The acrobatics of BQP

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 NPBQP ⊄ BQPPH, 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 BQPNP ⊄ PHBQP. • Relative to a random oracle, PP is not contained in the "QMA hierarchy" QMAQMAQMA…. • Relative to a random oracle, [EQUATION] 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 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.