Quantum hedging in two-round prover-verifier interactions

We consider the problem of a particular kind of quantum correlation that arises in some two-party games. In these games, one player is presented with a question they must answer, yielding an outcome of either 'win' or 'lose'. Molina and Watrous (arXiv:1104.1140) studied such a game that exhibited a perfect form of hedging, where the risk of losing a first game can completely offset the corresponding risk for a second game. This is a non-classical quantum phenomenon, and establishes the impossibility of performing strong error-reduction for quantum interactive proof systems by parallel repetition, unlike for classical interactive proof systems. We take a step in this article towards a better understanding of the hedging phenomenon by giving a complete characterization of when perfect hedging is possible for a natural generalization of the game in arXiv:1104.1140. Exploring in a different direction the subject of quantum hedging, and motivated by implementation concerns regarding loss-tolerance, we also consider a variation of the protocol where the player who receives the question can choose to restart the game rather than return an answer. We show that in this setting there is no possible hedging for any game played with state spaces corresponding to finite-dimensional complex Euclidean spaces.Comment: 34 pages, 1 figure. Added work on connections with other result

Quantum Turing Machines and Quantum Prover-Verifier Interactions

We present results on quantum Turing machines and on prover-verifier interactions. In our work on quantum Turing machines, we continue the line of research opened by Yao (1993), who proved that quantum Turing machines and quantum circuits are polynomially equivalent computational models: t ≥ n steps of a quantum Turing machine running on an input of length n can be simulated by a uniformly generated family of quantum circuits with size quadratic in t, and a polynomial-time uniformly generated family of quantum circuits can be simulated by a quantum Turing machine running in polynomial time. We then first revisit the simulation of quantum Turing machines with uniformly generated quantum circuits, and present a variation on the simulation method employed by Yao together with an analysis of it. This analysis reveals that the simulation of quantum Turing machines can be performed by quantum circuits having depth linear in t, rather than quadratic depth, and can be extended easily to many variants of quantum Turing machines, such as ones having multi-dimensional tapes. Our analysis is based on an extension of a method of Arrighi, Nesme, and Werner (2011) that allows for the localization of causal unitary evolutions, involving abstract lemmas that might be of independent interest. We also consider the more complex extension of our variant to the circuit simulation of multi-tape quantum Turing machines, where our variant provides a circuit with O(t^k) size and O(t^{k-1}) depth for the simulation of t steps of a machine with k tapes. This can be contrasted with the O(t^{k}) depth corresponding to the generalization of Yao's simulation by Nishimura and Ozawa (2002). Our usage of abstract techniques regarding the localization of causal unitary evolutions allows again for a simplification of the algebraic manipulation aspects of the construction. We also discuss the further extension to the case of oracle quantum Turing machines. In our work on prover-verifier interactions, we first consider a protocol under the name of perfect/conclusive quantum state exclusion. This means to be able to discard with certainty at least one out of n possible quantum state preparations by performing a measurement of the resulting state. When all the preparations correspond to pure states and there are no more of them than their common dimension, it is an open problem whether POVMs give any additional power for this task with respect to projective measurements. This is the case even for the simple case of three states in three dimensions, which is discussed by Caves, Fuchs and Schack (2002) as unsuccessfully tackled. In our work, we give an analytical proof that in this case POVMs do indeed not give any additional power with respect to projective measurements. We also discuss possible generalizations of our work, including an application of Quadratically Constrained Quadratic Programming that might be of special interest. We additionally consider the problem of quantum hedging, a particular kind of quantum correlation that arises between parallel instances of prover-verifier interactions. M. and Watrous (2012) studied a protocol that exhibited a perfect form of quantum hedging, where the risk for the prover of losing a first game can completely offset the corresponding risk for a second game. We take a step towards a better understanding of this hedging phenomenon by giving a characterization of the prover's optimal behavior for a natural generalization of this protocol. Furthermore, we discuss how the usage of the logarithmic utility principle to analyze prover-verifier interactions could justify further study of quantum hedging

Quantum Coin Hedging, and a Counter Measure

A quantum board game is a multi-round protocol between a single quantum player against the quantum board. Molina and Watrous discovered quantum hedging. They gave an example for perfect quantum hedging: a board game with winning probability < 1, such that the player can win with certainty at least 1-out-of-2 quantum board games played in parallel. Here we show that perfect quantum hedging occurs in a cryptographic protocol - quantum coin flipping. For this reason, when cryptographic protocols are composed, hedging may introduce serious challenges into their analysis. We also show that hedging cannot occur when playing two-outcome board games in sequence. This is done by showing a formula for the value of sequential two-outcome board games, which depends only on the optimal value of a single board game; this formula applies in a more general setting, in which hedging is only a special case

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

Quantum Speed-ups for Boolean Satisfiability and Derivative-Free Optimization

In this thesis, we have considered two important problems, Boolean satisfiability (SAT) and derivative free optimization in the context of large scale quantum computers. In the first part, we survey well known classical techniques for solving satisfiability. We compute the approximate time it would take to solve SAT instances using quantum techniques and compare it with state-of-the heart classical heuristics employed annually in SAT competitions. In the second part of the thesis, we consider a few classically well known algorithms for derivative free optimization which are ubiquitously employed in engineering problems. We propose a quantum speedup to this classical algorithm by using techniques of the quantum minimum finding algorithm. In the third part of the thesis, we consider practical applications in the fields of bio-informatics, petroleum refineries and civil engineering which involve solving either satisfiability or derivative free optimization. We investigate if using known quantum techniques to speedup these algorithms directly translate to the benefit of industries which invest in technology to solve these problems. In the last section, we propose a few open problems which we feel are immediate hurdles, either from an algorithmic or architecture perspective to getting a convincing speedup for the practical problems considered