  Entropy plays a crucial role in both physics and information science,
encompassing classical and quantum domains. In this work, we present the
Quantum Neural Entropy Estimator (QNEE), a novel approach that combines
classical neural network (NN) with variational quantum circuits to estimate the
von Neumann and Renyi entropies of a quantum state. QNEE provides accurate
estimates of entropy while also yielding the eigenvalues and eigenstates of the
input density matrix. Leveraging the capabilities of classical NN, QNEE can
classify different phases of quantum systems that accompany the changes of
entanglement entropy. Our numerical simulation demonstrates the effectiveness
of QNEE by applying it to the 1D XXZ Heisenberg model. In particular, QNEE
exhibits high sensitivity in estimating entanglement entropy near the phase
transition point. We expect that QNEE will serve as a valuable tool for quantum
entropy estimation and phase classification.
