Quantum query complexity of entropy estimation

Estimation of Shannon and R´enyi entropies of unknown discrete distributions is a fundamental problem in statistical property testing and an active research topic in both theoretical computer science and information theory. Tight bounds on the number of samples to estimate these entropies have been established in the classical setting, while little is known about their quantum counterparts. In this paper, we give the first quantum algorithms for estimating α- R´enyi entropies (Shannon entropy being 1-Renyi entropy). In particular, we demonstrate a quadratic quantum speedup for Shannon entropy estimation and a generic quantum speedup for α-R´enyi entropy estimation for all α ≥ 0, including a tight bound for the collision-entropy (2-R´enyi entropy). We also provide quantum upper bounds for extreme cases such as the Hartley entropy (i.e., the logarithm of the support size of a distribution, corresponding to α = 0) and the min-entropy case (i.e., α = +∞), as well as the Kullback-Leibler divergence between two distributions. Moreover, we complement our results with quantum lower bounds on α-R´enyi entropy estimation for all α ≥ 0. Our approach is inspired by the pioneering work of Bravyi, Harrow, and Hassidim (BHH) [13] on quantum algorithms for distributional property testing, however, with many new technical ingredients. For Shannon entropy and 0-R´enyi entropy estimation, we improve the performance of the BHH framework, especially its error dependence, using Montanaro’s approach to estimating the expected output value of a quantum subroutine with bounded variance [41] and giving a fine-tuned error analysis. For general α-R´enyi entropy estimation, we further develop a procedure that recursively approximates α-R´enyi entropy for a sequence of αs, which is in spirit similar to a cooling schedule in simulated annealing. For special cases such as integer α ≥ 2 and α = +∞ (i.e., the min-entropy), we reduce the entropy estimation problem to the α-distinctness and the dlog ne-distinctness problems, respectively. We exploit various techniques to obtain our lower bounds for different ranges of α, including reductions to (variants of) existing lower bounds in quantum query complexity as well as the polynomial method inspired by the celebrated quantum lower bound for the collision problem.