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 Rényi 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 $α>1$ and $0<α<1$, our algorithm framework estimates $α$-Rényi entropy $H_α(p)$ to within additive error $ε$ with probability at least $2/3$ using $\widetilde{\mathcal{O}}(n^{1-\frac{1}{2α}}/ε+ \sqrt{n}/ε^{1+\frac{1}{2α}})$ and $\widetilde{\mathcal{O}}(n^{\frac{1}{2α}}/ε^{1+\frac{1}{2α}})$ 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.