The Acrobatics of BQP
Scott Aaronson, DeVon Ingram, William Kretschmer
TL;DR
The paper addresses whether quantum computation can be tamed by classical relativization or by fixing randomness; by leveraging Raz–Tal’s Forrelation and new quantum-aware lower-bound tools, it shows a striking variety of oracle worlds where $\mathsf{BQP}$ acts independently from $\mathsf{P}$, $\mathsf{NP}$, and $\mathsf{PH}$. The authors introduce random-restriction techniques tailored to $\mathsf{QMA}$ queries and a block-sensitivity concentration for AC^0 circuits, plus sparse-oracle analogues of the Aaronson–Ambainis conjecture, to derive fine-grained separations. Among the key results are oracle separations such as $\mathsf{NP}^{\mathsf{BQP}} \not\subseteq \mathsf{BQP}^{\mathsf{NP}}$ and $\mathsf{P}=\mathsf{NP}$ but $\mathsf{BQP} \neq \mathsf{QCMA}$, as well as the existence of an oracle with $\mathsf{PH}$ infinite while $\mathsf{BQP}=\mathsf{P}^{\#P}$. These findings highlight the autonomy of quantum power under relativization and offer new tools that could inform nonrelativizing approaches to quantum supremacy and circuit lower bounds. The results extend the Raz–Tal program and open several directions for tighter control of $\mathsf{BQP}$ vs. $\mathsf{PH}$ and beyond.
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 ($\mathsf{BQP}$) 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, $\mathsfΣ_{k+1}^\mathsf{P}\not\subset\mathsf{BQP}^{\mathsfΣ_{k}^\mathsf{P}}$ 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.