References & Citations
Computer Science > Computational Complexity
Title: The Acrobatics of BQP
(Submitted on 19 Nov 2021 (v1), last revised 24 Apr 2024 (this version, v4))
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, $\mathsf{\Sigma}_{k+1}^\mathsf{P}\not\subset\mathsf{BQP}^{\mathsf{\Sigma}_{k}^\mathsf{P}}$ 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.
Submission history
From: William Kretschmer [view email][v1] Fri, 19 Nov 2021 19:40:05 GMT (54kb,D)
[v2] Mon, 28 Feb 2022 21:32:26 GMT (62kb,D)
[v3] Fri, 7 Oct 2022 18:33:41 GMT (63kb,D)
[v4] Wed, 24 Apr 2024 20:05:03 GMT (65kb,D)
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