  The von Neumann and quantum R\'enyi entropies characterize fundamental
properties of quantum systems and lead to theoretical and practical
applications in many fields. Quantum algorithms for estimating quantum
entropies, using a quantum query model that prepares the purification of the
input state, have been established in the literature. {However, constructing
such a model is almost as hard as state tomography.} In this paper, we propose
quantum algorithms to estimate the von Neumann and quantum $\alpha$-R\'enyi
entropies of an $n$-qubit quantum state $\rho$ using independent copies of the
input state. We also show how to efficiently construct the quantum circuits for
{quantum entropy estimation} using primitive single/two-qubit gates. We prove
that the number of required copies scales polynomially in $1/\epsilon$ and
$1/\Lambda$, where $\epsilon$ denotes the additive precision and $\Lambda$
denotes the lower bound on all non-zero eigenvalues. Notably, our method
outperforms previous methods in the aspect of practicality since it does not
require any quantum query oracles, which are usually necessary for previous
methods. Furthermore, we conduct experiments to show the efficacy of our
algorithms to single-qubit states and study the noise robustness. We also
discuss the applications to some quantum states of practical interest as well
as some meaningful tasks such as quantum Gibbs state preparation and
entanglement estimation.
