Abstract
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 P=NP but BQP≠QCMA. -Conversely, there exists an oracle relative to which BQP^NP⊂PH^BQP. -Relative to a random oracle, PP=PostBQP is not contained in the "QMA hierarchy" QMA^QMA^QMA^…. -Relative to a random oracle, Σ_k+1P⊂BQP^Σ_kP for every k. -There exists an oracle relative to which BQP=P^# P and yet PH is infinite. -There exists an oracle relative to which P=NP≠BQP=P^# 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 AC⁰ circuits, and a (provable) analogue of the Aaronson-Ambainis Conjecture for sparse oracles.