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URN: urn:nbn:de:0030-drops-146318
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### Quantum Sub-Gaussian Mean Estimator

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### Abstract

We present a new quantum algorithm for estimating the mean of a real-valued random variable obtained as the output of a quantum computation. Our estimator achieves a nearly-optimal quadratic speedup over the number of classical i.i.d. samples needed to estimate the mean of a heavy-tailed distribution with a sub-Gaussian error rate. This result subsumes (up to logarithmic factors) earlier works on the mean estimation problem that were not optimal for heavy-tailed distributions [Brassard et al., 2002; Brassard et al., 2011], or that require prior information on the variance [Heinrich, 2002; Montanaro, 2015; Hamoudi and Magniez, 2019]. As an application, we obtain new quantum algorithms for the (ε,δ)-approximation problem with an optimal dependence on the coefficient of variation of the input random variable.

### BibTeX - Entry

@InProceedings{hamoudi:LIPIcs.ESA.2021.50,
author =	{Hamoudi, Yassine},
title =	{{Quantum Sub-Gaussian Mean Estimator}},
booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
pages =	{50:1--50:17},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-204-4},
ISSN =	{1868-8969},
year =	{2021},
volume =	{204},
editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address =	{Dagstuhl, Germany},
URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14631},
URN =		{urn:nbn:de:0030-drops-146318},
doi =		{10.4230/LIPIcs.ESA.2021.50},
annote =	{Keywords: Quantum algorithm, statistical analysis, mean estimator, sub-Gaussian estimator, (\epsilon,\delta)-approximation, lower bound}
}

 Keywords: Quantum algorithm, statistical analysis, mean estimator, sub-Gaussian estimator, (ε,δ)-approximation, lower bound Seminar: 29th Annual European Symposium on Algorithms (ESA 2021) Issue date: 2021 Date of publication: 31.08.2021

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