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 $\alpha$-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 $\alpha$-R\'enyi entropy estimation for all $\alpha\geq 0$, 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 $\alpha=0$) and the min-entropy case (i.e., $\alpha=+\infty$), as well as the Kullback-Leibler divergence between two distributions. Moreover, we complement our results with quantum lower bounds on $\alpha$-R\'enyi entropy estimation for all $\alpha\geq 0$.