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Noise in Quantum and Classical Computation & Non-Locality Noise in Quantum and Classical Computation & Non-locality Falk Unger Noise in Quantum and Classical Computation & Non-locality ILLC Dissertation Series DS-2008-06 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/ Noise in Quantum and Classical Computation & Non-locality Academisch Proefschrift ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof. dr. D.C. van den Boom ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar te verdedigen in de Agnietenkapel op donderdag 18 september 2008, te 10.00 uur door Falk Peter Unger geboren te Dresden, Duitsland. Promotiecommissie: Promotor: prof. dr. H.M. Buhrman Overige Leden: prof. dr. R.E. Cleve prof. dr. R.D. Gill prof. dr. A. Schrijver dr. L. Torenvliet dr. R.M. de Wolf Faculteit der Natuurwetenschappen, Wiskunde en Informatica The investigations were financially supported by the EU project QAP (IST- 2005-15848) and by the Netherlands Organisation for Scientific Research (NWO) in the framework of the Innovational Research Incentives Scheme (in Dutch: Vernieuwingsimpuls), Vici-project 639.023.302. Copyright c 2008 by Falk Unger Printed and bound by PrintPartners Ipskamp. ISBN: 90-6196-547-0 The results in this thesis are based on the following articles Chapter 3 The results in this chapter are based on an unpublished manuscript: Falk Unger, Erasure noise threshold for fault-tolerant computation, unpublished Chapter 4 Julia Kempe, Oded Regev, Falk Unger and Ronald de Wolf, Upper bounds on the noise threshold for fault-tolerant quantum com- puting, ICALP 2008 Chapter 5 Harry Buhrman, Richard Cleve, Monique Laurent, Noah Linden, Lex Schrijver and Falk Unger, New Limits on Fault-Tolerant Quantum Computation, Proceedings of 47th IEEE FOCS, 2006 Chapter 6 Falk Unger, Noise threshold for universality of 2-input gates, Proceedings of IEEE International Symposium on Information Theory, 2007, accepted to IEEE Transactions on Information Theory Chapter 8 Gilles Brassard, Harry Buhrman, Noah Linden, Andr´eAllan Methot, Alain Tapp, and Falk Unger, Limit on nonlocality in any world in which communication complexity is not trivial, Physical Review Let- ters 96(25), 2006. Chapter 7 R. Cleve, W. Slofstra, F. Unger, and S. Upadhyay, Strong parallel repetition theorem for quantum XOR proof systems, In Special Issue of 22nd IEEE Conference on Computational Complexity, 2007. The author also coauthored the following papers, which are beyond the scope of this thesis Falk Unger, On small hard leaf languages, In Proceedings of the 30th In- • ternational Symposium on Mathematical Foundations of Computer Science, pages 781{792. Springer, 2005. Harry Buhrman, Leen Torenvliet, and Falk Unger, Sparse self-reducible • sets and polynomial size circut lower bounds, In Proceedings of the 23rd International Symposium on Theoretical Aspects of Computer Science, pages 455{468. Springer, 2006. H. Buhrman, M. Christandl, F. Unger, S. Wehner, and A. Winter, Impli- • cations of superstrong nonlocality for cryptography, Proceedings of the Royal Society A, 462(2071):1919{1932, 2006. v to my parents Brita and Hans-J¨urgen vi Contents Acknowledgments xi 1 Introduction 1 1.1 Limits on fault-tolerant computation . .2 1.1.1 Limits on fault-tolerant classical computation . .8 1.2 Entanglement and interactive proof systems . 10 1.2.1 Repetition of XOR games . 10 1.2.2 Limits on non-locality . 13 2 Preliminaries 17 2.1 Linear algebra notation . 17 2.2 Quantum states, operations and computation . 19 2.2.1 Quantum circuits and quantum computation . 21 2.3 Complexity classes . 23 2.4 Communication complexity . 25 2.5 Bloch sphere . 27 2.5.1 Pauli matrices . 27 2.5.2 Bloch-vector representation . 27 2.6 Semidefinite programming . 28 I Limits on noisy quantum and classical computation 31 3 Erasure noise 33 3.1 Erasure vs. depolarizing noise . 34 3.2 Circuit model . 35 3.3 Noise threshold . 37 3.4 Discussion . 41 vii 4 Perfect 1-qubit operations and noisy k-qubit unitaries 43 4.1 Introduction . 43 4.2 Model and results . 45 4.3 Preliminaries . 46 4.4 Proof of Theorem 4.2.1 . 48 4.4.1 Proof of Lemma 4.4.1 . 50 4.5 Arbitrary one-qubit gates and CNOT gates . 53 4.6 Discussion . 55 4.6.1 Comparison with other chapters . 55 4.6.2 Comments on results and open problems . 56 5 Perfect stabilizer operations and noisy 1-qubit unitaries 59 5.1 Introduction . 59 5.1.1 Organization . 61 5.2 Preliminaries and notation . 62 5.2.1 Noise . 62 5.2.2 Clifford group . 63 5.2.3 Stabilizer operations and the Gottesman-Knill Theorem . 