, as well as the g-factor tuning range based on statistics from measurements on SiMOS devices [42, 148].

219 CHAPTER 7

7.2.2 Exchange control in a linear array

Acknowledgement: The work presented in this section was originally drafted by the dissertation author and was further developed jointly between the researchers referenced in Appendix C.

The device schematics shown in Fig. 7.1 are configured with one gate elec- trode per quantum dot. The insulating oxide between the metal gates pro- duces a natural tunnel barrier between adjacent dots.

The exchange coupling between adjacent qubits Qj and Qj+1 is achieved by applying a differential voltage between gates Gj and Gj+1, to “detune” the electric potentials of the two dots in exactly the same method employed in Ch. 5 & 6. This form of control is illustrated in Fig. 7.3, demonstrating the necessity for adjustment of all gates simultaneously, via a self-consistent algorithm, to correct for the effect of cross-talk between gates when an ex- change operation is applied between a specific pair of qubits, or pairs of qubits. Exchange coupling is achieved by detuning relative electrochemical potential, however turning on J for one pair will require a similar shift in the electrochemical potentials of dots to the left and to the right. This is to pre- vent unintended exchange with neighbouring spins. A potential solution to applying exchange between any non-overlapping spin pairs simultaneously is illustrated in Fig. 7.3a. Here, the potential at each dot is set to one of two values, VA and VB. Starting from one end of the line of dots, potentials A or

220 CHAPTER 7

a)

VB ȟV=0

VA ȟV=0 ȟV ȟV ȟV=0 Potential energy

j-2 j-1 j j+1 j+2 j+3 Array position b) c)

(...,2,0,1,...) (...,1,1,1,...) (...,1,0,2,...) Exchange energy Chemical potential

Ԫ =0 Ԫ =0 Ԫ =0 Ԫ =0 (2,0) Detuning (0,2) (2,0) Detuning (0,2)

Figure 7.3: Exchange coupling via tilt control over detuning. a) Potential energy of the dot array where exchange control is modulated by nearest neighbour difference in electrostatic potential. b) Exchange is activated when neighbouring dots have different potentials, manifesting as a detuning in the (..., 1, 1, 1, ...) towards either (..., 2, 0, 1, ...)or(..., 1, 0, 2, ...) anticrossings. The blue (red) cross marks the detuning shift attributed to the corresponding blue (red) shift in nearest neighbour control voltage. c) The exchange energy J as a function of the detuning, with the corresponding blue (red) marker due to detuning shift.

B are assigned such that neighbouring dots have the same potential to turn off exchange, and different potentials to “detune” and activate exchange as shown in Fig 7.3b. As shown in Fig 7.3c, J is an exponential function of volt- age. Inhomogeneity in tunnel coupling can be handled by shifting voltages

221 CHAPTER 7

a) Potential energy j-2 j-1 j j+1 j+2 j+3 Array position b) c)

(...,2,0,1,...) (...,1,1,1,...) (...,1,0,2,...) Exchange energy Chemical potential

Ԫ =0 Ԫ =0 Ԫ =0 Ԫ =0 (2,0) Detuning (0,2) (2,0) Detuning (0,2)

Figure 7.4: Exchange coupling via symmetric control over inter-dot tunnel coupling t0. a) Potential energy of the dot array where tunnel coupling is modulating between quantum dots via exchange-gates [180]. b) Raising (low- ering) the potential barrier between dots results in decreasing (increasing) the tunnel coupling t0, and hence directly affects exchange energy between electrons. c) The exchange energy J(t0) as a function of the detuning in- creases as the tunnel coupling between dots is increased. as needed only slightly from this model, such that small detuning values will have negligible exchange, while large detuning is designed to match the tun- nel coupling of any dot pair. This results in a uniform exchange operation implemented in constant time for all dots. It is also possible to configure devices with an additional “exchange” gate between each pair of qubit control gates (labelled Gj in Fig. 7.1). Such an

222 CHAPTER 7 approach has been successfully implemented in Si/SiGe quantum dot de- vices [180, 172, 134]. The addition of the exchange gate gives control over the tunnel coupling t0 between quantum dots as shown in Fig 7.4a&b, allow- ing for the direct modulation of the exchange energy at the symmetry point seen in Fig 7.4c.

7.2.3 Simulated qubit performance

The controlled-phase entangling gate based on exchange was introduced in Sec. 1.7.6, and has been both analysed and experimentally demonstrated ex- tensively in the literature [113, 87, 200]. The example realised in SiMOS is performed via an adiabatic pulse on the electrostatic detuning towards the (0,2) charge-state anticrossing (tilt-control, as explained above). As seen in Ch. 6, if J is the dominant term in the Hamiltonian, the exchange operation

z implements SWAP operations. However, the combination of a non-zero B0 field and a g-factor difference δg splits the energy levels of the spin states |↑↓ and |↓↑. Here, the nonlinearity in the eigenvalue spectrum near the avoided crossing introduced by J results in a controllable phase shift which has pro- duced CZ and CNOT two-qubit gates between SiMOS quantum dots [113]. The duration and fidelity of the CZ gate depends in part on g-factor differences between dots, and it is important to characterise what values of δg are achievable. As discussed in Sec. 2.4.4, disorder perturbations at the Si/SiO2 interface lead to a stochastic and bias-dependent variation in g-factors as demonstrated by Fig 7.2. In the event where two neighbouring

223 CHAPTER 7 dots have small δg at zero detuning, the difference can be increased with Stark shifting by choosing whether to detune the dots towards (2,0) or (0,2) charge configuration, noting that this would yield a favourable configuration for detuning potentials A and B along the chain as seen in Fig. 7.3a.

Simulations of the decoupled CZ gate

Acknowledgement: The work presented in this section was developed jointly between the disser- tation author and the researchers referenced in Appendix C. In particular, the background theory and much of the simulation work was attributed to Dr. T. D. Ladd. Additional details can be found in Ref. [47].

From preliminary estimates, simple square pulsing of the CZ operation with observed values of δEZ can achieve a two-qubit gate fidelity above 99%, but substantially higher fidelity is accessible through pulse shaping. Adia- batic pulsing has several advantages, in particular, resilience to some noise processes. In the adiabatic limit, pulsing into the avoided crossing and back realises a combination of Zeeman phase shifts and a non-linear phase shift due to J(ε). The non-linear phase shift is given entirely by the time integral of J(ε), which is denoted here as ξ = J[ε(t)]dt/. The integral is over a sufficient time to fully capture a voltage pulse V(t) which brings J(ε)toand from a negligibly small value. The total adiabatic unitary evolution for the two spins is then given by

224 CHAPTER 7

ξ2 U(ξ)=exp − i σzˆ1σzˆ2 4  g¯[ε(t)]μ Bz − i B 0 dt(σ + σ ) 2 zˆ1 zˆ2 (7.1)   δE2[ε(t)] + J 2[ε(t)] −i Z dt(σ − σ ) . 2 zˆ1 zˆ2

The adiabatic limit is maintained by assuring that the frequency band- width of a J[ε(t)] pulse is much less than the minimum value of δEZ[(t)]/ as discussed in Sec. 6.5. If ξ = π for the overall pulse, a maximally entangled CZ gate is achieved. However, during this operation, local single-spin phase shifts are also generated. These phase shifts are substantial, and are subject to magnetic and charge noise, the latter due to the electric-field dependence of gj. Rather than attempting to compensate for these phases and accept errors due to low-frequency magnetic or charge noise, these phases can be decoupled via a similar method to a Hahn echo (Sec. 1.5.2) as demonstrated in Ref. [201] and illustrated in Fig. 7.5. Here, Xj denotes a π−pulse for qubit j,andthe J[ε(t)] pulse is divided into two identical halves, each with phase ξ = π/2. Then under perfectly adiabatic conditions,

π π π U X X U =exp −i P P U X X , (7.2) 2 1 2 2 1 1 2 CZ 1 2  where the Pj = Zj is a single qubit P -gate (Sec. 1.8.3).

225 CHAPTER 7

Figure 7.5: Quantum circuit illustration of the dynamical√ decoupling scheme for the adiabatic controlled-phase gate; the controlled- Z pulses on two qubits in two dots are implemented via adiabatic pulsing of exchange. The resulting operation includes single-qubit Z rotations; these are decoupled by the intervening single-qubit π pulses about X, which may implemented via a global ESR pulse. This figure is adopted from Ref. [47].

Two sources of error are expected to limit the fidelity of this CZ gate as shown in Fig. 7.6. Results from simulating the circuit illustrated in Fig. 7.5 are shown in Fig. 7.7. These simulations use a standard detuning model for J[ε(t)] and a linear model for δg[ε(t)], both informed by Ref. [113] and indicated in Fig. 7.6a. In the J(ε) model, the tunnel coupling tc between dots is assumed to decrease for very small ε, but saturates to a constant at large

∼ 2 detuning, where J(ε) tc /(ε0ε) [133]. This kind of model is similar to what is motivated by the results of Ch. 6. The chosen value and bias dependence of δg(ε) for this simulation are typical values from measurements of devices similar to the one in Ref. [113]. The simulation integrates the evolution due to these Gaussian pulses from

5σt to 5σt,whereσt is the root-mean-square temporal pulse width. As in- dicated in Fig. 7.6b, Gaussian pulses in ε(t) lead to sharply peaked pulses in J[ε(t)] for short σt and to smoother, broader pulses for long σt. These shapes are especially critical for the influence of charge noise, which is in-

226 CHAPTER 7

Figure 7.6: a) Functions of g-factor difference δg(ε) and exchange rate J(ε) versus detuning ε, employing typical parameters, similar to those demon- strated in Ref. [113]. b) Pulse shapes for δg[ε(t)] and J[ε(t)], plotted on the ∝ − 2 2 same scale as panel a), for a Gaussian detuning pulse ε exp( t /2σt )) for two different values of σt, including added voltage noise δV (t) with spectral 2 noise density SV (t)=A /f. These sample traces use a rather high value of A ∼ 30 V to allow the noise to be visible on this scale. This figure is adopted from Ref. [47]. troduced as a randomly sampled noisy voltage δV (t) which couples into ε(t) via device lever-arms. Ensembles of δV (t) functions are filtered from Gaus-

2 sian white noise to produce the noise spectral density SV (f)=A /f. This 1/f voltage noise mimics the expected influence of electric field noise from a variety of possible sources in a real device by modelling it as a single noisy voltage (often known as gate-referred voltage). A clear noise enhancement at the peak value of J[V (t)] is visible in Fig. 7.6b, especially for the shorter pulse (pale blue line); this is because the noise insensitivity I = J/|dJ/dV | rapidly decreases at high J for the

227 CHAPTER 7 detuning mode of operation [180]. The result of integrating the Schr¨odinger equation for these chosen pulse shapes is shown in Fig. 7.7, in which infidelity is given by the normalised trace-distance between the simulated, imperfect unitary and the ideal unitary of Eq. (7.2) under perfect adiabatic and noise- free conditions.

Figure 7.7: Simulated infidelity of the adiabatic CZ gate. The red line is produced from simulating with no charge noise but examining infidelity due to non-adiabatic behaviour. The cyan line considers strictly adiabatic evolu- tion but adds charge noise by Monte-Carlo integration using sampled volt- age noise as in Fig. 7.6a (A =5V for this example). The charge-noise- induced infidelity (cyan line) decreases due to exchange noise, for which 2 the primary trend indicated by the blue line follows 10(A/Ipeak) ,where Ipeak = J/|dJ/dV | is the insensitivity [180] at peak J. For longer pulses, the charge-noise-induced infidelity increases due to noise on the g-factor; the green trend plot is |δg[ε(t)]|2dt/(500 × 109rad/s). The thick grey line is the total infidelity, estimated as the sum of the diabaticity (red) and charge-noise (cyan) contributions. This figure is adopted from Ref. [47].

228 CHAPTER 7

The red curve of Fig. 7.7c shows the infidelity due to non-adiabatic be- haviour, which dominates at short pulse widths σt, but then falls rapidly with increasing σt. The cyan curve indicates infidelity due to randomly sampled 1/f charge noise. This contribution to gate error is decomposed into two sources. In the long-pulse limit, the limiting noise comes from charge-noise- induced fluctuations in g-factor, since this error increases with pulse length following a trend proportional to |δg[ε(t)]|2dt, as indicated by the green line in Fig. 7.7. At the minimum of infidelity relative to root-mean-square pulse-width σt the dominant noise source is an imperfect non-linear phase from the integral over a noisy J[ε(t)], which is dominated by charge noise at the peak of the exchange pulse. This noise source is therefore proportional

2 to (A/Ipeak) ,whereIpeak is the insensitivity at the peak of the exchange pulse. This contribution decreases for longer pulses which have a lower peak value of J, as indicated by the blue line in Fig. 7.7. This error source could be reduced with symmetric exchange pulsing if an additional gate electrode were available to modulate the tunnel barrier between dots [202, 180]. The minimum total infidelity occurs around σt ≈ 300 ns for these parameters, at

2 which, over a broad range of A, the minimum infidelity scales as A .Theσt providing this minimum varies approximately linearly with the constant and voltage-dependent g-factor differences, which vary between dot pairs, but the dependence on these parameters of the minimum fidelity reached at the op- timum pulse length is sub-linear, allowing a substantial range of g-factors with infidelity comparable to the simulation shown in Fig. 7.7.

