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}}|i \rangle $
, 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 2/3 using
$\widetilde {\mathcal {O}}\left({n^{1-\frac {1}{2\alpha }}/\epsilon + \sqrt {n}/\epsilon ^{1+\frac {1}{2\alpha }}}\right)$
and
$\widetilde {\mathcal {O}}\left({n^{\frac {1}{2\alpha }}/\epsilon ^{1+\frac {1}{2\alpha }}}\right)$
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.