UvA-DARE (Digital Academic Repository) Quantum algorithms and learning theory Arunachalam, S. Publication date 2018 Document Version Final published version License Other Link to publication Citation for published version (APA): Arunachalam, S. (2018). Quantum algorithms and learning theory. 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. 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UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:26 Sep 2021 Quantum Algorithms and Learning Theory Algorithms and Learning Quantum Quantum Algorithms and Learning Theory Srinivasan Arunachalam Srinivasan Arunachalam Quantum Algorithms and Learning Theory Srinivasan Arunachalam 517939-L-bw-Arunachalam Processed on: 12-4-2018 PDF page: 1 517939-L-bw-Arunachalam Processed on: 12-4-2018 PDF page: 2 Quantum Algorithms and Learning Theory 517939-L-bw-Arunachalam Processed on: 12-4-2018 PDF page: 3 ILLC Dissertation Series DS-2018-08 For further information about ILLC-publications, please contact Institute for Logic, Language and Computation Universiteit van Amsterdam Science Park 107 1098 XG Amsterdam phone: +31-20-525 6051 e-mail: [email protected] homepage: http://www.illc.uva.nl/ The investigations were supported by the ERC Consolidator Grant QPROGRESS. Copyright c 2018 by Srinivasan Arunachalam Printed and bound by Ipskamp Printing. ISBN: 978-94-028-0984-8 517939-L-bw-Arunachalam Processed on: 12-4-2018 PDF page: 4 Quantum Algorithms and Learning Theory Academisch Proefschrift ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof. dr. ir. K.I.J. Maex ten overstaan van een door het College voor Promoties ingestelde commissie, in het openbaar te verdedigen in de Agnietenkapel op woensdag 25 april 2018, te 14.00 uur door Srinivasan Arunachalam geboren te Bangalore, India 517939-L-bw-Arunachalam Processed on: 12-4-2018 PDF page: 5 Promotiecommisie Promotores: Prof. dr. R. M. de Wolf Universiteit van Amsterdam Prof. dr. H. M. Buhrman Universiteit van Amsterdam Overige leden: Prof. dr. E. M. Opdam Universiteit van Amsterdam Prof. dr. C. J. M. Schoutens Universiteit van Amsterdam Prof. dr. P. D. Gr¨unwald Universiteit Leiden Dr. M. Ozols Universiteit van Amsterdam Dr. A. Montanaro University of Bristol, UK Faculteit der Natuurwetenschappen, Wiskunde en Informatica 517939-L-bw-Arunachalam Processed on: 12-4-2018 PDF page: 6 v This thesis is based on the following papers. For the first five papers, the authors are ordered alphabetically and the co-authorship is shared equally. For the sixth paper, the authors are ordered based on their contribution. 1. [AW17c] Srinivasan Arunachalam and Ronald de Wolf. Optimizing the Number of Gates in Quantum Search. In Quantum Information & Compu- tation, 17(3&4):251-261, 2017. 2. [ABP18] Srinivasan Arunachalam, Jop Bri¨et, and Carlos Palazuelos. Quan- tum query algorithms are completely bounded forms. In Proceedings of the 9th Conference on Innovations in Theoretical Computer Science (ITCS), pages 3:1-3:21, 2018. 3. [AW17a] Srinivasan Arunachalam and Ronald de Wolf. Guest column: A survey of quantum learning theory. In SIGACT News, 48(2):41-67, 2017. 4. [AW17b] Srinivasan Arunachalam and Ronald de Wolf. Optimal quantum sample complexity of learning algorithms. In 32nd Computational Com- plexity Conference (CCC), pages 25:1-25:31, 2017. 5. [ACLW18] Srinivasan Arunachalam, Sourav Chakraborty, Troy Lee, and Ronald de Wolf. Two new results on quantum exact learning. Manuscript. 6. [GAW17] Andr´as Gily´en, Srinivasan Arunachalam, and Nathan Wiebe. Optimizing quantum optimization algorithms via faster quantum gradient computation. Preprint available at arXiv:1711.00465 [quant-ph]. In the course of his PhD, the author has additionally (co-)authored the fol- lowing articles that are not included in this thesis (most of the work in these projects was done for his Master’s degree). 1. [AMR17] Srinivasan Arunachalam, Abel Molina, and Vincent Russo. Quan- tum hedging in two-round prover-verifier interactions. In 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017). 2. [AGJO+15] Srinivasan Arunachalam, Vlad Gheorghiu, Tomas Jochym-O’ Connor, Michele Mosca, and Priyaa Varshinee Srinivasan. On the robust- ness of bucket brigade quantum RAM. In 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015). Also in New Journal of Physics, 17(12): 123010, 2015. 