229 CHAPTER 7

7.2.4 Tick-tock control protocol

Acknowledgement: The “tick-tock” protocol was not directly developed by the dissertation au- thor. It is summarised here as a means of discussing how the elements devel- oped in previous sections are applied for universal control and error correction of a linear array. See Ref. [47] for additional detail.

All of the necessary gates for a logical qubit can be produced by deliber- ate sequencing of the global ESR and exchange-driven CZ operations. The scheme is dubbed tick-tock control in Ref. [47] due to sequencing pulses into two alternating time intervals. These two intervals are separated by globally applied Hadamard gates as shown in Fig. 7.8.

Figure 7.8: Example of using tick-tock control to apply CNOTs to four spins, labelled Q1-Q4. a) Original control sequence consisting of global Hadamard gates that boarder tick (red) and tock (blue) intervals, with selectively ad- dressed CZ gates. Two Hadamard gates and a CZ merge to form a CNOT, with grouping shown by dashed boxes under the convention from Ref. [47]. b) Equivalent circuit diagram showing CNOTs on odd qubit intervals (Op1, Op2, & Op5) and on an even qubit intervals (Op3). Hadamard gates pair to identity when there is no CZ (Op4 & Op6). Figure is adopted from Ref. [47].

230 CHAPTER 7

The advantage of this protocol is that a CNOT gate can be applied be- tween any neighbouring spins by selectively applying a CZ pulse, where the Hadamard gates that transition between tick and tock intervals transform each CZ gate into a CNOT (see Sec. 1.7.6) as shown in Fig. 7.8. The CNOT is not a symmetric gate, hence the control orientation depend upon whether the CZ occurs in a tick or tock interval, as a manner of convention. The unpaired Hadamard gates at the beginning and end of the protocol seen in Fig. 7.8 can be ignored as the single-spin data qubits are in an arbitrary state at the beginning and end of the computation. Both the data and an- cilla qubits are are subject to the tick-tock control procedure.

Data initialisation and parity readout with tick-tock control

The “ancilla” qubits span the singlet-triplet basis. Within tick-tock control, this ancilla has the additional feature that it can be used in a “measurement gadget” to projectively measure a data spin in either X or Z basis, deter- mined by timing of exchange pulses. Only measurement operators that are purely X or Z type are considered, as this is sufficient for an encoding fam- ily known as Calderbank-Shor-Steane (CSS) error correction [15, 120, 203]. When one of the two ancilla qubits which form a prepared singlet state un- dergoes a CNOT with one of the data qubits, a conditional Pauli Z or X is applied to the participating spin in the singlet, depending upon which qubit is the control qubit. An X basis measurement is undertaken when the ancilla √ is the control, resulting in a conditional Z |S =(|↑↓ + |↓↑)/ 2 operation,

231 CHAPTER 7 which transforms the singlet one of the triplets. Similarly, a Z basis measure- ment is undertaken when the ancilla is the target in the CNOT, applying a √ conditional X |S =(|↓↓ − |↑↑)/ 2, which is also a triplet. Measuring the ancilla as a singlet or triplet performs projective measurement on the data spin in X or Z basis respectively. This measurement gadget is used to initialise data spins after loading data qubits in an arbitrary mixed state. The tick-tock protocol of periodic Hadamard gates is initiated, and measurement gadgets are used to prepare each data spin in either X or Z basis as needed for the computation. The measurement protocol can be extended to form a parity measurement as shown in Fig. 7.9 between to data qubits. As an example, measuring Z ⊗Z on two data spins results in the ancilla swapping between |S and X |S for each data spin in the |↑ state.

232 CHAPTER 7

Figure 7.9: Control sequence for parity-measurement experiment. a) Parity measurement circuit using a singlet ancilla and measurement. b) Tick-tock control sequence showing how the two-qubit gates in a) are decomposed into CZ entangling gates between the global Hadamards implemented by ESR. SWAP is decomposed into three CNOTs, and the CNOT-followed-by-SWAP gate is two CNOTs. This figure is adopted from Ref. [47].

233 CHAPTER 7

7.3 A logical qubit in one dimension

Acknowledgement: The protocols discussed in this section were not directly developed by the dissertation author. It is summarised here as a means of discussing how the elements and experimental methods developed in previous sections and chap- ters can be combined to achieve logical qubit encoding and error detection in a linear array of SiMOS quantum dot qubits. See Ref. [47] for additional details.

One of the main challenges for achieving a logical qubit in a linear array is that any encoding scheme requires two-qubit operations between qubits that cannot all be local (between adjacent qubits) for the linear arrangement. All non-local interactions must be mediated by SWAP gates as seen in Fig. 7.9. This constraint is at the core of what is known as linear nearest-neighbour (LNN) architectures. Many quantum codes do not adapt well to a linear geometry; for example, topological codes cannot have a threshold in one dimension [204, 205, 206]. However, past work has established methods for error correction in a linear or bilinear array of qubits by concatenating small codes (see Sec. 1.8.4 and Refs. [207, 208, 209]), and these methods are applied in this logical qubit proposal. Ref. [47] introduces a tile formalism as a strategy for building a logical qubit using nearest-neighbor gates in a linear array of qubits. This formal- ism is based on a set of “design rules” (see Appendix M) that prevent some

234 CHAPTER 7 of the pathological errors that can occur in LNN circuits. This set of instruc- tions is closely related to CSS codes [120, 203], and allows any standard-LNN encoded gate to be constructed solely from standard-LNN instructions. The encoding scheme using these instructions is summarised in Appendix N with a focus on the two-qubit repetition code. The encoded logic gates are grouped into blocks of instructions called “tiles”, which provide a simple scheme for scheduling instructions to operate a logical qubit as shown in Fig. 7.10. Here, the two-dimensional quantum circuit shows qubits spanning the vertical di- mension, while time flows to the right. Each tile is a sub-circuit consisting of nearest-neighbour gates on a small

Figure 7.10: Concatenation of distance-two repetition codes with tiles. A phase-flip idle tile (upper left) is encoded using the logical qubits of bit-flip codes. The individual elements of this idle are divided into separate tile instructions by dashed lines, and are numbered correspondingly to the bit- flip tiles of the (right) encoded idle operation. This figure is adopted from Ref. [47].

235 CHAPTER 7 set of adjacent data qubits and syndrome ancillas. The tiles naturally im- plement code concatenation by recursively building tiles at tier L from tiles at tier L − 1. The tiles bring syndrome ancillas into contact with all data qubits for error detection, with each tile moving the ancilla qubit(s) across a code block, opening the other side for an interleaved two-qubit gate. As discussed in Sec. 1.8.4 it was identified that the threshold theory for functional logical qubits depends upon the likelihood of errors and how ef- fectively they are corrected. Simulation methods and results summarised in Appendix. O can be used to determine these error thresholds, and act as a guide for which spin-control operations require further improvement in fidelity. The simulation results in Fig. 7.11 suggest a threshold for the two- qubit code around 10−4. The crossing of the level-one and level-four logical error rates occurs at p =9.5 × 10−5, while the crossing of level-two and level- four curves occurs at p =3.1 × 10−4. The simulations presented here and in Appendix O provide control fidelity targets for experiments to demonstrate a “signature” of error correction. This is the characteristic quadratic depen- dence of logical error rate with physical error rate when any single error is correctable. The results demonstrate that this signature can be detected even at error rates above threshold (up to 10−3 or higher), which allows an exper- iment to demonstrate the functionality of error correction by synthetically injecting errors. The components of the error correction scheme can be demonstrated in intermediate proof-of-concept experiments, as has been done in other qubit

236 CHAPTER 7

Figure 7.11: Simulated logical error rates for the two-qubit repetition code using concatenation. Dots represent Monte-Carlo error generation at the specified physical error rates. The solid curves are produced by malignant- set counting or sampling. This figure is adopted from Ref. [47]. technologies [210, 183, 185, 211, 212, 186, 187, 188, 116]. For the scheme introduced here, an experimental pathway is summarised in Fig. 7.12. Each of these experiments are summarised in Appendix P, and builds up from the most fundamental element of demonstrating the parity readout subroutine, to the execution of a fully error corrected logical qubit. Each fundamental experiment addresses the integration of different aspects of the extensibility criteria discussed in Sec. 7.1.

237 CHAPTER 7

Figure 7.12: Experimental pathway to a logical qubit in quantum dots. The middle columns indicate which criteria for an extensible logical qubit (see Sec. 7.1) are demonstrated by each experiment, denoted by green squares. In this context, “error threshold” demonstrations are achieved by demonstrat- ing functional error correction through purposefully inserting errors. The “fault tolerance” criterion indicates that any single bit-flip and/or phase-flip error is detectable. Each experiment is accompanied by a linear array device schematic, indicating the number and location of data and ancilla qubits. This figure is adopted from Ref. [47].

7.4 Discussion of results

This section summarises the discussion of results presented in Ref. [47]. Based on current quantum-dot technology as presented here and in ear- lier chapters of this dissertation, it is believed that the SiMOS quantum dot qubit platform is ready to begin developing a single logical qubit. However, it is noted that it is reasonable to assume further improvements in materi- als, fabrication, and certain experimental methods will occur alongside the experimental demonstrations outlined in the previous section. Specifically, the tick-tock protocol implements ESR control addressing all spins simultaneously. The microwave power necessary to perform global

238 CHAPTER 7

ESR control with high fidelity is dependent upon the need for spins with sufficient δg to perform fast CZ operations. This requires the ESR pulses to be non-selective despite the significant spread in resonance frequency of the individual qubits. However, once the g-factor spread has been characterised and, if needed, tuned, additional power is not required in order to add more spins. The prospects for scaling this proposal, and in particular handling the possibility of defective quantum dots, is an important consideration for ex- tensibility. It is noted that a single defective dot can disable an entire logical qubit when using LNN error correction. Current SiMOS technology has suf- ficient yield to reach 20 coupled dots in the near term. Beyond this scale, the LNN logical qubit could be a building block for a larger system. For exam- ple, it is possible to arrange short linear segments of dots that meet in three- or four-way junctions, such as in Ref. [48] enabling enough connectivity to route information around defective dots and tolerate imperfect yield. In such a scheme, linear segments of dots could be arranged in a grid pattern [51], enabling 2D connectivity at this scale for avoiding defects or implementing codes that are tolerant of defects [213, 214]. The focus of this proposal is on the SiMOS system, however, it certainly can be adapted to other semiconductor quantum dot systems. Confining the quantum dots in a Si/SiGe heterostructure rather than against a Si/SiO2 in- terface may reduce the effects of disorder and charge noise, at the expense of introducing smaller valley splittings (which may impair singlet initialisation

239 CHAPTER 7 and measurement). The controllable g-factor shifts in SiGe might be sub- stantially smaller than what is observed in SiMOS dots, possibly resulting in the integration of an induced magnetic field gradients [193, 215, 163]. This scheme may also be feasible using a heterostructure based on III-V semicon- ductors, which have no valley degeneracy and may be engineered to have high Stark-tunable g-factor shifts [216]. There are inevitably large numbers of nu- clear spins in III-V systems, requiring more reliance on dynamical decoupling. Encouragingly, dynamically decoupled coherence times approaching millisec- onds appear to be feasible [217], although a fully hyperfine-compensating modification to the control scheme would require additional design in this case. Finally, this scheme could be adapted to the problem of substitutional donors coupled to SiMOS-like dots or spin-shuttling channels, in which case its implementation would resemble the schemes indicated in Refs. [201, 218].