64 5.3 The power of Clifford circuits . 64 5.4 Simulating 1-qubit unitaries by Clifford gates . 68 5.5 Lower bound on θ^ .......................... 73 5.6 Discussion and extensions . 73 6 Classical 2-input gates 79 6.1 Introduction . 79 6.1.1 Noise threshold for fan-in 2 gates . 80 6.1.2 Outline of the proof . 81 6.1.3 Organization . 82 6.2 Definitions . 82 6.3 Bias reduction for noisy gates . 83 6.3.1 Technical lemmas . 84 6.3.2 Main Lemma . 85 6.4 Proof of Theorem 6.1.1 . 89 6.5 Discussion . 90 6.5.1 Circuits vs. formulas . 90 6.5.2 Choice of potential function . 90 II Entanglement and interactive proof systems 93 7 Parallel repetition of quantum XOR games 95 7.1 Introduction . 95 7.1.1 Motivation . 95 viii 7.1.2 Organization . 99 7.2 Background and definitions . 99 7.2.1 More on interactive proof systems . 99 7.2.2 XOR games . 101 7.2.3 XOR games and non-locality . 102 7.3 Characterization of quantum XOR games . 102 7.4 Additivity theorem . 105 7.5 Parallel repetition theorem . 107 7.6 Discussion . 110 7.6.1 Perfect parallel repetition of classical XOR games . 110 7.6.2 Parallel repetition of general games . 110 7.6.3 Feige-Lov´aszgames . 111 8 Limits on non-locality from communication complexity 113 8.1 Introduction . 113 8.2 Non-local boxes . 116 8.3 Main result . 118 8.4 Proof . 118 8.4.1 Distributed computation . 119 8.4.2 Bias Amplification . 120 8.5 Discussion . 123 8.5.1 One NLB for majority . 123 8.5.2 Fault-tolerance threshold . 123 A Some more facts about Linear algebra 125 B Convex hull of all 1-qubit Clifford operations 129 B.1 Preliminaries about the Clifford matrices . 129 B.2 The case F σ = 3 for some σ Sym(3) . 131 B.3 The case jR \C j 2 for all σ Sym(3)2 . 134 jRF \Cσj ≤ 2 C Classical entanglement-assisted communication complexity of in- ner product 137 D Tsirelson's vector characterization of XOR games 139 Bibliography 143 Index 151 Samenvatting 153 Abstract 157 ix Acknowledgments This thesis is the culmination of four very enjoyable years at CWI during which I was a PhD student in the INS4 group. I would like to thank all people at CWI for making it the place it is. In particular, I would like to thank my advisor Harry Buhrman for his guidance and the faith he had in me from the start. We had many scientific discussions, but I am also grateful for his advice on non-technical scientific matters. Almost literally the same can be said about Ronald de Wolf, who shared an office with me for four years, and Ben Toner who joined our office in 2006. They were always happy to help me with all kinds of small and big issues which come up during a PhD student's day. I would like to thank them for a lot of fun in the most \social office” at INS4. During my PhD I had the pleasure to work with many great people, who provided many interesting ideas and patiently explained many difficult things to me. Some of this work resulted in publications. I want to thank my co- authors Gilles Brassard, Harry Buhrman, Matthias Christandl, Richard Cleve, Julia Kempe, Monique Laurent, Noah Linden, Andre Methot, Oded Regev, Lex Schrijver, William Slofstra, Alain Tapp, Leen Torenvliet, Sarvagya Upadhyay, Stephanie Wehner, Andreas Winter and Ronald de Wolf for all the hard work they put into our joint work. I had many interesting scientific (and fortu- nately also non-scientific) conversations with other people, which have not re- sulted in publications. Among them were Scott Aaronson, Dorit Aharonov, Andris Ambainis, Manuel Ballester S´anchez, Hartwig Bosse, Jop Bri¨et,Serge Fehr, Daniel Gottesman, NebojˇsaGvozdenovi´c,Peter Harremo¨es,Aram Har- row, Sophie Laplante, Troy Lee, Lasse Leskel¨a,Ashwin Nayak, Tobias Osborne, Krzysztof Pietrzak, Sandu Popescu, David Poulin, Daniel Preda, Ben Reichardt, Renato Renner, Nitin Saxena, Pranab Sen, Christian Schaffner, Leonard Schul- man, Robert Spalek,ˇ Mario Szegedy, Ben Toner, John Tromp, Umesh Vazirani, Paul Vit´anyi, Shengyu Zhang and many other people who kindly shared their insights and expertise with me. I also managed to have something like a social life. Robert, Troy and Ronald xi I want to thank for being more than just friends to hang out with. I also had the fortune to live in five different places in Amsterdam which allowed me to stay with a sizeable number of flatmates: Nebojˇsa,Dennis, Matthieu, Eike, Hartwig, Lasse, Keit and Sumanta. You and all other friends I thank for the nice dinners, parties, Salsa, wind-surfing, etc. and all the things traditionally a family provides you with.
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