Quantum Proofs

Quantum information and computation provide a fascinating twist on the notion of proofs in computational complexity theory. For instance, one may consider a quantum computational analogue of the complexity class \class{NP}, known as QMA, in which a quantum state plays the role of a proof (also called a certificate or witness), and is checked by a polynomial-time quantum computation. For some problems, the fact that a quantum proof state could be a superposition over exponentially many classical states appears to offer computational advantages over classical proof strings. In the interactive proof system setting, one may consider a verifier and one or more provers that exchange and process quantum information rather than classical information during an interaction for a given input string, giving rise to quantum complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit some properties from their classical counterparts, they also possess distinct and uniquely quantum features that lead to an interesting landscape of complexity classes based on variants of this model. In this survey we provide an overview of many of the known results concerning quantum proofs, computational models based on this concept, and properties of the complexity classes they define. In particular, we discuss non-interactive proofs and the complexity class QMA, single-prover quantum interactive proof systems and the complexity class QIP, statistical zero-knowledge quantum interactive proof systems and the complexity class \class{QSZK}, and multiprover interactive proof systems and the complexity classes QMIP, QMIP*, and MIP*.Comment: Survey published by NOW publisher

Extended Nonlocal Games

The notions of entanglement and nonlocality are among the most striking ingredients found in quantum information theory. One tool to better understand these notions is the model of nonlocal games; a mathematical framework that abstractly models a physical system. The simplest instance of a nonlocal game involves two players, Alice and Bob, who are not allowed to communicate with each other once the game has started and who play cooperatively against an adversary referred to as the referee. The focus of this thesis is a class of games called extended nonlocal games, of which nonlocal games are a subset. In an extended nonlocal game, the players initially share a tripartite state with the referee. In such games, the winning conditions for Alice and Bob may depend on outcomes of measurements made by the referee, on its part of the shared quantum state, in addition to Alice and Bob's answers to the questions sent by the referee. We build up the framework for extended nonlocal games and study their properties and how they relate to nonlocal games.Comment: PhD thesis, Univ Waterloo, 2017. 151 pages, 11 figure

Gaussian resource theories and semidefinite programming hierarchies for quantum information

Determining which quantum tasks we can perform with currently available tools and devices is one of the most important goals of quantum information science today. To achieve this requires careful investigation of the capability of current quantum tools as well as development of classical protocols which can assist quantum tasks and amplify their abilities. In this thesis, we approach this problem through two different topics in quantum information theory: Gaussian resource theories and semidefinite programming hierarchies. In the first part of this thesis, we examine the possibility of implementing quantum information processing tasks in the Gaussian platform through the eyes of quantum resource theories. Gaussian states and operations are primary tools for the study of continuous-variable quantum information processing due to their easy accessibility and concise mathematical descriptions, although it has been discovered that they are subject to a number of limitations for advanced quantum information processing tasks. We explore the capability of the Gaussian platform further in the first part of this thesis. Firstly, we investigate whether introducing convex structure to the Gaussian framework can circumvent the known no-go theorem of Gaussian resource distillation. Surprisingly, we find that resource distillation becomes possible — albeit in a limited fashion — when convexity is introduced. Then, we consider the quantum resource theory of Gaussian thermal operations when catalysts are allowed, and examine the abilities of catalytic Gaussian thermal operations by characterising all possible state transformations under them. In the second part of this thesis, we address the problem of characterising quantum cor- relations via semidefinite programming hierarchies. In particular, we focus on characterising quantum correlations of fixed dimension, which is practically relevant to the field of semi- device-independent quantum information processing. Semidefinite programming is a special type of mathematical optimisation, and it is known that some important but difficult problems in quantum information theory admit semidefinite programming relaxations; these include the characterisation of general quantum correlations in the context of non-locality and the distinction of quantum separable states from entangled states. In this second part, we show how to construct a hierarchy of semidefinite programming relaxations for quantum correlations of fixed dimension and derive analytical bounds on the convergence speed of the hierarchy. For the proof, we make a connection to a variant of quantum separability problem and employ multipartite quantum de Finetti theorems with linear constraints.Open Acces

Improving Efficiency and Scalability of Sum of Squares Optimization: Recent Advances and Limitations

It is well-known that any sum of squares (SOS) program can be cast as a semidefinite program (SDP) of a particular structure and that therein lies the computational bottleneck for SOS programs, as the SDPs generated by this procedure are large and costly to solve when the polynomials involved in the SOS programs have a large number of variables and degree. In this paper, we review SOS optimization techniques and present two new methods for improving their computational efficiency. The first method leverages the sparsity of the underlying SDP to obtain computational speed-ups. Further improvements can be obtained if the coefficients of the polynomials that describe the problem have a particular sparsity pattern, called chordal sparsity. The second method bypasses semidefinite programming altogether and relies instead on solving a sequence of more tractable convex programs, namely linear and second order cone programs. This opens up the question as to how well one can approximate the cone of SOS polynomials by second order representable cones. In the last part of the paper, we present some recent negative results related to this question.Comment: Tutorial for CDC 201