240 Chapter 8

Discussions, conclusions and future work

The results presented in this thesis cover a broad scope regarding the SiMOS quantum dot qubit platform, from single qubit characterisation experiments, expansion into a two-qubit space, and discussion of how to facilitate scaling into larger quantum systems for demonstrations of fault tolerant encoded log- ical qubits. Key results presented are the expansion of the RB experiment for characterising qubit performance to include situations where non-Markovian noise sources are dominant, which is the case for many qubit platforms based in solid-state materials. Other important milestones presented in this disser- tation include the first reported instance of an experimentally demonstrated integrated device platform incorporating a SiMOS double quantum dot that is capable of single-spin addressability using electron spin resonance, with high-fidelity single-shot readout in the singlet-triplet basis. Full characterisa-

241 CHAPTER 8 tion of the system Hamiltonian revealed the ability to coherently control op- erations between the singlet state and both polarised and unpolarised triplet states, the latter providing a means to execute the fast two-qubit SWAP gate. Further studies presented in this dissertation include discussions on how the SiMOS quantum dot qubit platform can be practically scaled into intermediate demonstrations of encoded quantum information, error correc- tion and fault tolerance. The reports conclude with the outline of a recently proposed experimental pathway from small scale qubit systems into a fully error corrected logical qubit produced from a linear array of SiMOS quantum dots.

Non-exponential fidelity decays in randomised benchmarking

The contributions in this dissertation begin by focussing on the results of the randomised benchmarking experiment performed on a single SiMOS quan- tum dot qubit. The randomised benchmarking protocol performed on a single qubit has the advantages of the robustness of the fidelity measure against state preparation and measurement errors, and draws ease of exe- cution and scaling to larger qubit systems from the use of group theory to dictate the fundamental operations implemented. However, one key draw- back is the strict assumptions placed upon the physical system for correct functionality of the experiment. The protocol was originally developed to

242 CHAPTER 8 characterise uncorrelated Markovian noise, however, many solid-state qubit systems upon which the experiment has since been executed are generally subject to non-Markovian correlated noise. The single qubit randomised benchmarking experiment for the SiMOS quantum dot qubit device [42] demonstrated what can be observed when the strict assumptions of this experiment are violated. In this case, the standard 1/f α noise spectra as seen in semiconductor devices [78, 215, 134, 219], does not satisfy the requirement for a Markovian noise source. In actual fact, the improvements to coherence time measures [42] using the noise decoupling processes described in Sec. 1.5.2, is only testament to the fact that there is appreciable noise components operating at time-scales much longer than single qubit gate operation. The subsequent analysis of the single qubit RB results as presented in Ch. 3 (published as Ref. [58]) aims to expand upon the RB protocol to extract meaningful information from this kind of result. The discussion in Ch. 3 identifies the effects of a slowly varying, “random walk” of the fundamental control parameters within the single qubit Hamil- tonian by introducing the concept of the “instantaneous fidelity”. This pa- rameter is a descriptor of the system fidelity at any particular time instance, and experiences deviations from the ideal based on the random walk. Using this concept, the non-exponential behaviour extracted from the RB experi- ment can be attributed to an integration over an ensemble of measurements with a distribution of instantaneous fidelities, with the random walk man- ifesting as each experimental run sampling from a distribution of unitary

243 CHAPTER 8 miss-calibration errors. This quasistatic error model is often referred to as “DC noise” and is further explored in Refs. [220, 221, 222]. Via the concept of instantaneous fidelity, it can be inferred that the non-exponential decay can be broken down and fit by a sum of multiple exponentials. For simplicity, the methods of Ch. 3 focus on two equally weighted decay rates, one attributed to an average over instantaneous fidelities where the miss-calibration errors were large (demonstrating worst-case behaviour), while the other is an ensem- ble average over well behaved runs of the experiment. Thus, the additional information which can be drawn from this data is a fidelity representation of how poor the qubit control was as certain times over the course of data acquisition, while also providing a fidelity target representative of the best performance of the qubit. Since the methods and discussions of Ch. 3 were reported, the method of measuring two separate spin projections to remove SPAM terms and study more directly the underlying exponential behaviour has been discussed in more recent revisions of the RB experiment [223]. The process of reduc- ing the number of parameters has also found application in recent studies for decorrelating the preparation and measurement errors during studies utilising Bayesian inference within RB experiments [224]. Further, several other re- search initiatives have also demonstrated non-exponential fidelity decays pro- duced from RB. Methods involving RB on ensembles of electrons [225, 226] exhibited non-exponential decays due to sampling over a probability envelope

∗ of individual T2 times, which is likened to a statistical ensemble of measure-

244 CHAPTER 8 ments from a single electron undergoing a random walk during RB. These experiments were improved using SEL pulsing to narrow the linewidth of the ensemble [226], with the improved fidelities appearing better than the high fidelity tail of the non-exponential curves (see Fig. 11 in Ref. [226]). Other non-exponential decays were reported for a Si/SiGe quantum dot qubit in Ref. [215]. These experiments used the same dual measurement projection protocol as in Ch. 3 to remove SPAM elements, where the results and also highlighted the non-exponential decay on a log-linear plot. It was observed from both experiment and simulation that the decay deviated from a sin- gle exponential beyond a certain experiment run-time, consistent across all measurements (including interleaved benchmarking), thus only data accumu- lated before this time was utilised for the evaluation of gate error rates. This highlights an open ended question presented in Ch. 3, which states the need to define a set of stopping criteria for the RB sequence length under various noise conditions. The fact that realistic noise sources from devices were ca- pable of producing results which were drastically different from predictions lead to an expansion of theory in this area, including studies on time depen- dent noise [161], noise coherence [108], error correlation [220, 221, 227, 222] and state leakage [160, 228]. The 1/f α noise spectrum sampled by the qubit during the RB experi- ment exhibits non-Markovian components, being the elements which operate slower than the qubit Rabi frequency. This gives rise to temporal correlation in the gate errors within the RB experiment and produces gate-dependent

245 CHAPTER 8 errors. These elements are in direct violation of the standard assumptions of RB [153]. From concepts such as interleaved dynamical decoupling (proto- cols which are valid gate sequences and can be generated in RB), it is clear that temporally correlated elements can lead to unintentional correction of errors within RB sequences. This could result in an over-estimation of the RB fidelity, and is studied in Refs. [220, 221, 227, 222]. However, it was shown in the simulations of Ref. [107], that a 1/f spectrum yields a result which is within a factor of two of the correct error rate. Further, recent stud- ies on correlated noise have demonstrated that RB on one or more qubits has a deep connection to the Ising model and results in a long-range (power law) and short range (exponential) decay behaviour for noises with differ- ent spectra [221]. This study states that fitting RB experiments to a simple exponential decay model can be misleading, but corroborates the concept that this problem can be mitigated by fitting only the short sequence and asymptotic regime data, similar to the methods in Ch. 3. Since the publication of the results in Ch. 3, advancements in the field have taken explicit interest of more rigorous methods of quantum character- isation, validation, and verification (QCVV), which is still a major area of research and development. When it comes to benchmarking single qubit performance, state and process tomography experiments, as well as ran- domised benchmarking have been joined by processes such as purity bench- marking [108], adaptive tomography methods [229] and gate-set tomogra- phy [230, 231]. In the face of such processes and more modern understand-

246 CHAPTER 8 ings of the QCVV area, phenomenological attempts to explain deviation from ideal outputs, such as those presented in Ch. 3 could be considered “unten- able” [222] when attempting to extract QCVV results and compare them to metrics relevant to quantum error correction. Indeed, it is asserted in Ref. [225] that non-exponential RB decays imply that there exist error cor- rection/control procedures that are more favourable than simply comparing the average gate error rate to fault tolerance thresholds. Hence, even if the average gate error can be correctly estimated with RB for this case, it is pos- sible to learn nothing about the fault-tolerant behaviour of the system [232]. The ease at which these experiments can be performed, together with this somewhat ambiguous outlook for the results extracted from the RB experi- ment is the reason why the area continues to see new studies and expansions into broader areas of QCVV today.

Single-shot Pauli spin blockade in a SiMOS quantum dot qubit

In order to advance the studies of SiMOS quantum dot qubits beyond the control fidelity of a singular electron, designs which intentionally couple two or more quantum dots needed to be developed. The first instance of a SiMOS device such as these was reported in Ref. [113], which demonstrated the fun- damental elements of a two-qubit logical operation as well as the beginnings of the characterisation of the two-qubit Hamiltonian. These results demon-

247 CHAPTER 8 strated single- and two-qubit elements which can be combined to yield a uni- versal gate set for quantum computation on a single SiMOS device. Hitting this milestone spurred research into how the laboratory test-bench devices discussed in Ch. 2 could be expanded in a scalable manner to produce sys- tems capable of demonstrating fundamental elements of encoded quantum information in a linear array [47], and full scale quantum processors capa- ble of executing surface codes [50]. Outlined in many schemes which propose methods of operating scaled quantum systems (including these two which fo- cus on SiMOS) is a requirement for extracting the parity information of two data qubits [188, 187, 189, 47]. This process is often at the core of syndrome measurement in stabiliser codes [233, 211]. One scalable implementation of a parity readout subroutine is discussed in Ch. 7, and identifies core elements which had not been demonstrated in the SiMOS quantum dot qubit platform at the time of release. Namely, this is the single-shot readout of Pauli spin blockade (PSB). Since the release of Ref. [47] (results presented in Ch. 7), however, this mechanism has been demonstrated using a SiMOS quantum dot qubit device with poly-silicon electrodes [85], and a SiMOS device with metallic electrodes [60] (results presented in Ch. 4). Both of these mecha- nisms employ readout enhancement protocols utilising dot-reservoir charge states, which have also been explored within Refs. [164, 165, 166, 136, 80]. As discussed in Ch. 2, the device architecture used for these experiments was not designed for PSB and has both dots near symmetrically coupled to the SET island. Near-term SiMOS devices which intend to pursue and

248 CHAPTER 8 develop PSB as a readout mechanism (such as those presented in Ch. 2 and Appendix H) circumvent this issue though relocation and redesign of the SET. In order to enable further experiments, the enhancement mecha- nism described in Ch. 4 is employed to increase the visibility from 60.50% to 98.04%. Other enhancement mechanisms seen in the literature demon- strate similarly high readout visibilities for enhanced-PSB for dot-dot, dot- donor and donor-donor system produced from a variety of materials plat- forms [166, 165, 136, 171, 80, 85]. Each of these studies discuss a multitude of different methods though which enhancement is achieved. Further, dif- ferent mechanisms are observed depending on the location of the reservoir forming the charge state. The concept of utilising a charge state involving the reservoir seems counter-intuitive when focussing on PSB as a proposed (scalable) replace- ment for the reservoir-based readout mechanisms discussed in Ch. 2. Indeed, the enhancement-style readout mechanism presented in Ch. 4 would be con- sidered non-scalable for this reason. However, not all latching-based mech- anisms are incompatible with scalability, and recent concepts [234] work to demonstrate how latching can be performed within an array of dots at the expense of an additional (empty) dot per singlet-triplet ancilla. As discussed in Ch. 6, the overall visibility observed after readout of an experiment can be attributed to combined state-preparation and measure- ment processes which arise independently of the enhanced readout process. More explicitly, experiments which manipulate (1,1) states which are origi-

249 CHAPTER 8 nally prepared and then measured in (0,2), require mappings between these (1,1) and (0,2) states (see discussion in Ch. 6). The characterisation and calibration of these processes are crucial, yet remains poorly represented ex- perimentally in SiMOS devices in current literature. For the device presented here, any errors collected in the process of mapping from the standard PSB region to the enhancement region are difficult to disentangle due to poor readout visibility of the standard (0,2) PSB region. While the PSB enhancement protocol discussed in Ch. 4 may not be di- rectly scalable in its present form, the device studied was able to demonstrate elementary signals of single-shot PSB readout in a SiMOS double quan- tum dot. These studies produced understanding in the role of metastable charge states within these test-bench devices, insight into device architec- ture improvements, and provide a benchmark for improved readout via scal- able methods. This enhanced readout protocol also provides an avenue through which the full singlet-triplet Hamiltonian can be benchmarked in these SiMOS devices, which are the focus for Ch. 5 and Ch. 6 discussed below.

250 CHAPTER 8

Coupling between the singlet and polarised triplets within a SiMOS double quantum dot

For singlets prepared in the (0,2) state, the first accessible element of the system Hamiltonian is the bit-flipping term Δ introduced in Ch. 1. The origin of this is currently not well understood, but the possible origins are discussed in Ch. 5. The coupling term Δ is first observed in the standard spin-funnel [112] experiment, and the magnitude was further characterised at two relevant magnetic fields via single-passage Landau-Zener excitation

z experiments. At low-B0 external field (near to zero), the majority of the hybridised singlet is |(1, 1)S, and the coupling energy was measured to be

| | ± z Δ(θ) = 196 6.3kHz.ForB0 = 155 mT, the majority of the hybridised singlet is instead |(0, 2)S,and|Δ(θ)| =16.72±1.64 MHz. Here, θ represents the hybridisation of the (1,1) and (0,2) singlets. This preliminary demonstra- tion alludes to the ability to probe different relative coupling strengths of the two spin orbit mechanisms facilitated thought the use of a vector magnet, and these experiments are proposed as future work. The bit-flipping term Δ is also utilised to assist with benchmarking the exchange energy J within the system via Landau-Zener-St¨uckelberg interfer- ometric measurements [73, 71], where the relevant energy splitting is accessed through the Fourier transform of these results.

