The Acrobatics of BQP

One can fix the randomness used by a randomized algorithm, but there is no analogous notion of fixing the quantumness used by a quantum algorithm. Underscoring this fundamental difference, we show that, in the black-box setting, the behavior of quantum polynomial-time (BQP) can be remarkably decoupled from that of classical complexity classes like NP. Specifically: - There exists an oracle relative to which NP^{BQP} ? BQP^{PH}, resolving a 2005 problem of Fortnow. As a corollary, there exists an oracle relative to which ? = NP but BQP ? QCMA. - Conversely, there exists an oracle relative to which BQP^{NP} ? PH^{BQP}. - Relative to a random oracle, PP is not contained in the "QMA hierarchy" QMA^{QMA^{QMA^{?}}}. - Relative to a random oracle, ?_{k+1}^? ? BQP^{?_k^?} for every k. - There exists an oracle relative to which BQP = P^#P and yet PH is infinite. (By contrast, relative to all oracles, if NP ? BPP, then PH collapses.) - There exists an oracle relative to which ? = NP ? BQP = ?^#P. To achieve these results, we build on the 2018 achievement by Raz and Tal of an oracle relative to which BQP ? PH, and associated results about the Forrelation problem. We also introduce new tools that might be of independent interest. These include a "quantum-aware" version of the random restriction method, a concentration theorem for the block sensitivity of AC? circuits, and a (provable) analogue of the Aaronson-Ambainis Conjecture for sparse oracles