Quantum Machine Learning using the ZXW-Calculus Mark Koch

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Quantum Machine Learning

using the ZXW-Calculus

Mark Koch

Lady Margaret Hall

University of Oxford

A thesis submitted for the degree of

Master of Science in Advanced Computer Science

Trinity 2022

arXiv:2210.11523v1 [quant-ph] 18 Oct 2022

Word count: 15,793

Diagram count: 806

The word count was calculated using texcount via perl texcount.pl

-1 thesis.tex. Note that in diagram equations, each step is counted

as a separate diagram.

Abstract

The ﬁeld of quantum machine learning (QML) explores how quantum computers can

be used to more eﬃciently solve machine learning problems. As an application of

hybrid quantum-classical algorithms, it promises a potential quantum advantages in

the near term. In this thesis, we use the ZXW-calculus to diagrammatically analyse

two key problems that QML applications face.

First, we discuss algorithms to compute gradients on quantum hardware that are

needed to perform gradient-based optimisation for QML. Concretely, we give new

diagrammatic proofs of the common 2- and 4-term parameter shift rules used in the

literature. Additionally, we derive a novel, generalised parameter shift rule with 2n

terms that is applicable to gates that can be represented with nparametrised spiders

in the ZXW-calculus. Furthermore, to the best of our knowledge, we give the ﬁrst

proof of a conjecture by Anselmetti et al. by proving a no-go theorem ruling out

more eﬃcient alternatives to the 4-term shift rule.

Secondly, we analyse the gradient landscape of quantum ans¨atze for barren plateaus

using both empirical and analytical techniques. Concretely, we develop a tool that

automatically calculates the variance of gradients and use it to detect likely barren

plateaus in commonly used quantum ans¨atze. Furthermore, we formally prove the

existence or absence of barren plateaus for a selection of ans¨atze using diagrammatic

techniques from the ZXW-calculus.

Acknowledgements

First and foremost, I would like to thank my advisors Quanlong Wang and Richie

Yeung for their invaluable support and guidance throughout the writing of this

thesis. I am very grateful for their advice and many helpful discussions and ideas.

I would also like to thank Aleks Kissinger, as well as John van de Wetering and

Stephano Gogioso for sparking my interest in quantum computing and the ZX-

calculus through their lectures. In particular, I am thankful for the opportunity to

write this thesis under Aleks’ supervision.

Furthermore, I am very grateful to my family and friends both in Germany and

Oxford, who supported me throughout my studies. Together with the academic

community at my wonderful college Lady Margaret Hall, they provided a great

intellectual atmosphere that made the past year a truly unique experience. In par-

ticular, I would like to thank Nikhil Khatri for many inspiring discussions and for

proofreading this thesis.

Finally, I would like to thank the German Academic Exchange Service (DAAD) for

ﬁnancially supporting me during this year at Oxford.

Contents

Abstract iii

Acknowledgements iv

1 Introduction 1

1.1 MainContributions............................ 3

1.2 Structure of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Background 5

2.1 An Introduction to Quantum Theory . . . . . . . . . . . . . . . . . . 5

2.1.1 States ............................... 5

2.1.2 Unitary Evolution . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.3 Measurements .......................... 8

2.1.4 The Quantum Circuit Model . . . . . . . . . . . . . . . . . . 10

2.2 Quantum Machine Learning . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 TypesofAns¨atze......................... 12

2.2.2 Gradient-Based Optimisation . . . . . . . . . . . . . . . . . . 13

2.3 TheZXW-Calculus............................ 14

2.3.1 Generators and String Diagrams . . . . . . . . . . . . . . . . 15

2.3.2 Additional Notation . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.3 Rules ............................... 17

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QuantumMachineLearningusingtheZXW-CalculusMarkKochLadyMargaretHallUniversityofOxfordAthesissubmittedforthedegreeofMasterofScienceinAdvancedComputerScienceTrinity2022Wordcount:15;793Diagramcount:806Thewordcountwascalculatedusingtexcountviaperltexcount.pl-1thesis.tex.Notethatindiagramequations,eachste...

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Quantum Machine Learning using the ZXW-Calculus Mark Koch

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