251 CHAPTER 8

An integrated platform for quantum computation

The characterisation of the bit-flipping term Δ via Landau-Zener excitation experiments is an important result for the progress of the (0,2)-(1,1) Hamilto- nian characterisation. The results of Ch. 6 further benchmark this system by investigating the interactions between the |SH  and unpolarised triplet |T0. For this subset of states, the energy scales accessible are now the tunnel cou- pling tc and g-factor difference δg between the two dots. In order to probe these energy scales, the (1,1) ground state of this subset of states within the

(0,2)-(1,1) Hamiltonian must be prepared (excluding |T±). This is robustly achieved via diabatically ramping through the SH /T− anti-crossing medi- ated by Δ while maintaining a ramp rate ν which is adiabatic with respect to the tunnel coupling tc (as discussed in Refs. [112, 133] amongst others). For many experiments of this type seen in the literature, this semi-adiabatic procedure results in the preparation of a ground state dictated by δEZ which often manifests from external field gradients δB between the two quantum dots. This field gradient can be derived from a nuclear-spin bath [87] in both GaAs [112] and natural silicon [133], or from the addition of a micromag- net or nanomagnet structure [235, 179]. The mechanism implemented here (also seen in Ref. [85]) instead utilises δg, rather than any magnetic field gra- dient. However, this implies that the external applied field must yield δEZ which is orders of magnitude larger than the residual hyperfine field for ro- bust preparation of the same state over experimental repetition. For external magnetic fields which make δEZ appreciable from δg, the eigenstate prepared

252 CHAPTER 8 semi-adiabatically in the detuning region where J 0 is an element of the computational basis rather than the (1,1) singlet or T0 triplet. Either |↓↑ or |↑↓ is robustly prepared respective of whether g1 g2. Preparation of this computational basis state, which is an equal superpo- sition of the |(1, 1)S and |T0 states, provides an avenue for measuring the exchange energy J as a function of detuning within a region where J δEZ via the methods outlined in Ch. 6. Together with the spin-funnel and zero- field Landau-Zener-St¨ukelberg oscillations all combine to yield a consistent picture of the exchange coupling for ε from the (1,1)-(0,2) anti-crossing, out to the centre of the (1,1) region where J 0. For increased external field, the ESR spectrum observed excitations from the semi-adiabatic ground state to blockaded polarised triplet states as a function of . The energy gap fit from the resulting ESR funnel became a measure of not only this exchange

J, but also saturated to the δEZ for large detuning producing an estimate for δg for these qubits. The results of this Hammiltonian characterisation discussed in Ch. 6 al- tered what was originally known about the SiMOS quantum dot qubits.

Modelling the exchange function J(ε), and thus extracting tc, did not yield a good fit for constant tunnel coupling with respect to ε. Previous studies on

SiMOS quantum dots [113] were interpreted with fixed coupling tc between dots, measured to be the magnitude at the anti-crossing. From these studies, it has been suggested that the tunnel coupling is also a function of detun- ing ε, similar to the results found in Ref. [180], for a Si/SiGe heterostructure

253 CHAPTER 8 quantum dot device. The ability to robustly prepare a computational basis state from an ini- tial |(0, 2)S is not only useful for the characterisation experiments discussed above, but has further implications which extend to the operation of larger scale qubit systems. The experiments in Ch. 6 detail the capability to ro- bustly prepare the |↓↑ computational basis state, but demonstrates the abil- ity to coherently control an |↓↑ ↔ |↑↓ operation based on detuning ε.The effect of this operation is to exchange or “swap” the qubit information be- tween dots, and has been shown in Ch. 1 to be capable of producing the two-qubit element of a universal gate set. Further, it was also discussed in Ch. 7 that these operations can be useful for shuttling quantum states along a linear array. Thus, the results in Ch. 6 demonstrate the expansion of the toolbox available for the SiMOS quantum dot qubit devices beyond the exper- imentally implemented CPhase gate in Ref. [113]. The other core experiment presented in this chapter, which relies on robust preparation of the compu- tational basis state, is the observation of the ESR spectrum. Here, the two resonance peaks observed are attributed to addressing and rotating indepen- dently the two individual electrons within each quantum dot. While coherent control of these two electrons is not shown, this is the first example known by the author of individually addressing a singular electron within a prepared two-electron system via ESR techniques, where the resulting state is sub- sequently measured through PSB in a silicon qubit platform. This readout mechanism hosts several advantages when compared to the Elzerman-style

254 CHAPTER 8 reservoir readout more traditionally implemented in these SiMOS devices. A key factor of standard Elzerman scheme is the requirement of large exter- nal magnetic fields required to Zeeman split the electron states on a scale much larger than the reservoir thermal broadening. This external field is of- tenontheorderof1.4T, resulting in ESR driving frequencies of ∼ 40GHz. When using the PSB techniques described in Ch. 4, the limitations to the external field in the ESR experiment is based on the robust semi-adiabatic preparation of the computational basis state (based on δEZ). This allows the ESR experiment to be performed at external fields which are much smaller compared to the Elzerman scheme, and here the driving frequency was re- duced by almost a factor of 10, for an operating frequency of ∼ 4.2GHz. The integration of low-frequency ESR of individual spins with singlet-triplet based initialisation and readout holds promise for qubit architectures oper- ating at significantly lower magnetic fields and higher temperatures, while also implying cheaper overheads for supporting electronics. The combined results of Ch. 4, 5 & 6 demonstrate for the first time in a silicon device, experimentally combined single spin qubit control using electron spin resonance, with high-fidelity single-shot readout in the singlet- triplet basis. By characterising the relevant energy scales Δ, δEZ and t(ε)of the two-spin Hamiltonian, coherent manipulations between both the S/T− and S/T0 states were observed, the latter of which provides potential for a fast two-qubit SWAP gate at high exchange. The initialisation and readout of singlet-triplet states attests to the compatibility of the SiMOS quantum dot

255 CHAPTER 8 platform with parity readout based on spin-blockade, key for the realisation of future large-scale silicon-based quantum processors [50, 47].

Towards demonstrations of logical qubit operation with

SiMOS qubits

The results of the earlier chapters demonstrate a fully integrated platform which exhibits all required elements for the first few experimental logical qubit milestones discussed in Ch. 7. Indeed, the most fundamental experi- ment, the parity readout gadget, would simply require access to additional “data qubit” dots in the linear chain for attempts at execution. Utilising the fundamental building blocks for quantum computation presented within in this dissertation and from earlier works, the concepts outlined in Ch. 7 attempts to discuss several aspects which were missing from literature at the time of release. State-of-the-art experiments now involve operations on two to nine coherently controllable qubits [182, 183, 184, 185, 186, 113, 187, 188, 189], yet despite this rapidly accelerating progress, an extensible logi- cal qubit has not yet been demonstrated. While there are several discussions from many solid-state qubit platforms which are based on proposing scaled qubit architectures aiming to operate topological codes for the realisation of fully error corrected quantum computation [52, 218, 50], there has been little discussion within the community with regards to an experimental roadmap of smaller scale experiments which need to be achieved in order to reach this

256 CHAPTER 8 goal. More explicitly, these are discussions on how to evolve from single- and two-qubit test-bench devices still being studied in laboratories today, to the implementation of a singular error corrected logical qubit which operates in an extensible manner (let alone multiple logical qubits operating together). The discussions in Ch. 7 focus on how to build upon the current state-of- the-art in SiMOS quantum dot qubit devices, and strive towards experimen- tally demonstrating a singular error corrected logical qubit via implementa- tion of stabiliser codes. The larger qubit array must be capable of controlling the constituent qubits in parallel to be feasible and the fundamental build- ing blocks of the universal gate set for quantum computation are discussed at the device level accordingly. One potential issue in SiMOS technology is the variability of a number of fundamental elements from qubit-to-qubit, such as the electron g-factor. This is addressed through the proposed use of global-ESR techniques for single-qubit control, as well as self-decoupling two-qubit operations. Together with simulated error rates motivated from realistic noise sources seen in experiments, these protocols lay the foundation for the demonstration of a fault tolerant logical qubit. The simulated per- formance of the elementary operations are utilised along with the encoding schemes presented to produce the simulated performance of logical opera- tions generated from these schemes at each encoding layer and at multiple code distances. The results of these simulations suggest that it is feasible to observe the effects of error correction within experiments at current qubit control fidelities. Further, these experimental demonstrations are achievable

257 CHAPTER 8 near-term in the laboratory, by using larger scale test-bench devices and two layers of encoding. In order to reach this goal of operating an extensible logical qubit within the laboratory, a step-by-step experimental roadmap is proposed from the most fundamental element (i.e. the parity readout gad- get), which slowly increases operation complexity and device size in order to demonstrate the integration of extensibility criteria until a fully error cor- rected logical qubit is attained. The discussion in Ch. 7 focusses on the SiMOS quantum dot qubit plat- form, however, it can also be adapted to other semiconductor quantum dot systems. Quantum dots in a Si/SiGe heterostructures may reduce the effects of disorder and charge noise, but has the expense of introducing smaller val- ley splitting. Controllable g-factor shifts in SiGe may also be substantially smaller in comparison, possibly requiring integrated micro/nanomagnets for enhanced magnetic field gradients [193, 215, 163]. This scheme may also be feasible using a heterostructure based on III-V semiconductors, which have no valley degeneracy and can be engineered to have high Stark-tunable g-factor shifts [216]. However, the trade-off here is heavier reliance on dynamical decoupling due to background nuclear fields. As a final note, the linear chains presented in Ch. 7 demonstrate a singular extensible logical qubit, and have not been designed to be capable of execut- ing a topological code such as the surface code [49]. However, with additional work, these logical qubits could become the fundamental building block of a much larger construction such as the structures seen in Refs. [51, 48].

258 CHAPTER 8

Future work

There are many areas motivated by the results presented in this disserta- tion which are identified as potential avenues of future studies. For the randomised benchmarking experiment, the presence of non-Markovian noise sources as well as the ability to increase qubit performance can be gleaned

∗ by extensions to T2 via decoupling experiments (see Ref. [42]). In order to achieve better performance reflected from the RB experiment the effect of the random walk must be suppressed. One example of how this could be achieved is through the integration of subroutines capable of tracking parameters within the qubit Hamiltonian [163] with dynamic feedback on qubit control, thus intermittently tracking the random walk. A second pos- sibility is the application of self-refocusing elementary gate operations opti- mised against low frequency noise. These gates have been identified as being utilised as the fundamental control operations for scaled sequenced in Ch. 7. The design of these gates are accessible via pulse optimisation schemes such as GRAPE [236]. For this case, the amount of non-Markovian noise experi- enced by the qubit remains unchanged, however the qubit becomes decoupled from the effects by design. Both of these proposed techniques would greatly reduce the amount of the non-Markovian components captured within the RB experiment, and are left as possible future studies. With regards to the enhanced PSB reported in Ch. 4, valuable future work in this area could be to investigate and characterise, in a singular device with an asymmetric reservoir, the different enhancement conditions/modes

259 CHAPTER 8 observable at both the (0,2)-(1,1) anti-crossing, and the (2,0)-(1,1) anti- crossing. The combined works for SiMOS devices including Refs. [136, 85] and Ch. 4 suggest there are two different enhancement regions observable at each anti-crossing, for a total of four different enhancement protocols avail- able in SiMOS devices. Further, it was identified in the above discussion that the characterisation and calibration of these preparation and measurement using PSB, remains poorly represented experimentally for SiMOS devices in current literature. This leaves the earnest characterisation and optimisation of the preparation, mapping and readout processes with scalable PSB as an open area of research for SiMOS devices. It is noted that the errors attributed to the addition of the enhancement PSB is discussed in Ref. [136]. Pushing towards either scalable PSB in more contemporary SiMOS de- signs, or working towards schemes which utilise latching in a scalable man- ner are both areas of future research in this field. With regards to moving away from enhancement protocols, it is inferred from concepts of capacitive sensing within these devices that the reliance on a PSB readout mechanism which does not utilise enhancement protocols can result in some decrease in measurement visibility. However, this loss in visibility can potentially be compensated through other avenues including improved data processing such as “soft-averaging” [237, 238].

