License: Creative Commons Attribution-NoDerivs 3.0 Unported license (CC BY-ND 3.0)
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DOI: 10.4230/LIPIcs.STACS.2010.2450
URN: urn:nbn:de:0030-drops-24502
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2010/2450/
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Bravyi, Sergey ; Harrow, Aram W. ; Hassidim, Avinatan

Quantum Algorithms for Testing Properties of Distributions

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Abstract

Suppose one has access to oracles generating samples from two unknown probability distributions $p$ and $q$ on some $N$-element set. How many samples does one need to test whether the two distributions are close or far from each other in the $L_1$-norm? This and related questions have been extensively studied during the last years in the field of property testing.

In the present paper we study quantum algorithms for testing properties of distributions. It is shown that the $L_1$-distance $\|p-q\|_1$ can be estimated with a constant precision using only $O(N^{1/2})$ queries in the quantum settings, whereas classical computers need $\Omega(N^{1-o(1)})$ queries. We also describe quantum algorithms for testing Uniformity and Orthogonality with query complexity $O(N^{1/3})$. The classical query complexity of these
problems is known to be $\Omega(N^{1/2})$. A quantum algorithm for testing Uniformity has been recently independently discovered
by Chakraborty et al. \cite{CFMW09}.

BibTeX - Entry

@InProceedings{bravyi_et_al:LIPIcs:2010:2450,
  author =	{Sergey Bravyi and Aram W. Harrow and Avinatan Hassidim},
  title =	{{Quantum Algorithms for Testing Properties of Distributions}},
  booktitle =	{27th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{131--142},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-16-3},
  ISSN =	{1868-8969},
  year =	{2010},
  volume =	{5},
  editor =	{Jean-Yves Marion and Thomas Schwentick},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2010/2450},
  URN =		{urn:nbn:de:0030-drops-24502},
  doi =		{10.4230/LIPIcs.STACS.2010.2450},
  annote =	{Keywords: Quantum computing, property testing, sampling}
}

Keywords: Quantum computing, property testing, sampling
Collection: 27th International Symposium on Theoretical Aspects of Computer Science
Issue Date: 2010
Date of publication: 09.03.2010

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