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 (e,δ)-approximation problem with an optimal dependence on the coefficient of variation of the input random variable.