Identified in Ch. 5 was the presence of a Δ coupling term between the SH and T± states. Preliminary demonstrations of field dependence for this term

260 CHAPTER 8 alludes to the ability to probe different relative coupling strengths of the two spin orbit mechanisms facilitated thought the use of a vector magnet. Fur- ther, the Fourier transform of the Landau-Zener-St¨uckelberg interferomet- ric measurements is not the only method through which the exchange cou- pling can be measured. Other methods through which this Fourier transform could be resolved include spectroscopic experiments via microwave excitation across the SH /T− transitions. These experiments could be performed either using the integrated ESR control or via photon assisted tunnelling [173], and these alternate methods are left as potential future experiments which can corroborate the findings here. Other experiments which were not in- vestigated in this dissertation, but which are a logical next step, include the Landau-Zener-St¨uckelberg-Majorana interferometry experiments. Here, multiple-passage driving amplitude and frequency can be used to gather char- acterising information about the system. These methods have been recently applied to SOI CMOS systems [239], including investigations involving dot- reservoir charge states [240], and characterisation of Sisyphus resistance [241]. With regards to longer term goals achievable on the SiMOS platform, Ch. 7 presents a step-by-step experimental roadmap from the most funda- mental element (i.e. the parity readout gadget), and which slowly increases operation complexity and device size in to culminate in a demonstration of an extensible logical qubit. Further, it was mentioned in the above discussion that these logical qubits could become the fundamental building block of a much larger constructions. In order to achieve this, a method through which

261 CHAPTER 8 two logical qubits can interface between one-another must be engineered and integrated into the existing protocols for the error corrected logical qubit produced from an array of SiMOS quantum dot qubits.

262 Appendices

263 Appendix A

Acknowledgements: Part 1

Non-exponential Fidelity Decay in Random- ized Benchmarking with Low-Frequency Noise

The works presented in Ch. 3 and theory Sec. 1.6.2 was produced in collabo- ration between the author of this thesis and Dr. M. Veldhorst, Dr. R. Harper, Dr. C. H. Yang, Dr. S. D. Bartlett, Dr. S. T. Flammia, and Dr. A. S. Dzu- rak. as listed in Non-exponential Fidelity Decay in Randomized Benchmark- ing with Low-Frequency Noise, Physical Review A 92, 022326 (2015) (also referred to as Ref. [58]). Collectively, these authors thank Chris Ferrie and Chris Granade for helpful discussions, and acknowledge support from the Australian Research Council (CQC2T - CE11E0001017 and EQuS - CE11001013), the NSW Node of the Australian National Fabrication Facil- ity, the US Army Research Office (W911NF-13-1-0024, W911NF-14-1-0098, W911NF-14-1-0103 and W911NF-14-1-0133), and by iARPA via the MQCO

264 APPENDIX A program. Contributing authors Menno Veldhorst also acknowledges support from the Netherlands Organization for Scientific Research (NWO) through a Rubicon Grant, and Steve Flammia also acknowledges support from an ARC Future Fellowship (FT130101744).

Author Contributions

M.V and C.H.Y performed experiments, M.A.F performed noise modelling and simulations with the assistance of C.H.Y and M.V. The theoretical ideas related to the experimental design in the paper were developed in group discussions. Theoretical support for the experimental results was provided by R.H, S.D.B and S.T.F. Statistical analysis was performed by R.H, with the aid of S.D.B and S.T.F, including the introduction of alternative models, calculation and write up of the Akaike bounds. M.A.F wrote the manuscript with input from all authors.

265 Appendix B

Acknowledgements: Part 2

Integrated silicon qubit platform with single- spin addressability, exchange control and ro- bust single-shot singlet-triplet readout

The works presented in Ch. 4, 5, 6 and some elements of theory Sec. 1.7 was produced in collaboration with the author of this dissertation and Dr. K. W. Chan, Dr. B. Hensen, Mr. W. Huang, Dr. T. Tanttu, Dr. C. H. Yang, Dr. A. Laucht, Dr. M. Veldhorst, Dr. F. E. Hudson, Dr. K. M. Itoh, Dr. D. Culcer, Dr. T. D. Ladd, Dr. A. Morello, and Dr. A. S. Dzurak, as listed in M. A. Fogarty et. al. arXiv preprint arXiv:1708.03445v2 (also referred to as Ref. [60]). The authors thank Mark Gyure for helpful discussions and acknowl- edge support from the Australian Research Council (CE11E0001017 and CE170100039), the US Army Research Office (W911NF-13-1-0024 and W911NF- 17-1-0198) and the NSW Node of the Australian National Fabrication Facil-

266 APPENDIX B ity. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office or the U.S. Gov- ernment. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. M.V. and B.H. acknowledges support from the Netherlands Organi- zation for Scientific Research (NWO) through a Rubicon Grant. K.M.I. ac- knowledges support from a Grant-in-Aid for Scientific Research by MEXT, NanoQuine, FIRST, and the JSPS Core-to-Core Program.

Author Contributions

M.A.F and B.H performed experiments. M.V designed the device, fabricated by K.W.C and F.E.H with A.S.D’s supervision. K.M.I prepared and supplied the 28Si epilayer. W.H, T.T, C.H.Y and A.L contributed to the preparation of experiments. M.A.F, B.H and A.S.D designed the experiments, with T.D.L, W.H, D.C, K.W.C, T.T, C.H.Y, A.L, A.M contributing to results discussion and interpretation. M.A.F., B.H. and A.S.D. wrote the manuscript with input from all co-authors.

267 Appendix C

Acknowledgements: Part 3

A logical qubit in a linear array of semicon- ductor quantum dots

The works presented in Ch. 7 was produced in collaboration with the author of this dissertation and Dr. C. Jones, Dr. A. Morello, Dr. M. F. Gyure, Dr. A. S. Dzurak, and Dr. T. D. Ladd, as listed in C. Jones et. al. arXiv preprint arXiv:1608.06335v2 (also referred to as Ref. [47]). The authors thank Menno Veldhorst, Austin Fowler, and Jason Petta for helpful insights and discussions. M.F., A.M. and A.S.D. acknowledge support from the Australian Research Council (CE11E0001017) and the US Army Research Office (W911NF-13-1-0024).

268 APPENDIX C

Author Contributions

M.A.F, with the assistance of A.S.D and A.M, contributed drafts of funda- mental control operations and experimental state-of-the-art sections as a fea- sibility study for the SiMOS quantum dot qubit platform. M.A.F and T.D.L worked jointly on simulations of controlled-phase gate implementations. Con- trol protocols, encoding schemes and logical qubit simulated performance were developed by C.J, M.F.G and T.D.L. C.J wrote the manuscript with input from all authors.

269 Appendix D

Lanau-Zener Transitions

This appendix is included for completeness and acts a reference to give more detail and motivation to the Landau-Zener problem as introduced in Chap- ter 1 (Section 1.4.2). The mathematical proofs in this appendix follow those presented in Refs. [64, 71].

D.1 Wave function in the diabatic picture

For a wavefunction defined by: ⎛ ⎞

⎜α1⎟ |ψ(t) = ⎝ ⎠ (D.1) α2

270 APPENDIX D the time dependent Schr¨odinger equation

∂ i α = H(t)α (D.2) ∂t n n for action under a varying Hamiltonian H(t) results in the Eigenbasis also becomes time dependent. Where the wave function in the adiabatic picture is then given by the solution of the time-dependent Schr¨odinger equation [110,

63]:  i t α = C exp − E (t)dt (D.3) n n  n t0 For the time being, set  = 1 and re-insert it in the final result. Placing these solutions in the time-dependent Schr¨odinger equation for the qubit Hamilto- nian HQ in Eq. (1.14) of Ch. 1 yields the following differential equations:  t ˙ C1 = −iΔC2 exp i δEdt  t0 t (D.4) ˙ ∗ C2 = −iΔ C1 exp −i δEdt t0

where δE =(E1 − E2) is the difference in Eigenenergies and the time differ- ential and substitution of the above equations yields differential equations of the second order: ¨ ˙ 2 C1 − iδEC1 + |Δ| C1 =0 (D.5) ¨ ˙ 2 C2 + iδEC2 + |Δ| C2 =0

Following the assumptions of Ref. [66], we assume that a passage through the anti-crossing is made at constant energy level velocity ν for the decoupled

271 APPENDIX D system (when Δ = 0), such that:

ν = δE/t (D.6)

Substituting into Eq. (D.5), this produces two differential equations:

¨ ˙ 2 C1 − iνtC1 + |Δ| C1 = 0 (D.7) and ¨ ˙ 2 C2 + iνtC2 + |Δ| C2 = 0 (D.8) where, in order to progress, we are required to discuss the initial conditions at t = −∞ before solving the problem at t →∞, where the interaction under Δ ceases [64].

D.2 Non-adiabatic transitions: the solution

when t →∞

The following solution is presented in greater detail in Ref. [64]. The boundary conditions for this problem are chosen at liberty. For this solution, we choose |C1(−∞)| =1andC2(−∞) = 0. Further, the initial conditions which are used to simplify the system of equations includes the fact thatc ˙n → 0ast →∞due to the interaction under Δ ceasing as the energy levels are increasingly detuned. However, it the final population of

272 APPENDIX D

f | |2 these states Cn which is to be our solution. Due to the fact that Cn =1 we are required to solve only one of these differential equations, therefore we

f choose to solve for the final state C1 . The differential equation of interest in Eq. (D.7) can be approached by multiplying through by the parameter dt and integrating over the entire tC1 range (−∞ to ∞), producing the following:

f C ∞ ∞ 1 dC dt dt C¨ (t) iν 1 = |Δ|2 1 (D.9) 1 C1 −∞ t −∞ t C1(t)

Here, the LHS of this equation is simply the logarithmic integration of C1,

f f where the limits include the final value C1 resulting in iν ln(C1 )fortheLHS of Eq. (D.9) [64]. The integrals on the RHS are both solved via the path integration method and the residual theorem. For the first integral in the RHS we employ a path

γ1 which consists of a semicircular region in the upper complex plane of radius R closed by the real axis. The integral is solved as follows:

∞ dt R dt = lim (D.10) t R→∞ t −∞ −R dt π iReiθdθ = lim − lim (D.11) R→∞ t R→∞ Reiθ γ1  0 1 π iReiθdθ =2πi Res − lim (D.12) →∞ iθ t R 0 Re = πi (D.13)

273 APPENDIX D

Here, Resf(t) is the residual of complex function f(t). The second integral can be solved in the same way:

∞ dt C¨ (t) R dt C¨ (t) 1 = lim 1 (D.14) t C (t) R→∞ t C (t) −∞ 1 −R 1 dt C¨ (t) π iReiθdθ C¨ (t) = lim 1 − lim 1 (D.15) →∞ →∞ iθ R γ t C1(t) R 0 Re C1(t) 1   1 C¨ (t) π C¨ (t) =2πi Res 1 − lim idθ 1 (D.16) →∞ t C1(t) R 0 C1(t)

For the residue, we take into account the location of the pole at t =0.From Eq. (D.7): C¨ (0) 1 = −|Δ|2. (D.17) C1(0)

Therefore,   1 C¨ (t) 2πi Res 1 = −2πi|Δ|2. (D.18) t C1(t)

¨ Further, as time t →∞the interaction turns off, meaning C1(t) → 0. Re- ˙ garding Eq. (D.7), this implies that tC1(t) approaches a constant, which ˙ ¨ 2 further implies the function of C1(t) ∝ 1/t, and therefore C1(t) ∝ 1/t . Hence: C¨ (t) 1 1 ∝ (D.19) 2 C1(t) t which results in the integral

π C¨ (t) 1 lim idθ 1 ∝ lim → 0. (D.20) →∞ →∞ 2 R 0 C1(t) R R

274 APPENDIX D

Therefore, the total solution to the Eq. (D.9) becomes:

f | |2 − | |2 iν ln(C1 )=πi Δ 2πi Δ (D.21) and therefore the final population after the interaction is given by:

 −π|Δ|2 Cf =exp (D.22) 1 ν

And therefore the state probability indicating we have maintained the origi- nal state after the interaction has ceased is given by:

 −2π|Δ|2 P = |α |2 =exp (D.23) 1 ν

This transition probability is often denoted PLZ and is known as the Landau- Zener probability. There are several other methods by which the Landau-Zener problem has been solved (see Refs. [64, 71] for discussion). Originally the works of L. D. Landau (Ref. [65] and C. Zener (Ref. [66]) are credited with the original publications, however the solution can also be found in the works of E. Majorana (ref. [67]) and E. K. G. St¨ueckelberg (Ref. [72]).

