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