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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FSTTCS.2010.145
URN: urn:nbn:de:0030-drops-28603
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2010/2860/
Chakraborty, Sourav ;
Fischer, Eldar ;
Matsliah, Arie ;
de Wolf, Ronald
New Results on Quantum Property Testing
Abstract
We present several new examples of speed-ups obtainable by quantum algorithms in the context of property testing.
First, motivated by sampling algorithms, we consider probability distributions given in the form of an oracle $f:[n]\to[m]$. Here the probability $P_f(j)$ of an outcome $j$ in $[m]$ is the fraction of its domain that $f$ maps to $j$. We give quantum algorithms for testing whether two such distributions are identical or $epsilon$-far in $L_1$-norm. Recently, Bravyi, Hassidim, and Harrow showed that if
$P_f$ and $P_g$ are both unknown (i.e., given by oracles $f$ and $g$), then this testing can be done in roughly $sqrt{m}$ quantum queries to the functions. We consider the case where the second distribution is known, and show that testing can be done with roughly $m^{1/3}$ quantum queries, which we prove to be essentially optimal. In contrast, it is known that classical testing algorithms need about $m^{2/3}$ queries in the unknown-unknown case and about $sqrt{m}$ queries in the known-unknown case. Based on this result, we also reduce the query complexity of graph isomorphism testers with quantum oracle access.
While those examples provide polynomial quantum speed-ups, our third example gives a much larger improvement (constant quantum queries vs polynomial classical queries) for the problem of testing periodicity, based on Shor's algorithm and a modification of a classical lower bound by Lachish and Newman. This provides an alternative to a recent constant-vs-polynomial speed-up due to Aaronson.
BibTeX - Entry
@InProceedings{chakraborty_et_al:LIPIcs:2010:2860,
author = {Sourav Chakraborty and Eldar Fischer and Arie Matsliah and Ronald de Wolf},
title = {{New Results on Quantum Property Testing}},
booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)},
pages = {145--156},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-23-1},
ISSN = {1868-8969},
year = {2010},
volume = {8},
editor = {Kamal Lodaya and Meena Mahajan},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2010/2860},
URN = {urn:nbn:de:0030-drops-28603},
doi = {10.4230/LIPIcs.FSTTCS.2010.145},
annote = {Keywords: quantum algorithm, property testing}
}
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Keywords: |
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quantum algorithm, property testing |
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Collection: |
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IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010) |
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Issue Date: |
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2010 |
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Date of publication: |
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14.12.2010 |
