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Quantum Algorithms, Lower Bounds, and Time-Space Tradeoffs

Spalek, R.

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Citation for published version (APA): Spalek, R. (2006). Quantum Algorithms, Lower Bounds, and Time-Space Tradeoffs. ILLC.

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Quantum Algorithms, Lower Bounds,

and Time-Space Tradeoffs

Robert Špalek

Quantum Algorithms, Lower Bounds,

and Time-Space Tradeoffs

ILLC Dissertation Series DS-2006-04

For further information about ILLC-publications, please contact

Institute for Logic, Language and Computation Universiteit van Amsterdam Plantage Muidergracht 24

1018 TV Amsterdam phone: +31-20-525 6051 fax: +31-20-525 5206

e-mail: illc@science.uva.nl homepage: http://www.illc.uva.nl/

Quantum Algorithms, Lower Bounds,

and Time-Space Tradeoffs

Academisch Proefschrift

ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam

op gezag van de Rector Magnificus prof.mr. P.F. van der Heijden

ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar

te verdedigen in de Aula der Universiteit op donderdag 7 september 2006, te 12.00 uur

door

Robert Špalek

geboren te Žilina, Slowakije.

Promotiecommissie:

Promotor: Prof.dr. H.M. Buhrman

Co-promotor: Dr. R. de Wolf

Overige leden: Dr. P. Pudlák Prof.dr. A. Schrijver Prof.dr. M. Szegedy Prof.dr.ir. P.M.B. Vitányi

Faculteit der Natuurwetenschappen, Wiskunde en Informatica

Copyright c© 2006 by Robert Špalek

Cover design by Roman Kozub́ık. Printed and bound by PrintPartners Ipskamp.

ISBN-10: 90–5776–155–6 ISBN-13: 978–90–5776–155–3

The highest technique is to have no technique.

Bruce Lee

v

Contents

Acknowledgments xiii

1 Quantum Computation 1 1.1 Postulates of quantum mechanics . . . . . . . . . . . . . . . . . . 5

1.1.1 State space . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.2 Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.3 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.1.4 Density operator formalism . . . . . . . . . . . . . . . . . 12

1.2 Models of computation . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.1 Quantum circuits . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.2 Quantum query complexity . . . . . . . . . . . . . . . . . 17

1.3 Quantum search . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3.1 Searching ones in a bit string . . . . . . . . . . . . . . . . 20 1.3.2 Searching an unstructured database . . . . . . . . . . . . . 26 1.3.3 Amplitude amplification . . . . . . . . . . . . . . . . . . . 26

1.4 Quantum random walks . . . . . . . . . . . . . . . . . . . . . . . 28 1.4.1 Element distinctness . . . . . . . . . . . . . . . . . . . . . 28 1.4.2 Walking on general graphs . . . . . . . . . . . . . . . . . . 32

1.5 Quantum counting . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.5.1 General counting algorithm . . . . . . . . . . . . . . . . . 37 1.5.2 Estimating the Hamming weight of a string . . . . . . . . 39

1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2 Quantum Lower Bounds 41 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2 Distinguishing hard inputs . . . . . . . . . . . . . . . . . . . . . . 42 2.3 Adversary lower bounds . . . . . . . . . . . . . . . . . . . . . . . 44 2.4 Applying the spectral method . . . . . . . . . . . . . . . . . . . . 46 2.5 Limitations of the spectral method . . . . . . . . . . . . . . . . . 49

vii

2.6 Polynomial lower bounds . . . . . . . . . . . . . . . . . . . . . . . 51 2.7 Applying the polynomial method . . . . . . . . . . . . . . . . . . 53 2.8 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

I Algorithms 59

3 Matching and Network Flows 61 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.3 Finding a layered subgraph . . . . . . . . . . . . . . . . . . . . . . 64 3.4 Bipartite matching . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.5 Non-bipartite matching . . . . . . . . . . . . . . . . . . . . . . . . 67 3.6 Integer network flows . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4 Matrix Verification 75 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2 Previous algorithm for matrix verification . . . . . . . . . . . . . . 76 4.3 Algorithm for matrix verification . . . . . . . . . . . . . . . . . . 78 4.4 Analysis of the algorithm . . . . . . . . . . . . . . . . . . . . . . . 78

4.4.1 Multiplication by random vectors . . . . . . . . . . . . . . 78 4.4.2 Analysis of Matrix Verification . . . . . . . . . . . . . . . . 82 4.4.3 Comparison with other quantum walk algorithms . . . . . 84

4.5 Fraction of marked pairs . . . . . . . . . . . . . . . . . . . . . . . 84 4.5.1 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.5.2 Proof of the main lemma . . . . . . . . . . . . . . . . . . . 87 4.5.3 The bound is tight . . . . . . . . . . . . . . . . . . . . . . 88

4.6 Algorithm for matrix multiplication . . . . . . . . . . . . . . . . . 88 4.7 Boolean matrix verification . . . . . . . . . . . . . . . . . . . . . . 92 4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

II Lower Bounds 93

5 Adversary Lower Bounds 95 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.3 Equivalence of adversary bounds . . . . . . . . . . . . . . . . . . . 101

5.3.1 Equivalence of spectral and weighted adversary . . . . . . 104 5.3.2 Equivalence of primal and dual adversary bounds . . . . . 109 5.3.3 Equivalence of minimax and Kolmogorov adversary . . . . 112

viii

5.4 Limitation of adversary bounds . . . . . . . . . . . . . . . . . . . 112 5.5 Composition of adversary bounds . . . . . . . . . . . . . . . . . . 114

5.5.1 Adversary bound with costs . . . . . . . . . . . . . . . . . 114 5.5.2 Spectral norm of a composite spectral matrix . . . . . . . 115 5.5.3 Composition properties . . . . . . . . . . . . . . . . . . . . 118

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6 Direct Product Theorems 123 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.2 Classical DPT for OR . . . . . . . . . . . . . . . . . . . . . . . . 128

6.2.1 Non-adaptive algorithms . . . . . . . . . . . . . . . . . . . 128 6.2.2 Adaptive algorithms . . . . . . . . . . . . . . . . . . . . . 129 6.2.3 A bound for the parity of the outcomes . . . . . . . . . . . 130 6.2.4 A bound for all functions . . . . . . . . . . . . . . . . . . . 130

6.3 Quantum DPT for OR . . . . . . . . . . . . . . . . . . . . . . . . 131 6.3.1 Bounds on polynomials . . . . . . . . . . . . . . . . . . . . 131 6.3.2 Consequences for quantum algorithms . . . . . . . . . . . 135 6.3.3 A bound for all functions . . . . . . . . . . . . . . . . . . . 139

6.4 Quantum DPT for Disjointness . . . . . . . . . . . . . . . . . . . 139 6.4.1 Razborov’s technique . . . . . . . . . . . . . . . . . . . . . 139 6.4.2 Consequences for quantum protocols . . . . . . . . . . . . 141

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7 A New Adversary Method 143 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.2 Quantum DPT for symmetric functions . . . . . . . . . . . . . . . 145 7.3 Measurement in bad subspaces . . . . . . . . . . . . . . . . . . . . 149 7.4 Total success probability . . . . . . . . . . . . . . . . . . . . . . . 152 7.5 Subspaces when asking one query . . . . . . . . . . . . . . . . . . 153 7.6 Norms of projected basis states . . . . . . . . . . . . . . . . . . . 160 7.7 Change of the potential function . . . . . . . . . . . . . . . . . . . 164 7.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

8 Time-Space Tradeoffs 167 8.1 Introduction .