3. [AJR15] Srinivasan Arunachalam, Nathaniel Johnston, and Vincent Russo Is absolute separability determined by the partial transpose? In Quantum Information & Computation, 15(7&8):694-720, 2015. 517939-L-bw-Arunachalam Processed on: 12-4-2018 PDF page: 7 517939-L-bw-Arunachalam Processed on: 12-4-2018 PDF page: 8 Contents Acknowledgments ix 1 Overview 1 1.1 Introduction .............................. 1 1.2 Quantum algorithms ......................... 3 1.3 Learning in a quantum world .................... 6 Part One: Quantum algorithms 2 Preliminaries and query complexity 13 2.1 Mathematical objects of interest ................... 14 2.2 Quantum information ......................... 16 2.3 Query models ............................. 20 2.4 Lower bound methods for quantum query complexity ....... 25 2.5 Quantum search in a database .................... 30 3 Gate complexity of quantum search 37 3.1 Introduction .............................. 38 3.2 Overview of the proof ......................... 39 3.3 Gate complexity of exact amplitude amplification ......... 41 3.4 Improving the gate complexity for quantum search ........ 43 3.5 Conclusion and future work ..................... 52 4 Refining the polynomial method 53 4.1 Introduction .............................. 54 4.2 Our results .............................. 56 4.3 Preliminaries ............................. 60 4.4 Characterizing quantum query algorithms ............. 64 vii 517939-L-bw-Arunachalam Processed on: 12-4-2018 PDF page: 9 viii Contents 4.5 Separations for quartic polynomials ................. 70 4.6 Short proof of Theorem 4.1.1 ..................... 77 4.7 Conclusion and future work ..................... 81 5 Quantum gradient-based optimization 83 5.1 Introduction .............................. 84 5.2 Gradient-based optimization ..................... 85 5.3 Quantum gradient calculation algorithm .............. 90 5.4 Other results ............................. 101 5.5 Conclusion and future work ..................... 104 Part Two: Learning in a quantum world 6 Survey of quantum learning theory 109 6.1 Introduction .............................. 110 6.2 Quantum subroutines ......................... 113 6.3 Learning models ............................ 114 6.4 Results on query complexity ..................... 117 6.5 Results on sample complexity .................... 125 6.6 The learnability of quantum states ................. 127 6.7 Time complexity ........................... 132 6.8 Conclusion and future work ..................... 137 7 Quantum sample complexity 139 7.1 Sample complexity and VC dimension ............... 140 7.2 Our results .............................. 142 7.3 Preliminaries ............................. 146 7.4 Information-theoretic lower bounds ................. 150 7.5 A lower bound by analysis of state identification .......... 158 7.6 Additional results. .......................... 168 7.7 Conclusion and future work ..................... 171 Bibliography 173 Abstract 195 Samenvatting 199 517939-L-bw-Arunachalam Processed on: 12-4-2018 PDF page: 10 Acknowledgments First and foremost, I would like to thank my advisor Ronald de Wolf, without whom this thesis would definitely have not been possible. Apart from his patient supervision and beautiful insights into various problems that we worked on, most importantly (and I can’t stress this enough) he had faith in me. The first two years of my PhD were slow, depressing, felt never-ending and needless to say, scientifically unproductive. When I thought it was time to quit, Ronald continued encouraging me, reiterating that things were going fine. Since you are reading this, I guess Ronald was right after all and his faith in me was justified maybe. I really thank him for this. There were two memorable moments in these four years. First, when Ronald went nuts when I was sloppy (and didn’t know!) when using the Kleene star notation1 and this emphasized the importance of rigor in his style of research. Second, when Ronald finally agreed to work on quantum learning theory with me and those collaborations contributed to a major part of this thesis. Overall, it has been an amazing learning experience being a student and collaborator of Ronald. Thanks for the opportunity. I am grateful to the members of my PhD committee, Harry Buhrman, Eric Op- dam, Kareljan Schoutens, Peter