On the Complexity of Zero Gap MIP*

The class $\mathsf{MIP}^*$ is the set of languages decidable by multiprover interactive proofs with quantum entangled provers. It was recently shown by Ji, Natarajan, Vidick, Wright and Yuen that $\mathsf{MIP}^*$ is equal to $\mathsf{RE}$, the set of recursively enumerable languages. In particular this shows that the complexity of approximating the quantum value of a non-local game $G$ is equivalent to the complexity of the Halting problem. In this paper we investigate the complexity of deciding whether the quantum value of a non-local game $G$ is exactly $1$. This problem corresponds to a complexity class that we call zero gap $\mathsf{MIP}^*$, denoted by $\mathsf{MIP}^*_0$, where there is no promise gap between the verifier's acceptance probabilities in the YES and NO cases. We prove that $\mathsf{MIP}^*_0$ extends beyond the first level of the arithmetical hierarchy (which includes $\mathsf{RE}$ and its complement $\mathsf{coRE}$), and in fact is equal to $\Pi_2^0$, the class of languages that can be decided by quantified formulas of the form $\forall y \, \exists z \, R(x,y,z)$. Combined with the previously known result that $\mathsf{MIP}^{co}_0$ (the commuting operator variant of $\mathsf{MIP}^*_0$) is equal to $\mathsf{coRE}$, our result further highlights the fascinating connection between various models of quantum multiprover interactive proofs and different classes in computability theory.Comment: Fixed typos and edited protocol to more smoothly follow from reference

Repetition in Words

The main topic of this thesis is combinatorics on words. The field of combinatorics on words dates back at least to the beginning of the 20th century when Axel Thue constructed an infinite squarefree sequence over a ternary alphabet. From this celebrated result also emerged the subfield of repetition in words which is the main focus of this thesis. One basic tool in the study of repetition in words is the iteration of morphisms. In Chapter 1, we introduce this tool among other basic notions. In Chapter 2, we see applications of iterated morphisms in several examples. The second half of the chapter contains a survey of results concerning Dejean's conjecture. In Chapter 3, we generalize Dejean's conjecture to circular factors. We see several applications of iterated morphism in this chapter. We continue our study of repetition in words in Chapter 4, where we study the length of the shortest repetition-free word in regular languages. Finally, in Chapter 5, we conclude by presenting a number of open problems

Some Aspects of Noncommutativity in Polynomial Optimization

Most combinatorial optimization problems from theoretical computer science have a natural framing as optimization of polynomials in commuting variables. Noncommutativity is one of the defining features of quantum mechanics. So it is not surprising that noncommutative polynomial optimization plays an equally important role in quantum computer science. Our main goal here is to understand the relative hardness of commutative versus noncommutative polynomial optimization. At a first glance it might seem that noncommutative polynomial optimization must be more complex. However this is not always true and this question of relative hardness is substantially more subtle than might appear at the outset. First in this thesis we show that the general noncommutative polynomial optimization is complete for the class $\Pi_2$; this class is in the second level of the arithmetical hierarchy and strictly contains both the set of recursively enumerable languages and its complement. On the other hand, commutative polynomial optimization is decidable and belongs to \PSPACE. We then provide evidence that for polynomials arising from a large class of constraint satisfaction problems the situation is reversed: the noncommutative polynomial optimization is an easier computational problem compared to its commutative analogue. A second question we are interested in is about whether we could extract good commutative solutions from noncommutative solutions? This brings us to the second theme of this thesis which is about understanding the algebraic structure of the solutions of noncommutative polynomial optimization. We show that this structural insight then could shed light on the optimal commutative solutions and thereby paves the path in understanding the relationships between the commutative and noncommutative solutions. Here we first use the sum-of-squares framework to understand the algebraic relationships that are present between operators in any optimal noncommutative solution of a class of polynomial optimization problems arising from certain constraint satisfaction problems. We then show how we can design approximation algorithms for these problems so that some algebraic structures of our choosing is present. Finally we propose a rounding scheme for extracting good commutative solutions from noncommutative ones

A generalization of CHSH and the algebraic structure of optimal strategies

Self-testing has been a rich area of study in quantum information theory. It allows an experimenter to interact classically with a black box quantum system and to test that a specific entangled state was present and a specific set of measurements were performed. Recently, self-testing has been central to high-profile results in complexity theory as seen in the work on entangled games PCP of Natarajan and Vidick (FOCS 2018), iterated compression by Fitzsimons et al. (STOC 2019), and NEEXP in MIP* due to Natarajan and Wright (FOCS 2019). In this work, we introduce an algebraic generalization of CHSH by viewing it as a linear constraint system (LCS) game, exhibiting self-testing properties that are qualitatively different. These provide the first example of non-local games that self-test non-Pauli operators resolving an open questions posed by Coladangelo and Stark (QIP 2017). Our games also provide a self-test for states other than the maximally entangled state, and hence resolves the open question posed by Cleve and Mittal (ICALP 2012). Additionally, our games have $1$ bit question and $\log n$ bit answer lengths making them suitable candidates for complexity theoretic application. This work is the first step towards a general theory of self-testing arbitrary groups. In order to obtain our results, we exploit connections between sum of squares proofs, non-commutative ring theory, and the Gowers-Hatami theorem from approximate representation theory. A crucial part of our analysis is to introduce a sum of squares framework that generalizes the \emph{solution group} of Cleve, Liu, and Slofstra (Journal of Mathematical Physics 2017) to the non-pseudo-telepathic regime. Finally, we give the first example of a game that is not a self-test. Our results suggest a richer landscape of self-testing phenomena than previously considered.Comment: Incorporated reviewers comments and fixed typo

Decision algorithms for Fibonacci-automatic Words, I: Basic results

We implement a decision procedure for answering questions about a class of infinite words that might be called (for lack of a better name) “Fibonacci-automatic”. This class includes, for example, the famous Fibonacci word f = f0f1f2··· = 01001010··· , the fixed point of the morphism 0 → 01 and 1 → 0. We then recover many results about the Fibonacci word from the literature (and improve some of them), such as assertions about the occurrences in f of squares, cubes, palindromes, and so forth

Decision algorithms for Fibonacci-automatic Words, I: Basic results

We implement a decision procedure for answering questions about a class of infinite words that might be called (for lack of a better name) “Fibonacci-automatic”. This class includes, for example, the famous Fibonacci word f = f0f1f2··· = 01001010··· , the fixed point of the morphism 0 → 01 and 1 → 0. We then recover many results about the Fibonacci word from the literature (and improve some of them), such as assertions about the occurrences in f of squares, cubes, palindromes, and so forth