A Quantum Algorithm Framework for Discrete Probability Distributions with Applications to Rényi Entropy Estimation

Estimating statistical properties is fundamental in statistics and computer science. In this paper, we propose a unified quantum algorithm framework for estimating properties of discrete probability distributions, with estimating R\'enyi entropies as specific examples. In particular, given a quantum oracle that prepares an $n$-dimensional quantum state $\sum{i=1}^{n}\sqrt{p{i}}|i\rangle$, for $\alpha>1$ and $0<\alpha<1$, our algorithm framework estimates $\alpha$-R\'enyi entropy $H_{\alpha}(p)$ to within 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 $\epsilon$ as well as the joint dependence between $n$ and $1/\epsilon$. Technically, our quantum algorithms combine quantum singular value transformation, quantum annealing, and variable-time amplitude estimation. We believe that our algorithm framework is of general interest and has wide applications.