Oracle Separation of BQP and PH

Ran Raz, Avishay Tal
2022 Journal of the ACM  
We present a distribution \({\mathcal {D}} \) over inputs in { ± 1} 2 N , 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 \({\mathcal {D}} \) and the uniform distribution with advantage \(\Omega(1/\log N)\) . (2) No Boolean circuit of quasi-polynomial size and constant depth distinguishes between \({\mathcal {D}} \) and the uniform distribution with advantage better than
more » ... sqrt {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 .
doi:10.1145/3530258 fatcat:s5e7xuaplrh5zj4p3fxwsholgu