UNIVERSITY OF CALIFORNIA, SAN DIEGO The Complexity of the Consistency and N-representability Problems for Quantum States A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Computer Science by Yi-Kai Liu Committee in charge: Professor Russell Impagliazzo, Chair Professor David Meyer, Co-chair Professor Sanjoy Dasgupta Professor Daniele Micciancio Professor Jeffrey Remmel 2007 Copyright Yi-Kai Liu, 2007 All rights reserved. The dissertation of Yi-Kai Liu is approved, and it is accept- able in quality and form for publication on microfilm: Co-chair Chair University of California, San Diego 2007 iii TABLE OF CONTENTS SignaturePage ................................... iii TableofContents .................................. iv Acknowledgements ................................. vi VitaandPublications................................ vii Abstract ....................................... viii 1 Introduction..................................... 1 1. Overview..................................... 1 2. Quantumcomputation ............................. 5 3. QuantumComplexityClasses . 8 4. TheLocalHamiltonianProblem . 9 5. Promise Problems and Polynomial Time . 10 6. DensityMatrices ................................ 11 7. ConsistencyofLocalDensityMatrices . 14 2 Consistency of Local Density Matrices is QMA-complete . ........ 16 1. Introduction ................................... 16 2. ConsistencyisinQMA ............................. 19 3. Convex Optimization using a Membership Oracle . ..... 21 1. ASimpleReduction............................. 26 2. Round-offErrors............................... 34 3. TheEllipsoidMethod............................ 37 4. Algorithms using Random Walks . 39 4. ConsistencyisQMA-hard ........................... 41 1. TheBasicIdea................................ 42 2. How to represent (ρ1,...,ρm) ....................... 43 3. NumericalPrecision............................. 46 5. Discussion .................................... 51 3 N-representabilityisQMA-complete . 52 1. Introduction ................................... 52 2. Fermions..................................... 54 1. Second-QuantizedOperators . 55 2. Two-ParticleObservables. 58 3. The N-representability and Fermionic Local Hamiltonian problems .... 61 4. OurResults ................................... 63 5. Fermionic Local Hamiltonian is QMA-hard . 64 6. N-representabilityisQMA-hard . 66 1. Convex Optimization with a Membership Oracle . 66 2. N-representabilityisQMA-hard . 67 3. Bounds on the Geometry of K ....................... 70 iv 7. FermionicProblemsinQMA. 75 1. Pure-state N-representabilityisinQMA(2). 77 8. Discussion .................................... 79 1. Related Work in Quantum Information . 79 2. Applications to Quantum Chemistry . 80 4 The Consistency Problem for 1-D and Stoquastic Systems . ....... 83 1. Introduction ................................... 83 2. Reductions from Consistency to Local Hamiltonian . ....... 86 3. Consistencyfor1-DSystems . 91 4. Consistency for Stoquastic Systems . 92 1. Reducing from Stoquastic Local Hamiltonian to Stoquastic Consistency 94 2. Reducing from Stoquastic Consistency to Stoquastic Local Hamiltonian 95 5 Gibbs States and the Consistency of Local Density Matrices .......... 98 1. Introduction ................................... 98 2. Proofsofourresults............................... 101 1. Thepartitionfunction . .101 2. Proofoftheorem2 .............................102 3. Proofoftheorem5.1 ............................105 Figure5.1: Asingle-qubitexample . 107 6 Conclusions .....................................108 Bibliography.....................................110 v ACKNOWLEDGEMENTS The path that led to this dissertation was neither easy nor predictable. I am especially grateful to the following people (listed in alphabetical order) who helped me along the way: Dorit Aharonov, Andrew Childs, Matthias Christandl, Sanjoy Dasgupta, Russell Impagliazzo, David Meyer, John Preskill and Frank Verstraete. I have also benefitted from many conversations with the other graduate stu- dents in the theory group, including Chris Calabro, Sashka Davis, Ragesh Jaiswal, Kirill Levchenko, Vadim Lyubashevsky and Nathan Segerlind. Thanks to my various friends who do interesting things other than computer science. And thanks to my parents, for always being there. I was supported by a Quantum Computing Graduate Research Fellowship (QuaCGR), provided by the US Army Research Office (ARO) and the Disruptive Tech- nology Office (DTO). Without their financial support, this work probably would not have taken place. Chapter 2 contains some material that was previously published in the paper: Y.