  Can near-term gate model based quantum processors offer quantum advantage for
practical applications in the pre-fault tolerance noise regime? A class of
algorithms which have shown some promise in this regard are the so-called
classical-quantum hybrid variational algorithms. Here we develop a low-depth
quantum algorithm to generative neural networks using variational quantum
circuits. We introduce a method which employs the quantum approximate
optimization algorithm as a subroutine in order produce then sample low-energy
distributions of Ising Hamiltonians. We sample these states to train neural
networks and demonstrate training convergence for numerically simulated noisy
circuits with depolarizing errors of rates of up to $4\%$.
