Combinatorial Structures in Quantum Information University College London Joshua Lockhart 2 For Sarah. Declaration I, Joshua Lockhart confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. 3 4 My appreciation to Simone Severini for his advice, candour and friendship throughout the PhD. Thanks to Otfried Gühne for his insight on all things entanglement. Thanks to David Roberson and Laura Manˇcinska for their encouragement, mentorship, and hospitality. Many thanks also to Toby Cubitt for his open office door, and to Carlos González-Guillén for his supply of counter-examples to conjectures. Thanks to Scott Aaronson, Aram Harrow, Ashley Montanaro, Fernando Brandão, László Babai, Andreas Winter, Giannicola Scarpa, and Will Matthews for help- ful discussions. To all those I have shared an office with, Dan Dervovic, Dimitris Stamos, Giu- lia Luise, Alexander Botev, Nadish de Silva, Leonard Wossnig, Smudge the rabbit, Andrea Rochetto, James Watson, Ed Grant, Carlo Sparaciari, Thomas Galley, Alexandru Paraschiv and Octavio Zapata: thank you for all the good times. Thanks to Dan in particular for going above and beyond to maintain the cookie culture in Research Office 3.14. Big thanks to Cohort One of UCL’s Delivering Quantum Technologies CDT for your support and friendship. Finally, I thank Sarah Lockhart for her unwavering support over these four years. Contents 1 Introduction 13 1.1 Quantum preliminaries . 19 2 Quantum state isomorphism 25 2.1 Graph isomorphism . 28 2.2 Probabilistic and interactive proofs . 32 2.2.1 Proofs with private coins . 35 2.2.2 Proofs with public coins . 37 2.3 Isomorphisms of strings and states . 41 2.3.1 Permutations and S TRING I SOMORPHISM ........ 42 2.3.2 Permutations of quantum states and S TATE I SOMORPHISM 43 2.3.3 Quantum complexity theory . 45 2.3.4 Quantum interactive proofs and zero knowledge . 54 2.3.5 Statistical zero knowledge . 58 2.4 Summary . 63 3 Grid states 67 3.1 Chapter Overview . 69 3.2 Graphs and quantum entanglement . 73 3.2.1 Subsequent work . 78 3.2.2 Discussion of literature . 81 3.3 Preliminaries . 82 3.3.1 Grid-labelled graphs and grid-states . 83 3.3.2 Local isomorphism . 84 3.3.3 Local operations and classical communication . 88 5 6 CONTENTS 3.4 The Degree Criterion . 92 3.4.1 Extensions and LE-isomorphism . 99 3.4.2 Decompositions . 101 3.5 3 × 3 graphs that satisfy the degree criterion . 105 3.5.1 Edge contributions . 108 3.5.2 3 × 3 degree criterion with 2 diagonal edges . 111 3.5.3 3 × 3 degree criterion with 3 diagonal edges . 113 3.5.4 3 × 3 degree criterion with 4 and 5 diagonal edges . 114 3.5.5 Bound entanglement . 121 3.6 Matrix Realignment Criterion . 123 3.6.1 Realignment of combinatorial Laplacian matrices . 124 3.6.2 Failure of the matrix realignment criterion . 131 3.6.3 Applying the matrix realignment criterion to B4 and B5.. 135 3.7 Range criterion . 137 3.7.1 Row and column surgery . 137 3.7.2 Bound entangled graphs via the range criterion . 141 3.8 Summary . 149 4 Discussion 153 A Grid States Miscellany 181 A.1 Entanglement of B5 .......................... 181 A.2 Counting graphs that satisfy the degree criterion . 185 A.3 Proof of 6; 7; 8; 9 edge lemma . 191 Abstract This work is an exploration of how graphs and permutations can be applied in the context of quantum information processing. In Chapter 2 we consider problems about the permutations of the subsystems of a quantum system. Explicitly, we attempt to understand the problem of determining if two quantum states of N qubits are isomorphic: if one can be obtained from the other by permuting its subsystems. We show that the well known graph isomorphism problem is a special case of state isomorphism. We also show that the complement of state isomorphism, the problem of determining if two states are not isomorphic, can be verified by a quantum interactive proof system, and that this proof system can be made statistical zero knowledge. We also consider the complexity of isomorphism problems for stabilizer states, and mixed states. In Chapter 3 we work with a special class of quantum states called grid states, in an effort to develop a toy model for mixed state entanglement. The key idea with grid states is that they can be represented by what we call a grid-labelled graph, literally, a graph forced to have vertices on a two dimensional grid. We show that whether or not a grid state is entangled can sometimes be determined solely from the structural properties of its corresponding grid-labelled graph. We use the grid state framework to build families of bound entangled states, suggesting that even in this restricted setting detecting entanglement is non-trivial and will require more than a single entanglement criterion. 