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Estimating statistical properties is fundamental in statistics and computerscience. In this paper, we propose a unified quantum algorithm framework forestimating properties of discrete probability distributions, with estimatingR\'enyi entropies as specific examples. In particular, given a quantum oraclethat prepares an $n$-dimensional quantum state$\sum_{i=1}^{n}\sqrt{p_{i}}|i\rangle$, for $\alpha>1$ and $0<\alpha<1$, ouralgorithm framework estimates $\alpha$-R\'enyi entropy $H_{\alpha}(p)$ towithin additive error $\epsilon$ with probability at least $2/3$ using$\widetilde{\mathcal{O}}(n^{1-\frac{1}{2\alpha}}/\epsilon +\sqrt{n}/\epsilon^{1+\frac{1}{2\alpha}})$ and$\widetilde{\mathcal{O}}(n^{\frac{1}{2\alpha}}/\epsilon^{1+\frac{1}{2\alpha}})$queries, respectively. This improves the best known dependence in $\epsilon$ aswell as the joint dependence between $n$ and $1/\epsilon$. Technically, ourquantum algorithms combine quantum singular value transformation, quantumannealing, and variable-time amplitude estimation. We believe that ouralgorithm framework is of general interest and has wide applications.

A Unified Quantum Algorithm Framework for Estimating Properties of Discrete Probability Distributions