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On the Relation Between Quantum Computation and Classical Statistical Mechanics On the relation between quantum computation and classical statistical mechanics by Joseph Geraci A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto Copyright c 2008 by Joseph Geraci ! Abstract On the relation between quantum computation and classical statistical mechanics Joseph Geraci Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2008 We provide a quantum algorithm for the exact evaluation of the Potts partition function for a certain class of restricted instances of graphs that correspond to irreducible cyclic codes. We use the same approach to demonstrate that quantum computers can provide an exponential speed up over the best classical algorithms for the exact evaluation of the weight enumerator polynomial for a family of classical cyclic codes. In addition to this we also provide an efficient quantum approximation algorithm for a function (signed-Euler generating function) closely related to the Ising partition function and demonstrate that this problem is BQP-complete. We accomplish the above for the Potts partition function by using a series of links between Gauss sums, classical coding theory, graph theory and the partition function. We exploit the fact that there exists an efficient approximation algorithm for Gauss sums and the fact that this problem is equivalent in complexity to evaluating discrete log. A theorem of McEliece allows one to turn the Gauss sum approximation into an exact evaluation of the Potts partition function. Stripping the physics from this result leaves one with the result for the weight enumerator polynomial. The result for the approximation of the signed-Euler generating function was accomplished by fashioning a new mapping between quantum circuits and graphs. The mapping provided us with a way of relating the cycle structure of graphs with quantum circuits. Using a slight variant of this mapping, we present the final result of this thesis which presents a way of testing families of quantum circuits for their classical simulatability. We thus provide an efficient way of deciding whether a quantum circuit provides any additional computational power over classical computation and this is achieved by exploiting the fact that planar instances of the Ising partition function (with no external magnetic field) can be efficiently classically computed. ii Acknowledgements I would like to first off thank my supervisor Daniel Lidar 1 who provided me with opportunity beyond my expectations and a project with so much potential. The first portion of this thesis would surely have ended up in the Pacific ocean if it were not for him, as he continued to have faith in my vision long after I had lost it. He was an excellent supervisor that led me but left me to create as I saw fit. I not only learned a great amount of physics from him, but his example led me to learn and aspire to what it takes to be a great scientist. And he gave me the opportunity to dwell in the sub-concious of the USA, better known as Los Angeles. The experience still lingers within me as the residue of a warm dream with images of beaches, dolphins, mountains, freeways, guns and palm trees. My choice to enter graduate level mathematics was due to my Master’s supervisor, I.M. Sigal, who saw in me the potential to pursue a career in mathematics. I would have surely never have continued in mathematics if it wasn’t for his help and inspiration. He was an excellent mentor and he inspired me to finally push myself. His greatness still inspires me and my gratitude goes out to him. Another professor of mathematics who inspired me was Man-Duen Choi. He was an excellent instructor and I regret never taking full advantage of having access to a mathematician of such a calibre when I was an undergraduate, being that I was a lazy sod. He was there for me when, during my Ph.D., I was going through some personal difficulties and helped my immensely. I thank him very much. One mathematician was with me throughout my whole mathematical education, and that is Professor Catherine Sulem. Recently she described me during my earlier days as a “papillon”. The truth is that I was undisciplined and distracted as an undergrad. Dr. Sulem’s guidance however helped me to avoid disaster. Her excellence in mathematics and teaching was also a great inspiration to me. I am indebted to her. I would like to thank my fellow graduate students and the other friends I have made (here and in California) who have made this experience