Quantum Computing and Communication Complexity Ronald de Wolf Quantum Computing and Communication Complexity ILLC Dissertation Series 2001-06 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 Computing and Communication Complexity Academisch Proefschrift ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof.dr. J.J.M. Franse ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar te verdedigen in de Aula der Universiteit op donderdag 6 september 2001, te 12.00 uur door Ronald Michiel de Wolf geboren te Zaandam. Promotiecommissie: Promotores: Prof.dr. H.M. Buhrman Prof.dr.ir. P.M.B. Vit´anyi Overige leden: Prof.dr. R.E. Cleve Dr. M. Santha Prof.dr. A. Schrijver Dr. L. Torenvliet Faculteit der Natuurwetenschappen, Wiskunde en Informatica Copyright c 2001 by Ronald de Wolf ° ISBN: 90–5776–068–1 Se non `evero, `eben trovato. v Contents Acknowledgments xiii 1 Quantum Computing 1 1.1 Introduction.............................. 1 1.2 QuantumMechanics ......................... 2 1.2.1 Superposition ......................... 3 1.2.2 Measurement ......................... 3 1.2.3 Unitaryevolution....................... 4 1.3 QuantumMemory .......................... 5 1.4 QuantumComputation........................ 7 1.4.1 Classical circuits . 7 1.4.2 Quantumcircuits ....................... 8 1.5 TheEarlyAlgorithms ........................ 10 1.5.1 Deutsch-Jozsa......................... 11 1.5.2 Bernstein-Vazirani . 13 1.5.3 Simon ............................. 13 1.6 Shor’sFactoringAlgorithm. 15 1.6.1 Reduction from factoring to period-finding . 15 1.6.2 The quantum Fourier transform . 16 1.6.3 Period-finding, easy case: r divides q ............ 17 1.6.4 Period-finding, hard case: r does not divide q ....... 19 1.7 Grover’sSearchAlgorithm . 19 1.8 Summary ............................... 21 I Query Complexity 23 2 Lower Bounds by Polynomials 25 vii 2.1 Introduction.............................. 25 2.2 Boolean Functions and Polynomials . 27 2.2.1 Booleanfunctions. 27 2.2.2 Multilinear polynomials . 27 2.2.3 Representing and approximating functions . 28 2.3 QueryComplexity .......................... 31 2.3.1 Deterministic ......................... 31 2.3.2 Randomized.......................... 31 2.3.3 Quantum ........................... 33 2.4 Degree Lower Bounds on Query Complexity . 34 2.5 Polynomial Relation for All Total Functions . 36 2.5.1 Certificate complexity and block sensitivity . 36 2.5.2 Polynomial bound for QE(f) and Q0(f) .......... 39 2.5.3 Polynomial bound for Q2(f)................. 41 2.6 SymmetricFunctions . 42 2.6.1 Tightbounds ......................... 43 2.6.2 OR............................... 44 2.6.3 PARITY............................ 45 2.6.4 MAJORITY.......................... 46 2.7 MonotoneFunctions ......................... 46 2.7.1 Improvements of the general bounds . 46 2.7.2 Tight bounds for zero-error . 47 2.7.3 Monotonegraphproperties. 49 2.8 Summary ............................... 51 3 Bounds for Quantum Search 53 3.1 Introduction.............................. 53 3.2 Tight Bounds for Unordered Search . 55 3.3 Application to Success Amplification . 60 3.4 LowerBoundforOrderedSearching. 61 3.4.1 Intuition............................ 61 3.4.2 Simulating queries to an ordered input . 62 3.4.3 Lower bound for ordered search . 65 3.5 Summary ............................... 66 4 Element Distinctness and Related Problems 67 4.1 Introduction.............................. 67 4.2 Finding Claws if f and g ArenotOrdered ............. 69 4.3 Finding Claws if f isOrdered .................... 72 4.4 Finding Claws if both f and g AreOrdered ............ 72 4.5 FindingaTriangleinaGraph . 74 4.6 Summary ............................... 75 viii 5 Average-Case and Non-Deterministic Query Complexity 77 5.1 Introduction.............................. 77 5.2 Average-Case Complexity: Definitions . 78 5.3 Average-Case: Deterministic vs. Bounded-Error . 80 5.4 Average-Case: Randomized vs. Quantum . 81 5.4.1 Thefunction ......................... 81 5.4.2 Quantumupperbound . 82 5.4.3 Classical lower bound . 84 5.4.4 Worst-case quantum complexity of f ............ 85 5.5 Further Average-Case Results . 86 5.5.1 Super-exponential gap for non-uniform µ .......... 86 5.5.2 Generalbounds ........................ 86 5.5.3 MAJORITY.......................... 87 5.5.4 PARITY............................ 90 5.6 Non-Deterministic Complexity: Definitions . 90 5.7 Non-Deterministic Complexity: Characterization and Separation . 92 5.7.1 Non-deterministic polynomials . 93 5.7.2 Characterization of N(f) and NQ(f)............ 94 5.7.3 Separations .......................... 