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

Abstract

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 ${\sum }_{i=1}^n\sqrt{p_i}|i⟩$, for $\alpha >1$ and $0<\alpha <1$, our algorithm framework estimates $\alpha$-R\'enyi entropy $H_{\alpha }(p)$ to within additive error $ϵ$ 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 $\tilde{\mathcal{O}}(n^{\frac{1}{2\alpha }}/{ϵ}^{1+\frac{1}{2\alpha }})$ queries, respectively. This improves the best known dependence in $ϵ$ as well as the joint dependence between $n$ and $1/ϵ$. 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.