a great one. Many thanks go to Itamar Halevy and Ravi Minhas for their guidance and friendship. I must give a special thanks to the whole staff at the University of Toronto mathematics department especially Marie Bachtis and Ida Bulat. They have put up with me gracefully for years and have helped me in so many ways that it is impossible to thank them enough. My parents Peter and Franca Geraci, have always encouraged me in my studies . They, along with my brother Tony Geraci and his wife Mary, provided a great support system for me while I was away. Their visits were appreciated more than they could know. I thank them all. Lastly, a special thanks to my amazing and generous mother-in-law LouAnn Leon, who enabled me to an extent that I had never ever expected. I thank her with all of my being and consider myself blessed to have her in my life. This and any future accomplishments I enjoy will be imbued with her spirit. 1Thanks goes to the people at ARO/DTO (grant W911NF-05-1-0440) for their financial support. iii Dedication This thesis is dedicated to my lovely wife Summer Nudel, without whom I would have never been able to cross the desert, both literally and figuratively, that lay ahead of me. I will always be indebted to you for the transformation that my life underwent over the last ten years and I thank you and love you with full surrender. iv Contents 1 Introduction 1 1.1 Quantum Computation . 1 1.1.1 Models of Quantum Computation . 4 1.2 A few definitions from Complexity Theory . 7 1.3 Statistical Physics . 9 1.3.1 Ising spin model . 10 1.3.2 The Potts Model . 11 1.4 My contributions . 12 2 Review of previous work 15 2.1 Relation between the Potts partition function and knot invariants . 15 2.2 Graphs, quantum circuits and classical simulations . 18 3 A review of some key concepts from coding theory 21 3.1 Introduction . 21 3.2 Cyclic Codes . 21 3.2.1 A Pause for Cyclotomic Cosets . 22 3.2.2 An application of cyclotomic cosets . 23 3.3 Codes continued . 24 3.4 Gauss sums and their relationship to the weight spectrum of linear codes . 25 4 An evaluation of the Weight Enumerator via Quantum Computation 31 4.0.1 A Theorem on the evaluation of certain Weight Enumerators . 31 4.0.2 Overview of the Algorithm to Obtain the Exact Weight Enumerator of a Code in ICQ! 33 5 A quantum algorithm for the Potts partition function 35 5.1 Structure of this chapter . 35 5.2 A Theorem about QC and instances of the Potts Model . 35 v 5.2.1 Main Theorem . 35 5.2.2 Background . 37 5.2.3 The relationship to linear codes . 40 5.2.4 Testing the graph for membership in the ICCC! class . 41 5.2.5 Proof of the Main Theorem . 41 5.2.6 Proof of the Corrolary . 48 5.2.7 Reducing the Computational Cost of the Algorithm via Permutation Symmetry . 48 5.3 Classical and Quantum Complexity of the Scheme . 49 5.4 Detailed Summary . 50 5.5 Examples and Discussion . 52 5.5.1 Example . 52 5.5.2 Degenerate Cyclic Codes . 55 5.6 Conclusions, Future Directions and Critical Analysis . 56 6 Additive Approximation of the Signed-Euler Generating Function 59 6.1 Introduction . 59 6.1.1 Generating function of Eulerian subgraphs . 59 6.2 QWGTs and their relation to the Ising partition function . 60 6.3 A relationship between hypergraphs and quantum circuits via QWGTs . 62 6.3.1 The Mapping . 63 6.4 BQP-completeness . 65 6.4.1 Examples . 69 6.5 Future work: Approximating the Ising partition function . 70 6.6 Conclusion . 71 7 On classically simulatable quantum circuits 73 7.1 Introduction . 73 7.1.1 Definitions from Graph Theory . 74 7.2 Once again - The Mapping . 75 7.3 Determination of the edge interaction distribution and consequences . 76 7.4 Proof of the main theorem . 78 7.4.1 “The Test” and consequences for the structure of quantum circuits . 82 7.4.2 Quantum circuits corresponding to a class of sparse graphs . 83 7.5 The Next Step . 86 7.5.1 On the existence of edge interactions . 86 vi 7.5.2 Computing the Ising partition function . 87 7.6 Conclusion and Critical Analysis . 89 7.7 Proof of the Lemma . 89 8 Conclusion 93 9 Appendix 97 9.1 A Classical Algorithm for the Computation of Coset Leaders and Coset Size . 97 9.2 Matroids . 98 9.2.1 Generator matrix of a cyclic code and the cycle matroid matrix . 99 9.3 Characters . 100 9.4 Discrete Log . 100 9.5 Samples of Mathematica Notebooks . 101 Bibliography 123 vii viii Chapter 1 Introduction 1.1 Quantum Computation A large portion of the people who will ever attempt to read any part of this thesis will most likely know very little about quantum computation.
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