  Entropy measures quantify the amount of information and correlation present
in a quantum system. In practice, when the quantum state is unknown and only
copies thereof are available, one must resort to the estimation of such entropy
measures. Here we propose a variational quantum algorithm for estimating the
von Neumann and R\'enyi entropies, as well as the measured relative entropy and
measured R\'enyi relative entropy. Our approach first parameterizes a
variational formula for the measure of interest by a quantum circuit and a
classical neural network, and then optimizes the resulting objective over
parameter space. Numerical simulations of our quantum algorithm are provided,
using a noiseless quantum simulator. The algorithm provides accurate estimates
of the various entropy measures for the examples tested, which renders it as a
promising approach for usage in downstream tasks.
