  Entropy is a measure of the randomness of a system. Estimating the entropy of
a quantum state is a basic problem in quantum information. In this paper, we
introduce a time-efficient quantum approach to estimating the von Neumann
entropy $S(\rho)$ and R\'enyi entropy $S_\alpha(\rho)$ of an $N$-dimensional
quantum state $\rho$, given access to independent samples of $\rho$.
Specifically, we provide the following:
  1. A quantum estimator for $S(\rho)$ with time complexity $\tilde O(N^2)$,
improving the prior best time complexity $\tilde O(N^6)$ by Acharya, Issa,
Shende, and Wagner (2020) and Bavarian, Mehraba, and Wright (2016).
  2. A quantum estimator for $S_\alpha(\rho)$ with time complexity $\tilde
O(N^{4/\alpha-2})$ for $0<\alpha<1$ and $\tilde O(N^{4-2/\alpha})$ for
$\alpha>1$, improving the prior best time complexity $\tilde O(N^{6/\alpha})$
for $0<\alpha<1$ and $\tilde O(N^6)$ for $\alpha>1$ by Acharya, Issa, Shende,
and Wagner (2020), though at a cost of a slightly larger sample complexity.
  Moreover, these estimators are naturally extensible to the low-rank case. We
also provide a sample lower bound for estimating $S_{\alpha}(\rho)$.
  Technically, our method is quite different from the previous ones that are
based on weak Schur sampling and Young diagrams. At the heart of our
construction, is a novel tool called samplizer, which can "samplize" a quantum
query algorithm to a quantum algorithm with similar behavior using only samples
of quantum states; this suggests a unified framework for estimating quantum
entropies. Specifically, when a quantum oracle $U$ block-encodes a mixed
quantum state $\rho$, any quantum query algorithm using $Q$ queries to $U$ can
be samplized to a $\delta$-close (in the diamond norm) quantum algorithm using
$\tilde\Theta(Q^2/\delta)$ samples of $\rho$. Moreover, this samplization is
proven to be optimal, up to a polylogarithmic factor.
