Purdue University Purdue e-Pubs Open Access Dissertations Theses and Dissertations 12-2016 Combinatorial algorithms for perturbation theory and application on quantum computing Yudong Cao Purdue University Follow this and additional works at: https://docs.lib.purdue.edu/open_access_dissertations Part of the Computer Sciences Commons, and the Quantum Physics Commons Recommended Citation Cao, Yudong, "Combinatorial algorithms for perturbation theory and application on quantum computing" (2016). Open Access Dissertations. 908. https://docs.lib.purdue.edu/open_access_dissertations/908 This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information. Graduate School Form 30 Updated PURDUE UNIVERSITY GRADUATE SCHOOL Thesis/Dissertation Acceptance This is to certify that the thesis/dissertation prepared By Yudong Cao Entitled Combinatorial Algorithms for Perturbation Theory and Application on Quantum Computing For the degree of Doctor of Philosophy Is approved by the final examining committee: Sabre Kais Chair Mikhail Atallah Co-chair David Gleich Ahmed Sameh Samuel Wagstaff To the best of my knowledge and as understood by the student in the Thesis/Dissertation Agreement, Publication Delay, and Certification Disclaimer (Graduate School Form 32), this thesis/dissertation adheres to the provisions of Purdue University’s “Policy of Integrity in Research” and the use of copyright material. Approved by Major Professor(s): Sabre Kais Approved by: William J. Gorman 8/24/2016 Head of the Departmental Graduate Program Date COMBINATORIAL ALGORITHMS FOR PERTURBATION THEORY AND APPLICATION ON QUANTUM COMPUTING A Dissertation Submitted to the Faculty of Purdue University by Yudong Cao In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2016 Purdue University West Lafayette, Indiana ii Dedicated to my family and my teachers iii ACKNOWLEDGMENTS I would like to first of all thank my advisor Sabre Kais for his consistent and generous support over the past five years of my graduate studies. Also I would like to thank my collaborators, mainly Anargyros Papageorgiou, Joseph Traub, Jiangfeng Du, Felipe Herrera, Ryan Babbush, Jacob Biamonte, Youhan Fang and Daniel Na- gaj. During my pursuit of PhD, I have also benefited from the teaching and helpful discussions with the faculties of Computer Science at Purdue, particularly my com- mittee members: Mike Atallah, David Gleich, Ahmed Sameh and Samuel Wagstaff. In various group meetings, conferences, visits and email communications I have also enjoyed helpful discussions with many great people: Sergio Boixo, Vasil Denchev, An- drew Landahl, Daniel Lidar, James Whitfield, Nathan Wiebe and Zoltan Zimboras. I would also like to thank Qatar Foundation and HBKU for their generous acco- modations during my visits to Doha. Finally I acknowledge financial support from National Science Foundation Center for Quantum Information and Computation for Chemistry, Award number CHE-1037992. iv PREFACE This dissertation is interdisciplinary between physics and computer science. The presentation assumes knowledge in basic elements of theoretical computer science but none in physics. The mathematics used is mostly linear algebra. There are in total four chapters. Chapter 1 serves to motivate the central problem that the dissertation concerns. The remaining chapters and appendices are original results obtained in the past three years of research. Most contents in the three chapters have been published; but they also include previously unpublished materials and findings that facilitate the discussions. The main problem concerned in this dissertation is to use perturbation theory for reducing many-body quantum interactions to realistic two-body ones. Many-body interactions are extremely difficult to implement in experimental conditions, while two-body interactions are far more technologically feasible to realize (for instance D-Wave Systems Inc. has manufactured programmable quantum devices based on two-body interactions to a rather impressive scale). Chapter 1 is intended to ar- gue that 1) many-body interactions arise in a wide variety of contexts in quantum computation, 2) quantum complexity theory offers a powerful set of tools, called per- turbative gadgets, for reducing many-body interactions to two-body ones and 3) these tools have inherent drawbacks from the perspective of physical realization; and it is the purpose of this dissertation to propose methods for overcoming these drawbacks. In addition, I would like to use Chapter 1 as an opportunity to introduce quantum mechanics and provide simple intuitions on why it is difficult to simulate on classical computers, thus motivating the subject of quantum computing and at the same time provide the