Quantum query complexity through the ens of communication complexity and exact learning

Abstract

Query complexity, both classical and quantum, are important and well-established notions of computation. In this thesis, we will investigate query complexity in both classical and quantum worlds and study their relationship with other closely related models of computation. Other than the query model, we will consider a few models of computation, namely model of computational learning, communications model and local query model for graphs. We will try to understand the power of these models for various classes of functions. We will be investigating how various complexity measures (defined as per different models of computation) relate to each other. In this endeavor we will be working both upper bound (i.e. designing efficient algorithms in the model of interest) and lower bound (i.e. proving hardness of computing interesting functions in the model of interest). As a part of our investigation, we use various mathematical tools like Fourier analysis, linear algebra, and geometry. Some questions that have motivated the work in this thesis are: • How does structural simplicity affect its computational complexity in various models of computation? Does Fourier analytical simplicity lead to easier computation? Does geometric simplicity imply efficient communication? • How does the behavior of quantum computing differ from its classical counterpart? What is the relation between classical and quantum query complexity of learning a function? Does the same relationship between a pair of classical complexity measures also hold in the quantum world? Understanding the complexity measures that we study and the relationship between these measures has been an ongoing work for several decades. In this thesis, we push the boundary of our knowledge a bit further. The results in this thesis have been divided into three parts. Part I In this part, we study quantum query complexity from the view of quantum learning theory. First, we first study the model of exact learning using uniform quantum examples. Each such quantum example can be generated by making one query to the function being learned. In this model we are interested in learning the class of Boolean functions whose Fourier sparsity is bounded, that is, the class of Fourier-sparse Boolean functions. For this class of functions we give a quantum learning algorithm which improves upon the tight classical algorithm by Haviv and Regev, 2016. Then, we consider the model of exact active learning. In this model, a learner is given access to the function via membership queries. We study the relationship between the number of classical and quantum membership queries needed to exactly learn a class of Boolean functions, where improve upon the previous best known relationship by Servedio and Gortler, 2004. Finally, we study Chang’s lemma, a fundamental result in additive combinatorics. Our main result here is a refinement of Chang’s lemma for Fourier-sparse Boolean functions. We also investigate how this lemma is connected to quantum learning theory. Part II This part is dedicated to the study of the relation between quantum query and quantum communication complexity. It is well known, in the classical world, that a query algorithm for a function can be simulated to give a communication protocol of a closely related communication problem with only a constant overhead. The best known such simulation theorem in the quantum world, due to Buhrman, Cleve and Wigderson, 1998, has an overhead that is logarithmic in input size of the function. We construct the first total Boolean function that witnesses this logarithmic gap. This closes a long line of work. We also give a general recipe of constructing functions that witness separation between quantum query-to-communication simulation. Finally, we explore the role of symmetry on this simulation problem. Part III This part of the thesis is devoted to classical query and communication complexity. In the first chapter of this part, we explore the role of geometric simplicity, quantified by bounded Vapnik–Chervonenkis Dimension (VC Dimension) of set systems, in communication complexity. Our work is motivated by the work of H˚astad and Wigderson, 2007, who considered the Disjointness problem, a canonical problem in communication complexity, when inputs to the problem are promised to be sets of bounded cardinality. Indeed, set systems of bounded VC Dimension are a generalization of set systems with bounded cardinality, with a geometric motivation. Our main result here it to show that geometric simplicity does not imply efficient communication. In the second chapter of this part, we consider the problem of estimating the size of the global minimum cut in the local query model, a fundamental and well-studies model in graph property testing. We give an algorithm for this problem which almost matches the known lower bound by Eden and Rosenbaum, 2018. This resolves the query complexity of estimating global minimum cut in the local query model.