Quantum Algorithms, Lower Bounds, and Time-Space Tradeoffs Robert Spalekˇ 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: [email protected] 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 Spalekˇ geboren te Zilina,ˇ Slowakije. Promotiecommissie: Promotor: Prof.dr. H.M. Buhrman Co-promotor: Dr. R. de Wolf Overige leden: Dr. P. Pudl´ak Prof.dr. A. Schrijver Prof.dr. M. Szegedy Prof.dr.ir. P.M.B. Vit´anyi Faculteit der Natuurwetenschappen, Wiskunde en Informatica Copyright c 2006 by Robert Spalekˇ Cover design by Roman Kozub´ık. 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 .............................. 167 8.2 Preliminaries ............................. 170 8.3 Time-space tradeoff for sorting .................... 171 8.4 Time-space tradeoffs for matrix products .............. 172 8.4.1 Construction of a hard matrix ................ 172 8.4.2 Boolean matrix products ................... 173 8.5 Communication-space tradeoffs ................... 175 8.6 Time-space tradeoff for linear inequalities .............. 178 8.6.1 Classical algorithm ...................... 178 ix 8.6.2 Quantum algorithm ...................... 178 8.6.3 Matching quantum lower bound ............... 181 8.7 Summary ............................... 183 Bibliography 185 List of symbols 199 Samenvatting 201 Abstract 205 x List of Figures 1.1 Double-slit experiment ........................ 3 1.2 Classical circuit computing the parity of 4 bits ........... 15 1.3 Quantum circuit computing the parity of n bits .......... 17 1.4 Quantum algorithm Grover search [Gro96] ........... 21 1.5 The Grover iteration G drawn as a rotation in two dimensions .. 22 1.6 Quantum algorithm Generalized Grover search [BBHT98] . 23 1.7 Quantum algorithm Older Element distinctness [BDH+01] . 28 1.8 Quantum algorithm Element distinctness [Amb04] ...... 29 1.9 Quantum algorithm Decide Marked Vertices [Sze04] ..... 33 1.10 Quantum algorithm Quantum Counting [BHMT02] ...... 37 3.1 Quantum algorithm Find Layered Subgraph .......... 64 3.2 Classical algorithm [HK73] Find Bipartite Matching ..... 66 3.3 Vertex-disjoint augmenting paths in an example layered graph .. 66 3.4 Classical algorithm [Edm65, Gab76] Find Augmenting Path . 69 3.5 Progress of Find Augmenting Path on an example graph ... 70 3.6 Quantum algorithm Find Blocking Flow ............ 72 4.1 Quantum algorithm Older Matrix Verification [ABH+02] . 77 4.2 Quantum algorithm Matrix Verification ............ 79 4.3 Quantum algorithm Matrix Multiplication .......... 90 7.1 States and subspaces used in the adversary-type DPT ....... 148 8.1 Quantum algorithm Bounded Matrix Product ........ 179 xi Acknowledgments First, I would like to thank my advisor Harry Buhrman. After supervising my Master’s Thesis, he offered me a PhD position. He was always a good source of interesting new problems, and let me work on topics that I liked. Second, I want to thank my co-advisor Ronald de Wolf. He spent much time in discussions with me, proofread all my papers and always had plenty of comments, refined my style, corrected my English, and drank many bottles of good wine with me. I thank my coauthors: Andris Ambainis (2×), Harry Buhrman, Peter Høyer (3×), Hartmut Klauck, Troy Lee, Mario Szegedy, and Ronald de Wolf (2×), for the wonderful time spent in our research. It was very exciting to collaborate on frontier research. I also thank the members of my thesis committee: Pavel Pudl´ak, Lex Schrijver, Mario Szegedy, and Paul Vit´anyi. A lot of my results was done in different time-zones. I thank the following people for their hospitality, for fruitful scientific discussions, or both. My first steps overseas went to the Rutgers University in New Jersey in the autumn 2004, and I thank Andr´eMadeira, Martin P´al, Peter Richter, Xiaoming Sun, and Mario Szegedy and his family, among others for carrying me everywhere in their cars. In the summer 2005, I visited two research groups in Canada. I thank Scott Aaronson, Andris Ambainis, Elham