conceptual machinery necessary for the developments of later chapters. Chapter 2 improves the existing constructions of perturbative gadgets. There are also new gadgets that were discovered during the research (Sections 2.8 and 2.6), v which could potentially be of interest. Compared with the published the version [1], the chapter also contains an unpublished Section 2.7. Chapter 3, which is published in [2], proposes a new gadget construction that is entirely different from the exist- ing constructions in the sense that it circumvents the need for strong interactions, which is a common downside of the perturbative gadgets. Apart from its experimen- tal implications, the gadget construction in Chapter 3 is also used in an important theoretical development which shows a counterexample to the area law conjecture in condensed matter physics [3]. The title of this dissertation focuses on combinatorics of perturbation theory and mentions quantum computing as an application. Although Chapters 2 and 3 appear to be entirely devoted to perturbative gadgets, whose primary application is in quantum computing and quantum complexity theory, a hidden theme that develops at their core is in fact a continuously deepened understanding of perturbation theory. In Chapter 2 we are able to improve some existing gadget constructions by a careful examination of the perturbation series. In Chapter 3, in order to prove that the perturbation series expansion converges (Section 3.3.2), we adopt combinatorial analyses of perturbation series that are more involved than those in Chapter 2. It is these analyses that uncovered the association between the perturbative expansion and Motzkin walks, paving the way for more general algorithms in Chapter 4. Essentially, improving perturbative gadgets boils down to finding a tighter upper bound to the norm of the perturbation series from a certain order to infinity. Chapters 2 and 3 deal with this task for specific gadget constructions while the algorithms in Chapter 4 deal with far more general settings (Section 4.2.1). In this sense, Chapters 2 and 3 build up to Chapter 4, which is highlighted in the title as the strongest result of the dissertation. Preliminary version of Chapter 4 is available online [4]. However, in the dissertation there are additional rigorous analyses (Section 4.5.2) which provide evidence as to why our algorithms are able to find tight upper bound to the norm of terms at arbitrary order in the perturbation series. vi TABLE OF CONTENTS Page LIST OF TABLES :::::::::::::::::::::::::::::::: vii LIST OF FIGURES ::::::::::::::::::::::::::::::: viii SYMBOLS :::::::::::::::::::::::::::::::::::: ix ABBREVIATIONS :::::::::::::::::::::::::::::::: xi NOMENCLATURE ::::::::::::::::::::::::::::::: xii ABSTRACT ::::::::::::::::::::::::::::::::::: xv 1 Introduction :::::::::::::::::::::::::::::::::: 1 1.1 Overview ::::::::::::::::::::::::::::::::: 1 1.2 Quantum mechanics :::::::::::::::::::::::::: 9 1.3 Perturbation theory ::::::::::::::::::::::::::: 14 1.3.1 Rayleigh-Schr¨odingerformalism :::::::::::::::: 15 1.3.2 Self-energy expansion :::::::::::::::::::::: 19 1.4 Quantum computing :::::::::::::::::::::::::: 23 1.4.1 Gate model ::::::::::::::::::::::::::: 25 1.4.2 Adiabatic model :::::::::::::::::::::::: 27 1.4.3 Measurement-based model ::::::::::::::::::: 33 1.5 Quantum simulation :::::::::::::::::::::::::: 34 1.5.1 Molecular Hamiltonian ::::::::::::::::::::: 35 1.5.2 Second quantization :::::::::::::::::::::: 40 1.5.3 Mapping to many-body qubit systems :::::::::::: 43 1.6 Quantum Hamiltonian complexity ::::::::::::::::::: 44 1.6.1 Local Hamiltonian and QMA ::::::::::::::::: 45 1.6.2 Perturbative gadgets :::::::::::::::::::::: 47 1.7 Summary :::::::::::::::::::::::::::::::: 50 vii Page 2 Improved perturbative gadgets :::::::::::::::::::::::: 51 2.1 Overview ::::::::::::::::::::::::::::::::: 52 2.2 Improved subdivision gadget :::::::::::::::::::::: 53 2.3 Parallel subdivision and k- to 3-body reduction ::::::::::: 60 2.4 Improved 3- to 2-body gadget ::::::::::::::::::::: 69 2.5 Parallel 3- to 2-body gadget :::::::::::::::::::::: 79 2.6 Creating 3-body gadget from local X terms :::::::::::::: 88 2.7 Alternative construction for k- to 2-body reduction ::::::::: 91 2.7.1 Numerical examples :::::::::::::::::::::: 96 2.7.2 Error analysis :::::::::::::::::::::::::: 99 2.7.3 Gap scaling ::::::::::::::::::::::::::: 99 2.7.4 Connection between Bloch formalism and self-energy :::: 101 2.8 YY gadget :::::::::::::::::::::::::::::::: 106 2.9 Conclusion :::::::::::::::::::::::::::::::: 110 3 Perturbative gadgets without strong interactions :::::::::::::: 111 3.1 Overview ::::::::::::::::::::::::::::::::: 111 3.2 Effective interactions based on perturbation