  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
R\'enyi 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 $\alpha$-R\'enyi entropy $H_{\alpha}(p)$ to
within additive error $\epsilon$ with probability at least $2/3$ using
$\widetilde{\mathcal{O}}(n^{1-\frac{1}{2\alpha}}/\epsilon +
\sqrt{n}/\epsilon^{1+\frac{1}{2\alpha}})$ and
$\widetilde{\mathcal{O}}(n^{\frac{1}{2\alpha}}/\epsilon^{1+\frac{1}{2\alpha}})$
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.