275 Appendix E

No-cloning theorem for quantum information

This appendix follows the arguments presented in Ref. [15] regarding the concept of cloning quantum information. Suppose we have two qubits Q1 and Q2 within a quantum device M.Q1 possesses an unknown pure qubit state |ψ, which is to be copied via some unitary process U onto the state of Q2. The target qubit Q2 is initialised in an arbitrary pure state |s, resulting in the initial state of M as

|ψ⊗|s . (E.1)

The copying process can be written as

|ψ⊗|s −→U U(|ψ⊗|s)=|ψ⊗|ψ . (E.2)

276 APPENDIX E

Suppose this process holds for another unknown pure qubit state |φ, This results in:

U(|ψ⊗|s)=|ψ⊗|ψ (E.3)

U(|φ⊗|s)=|φ⊗|φ . (E.4)

If we take the inner product of these two states, we get

φ|ψ =(φ|ψ)2 (E.5) which has only two solutions:

1. |ψ = |φ (E.6)

2. |ψ⊥|φ (E.7)

Therefore, the machine M can only clone states which are orthogonal to one another, leading to the conclusion that cloning of general quantum states is impossible.

277 Appendix F

The SiMOS fabrication process

The SiMOS devices studied in this thesis were fabricated at the NSW node of the Australian National Fabrication Facility (ANFF). The overall fabrication process for these devices is categorised into two sections: microfabrication stages attributed to substrate preparation, and nanofabrication stages which involve patterning the metallic electrodes which form the muli-level gated stack.

F.1 Microfabrication stages

Contemporary SiMOS devices employ a semiconductor substrate which con- sists of a natural silicon handle wafer capped with an 900 nm thick epitaxi- ally grown 28Si epilayer with background concentration of 800 ppm 29Si (See Ref. [175] for further details).

278 APPENDIX F

Surface oxides

Field oxides of approximately 200nm thickness are thermally grown in a wet-oxidation furnace (≈ 1000◦C). To mitigate the number of traps in the qubit region, a window is opened in the field oxide where a high quality

◦ thin-oxide (5 − 10nm) is grown in an ultra-dry furnace (800 C)withO2 and dichloroethylene (DCE) to reduce impurities.

Doped regions

High concentrations of ≈ 1×1020cm3 phosphorous dopant are used to diffuse n-type ohmic contact regions for source, drain and electron reservoirs. Re- gions of 1×1017cm3 boron dopant are used to produce “channel stoppers” to block any potential leakage paths between the ohmic regions. These channel stoppers border each individual device pixel, with separation channels run- ning from the border to the thin-oxide to create a separated region for each ohmic contact.

F.2 Nanofabrication stages

The nanofabrication process flow is centred around electron beam lithography (EBL) which is used to define the metallic electrodes of each device. The multi-layered gate stack technology relies on the self-limiting nature of the thermal oxidation of Al gates defined to form electrical isolation between each layer.

279 APPENDIX F

For each individual layer, the EBL process flow has six stages as follows:

1. EBL patterning

Electron beam lithography is a process in which patterns can be transferred onto a surface by exposing regions of a resist material with a focused electron beam. Polymethyl-methacrylate (PMMA), used as the resist material, is spun onto the chipset and cured through baking on a hotplate. The common thickness of the final resist layer is 150nm. Resist exposure is carried out in a Raith 150TWO EBL system located at ANFF-NSW. The exposure process is divided into two sub-processes, one which is optimised for fine features in the thin-oxide regions, and the other for rapid writing of large features including electrode fan-out and bond-pads. Due to the multi-layer nature of the devices, it is critical that each sub- sequent layer is precisely aligned to the last. This is achieved via the Raith 150TWO automatic write-field alignment function, where the machine scans for specially placed high-contrast Ti/Pt alignment markers.

2. Development

The pattern is developed by immersing the chipset in MIBK solution with applied ultrasonic agitation. This dissolves only the PMMA material exposed during the EBL stage, uncovering host wafer beneath.

280 APPENDIX F

3. Metal deposition

Gate electrodes employ a Lesker thermal evaporator with a rotating stage to uniformly deposit aluminium onto the chip. EBL alignment markers use a Lesker PVD75 e-beam evaporator to deposit, 150 A˚ of titanium followed by 650 A˚ of platinum.

4. Metal lift-off

The chipset is immersed for 2-3 hours in N-Methyl-2-pyrrolidone (NMP) solution at 80◦C, followed by ultrasonic agitation. The chipset is then rinsed in acetone, followed by isopropanol (IPA) and dried in an N2 environment.

5. Plasma ashing

In order to clean the surface of the chipset of any existing organic compounds (such as residual PMMA resist) an oxygen plasma is applied after each lift- off stage. This process is also undertaken before the first layer EBL stage to clean the host wafer.

6. Thermal Oxidation

Electrical insulation between each metallic layer is achieved through oxida- tion of the Al electrodes. The chipset is placed on a hotplate for 15 min at 150◦C to create an oxide of ∼ 5nm thickness on the exposed Al surface.

281 APPENDIX F

Forming gas anneal

After the final metal layer is deposited the chipset undergoes an anneal pro-

◦ cess for 25 minutes in forming gas (95%N2 :5%H2)at400 C. This addresses the problem of dangling bonds in the silicon-oxide, which can be produced during the EBL stages. These dangling bonds are terminated by the intro- duced hydrogen atoms to reduce the chance of forming charge traps near the qubit device.

Protective resist and storage

After the completion of the full fabrication procedure, a final layer of PMMA is spun onto the chipset to form a protective layer during storage. The chipset is deposited in a gel-pack before storage in a Nitrogen chamber.

282 Appendix G

Experimental equipment and protocols

This section details the general experimental methods, equipment and pro- cesses used to enable qubit experiments.

G.1 Device packaging

After the fabrication process is completed (details given in Appendix F), the chipset is diced using a dicing saw into either 2 × 2or2× 1 sample devices per single chip. These samples are then washed in an acetone bath, followed by IPA and dried in an N2 environment. The cleaned device is mounted in the RF enclosure with PMMA A5 resist used as an adhesive material. The RF enclosure seen in Fig. G.1 supports

283 APPENDIX G

Figure G.1: RF enclosure used to support the sample device. High bandwidth MMCX and SMA connectors are used for the device pins and microwave transmission line respectively. micro-miniature coaxial (MMCX) connectivity and an SMA connector for passing microwave pulses to the on-chip ESR line. Electrical connections between the bondpads of a singular sample device and the printed circuit board housed within the RF enclosure are made with an Kulicke & Soffa - 4523 Aluminium Wedge Bonder. During bonding and transport each MMCX port is grounded to the surrounding enclosure using shorted connectors. The SMA connector has attached an SMA 50Ω ground- ing termination.

284 APPENDIX G

G.2 Cryogenic equipment and measurements

There are two core pieces of cryogenics equipment which are used as platforms for device characterisation and measurement. These include the liquid helium dewar and the dilution refrigerator. The Liquid helium dewar acts as a cryogenic test-bench, where the functionality of each device can be verified before it is mounted in a dilution refrigerator.

G.2.1 Liquid helium dewar

Cryofab CMSH series liquid helium dewars are used as the first stage of sample testing. The devices are mounted to a specially designed dipping stick which houses the RF enclosure and provides external connectivity to the sample via a BNC-loom-MMCX wiring setup as shown in Fig. G.2. The dipping stick is inserted into the dewar and allows the enclosure to be in contact with liquid helium. The liquid helium temperature sits at the boiling point of 4K, and once the dipping stick system reaches a thermal equilibrium, the enclosure and sample are cooled to this temperature. The sample sitting at 4K is considered too hot to be capable of performing qubit experiments, however there are a large number of preliminary testing procedures which can be carried out at this temperature to ensure the sample functions correctly. The first stage of testing characterises the wafer stock and sample lithography/lift- off in the form of leakage tests. These functional tests include:

285 APPENDIX G

Figure G.2: Liquid helium dewar device dipping stick schematic. The dip- ping stick is used to control submersion of the RF enclosure in the helium dewar, while providing a means for electrical connection to room temperature measurement equipment.

Ohmic-to-substrate leakage tests

Ohmic-to-ohmic leakage tests

Ohmic-to-gate leakage tests

Gate-to-gate leakage tests.

These leakage tests are often conducted using a Kiethley 236 Source Mea- surement Unit (SMU). If the device successfully passes each of the above leakage tests, the next stage of testing involves ensuring correct functionality of the lithographic gates. These tests are:

SET sensor channel formation

286 APPENDIX G

ˆ SET barrier pinch-off

ˆ Observation of coulomb blockade

ˆ Gate electrode turn-on tests and capacitive lever arm assessment

ˆ Quantum dot stability diagrams.

Even though complete qubit control is difficult due to high operating temperatures, the 4K bath temperature is low enough to construct func- tional SET sensor devices and begin to assess electron tunnelling events. If the sample is able to successfully produce an SET sensor, the quantum dots themselves are assessed. If the sample demonstrates operation which is sta- ble enough to produce quantum dot stability diagrams illustrating electron occupancies down to the last electron, the device graduates to the next stage of testing.

G.2.2 Dilution refrigerator

If the sample passes the initial testing procedures at liquid helium temper- atures, the RF enclosure is transferred to an Oxford Instruments Kelvinox K100 dilution refrigerator system. The dilution refrigeration circuit [242] allows the device to be cooled down to a base temperature of 30mK where conditions are more favourable for the observation and coherent manipulation of quantum states. Built into the dilution refrigerator is a superconducting

z magnet, capable of producing stable magnetic fields of up to B0 = 5 Tesla.

287 APPENDIX G

The large amount of supporting hardware employed within these measure- ments are discussed in the next section.

G.3 Electronic hardware

An electrical connectivity diagram for the dilution refrigerator is illustrated in Fig. G.3, where the tables G.1&G.2 give the details of each piece of equip-

Computer Shielded Room Equipment Rack GPIB Card GPIB MW Source Magnet PS TRIG OUT GPIB

Oscilloscope GPIB Dot Lock-In Femto Amp CH1 CH2 X2 X1 EXT D C B A ECLK A B OUT OUT REF IN OUT OUT IN

GPIB Sensor Lock-In Fridge Insert CH1 CH2 A B OUT OUT REF IN OUT ESR DRAIN

FAST Pulse Blaster LINES GPIB CH1 CH2 CH3 CH4 SIM Module 1 SLOW SIM BUS LINES 8

GPIB SIM Module 2 SIM BUS FILTER J-FET ARB Studio 4 EXT TRIGGER IN OUT IN DPA DPB CLK OUT CH1 CH2 CH3 CH4 DIVIDER BLOCK

Figure G.3: Hardware connection diagram for equipment at room temper- ature, external to the dilution refrigerator. Diagram illustrates hardware installed within the measurement computer, hardware inside/outside EM shielded room and hardware installed in the equipment racks. ment. The equipment is separated into two key categories, those which are located outside the EM shielded room (Table G.1), and those which are lo-

288 APPENDIX G cated inside the shielded room (Table G.2). A wall panel with BNC connec- tivity is used to pass signals from the experimental setup to the measurement computer outside the EM shield.

Table G.1: Room temperature electronics used to support dilution refriger- ator measurements. These pieces of equipment are located outside the EM shielded room. Equipment Model Description Oscilloscope Alazar DSO v1.1.72 Digital oscilloscope internal to ATS9440-128M - measurement computer S940241 GPIB Card Aligent 82351A GPIB card internal to computer Pulse blaster SpinCore Used as a trigger source for all pulseblasterESR- downstream electronics PRO (dv2) ARB Studio LeCroy ArbStudio Arbitrary waveform generator for 1104 gate voltage pulsing (4 channels) Magnet PS Oxford instruments Power supply for single coil super- IPS 120-10 conducting magnet

A similar electronic setup is used for the liquid helium dipping station (Sec G.2.1), however only a measurement computer (with oscilloscope, pulse blaster and GPIB capability), Femto amplifier, divider block, SRS Lock-in and SIM components are used. There is no magnet capabilities, microwave pulsing or fast-gate pulsing (ARB studio) required. Further, the use of an EM shielded room is not utilised for measurements in the helium dewar.

289 APPENDIX G

Table G.2: Room temperature electronics used to support dilution refriger- ator measurements. These pieces of equipment are located inside the EM shielded room, with the dilution refrigerator.

Equipment Model Description MW Source Agilent E8257D Analogue signal generator (40GHz) Dot Lock-in Stanford Research Lock-in signal supplied to dot top Systems (SRS) 830 gate (see Ref. [167] for methods) Lock-in Amplifier Sensor Lock-in SRS 830 Lock-in Lock-in signal supplied to sensor Amplifier source SIM module SRS Small In- Platform for eight SRS instru- 1&2 strumentation ments which share a mainframe Modules (SIM) and computer interface. 900 Mainframe Voltage SRS SIM928 Iso- Clean 20 VDC, (10 mA) source Sources lated Voltage with millivolt resolution. The out- Source put circuit is optically isolated from earth-referenced charging cir- cuitry. JFET SRS SIM911 JFET Low-noise, programmable pream- & BJT preampli- plifier designed for small signal ap- fier plications. Filter SRS SIM965 Bessel Analogue High-pass/low-pass fil- & Butterworth fil- ter designed for signal condition- ter ing applications. Femto DLPCA-200 Variable gain, low noise tran- simpedance amplifier. Divider block In-house produced Voltage dividers/combiners for electronics stepping down VDC, combin- ing VDC + ARB signals, or VDC + SRS830 output.

