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 ($\mathsf{BQP}$) can be remarkably decoupled from that of classical complexity classes like $\mathsf{NP}$. Specifically: -There exists an oracle relative to which $\mathsf{NP^{BQP}}\not\subset\mathsf{BQP^{PH}}$, resolving a 2005 problem of Fortnow. As a corollary, there exists an oracle relative to which $\mathsf{P}=\mathsf{NP}$ but $\mathsf{BQP}\neq\mathsf{QCMA}$. -Conversely, there exists an oracle relative to which $\mathsf{BQP^{NP}}\not\subset\mathsf{PH^{BQP}}$. -Relative to a random oracle, $\mathsf{PP}=\mathsf{PostBQP}$ is not contained in the "$\mathsf{QMA}$ hierarchy" $\mathsf{QMA}^{\mathsf{QMA}^{\mathsf{QMA}^{\cdots}}}$. -Relative to a random oracle, $\mathsf{\Sigma}_{k+1}^\mathsf{P}\not\subset\mathsf{BQP}^{\mathsf{\Sigma}_{k}^\mathsf{P}}$ for every $k$. -There exists an oracle relative to which $\mathsf{BQP}=\mathsf{P^{\# P}}$ and yet $\mathsf{PH}$ is infinite. -There exists an oracle relative to which $\mathsf{P}=\mathsf{NP}\neq\mathsf{BQP}=\mathsf{P^{\# P}}$. To achieve these results, we build on the 2018 achievement by Raz and Tal of an oracle relative to which $\mathsf{BQP}\not \subset \mathsf{PH}$, 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 $\mathsf{AC^0}$ circuits, and a (provable) analogue of the Aaronson-Ambainis Conjecture for sparse oracles.

翻译：我们可以固定随机化算法使用的随机性，但不存在类似的方法来固定量子算法使用的量子性。为强调这一根本差异，我们证明在黑盒设定下，量子多项式时间类（$\mathsf{BQP}$）的行为可与经典复杂度类（如$\mathsf{NP}$）显著解耦。具体而言：-存在一个预言机使得 $\mathsf{NP^{BQP}}\not\subset\mathsf{BQP^{PH}}$，解决了Fortnow 2005年的问题。作为推论，存在一个预言机使得 $\mathsf{P}=\mathsf{NP}$ 但 $\mathsf{BQP}\neq\mathsf{QCMA}$。-反之，存在一个预言机使得 $\mathsf{BQP^{NP}}\not\subset\mathsf{PH^{BQP}}$。-相对于随机预言机，$\mathsf{PP}=\mathsf{PostBQP}$ 不包含在“$\mathsf{QMA}$ 层级” $\mathsf{QMA}^{\mathsf{QMA}^{\mathsf{QMA}^{\cdots}}}$ 中。-相对于随机预言机，对每个 $k$ 有 $\mathsf{\Sigma}_{k+1}^\mathsf{P}\not\subset\mathsf{BQP}^{\mathsf{\Sigma}_{k}^\mathsf{P}}$。-存在一个预言机使得 $\mathsf{BQP}=\mathsf{P^{\# P}}$ 但 $\mathsf{PH}$ 是无穷的。-存在一个预言机使得 $\mathsf{P}=\mathsf{NP}\neq\mathsf{BQP}=\mathsf{P^{\# P}}$。为达成这些结果，我们基于2018年Raz和Tal关于存在预言机使 $\mathsf{BQP}\not \subset \mathsf{PH}$ 的成果及相关Forrelation问题的研究，并引入了可能具有独立意义的新工具，包括随机限制方法的“量子感知”版本、$\mathsf{AC^0}$ 电路块敏感性的集中定理，以及（可证明的）稀疏预言机下Aaronson-Ambainis猜想的类比。