License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2022.20
URN: urn:nbn:de:0030-drops-165820
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16582/
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Aaronson, Scott ; Ingram, DeVon ; Kretschmer, William

The Acrobatics of BQP

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LIPIcs-CCC-2022-20.pdf (0.8 MB)

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 ? = NP but BQP ≠ QCMA.
- Conversely, there exists an oracle relative to which BQP^{NP} ⊄ PH^{BQP}.
- Relative to a random oracle, PP is not contained in the "QMA hierarchy" QMA^{QMA^{QMA^{⋯}}}.
- Relative to a random oracle, Σ_{k+1}^? ⊄ BQP^{Σ_k^?} 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 ? = NP ≠ BQP = ?^#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.

BibTeX - Entry

@InProceedings{aaronson_et_al:LIPIcs.CCC.2022.20,
  author =	{Aaronson, Scott and Ingram, DeVon and Kretschmer, William},
  title =	{{The Acrobatics of BQP}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{20:1--20:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16582},
  URN =		{urn:nbn:de:0030-drops-165820},
  doi =		{10.4230/LIPIcs.CCC.2022.20},
  annote =	{Keywords: BQP, Forrelation, oracle separations, Polynomial Hierarchy, query complexity}
}

Keywords: BQP, Forrelation, oracle separations, Polynomial Hierarchy, query complexity
Collection: 37th Computational Complexity Conference (CCC 2022)
Issue Date: 2022
Date of publication: 11.07.2022

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