290 Appendix H

Configuration regimes of Architecture 3

Acknowledgement: The device architecture presented in this appendix and its intended operation was theorised and designed in collaboration with Mr.A.West. The device was fabricated and imaged by Mr. A. West at ANFF-NSW, via the standard processes as presented in Appendix F.

As stated in Sec. 2.5.3, the third architecture presented in this thesis is intended to be configured into two vastly different operation regimes. These two regimes are definable by the number of 2DEG reservoirs tunnel coupled to the quantum dot region.

291 APPENDIX H

H.1 Single-reservoir configuration

The current standard configuration for the SiMOS quantum dot devices pre- sented from the research team of A. S. Dzurak has been a line of quantum dots which are coupled to a single electron reservoir. This style of architec- ture is presented in Refs [126, 42, 113, 149]. This is the natural configuration for the architecture illustrated in Fig H.1a, and is achieved by operating CB1, CB2 and B1 together in a regime of high electrostatic confinement.

Figure H.1: The accumulated 2DEG regions and dots are shown for the de- vice tuned to a) single-reservoir operation and b) dual-reservoir operation. Dashed lines indicate electrostatic isolation barriers between key ohmic re- gions connected to sensor, dot or fan-out 2DEGs.

As discussed briefly in Sec. 2.5.3, these gates are integral to the electro- static isolation of the ohmic regions and their extended 2DEGs. As illustrated by the dashed lines of Fig H.1a, there are three such regions:

1. The dot reservoir 2DEG extended under electrode R, isolated by the

292 APPENDIX H

combination of the three confinement gates CB1, CB2 and B1 (red region).

2. The sensor 2DEGs, source and drain, isolated from SG by the fan-out of B1. The source and drain are isolated in the conventional way, used in all previous device architectures (purple region).

3. The screening 2DEG, which extends from the SG ohmic region by the SG electrode, runs below the accumulation electrodes B2 and G1-G4. This 2DEG feeds the dynamic charging and discharging of the fan-out 2DEGs for dot defining gates during qubit experiments. This region is isolated from other ohmics on the chip through depletion of B1 and CB2 electrodes (green region).

H.2 Dual-reservoir configuration

Electron transport through a line of quantum dots separating a source and drain electron reservoir were some of the first experiments utilised to inves- tigate Pauli spin blockade effects (see Ref. [135] for the discussion in context of SiMOS quantum dots). In order to operate in this dual-reservoir regime, the device can be re-configured as shown in Fig H.1b. Here, the SET is bro- ken down into an electron reservoir on the source ohmic side by removing confinement due to SLB, while the remaining electrode SRB acts to isolate the drain ohmic.

293 APPENDIX H

The gate B1 is re-configured to act as a tunnel barrier for electrons be- tween the source reservoir and the dot under gate G1. A consequence of this is the connection of the 2DEG on the drain side of the device being con- nected to the fan-out 2DEGs, potentially creating a short between D and SG. To rectify this, the SG electrode potential must be set below the threshold voltage such that no 2DEG extends from the SG ohmic towards the device. The resulting configuration separates the experimental 2DEG reservoirs from the current transients experienced in the fan-out 2DEG through the red dashed isolation line shown in Fig H.1b, produced by a combination of SRB and CB2.

294 Appendix I

Additional device characterisation

This appendix holds additional characterisation details for the device pre- sentedinCh.4,5and6.

I.1 Device lever arms

To convert pulsed gate voltages on G1, G2 to detuning ε = μ2 −μ1,the(0, 2), (1, 1), (2, 0) part of the charge stability diagram in Fig. 4.3 is employed to model the double dot using the constant interaction model (Eq. (32) on page

1245 of Ref. [39]). Leaving the unknown average charging energy EC as a free parameter, quantum dot detuning ε is solved for self-consistently, where

 ∼ α2VG2 + α1VG1,withα2 = −6.1EC/V and α1 =3.7EC/V extracted.

295 APPENDIX I

I.2 Excited states

Typical charging energies for this device design is 10−20 meV, and a value of

EC = 10 meV is chosen for ε in these experiments. A transformation of the three level pulse seen in Fig. 4.2a of Ch. 4 is presented in Fig. I.1, allowing for the energy of the first excited state EES in G2 to be measured [133] with respect to EC.

0.2

(1,2)

) EVS2 C 0.1 ሻሺͳȀ

1 E (0,2) VS2 (1,1) ൅Ɋ 2 Ɋ 0.0 ጟ ൌǦሺ ȟ

(0,1)

-0.1 -0.1 0 -0.1 0.2

ԪൌɊ2ǦɊ1ሺͳȀC)

Figure I.1: Transformation of three level pulse sequence from Fig. 4.2a to illustrate detuning ε and average energy E¯. Valley splitting for dot G2can be measured with respect to charging energy EC via the splitting of standard- PSB and Latched-PSB. Image reproduced from Ref. [60]

This energy is given by the blockade width, measured as EES =(0.0172±

0.002)EC. These values are similar to valley splitting energy EVS measured

296 APPENDIX I on previous SiMOS devices [167, 59]. For the latched region the blockade is lifted via the presence of this same excited state, allowing the G1 electron to shuttle to G2, occupying this excited state. The physical detuning measured from truncation of the metastable re- gions corresponds directly to an excited state energy based on the lever arm analysis from the previous section. This is shown on the energy diagram of Fig. I.2.

ۄ-+Tm+,v(2,0)| ۄ-+Tm-,v(2,0)|

ۄ-+Sv(2,0)| ۄ-+Tm0,v(2,0)|

|(1,1)T

ۄ --m+,v --v ۄ T(1,1)| |(2,0)S m-,v-- ۄ Energy

ԪVS(QD2) ۄ--Tm0,v(1,1)| ۄ--Sv(1,1)| 0 Detuning Ԫ

Figure I.2: Energy diagram showing the first excited state in parameter and energy space. This includes the state mixture which allows for the lifting of the metastable charge state and the truncation of the PSB window.

This energy diagram follows from research into the valley degrees of free- dom in silicon based devices presented in Refs. [170, 243].

297 Appendix J

Chronological experimental pathway for |Δ(θ)|

As mentioned in the main text, the experimental pathway to achieve the fittings for the single passage LZ measurements was convoluted, with esti- mations of |Δ(θ)| feeding forward into pulse sequence designs for experiments which were able to provide results measuring independent Hamiltonian pa- rameters. These results produced the Hamiltonian models presented in Ch. 6. This model was then used to feed back into more accurate measures of |Δ(θ)|. This was achievable based on the triplet population with respect to the voltage rate of change being relatively independent of the magnitude of the external magnetic fields. The reasoning for this is complicated, and is the focus of future studies, but is directly observable from Fig. J.1. Taking this fact into account, despite not knowing an accurate value for

298 APPENDIX J

z B0= 155mT z

T 0.8 B0= 0mT P

0.6

0.4 Triplet Probability Triplet 0.2 101 10 3 105 Voltage ramp rate (V/s) Figure J.1: Results of the single passage experiment including curve fittings which illustrate little alteration to triplet population with respect to the voltage rate of change, despite drastic change in magnetic fields.

|Δ(θ)|, the ramp rate with respect to the control voltages (and therefore ε) required to produce semi-adiabatic and diabatic passages through the S/T− anti-crossing could be deduced and were taken into account during the pulse design phase of proceeding experiments.

299 Appendix K

z Fitting Bos from the spin funnel experiment

As stated in Ch. 5.4.2, the offset in the applied field of the solenoid can further be deduced from this spin funnel experiment, however, this process requires more sophisticated models for exchange as presented in Ch. 6.4. The model for tunnel coupling presented in Eq. (6.1) produces a phe- nomenological fit to the exchange coupling motivated by simulation of de- vice electrostatics[134, 180]. This fit is employed in Ch. 6.4 to describe the system Hamiltonian of Eq. (1.108), and it is desired to use the model here for accuracy and consistency.

300 APPENDIX K

The exchange coupling is given by the following models:

1 1 J(ε)=− ε + ε2 +4t2(ε)(K.1) 2 2 c  tc(ε)=Λ exp(2ϕ(ε)) + 1 − exp(ϕ(ε)) (K.2)

ϕ(ε)=ε/ζ (K.3) from Ch. 1.7.4, Eq. (6.1) and Ch. 6.4. In order to accurately fit the funnel field offset, both the positive and negative field data must play a role. Therefore, finding and fitting the inverse function of the exchange model to the data is used. This inverse function of exchange results in the following functional form

εfit = f(Bfit), (K.4)

where f(Bfit) becomes quite complicated due to the non-trivial form of the tunnel coupling as presented above. An alternative form for the tunnel coupling model is given by the standard WKB solution due to the simple form of ϕ(ε)[180]. This model is

tc(ε)=t0 exp(−ϕ(ε)), (K.5)

√ where t0 =( 2−1)Λ. Note, this simplification does not hold for ϕ(ε)=K1 + K2ε as presented in Refs. [134, 180].

301 APPENDIX K

Making the substitution |B| = J, the inverse function, which outputs a detuning in ε, is then found to be

  t2 |B| f (B,t ,ζ)=ζW 0 exp −|B|, (K.6) ε 0 |B|ζ ζ where W (...) is the Lambert-W function.

20

10 (mT) z 0 B 0

-10 Applied field

-20-0.4 -0.2 0 0.2 0.4 0.6 Detuning ፴ (meV) Figure K.1: Differentiated spin funnel map (original shown in Fig. 5.4) where the SH /T− anti-crossing location is extracted and shown in red. The inverse z function for B0 (ε) (degenerate with exchange energy J(ε)) is employed to fit z the tunnel coupling model and offset magnetic field BOS

In order to employ this function as a fitting tool, the exchange must be ex- tracted from the funnel. As shown in Fig. K.1, the exchange can be extracted by fitting a Gaussian envelope to the primary peak of the differentiated spin funnel data of Fig. 5.4. These extracted data points are shown in red. The fitting results of these data points to the function in Eq. (K.6) is

302 APPENDIX K shown in blue, which is a single fitting result which contains both the positive

z and negative B0 field data. The fitting results are summarised in Table K.1.

Table K.1: Key fitting results from spin funnel analysis including confidence intervals. Fit Parameter Fit Result 95% confidence interval

t0 4.12 GHz ±0.09 GHz ζ 7.28 meV ±0.66 meV z ± Bos -1.04 mT 0.06 mT

303 Appendix L

Additional spin-funnel data

As stated in Ch. 6.4.2, the spin-funnel data used for Hamiltonian parameter fitting is presented in Fig. 6.5 was collected at a different instance compared to the data presented in Fig. 5.4. All data collated in Fig. 6.5 was taken during the same device cool down, using the same experimental set-up and dilution refrigerator for all experiments. This data is shown in Fig. L.1. The device was transported to a different dilution refrigeration unit (which possessed a magnet capable of reversing polarity), where the full spin funnel data as illustrated in Fig. 5.4, and the single passage LZ data from Fig. 5.2b for both low and high magnetic fields were collected.

304 APPENDIX L

0

0.1 (meV) Ԫ 0.2

Detuning 0.3 0.0 0.1 0.2 PT 0246 z Applied Field BͲ (mT)

Figure L.1: The spin-funnel used for data collation in Fig. 6.5 in Sec. 6.4.2. Data is taken for the same device cooldown.

305 Appendix M

Instruction set and design rules for linear nearest-neighbour error correction

Ref. [47] introduces a tile formalism as a strategy for building a logical qubit using nearest-neighbor gates in a linear array of qubits. This formalism is based on a set of “design rules” that prevent some of the pathological errors that can occur in LNN circuits. The instruction set is restricted to a small set, called the “standard-LNN,” set shown in Fig. M.1. This set of instructions is closely related to CSS codes [120, 203], as standard-LNN instructions are sufficient to encode, decode, and detect er- rors for any CSS code. Further, any standard-LNN encoded gate can be constructed solely from standard-LNN instructions, making this set a natu-

306 APPENDIX M

Figure M.1: Standard-LNN instructions and their circuit-diagram symbols. All of the instructions are available at the hardware level using tick-tock control. The five two-qubit gates are all combinations of CNOT gates. The instruction “free” differs from “idle” in that a free qubit (temporarily) has undefined state and carries no information, whereas an idle qubit does carry information. At the hardware level, the distinction may only be a labelling, but the instructions will have different encoded representations after code concatenation. This figure is adopted from Ref. [47]. ral choice when using code concatenation. The following circuit design rules are used to restrict the possible error events that can occur:

1. Only standard-LNN instructions at tier L − 1 are used to encode all standard-LNN instructions at tier L, instigated at the first tier, which includes the tick-tock control.

