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

• URL: http://arxiv.org/abs/2212.01571v2
• Date: Wed, 3 Apr 2024 11:13:38 GMT
• Title: A Quantum Algorithm Framework for Discrete Probability Distributions with Applications to Rényi Entropy Estimation
• Authors: Xinzhao Wang, Shengyu Zhang, Tongyang Li,
• Abstract summary: We propose a unified quantum algorithm framework for estimating properties of discrete probability distributions. Our framework estimates $alpha$-R'enyi entropy $H_alpha(p)$ to within additive error $epsilon$ with probability at least $2/3$.
• Score: 13.810917492304565
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $\sum_{i=1}^{n}\sqrt{p_{i}}|i\rangle$, 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.

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