Oracle Separation of BQP and PH

Abstract

We present a distribution 𝓓 over inputs in {± 1}^2N, such that: (1) There exists a quantum algorithm that makes one (quantum) query to the input, and runs in time O(log N), that distinguishes between 𝓓 and the uniform distribution with advantage Ω (1/log N). (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 relative to which BQP is not contained in PH.