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 ${\sum }_{i=1}^n\sqrt{p_i}\mathrm{\mid }i⟩$, for $\alpha >1$ and $0<\alpha <1$, our algorithm framework estimates $\alpha$-Rényi entropy $H_{\alpha }(p)$ to within additive error $\epsilon$ with probability at least $2\mathrm{/}3$ using $\tilde{\mathcal{O}}(n^{1-\frac{1}{2\alpha }}\mathrm{/}\epsilon +\sqrt{n}\mathrm{/}{\epsilon }^{1+\frac{1}{2\alpha }})$ and $\tilde{\mathcal{O}}(n^{\frac{1}{2\alpha }}\mathrm{/}{\epsilon }^{1+\frac{1}{2\alpha }})$ queries, respectively. This improves the best known dependence in $\epsilon$ as well as the joint dependence between $n$ and $1\mathrm{/}\epsilon$. 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.