7 8 CONTENTS Impact Statement The thesis focuses on two ways in which mathematical tools more traditionally associated with classical computer science can be brought to bear on questions to do with quantum computing. Quantum computers are known to provide exponential speed up over the best known classical algorithms for a number of well known problems including integer factorisation and solving linear systems of equations. The first section is immersed in the theory of quantum computational complexity, with the aim of bringing to it fresh ideas from classical computer science. We consider a quantum generalisation of the graph isomorphism problem: the prob- lem of determining if two quantum states are equivalent up to rearrangements of their constituent subsystems. We show that this problem, which we call the state isomorphism problem, has a number of similarities to its classical counterpart in terms of its complexity classification. This work was presented as a poster at a number of quantum computing conferences, with highlights being QIP and TQC, suggesting that it is of interest to the broader theoretical quantum computing community. It is not clear which aspect of quantum mechanics is responsible for quantum algorithmic speed up. One candidate phenomena is quantum entanglement, the so called “spooky” way in which quantum particles can be correlated with one another, and affect each others state despite potentially being astronomically far apart. In this latter part of the thesis, we consider a special class of quantum states that can be represented by graphs on a grid. We show that properties relating to the entanglement of such a state can be deduced by considering structural properties of its corresponding graph. The work in this direction has resulted in a publication in Physical Review A. The work is intended to be inter-disciplinary, 9 10 CONTENTS with the aim of fostering collaboration between the quantum information and combinatorics communities. The work was presented as a talk at Eurocomb 2017 and the British Colloquium for Theoretical Computer Science. Complexity Class Glossary P: Polynomial time NP: Non-deterministic Polynomial time PSPACE: Polynomial Space BPP: Bounded error Probabilistic Polynomial time BQP: Bounded error Quantum Polynomial time QMA: Quantum Merlin Arthur QCMA: Quantum Merlin Arthur with classical certificate QAM: Quantum Arthur Merlin QCAM: Quantum Arthur Merlin with classical certificate IP: Interactive Proofs QIP: Quantum Interactive Proofs QSZK: Quantum Statistical Zero Knowledge QCSZK: Quantum Statistical Zero Knowledge with classical communication MA: Merlin Arthur AM: Arthur Merlin 11 12 CONTENTS Chapter 1 Introduction A computer that can access advanced processing power by tapping into the realm of quantum physics is an idea that could have come straight from the pages of a science fiction novel. However, at the time of writing a Canadian startup have received $9 million in funding to investigate quantum machine learning [1], China and the USA are engaged in what Bloomberg calls a “Quantum Computing Arms Race” [2], and Microsoft have just released a Visual Studio plugin for Q#, their quantum programming language [3]. It appears that science fiction is well on its way to becoming fact. Despite this recent explosion of activity, this field is not a new one. The idea of a quantum computer is rooted in Feynman’s 1981 talk “Simulating Physics with Computers” [4], in which he proposes that in order to provide an accurate simulation of a quantum mechanical system, a computer must operate according to quantum laws. Four years later, David Deutsch mapped out some of the the- oretical underpinnings of such a computing device [5], before showing in 1992 with Richard Jozsa [6] that they could outperform classical computers at certain, rather contrived, tasks. It wasn’t until 1994, when Peter Shor [7] came up with a quantum algorithm that could efficiently find the prime factors of large numbers, that the broader scientific community started to pay attention. Here was a new kind of computing device that, if built, would allow its users to break the RSA cryptosystem [8], built on the assumption that this exact task was impractical. RSA formed, and 13 14 CHAPTER 1. INTRODUCTION still forms, the backbone of secure Internet communications, so the implications of such a device being built would be catastrophic. Another key result came in 1996, when Lov Grover [9] showed that quantum computers are inherently better than classical computers at a foundational problem in information processing: searching through unordered lists of data. Suppose you have a list of N elements, and you are tasked with finding the location of a particular item on that list. In the worst case, the marked item could be the last element: this would require you to check all entries of the list, O(N) queries.