License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.TQC.2016.7
URN: urn:nbn:de:0030-drops-66882
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6688/
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Ben-David, Shalev

The Structure of Promises in Quantum Speedups

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LIPIcs-TQC-2016-7.pdf (0.5 MB)

Abstract

In 1998, Beals, Buhrman, Cleve, Mosca, and de Wolf showed that no super-polynomial quantum speedup is possible in the query complexity setting unless there is a promise on the input. We examine several types of "unstructured" promises, and show that they also are not compatible with super-polynomial quantum speedups. We conclude that such speedups are only possible when the input is known to have some structure.

Specifically, we show that there is a polynomial relationship of degree 18 between D(f) and Q(f) for any Boolean function f defined on permutations (elements of [n]^n in which each alphabet element occurs exactly once). More generally, this holds for all f defined on orbits of the symmetric group action (which acts on an element of [M]^n by permuting its entries). We also show that any Boolean function f defined on a "symmetric" subset of the Boolean hypercube has a polynomial relationship between R(f) and Q(f) - although in that setting, D(f) may be exponentially larger.

BibTeX - Entry

@InProceedings{bendavid:LIPIcs:2016:6688,
  author =	{Shalev Ben-David},
  title =	{{The Structure of Promises in Quantum Speedups}},
  booktitle =	{11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016)},
  pages =	{7:1--7:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-019-4},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{61},
  editor =	{Anne Broadbent},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6688},
  URN =		{urn:nbn:de:0030-drops-66882},
  doi =		{10.4230/LIPIcs.TQC.2016.7},
  annote =	{Keywords: Quantum computing, quantum query complexity, decision tree complexity, lower bounds, quantum adversary method}
}

Keywords: Quantum computing, quantum query complexity, decision tree complexity, lower bounds, quantum adversary method
Collection: 11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016)
Issue Date: 2016
Date of publication: 22.09.2016

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