  We extend the ability of unitary quantum circuits by interfacing it with
classical autoregressive neural networks. The combined model parametrizes a
variational density matrix as a classical mixture of quantum pure states, where
the autoregressive network generates bitstring samples as input states to the
quantum circuit. We devise an efficient variational algorithm to jointly
optimize the classical neural network and the quantum circuit for quantum
statistical mechanics problems. One can obtain thermal observables such as the
variational free energy, entropy, and specific heat. As a by product, the
algorithm also gives access to low energy excitation states. We demonstrate
applications to thermal properties and excitation spectra of the quantum Ising
model with resources that are feasible on near-term quantum computers.
