论文标题
用于离散概率分布的量子算法框架，并应用于Rényi熵估计
A Quantum Algorithm Framework for Discrete Probability Distributions with Applications to Rényi Entropy Estimation
论文作者
论文摘要
估计统计属性是统计和计算机科学的基础。在本文中，我们提出了一个统一的量子算法框架，用于估计离散概率分布的性质，并将Rényi熵估计为特定示例。 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错误的$ $ $ $ $ 2/3 $使用$ \ widetilde {\ Mathcal {o}}}}（n^{1- \ frac {1} {2α}} {2α}}/ε+ \ sqrt {n}/ε^ $ \ widetilde {\ Mathcal {o}}（n^{\ frac {1} {2α}}}/ε^{1+ \ frac {1} {1} {2α}}}}}）$查询。这改善了$ε$的最著名依赖性以及$ n $和$ 1/ε$之间的联合依赖性。从技术上讲，我们的量子算法结合了量子奇异值转换，量子退火和可变的时间振幅估计。我们认为，我们的算法框架具有普遍的兴趣，并且具有广泛的应用。
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.