95 5.7.4 Relation of NQ(f) to other complexities . 96 5.8 Summary ............................... 97 II Communication and Complexity 99 6 Quantum Communication Complexity 101 6.1 Introduction.............................. 101 6.2 QuantumCommunication . 102 6.3 The Model of Communication Complexity . 104 6.3.1 Classical............................ 104 6.3.2 Quantum ........................... 106 6.4 QuantumUpperBounds . 108 6.4.1 Initialsteps .......................... 108 6.4.2 Buhrman, Cleve, and Wigderson . 108 6.4.3 Raz .............................. 110 6.5 SomeApplications .......................... 111 6.6 OtherDevelopments . 112 6.7 Summary ............................... 113 7 Lower Bounds for Quantum Communication Complexity 115 7.1 Introduction.............................. 115 7.2 Lower Bounds for Exact Protocols . 117 7.3 A Lower Bound Technique via Polynomials . 121 ix 7.3.1 Decompositions and polynomials . 121 7.3.2 Symmetricfunctions . 123 7.3.3 Monotonefunctions. 126 7.4 Lower Bounds for Bounded-Error Protocols . 127 7.5 Non-Deterministic Complexity . 133 7.5.1 Somedefinitions. 133 7.5.2 Equality to non-deterministic rank . 134 7.5.3 Exponential quantum-classical separation . 135 7.6 OpenProblems ............................ 136 7.7 Summary ............................... 137 8 Quantum Fingerprinting 139 8.1 Introduction.............................. 139 8.2 Simultaneous Message Passing . 141 8.3 ShortNear-OrthogonalQuantumStates . 145 8.4 TheStateDistinguishingProblem . 146 8.5 ExactlyOrthogonalQuantumStates . 148 8.6 Exact Fingerprinting with a Quantum Key . 150 8.7 QuantumDataStructures . 151 8.7.1 Thequantumcase ...................... 151 8.7.2 Comparison with the classical case . 153 8.8 Summary ............................... 154 9 Private Quantum Channels 155 9.1 Introduction.............................. 155 9.2 Preliminaries ............................. 157 9.2.1 Mixed states and superoperators . 157 9.2.2 Von Neumann entropy . 158 9.3 Definition of Private Quantum Channel . 159 9.4 ExamplesandPropertiesofPQCs. 160 9.5 LowerBoundontheEntropyofPQCs . 164 9.6 Summary ............................... 167 A Some Useful Linear Algebra 169 A.1 Some Terminology and Notation . 169 A.2 UnitaryMatrices ........................... 170 A.3 Diagonalization and Singular Values . 170 A.4 Trace.................................. 172 A.5 TensorProducts............................ 172 A.6 Rank.................................. 173 A.7 DiracNotation ............................ 173 Bibliography 175 x Samenvatting 189 Abstract 193 List of Symbols 197 Index 199 xi Acknowledgments First and foremost I want to thank Harry Buhrman, who in the last four years has been a very creative and interminable source of interesting problems, as well as of clever ideas for solving those problems. Most of this thesis is owed to him, either directly or indirectly. Secondly I want to thank Paul Vit´anyi, who gave me the PhD position that resulted in this thesis and who has been a source of good advice during these years, scientific and otherwise. I also thank him for a very careful reading of this thesis. I further want to thank Shan-Hwei Nienhuys-Cheng for making me a scientist and for gently pushing me towards the area of complexity theory after the work we did together on Inductive Logic Programming during my undergraduate years in Rotterdam. Thanks also to the people who were willing to take place in the thesis committee: Richard Cleve, Miklos Santha, Lex Schrijver, and Leen Torenvliet. I thank the University of Amsterdam (ILLC) for hiring me and the Centre for Mathematics and Computer Science (CWI) for hosting me. I also thank the co-authors of the various papers that make up this thesis: Harry Buhrman (7 x), Richard Cleve (3 x), Andris Ambainis (2 x), Michele Mosca (2 x), Robert Beals, Christoph D¨urr, Mark Heiligman, Peter Høyer, Fr´ed´eric Magniez, Miklos Santha, Alain Tapp, John Watrous, and Christof Zalka. Furthermore I want to thank the following people for many pleasant, interest- ing, and useful scientific discussions throughout the period of my PhD-ship, many of which contributed to the papers on which this thesis is based: Dorit Aharonov, Krzysztof Apt, Johan van Benthem, P. Oscar Boykin, Wim van Dam, Jordi Del- gado, David Deutsch, David DiVincenzo, Yevgeniy Dodis, Lance Fortnow, P´eter G´acs, Mart de Graaf, Lov Grover, Vesa Halava, Edith and Lane Hemaspaandra, Mika Hirvensalo, Hoi-Kwong Lo, Rashindra Manniesing, Dieter van Melkebeek, Ashwin Nayak, Noam Nisan, Michael Nielsen, Ramamohan Paturi, John Preskill, Hein R¨ohrig, John Smolin, Mario Szegedy, Barbara Terhal,