UvA-DARE (Digital Academic Repository) Quantum Computing and Communication Complexity de Wolf, R.M. Publication date 2001 Document Version Final published version Link to publication Citation for published version (APA): de Wolf, R. M. (2001). Quantum Computing and Communication Complexity. Institute for Logic, Language and Computation. General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:30 Sep 2021 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 Magni¯cus 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 `e vero, `e ben trovato. v Contents Acknowledgments xiii 1 Quantum Computing 1 1.1 Introduction . 1 1.2 Quantum Mechanics . 2 1.2.1 Superposition . 3 1.2.2 Measurement . 3 1.2.3 Unitary evolution . 4 1.3 Quantum Memory . 5 1.4 Quantum Computation . 7 1.4.1 Classical circuits . 7 1.4.2 Quantum circuits . 8 1.5 The Early Algorithms . 10 1.5.1 Deutsch-Jozsa . 11 1.5.2 Bernstein-Vazirani . 13 1.5.3 Simon . 13 1.6 Shor's Factoring Algorithm . 15 1.6.1 Reduction from factoring to period-¯nding . 15 1.6.2 The quantum Fourier transform . 16 1.6.3 Period-¯nding, easy case: r divides q . 17 1.6.4 Period-¯nding, hard case: r does not divide q . 19 1.7 Grover's Search Algorithm . 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 Boolean functions . 27 2.2.2 Multilinear polynomials . 27 2.2.3 Representing and approximating functions . 28 2.3 Query Complexity . 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 Certi¯cate 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 Symmetric Functions . 42 2.6.1 Tight bounds . 43 2.6.2 OR . 44 2.6.3 PARITY . 45 2.6.4 MAJORITY . 46 2.7 Monotone Functions . 46 2.7.1 Improvements of the general bounds . 46 2.7.2 Tight bounds for zero-error . 47 2.7.3 Monotone graph properties . 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 Ampli¯cation . 60 3.4 Lower Bound for Ordered Searching . 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 Are not Ordered . 69 4.3 Finding Claws if f is Ordered . 72 4.4 Finding Claws if both f and g Are Ordered . 72 4.5 Finding a Triangle in a Graph . 74 4.6 Summary . 75 viii 5 Average-Case and Non-Deterministic Query Complexity 77 5.1 Introduction . 77 5.2 Average-Case Complexity: De¯nitions . 78 5.3 Average-Case: Deterministic vs. Bounded-Error . 80 5.4 Average-Case: Randomized vs. Quantum . 81 5.4.1 The function . 81 5.4.2 Quantum upper bound . 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 ¹ . 86 5.5.2 General bounds . 86 5.5.3 MAJORITY . 87 5.5.4 PARITY . 90 5.6 Non-Deterministic Complexity: De¯nitions . 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 Quantum Communication . 102 6.3 The Model of Communication Complexity . 104 6.3.1 Classical . 104 6.3.2 Quantum . 106 6.4 Quantum Upper Bounds . 108 6.4.1 Initial steps . 108 6.4.2 Buhrman, Cleve, and Wigderson . 108 6.4.3 Raz . 110 6.5 Some Applications . 111 6.6 Other Developments . 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 Symmetric functions . 123 7.3.3 Monotone functions . 126 7.4 Lower Bounds for Bounded-Error Protocols . 127 7.5 Non-Deterministic Complexity . 133 7.5.1 Some de¯nitions . 133 7.5.2 Equality to non-deterministic rank . 134 7.5.3 Exponential quantum-classical separation . 135 7.6 Open Problems . 136 7.7 Summary . 137 8 Quantum Fingerprinting 139 8.1 Introduction . 139 8.2 Simultaneous Message Passing . 141 8.3 Short Near-Orthogonal Quantum States . 145 8.4 The State Distinguishing Problem . 146 8.5 Exactly Orthogonal Quantum States . 148 8.6 Exact Fingerprinting with a Quantum Key . 150 8.7 Quantum Data Structures . 151 8.7.1 The quantum case . 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 De¯nition of Private Quantum Channel . 159 9.4 Examples and Properties of PQCs . 160 9.5 Lower Bound on the Entropy of PQCs . 164 9.6 Summary . 167 A Some Useful Linear Algebra 169 A.1 Some Terminology and Notation . 169 A.2 Unitary Matrices . 170 A.3 Diagonalization and Singular.