Quantum Sub-Gaussian Mean Estimator

Author Yassine Hamoudi

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Yassine Hamoudi
  • Université de Paris, IRIF, CNRS, F-75013 Paris, France


The authors want to thank the anonymous referees for their valuable comments and suggestions which helped to improve this paper.

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Yassine Hamoudi. Quantum Sub-Gaussian Mean Estimator. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 50:1-50:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Quantum algorithm
  • statistical analysis
  • mean estimator
  • sub-Gaussian estimator
  • (ε,δ)-approximation
  • lower bound


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