A Unified Quantum Algorithm Framework for Estimating Properties of Discrete Probability Distributions

by   Tongyang Li, et al.

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 ∑_i=1^n√(p_i)|i⟩, 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 𝒪(n^1-1/2α/ϵ + √(n)/ϵ^1+1/2α) and 𝒪(n^1/2α/ϵ^1+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.


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