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URN: urn:nbn:de:0030-drops-28603
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### New Results on Quantum Property Testing

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### 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},