BQP and the Polynomial Hierarchy

TL;DR

This paper provides evidence that quantum computers can solve certain problems outside the polynomial hierarchy by relating quantum complexity to circuit complexity and pseudorandomness, including new oracle separations.

Contribution

It introduces an oracle relation problem separating BQP from PH and formulates the Generalized Linial-Nisan Conjecture linking BQP and PH separations.

Findings

Existence of an oracle relation problem solvable in BQP but not in PH

Existence of a quantum logarithmic time problem not in AC0

Formulation of the Generalized Linial-Nisan Conjecture

Abstract

The relationship between BQP and PH has been an open problem since the earliest days of quantum computing. We present evidence that quantum computers can solve problems outside the entire polynomial hierarchy, by relating this question to topics in circuit complexity, pseudorandomness, and Fourier analysis. First, we show that there exists an oracle relation problem (i.e., a problem with many valid outputs) that is solvable in BQP, but not in PH. This also yields a non-oracle relation problem that is solvable in quantum logarithmic time, but not in AC0. Second, we show that an oracle decision problem separating BQP from PH would follow from the Generalized Linial-Nisan Conjecture, which we formulate here and which is likely of independent interest. The original Linial-Nisan Conjecture (about pseudorandomness against constant-depth circuits) was recently proved by Braverman, after being open for twenty years.