2. Two-qubit gates are never to be performed between two data qubits within the same code block. Allowable pairs for two-qubit gates are a data qubit and ancilla or two data qubits from different blocks. The

307 APPENDIX M

exception is encoding or decoding a single block.

3. Codes with weight-two stabilisers use a single ancilla qubit for stabiliser measurement. Here, a single failure causes at most one data error.

The first rule is motivated by the observation that the standard-LNN set is directly related to concatenation of CSS codes. The second rule prevents a single gate failure from introducing a weight-two error into a single code block, while the final rule similarly ensures that a single failure in the syn- drome extraction circuit will introduce at most one error into a data block. For more detailed discussion, see Ref. [47] or Sec. 1.8.4.

308 Appendix N

Encoding and concatenation schemes for linear nearest-neighbour error correction

The full encoding scheme is described in Ref. [47], but is summarised here with a focus on the two-qubit repetition code. This code is a simple codes that satisfy the design rules, but can also be concatenated to increase code dis- tance. Implementing the standard-LNN instruction set makes the presented figures interchangeable between different layers of concatenation. The en- coded logic gates are grouped into blocks of instructions called “tiles”, which provide a simple scheme for scheduling instructions to operate a logical qubit

309 APPENDIX N as shown in Fig. N.1 & 7.10. Here, the two-dimensional quantum circuit shows qubits spanning the vertical dimension, while time flows to the right. The tile size for a given code is set by the most complex circuit for an encoded standard-LNN instruction. This is the implementation of a two- qubit gate, where the CNOT tile for the two-qubit, bit-flip code is shown in Fig. 7.10, labelled blocks 5 & 6. In the tile formalism, a qubit is measured and prepared in the the same time as allotted for a two-qubit gate. Since the measure-and-prepare joint instruction acts on one encoded block, it occupies

Figure N.1: Half tiles for encoded measurement, preparation and state injec- tion for the two-qubit code. Dashed lines in these figures denote a qubit that is free, meaning its state is unimportant (and unencoded at lower layers). a) Measurement and preparation in Z basis for the bit-flip code. b) measure- ment and preparation in X basis√ for the bit-flip code. This requires a CNOT for encoding a |+ =(|0 + |1)/ 2 state. c) State injection for the bit-flip code. d) State injection for the phase-flip code. This figure is adopted from Ref. [47].

310 APPENDIX N a half-tile, as shown in Fig. N.1a&b. The half-tiles in this figure can be further decomposed into quarter tiles as seen bordering Fig. 7.10. Each half-tile instruction must be matched with another half tile to form a complete diamond-shaped tile. The bottom half of this full diamond is the horizontal mirror image of a half-tile shown in Fig. N.1. This figure shows measurement and preparation in the same basis, but measurement in X basis combined with prepare in Z basis can be achieved by combining the appropriate operations (See Ref. [47] for details). Fig. 7.10 illustrates the interlocking tiles are a simple, yet effective tool for instruction scheduling. Each tile is a sub-circuit consisting of nearest- neighbour gates on a small set of adjacent data qubits and syndrome ancillas. The tiles naturally implement code concatenation by recursively building tiles at tier L from tiles at tier L − 1. The tiles fit together perfectly in space and time, so they provide a simple method to efficiently construct concatenated LNN circuits. The tiles bring syndrome ancillas into contact with all data qubits for error detection, with each tile moving the ancilla qubit(s) across a code block, opening the other side for an interleaved two- qubit gate. Represented in Fig. 7.10 is the concatenation of a phase-flip code on top of a bit-flip code. Starting with a phase-flip encoded idle (upper left in figure), each instruction is replaced with its appropriate tile in the bit-flip code. Further details such as expanding protocols to three- and four-qubit encoding schemes are given in Ref. [47]

311 Appendix O

Simulated logical qubit performance of a linear array of SiMOS quantum dots

As discussed in Sec. 1.8.4 it was identified that the threshold theory for func- tional logical qubits depends upon the likelihood of errors and how effectively they are corrected. This section summarises the simulation methods and re- sults presented in Ref. [47] used for determining these error thresholds, and act as a guide for which spin-control operations require further improvement in fidelity. The simulations use a simplified error model consisting of independent Pauli errors applied after every operation, following common conventions shown in literature [244, 245, 246, 247, 248, 233, 249]. In order to gain a

312 APPENDIX O threshold for error correction, the simulation of logical error rates for several different code distances must be undertaken. Concatenation is necessary for distance-two codes like the two-qubit and four-qubit codes, only errors in a single layer of encoding can be detected [244, 250, 251]. For the simulations, a logical CNOT gate is implemented in alternating layers of bit-flip and phase-flip concatenation. Each simulation inserts ran- domly generated Pauli errors into the circuit for an encoded CNOT at one to four layers of concatenation (two-qubit code) and one to two layers (four- qubit code). The error model is depolarising noise (see Sec. 1.5.3) following every gate (or randomly negating a measurement). Two methods of estimating logical failure rates are employed, Monte- Carlo sampling and malignant-set sampling [208, 248]. Both of these are shown in Fig. 7.11 in the main text and in Fig. O.1. For Monte-Carlo sampling, errors are generated independently for each gate according to a physical error parameter and compared to the number of logical failures. In malignant-set sampling, configurations of k errors are created, where the fraction of configurations leading to logical failure are counted. The logical failure rate is then given by

N P(Fail) = P(Fail|k errors)P(k errors), (O.1) k=1 where N is the number of gates. Since each gate has an error with the same

313 APPENDIX O

Figure O.1: Simulated logical error rates for the four-qubit code using concatenation. As in Fig. 7.11, both Monte Carlo error generation and malignant-set counting are employed. This figure is adopted from Ref. [47]. probability p, the second term is simply the Bernoulli distribution, ⎛ ⎞ ⎜ ⎟ ⎜N⎟ ⎜ ⎟ k N−k P(k errors) = ⎜ ⎟ p (1 − p) . (O.2) ⎝ ⎠ k

The simulation results in Fig. 7.11 suggest a threshold for the two-qubit code around 10−4. The crossing of the level-one and level-four logical error rates occurs at p =9.5 × 10−5, while the crossing of level-two and level-four curves occurs at p =3.1 × 10−4. These simulation results are compared to that of the four-qubit code (de- tails for encoding schemes given in Ref. [47]). Shown in Fig. O.1 is a crossing

314 APPENDIX O of the logical-error-rate curves for layers one and two of concatenation at 3.8 × 10−4. These results are consistent with Fig. 7.11, suggesting that a logical qubit in an LNN architecture would require error rates around 10−4.Intermsof the resources for the encoded CNOTs for each; the two-qubit code requires four layers of concatenation to correct a single fault, while the four-qubit codes only require two layers. This amounts to the concatenated four-qubit code requiring only 20% of the number of gates compared to a CNOT in the concatenated two-qubit code, while the error threshold given by each is very similar. The simulations presented here provide control fidelity targets for exper- iments to demonstrate a “signature” of error correction. This is the char- acteristic quadratic dependence of logical error rate with physical error rate when any single error is correctable. The results shown in Fig. 7.11 & O.1 demonstrate that this signature can be detected even at error rates above threshold (up to 10−3 or higher), which allows an experiment to demonstrate the functionality of error correction by synthetically injecting errors.

315 Appendix P

Experimental pathway to a logical qubit

The components of the error correction scheme can be demonstrated in in- termediate proof-of-concept experiments, as has been done in other qubit technologies [210, 183, 185, 211, 212, 186, 187, 188, 116]. This section sum- marises the sequence of experiments for developing a logical qubit in quan- tum dots and follows those which are presented in Fig. 7.12. Each of the experiments outlined in this figure are summarised in the following sections.

P.1 Parity Measurements

This parity measurement was first introduced in Sec. 7.2.4, and is illustrated in Fig. 7.9. This experiment requires four qubits to implement a two-qubit

316 APPENDIX P code where a two-qubit ancilla detects either one bit-flip or one phase-flip error (depending on choice of encoding). This type of experiment can be supported on the architectures introduced in Sec. 2.5, using non-extensible control techniques such as single qubit ESR. The functionality of such an experiment can be assessed through error injection into data qubits [210, 185, 211, 212, 186, 187], where there are three indicators which show the experiment is working. These are:

1. Injected error increases the number of parity flips measured.

2. Measuring individual data qubits should correlate with parity measure- ments [189].

3. For the bit-flip code, the probability of measuring |11 after initialising |00 should be suppressed when no parity flips are detected (this process requires two independent bit-flips to occur).

The third method of verification requires post-selection of measured data (when no parity flip is seen) and is useful for other experiments discussed here. Through initialising spins which are sensitive to (for example) bit-flips, this result is a signature for the correction of that type of error.

P.2 Correcting one type of error

Extending the above system to include an additional data qubit (now five dots in total, see Fig. 7.12), allows for the three-qubit repetition code to be

317 APPENDIX P executed. Depending upon encoding, this experiment is capable of detecting a single bit-flip (phase-flip) error. This is achieved through the execution of two parity measurements on the single ancilla, sequentially assessing the

Z1Z2 (X1X2)andZ2Z3 (X2X3) stabilisers as illustrated in Fig P.1.

Figure P.1: Idle half tile for the three-qubit repetition code with phase-flip error detection. The lateral compression is denoted by the curved white slash. The left side of the tile has either idle or free segments that pad width to match the CNOT tile. This figure is adopted from Ref. [47].

The three-qubit repetition code demonstrates the “error threshold” cri- terion through performing functional error correction when these errors are purposefully inserted into the data qubits. The three-qubit-code experiment incorporates parity measurement as a subroutine, required to measure each of the two stabilisers twice. The duplication of the stabiliser is required in order to catch an instance where two errors are emitted by a SWAP gate, which can be propagated through a transversal CNOT following that SWAP operation (see “Appendix A” of Ref. [47] for further details).

318 APPENDIX P

P.3 Correcting a single error of any type

There are a number of experiments which fall under this category, each fo- cussing on different elements of the criteria of an extensible logical qubit. The common theme of these experiments is the ability to detect a either a single bit-flip or phase-flip error, thus satisfying the “fault tolerance” criterion of an extensible logical qubit.

Minimal four-qubit code The smallest demonstration of detecting any single-qubit error in the data qubits is a four-qubit code with a single ancilla (for a total of six dots as shown in Fig. 7.12). The resulting tiles are enlarged in order to reuse a single ancilla to measure two stabiliser generators.

Four-qubit code with parallel measurement

Expanding upon this experiment is a more compact tile formalism of the four-qubit code (presented in Ref. [47]) which employs two ancillas (for a total of eight dots). The additional ancilla qubit facilitates the “parallel measurement” criteria for the detection of a single error of any type.

Concatenated error detection

Using twelve qubits the distance-two bit-flip and phase-flip codes can be con- catenated together, as illustrated in Fig. 7.10. Under concatenation, this ex-

319 APPENDIX P periment can be thought of as three copies of the parity experiment (Fig. 7.9) integrated together. The 12-dot experiment demonstrates two criteria for ex- tensibility: concatenation and measurement parallelism. The code can detect at least one error of any type, which could realise error correction if the code were concatenated again to distance four.

P.4 Demonstrating a LNN logical qubit

The logical qubit demonstrations proposed in Ref. [47] are based on the nine- qubit Shor code [119, 7]. The physical implementation is the concatenation of three-qubit bit-flip and phase-flip codes.

Minimal logical qubit This minimal design is a compression into a total of fourteen dots, comprised of nine data spins, three auxiliary data spins (used as an encoded block for measurement of the second-level syndrome), and one two-spin ancilla to measure the first-level syndrome in all four blocks. The number of physical qubits (13 - two of which are singlet-triplet qubits) is the same as the smallest surface code [50, 252, 253]. The additional idle times and SWAP operations required to move the single ancilla along the chain will penalise the code performance, however the signature of error detection can be detected in the syndrome measurements even above the error-correction threshold [210, 183, 211, 212, 186, 187].

320 APPENDIX P

This experiment is capable of demonstrating the incorporation of three of the four extensibility criteria in a single logical qubit, including syndrome measurement with an ancilla, code concatenation, and fault tolerance.

Logical qubit with parallel measurement The final experiment discussed integrates the fourth and final criterion; mea- surement parallelism. This is another implementation of the nine-qubit Shor code, however the additional ancillas increase the total number of dots to twenty. Four copies of the 5-dot experiment (which detects one error type) are integrated together in a standard implementation of code concatenation, where each block has a dedicated measurement ancilla.

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