 
Summary: BQP and the Polynomial Hierarchy
Scott Aaronson
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 nonoracle 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 LinialNisan Conjecture, which we formulate here and which is likely of
independent interest. The original LinialNisan Conjecture (about pseudorandomness against
constantdepth circuits) was recently proved by Braverman, after being open for twenty years.
Contents
1 Introduction 2
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Our Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 In Defense of Oracles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
