On Quantum Computing and Pseudorandomness

Abstract

The relationship between efficient verification and quantum computing is one of the most important and least well-understood questions in the theory of computation. In particular, is there a problem that can be solved efficiently on a quantum computer that cannot be verified? In this thesis we give evidence that BQP ⊄ PH, relating the classes of languages decidable with a quantum computer to a generalization of NP. In so doing we connect a question in pseudorandomness, first studied [BSW03] to the problem of finding an oracle relative to which BQP ⊄ PH. The primary technical challenge is to construct a unitary matrix, realized by an efficient quantum circuit and whose rows are supported on nearly disjoint subsets. Using this matrix and assuming the validity of the aforementioned question in pseudorandomness, we show an instantiation of the Nisan-Wigderson pseudorandom generator that can be broken with quantum computers, but not with the relevant mode of classical computation.