Oracle separation of BQP and PH

Abstract

We present a distribution D over inputs in {−1,1}2N, such that: (1) There exists a quantum algorithm that makes one (quantum) query to the input, and runs in time O(logN), that distinguishes between D and the uniform distribution with advantage Ω(1/logN). (2) No Boolean circuit of quasi-polynomial size and constant depth distinguishes between D and the uniform distribution with advantage better than polylog(N)/√N.

By well known reductions, this gives a separation of the classes Promise-BQP and Promise-PH in the black-box model and implies an oracle O relative to which BQPO ⊈PHO.