-K. Liu, “Consistency of Local Density Matrices is QMA-complete,” Proc. RANDOM 2006, pp.438-449, Springer-Verlag (2006). The dissertation author was the primary investigator and author of this paper. Chapter 3 contains some material that was previously published in the paper: Y.-K. Liu, M. Christandl and F. Verstraete, “N-representability is QMA-complete,” Phys. Rev. Lett. 98, 110503 (2007). The dissertation author was the primary investiga- tor and author of this paper. vi VITA 2002 A.B., Princeton University 2007 Ph.D., University of California, San Diego PUBLICATIONS Y.-K. Liu, M. Christandl and F. Verstraete, “N-representability is QMA-complete,” Phys. Rev. Lett. 98, 110503 (2007); Arxiv preprint: quant-ph/ 0609125. Y.-K. Liu, “Consistency of Local Density Matrices is QMA-complete,” Proc. RANDOM 2006, pp. 438-449; Arxiv preprint: quant-ph/ 0604166. Y.-K. Liu, V. Lyubashevsky and D. Micciancio, “On Bounded Distance Decoding for General Lattices,” Proc. RANDOM 2006, pp.450-461. Y.-K. Liu, “Gibbs States and the Consistency of Local Density Matrices,” poster at the SQuInT workshop, Albuquerque, NM, Feb. 17-19, 2006; Arxiv preprint: quant-ph/ 0603012. A. Blanc, Y.-K. Liu and A. Vahdat, “Designing Incentives for Peer-to-Peer Routing,” Proc. INFOCOM 2005, pp.374-385; a preliminary version appeared in P2PEcon 2004. vii ABSTRACT OF THE DISSERTATION The Complexity of the Consistency and N-representability Problems for Quantum States by Yi-Kai Liu Doctor of Philosophy in Computer Science University of California, San Diego, 2007 Professor Russell Impagliazzo, Chair Professor David Meyer, Co-chair Quantum mechanics has important consequences for machines that store and manip- ulate information. In particular, quantum computers might be more powerful than classical computers; examples of this include Shor’s algorithm for factoring and discrete logarithms, and Grover’s algorithm for black-box search. Because of these theoretical results, and the possibility that we may eventually succeed in building scalable quantum computers, it is interesting to study complexity classes based on quantum computation. QMA (Quantum Merlin-Arthur) is the quantum analogue of the class NP. There are a few QMA-complete problems, most of which are variants of the “Local Hamiltonian” problem introduced by Kitaev. In this dissertation we show some new QMA-complete problems which are very different from those known previously, and have applications in quantum chemistry. The first one is “Consistency of Local Density Matrices”: given a collection of density matrices describing different subsets of an n-qubit system (where each subset has constant size), decide whether these are consistent with some global state of all n qubits. This problem was first suggested by Aharonov. We show that it is QMA-complete, via an oracle reduction from Local Hamiltonian. Our reduction is based on algorithms for convex optimization with a membership oracle, due to Yudin and Nemirovskii. Next we show that two problems from quantum chemistry, “Fermionic Local Hamiltonian” and “N-representability,” are QMA-complete. These problems involve viii systems of fermions, rather than qubits; they arise in calculating the ground state en- ergies of molecular systems. N-representability is particularly interesting, as it is a key component in recently developed numerical methods using the contracted Schrodinger equation. Although these problems have been studied since the 1960’s, it is only re- cently that the theory of quantum computation has provided the right tools to properly characterize their complexity. Finally, we study some special cases of the Consistency problem, pertaining to 1-dimensional and “stoquastic” systems. We also give an alternative proof of a result due to Jaynes: whenever local density matrices are consistent, they are consistent with a Gibbs state. ix 1 Introduction 1.1 Overview Beginning in the 1980’s, the field of quantum mechanics was reinvigorated by a new idea: that quantum mechanics has important consequences for machines that store and manipulate information. In particular, it appeared that quantum computers might be more powerful than classical computers. This opened up a new direction in com- puter science, and led to discoveries such as Shor’s algorithm for factoring and discrete logarithms [76], Grover’s algorithm for black-box search [39], and the first schemes for fault-tolerant quantum computation [75]. Since then, the field of quantum computation has developed rapidly, and there is considerable interest in building practical quantum computers and finding new quantum algorithms. (See [68] for a survey of this area, as it stood in 2000.) In this dissertation we study complexity classes based on quantum computation. On one hand, this is motivated by the possibility that we may eventually succeed in building scalable quantum computers