Quantum Query Complexity of Entropy Estimation

Estimation of Shannon and Rényi entropies of unknown discrete distributions is a fundamental problem in statistical property testing. In this paper, we give the first quantum algorithms for estimating α-Rényi entropies (Shannon entropy being 1-Rényi entropy). In particular, we demonstrate a quadratic quantum speedup for Shannon entropy estimation and a generic quantum speedup for α-Rényi entropy estimation for all α ≥ 0 values, including tight bounds for the Shannon entropy, the Hartley entropy (α = 0), and the collision entropy (α = 2). We also provide quantum upper bounds for estimating min-entropy (α = +∞) as well as the Kullback-Leibler divergence. We complement our results with quantum lower bounds on α-Rényi entropy estimation for all α ≥ 0 values. Our approach is inspired by the pioneering work of Bravyi, Harrow, and Hassidim (BHH); however, with many new technical ingredients: 1) we improve the error dependence of the BHH framework by a fine-tuned error analysis together with Montanaro's approach to estimating the expected output of quantum subroutines for α = 0, 1; 2) we develop a procedure, similar to cooling schedules in simulated annealing, for general α ≥ 0, and 3) in the cases of integer α ≥ 2 and α = +∞, we reduce the entropy estimation problem to the α-distinctness and [log n]-distinctness problems, respectively.