  We investigate the use of variational wave-functions that mimic stochastic
recurrent neural networks, specifically, unrestricted Boltzmann machines, as
guiding functions in projective quantum Monte Carlo (PQMC) simulations of
quantum spin models. As a preliminary step, we investigate the accuracy of such
unrestricted neural network states as variational Ans\"atze for the ground
state of the ferromagnetic quantum Ising chain. We find that by optimizing just
three variational parameters, independently on the system size, accurate
ground-state energies are obtained, comparable to those previously obtained
using restricted Boltzmann machines with few variational parameters per spin.
Chiefly, we show that if one uses optimized unrestricted neural network states
as guiding functions for importance sampling the efficiency of the PQMC
algorithms is greatly enhanced, drastically reducing the most relevant
systematic bias, namely that due to the finite random-walker population. The
scaling of the computational cost with the system size changes from the
exponential scaling characteristic of PQMC simulations performed without
importance sampling, to a polynomial scaling, even at the ferromagnetic quantum
critical point. The important role of the protocol chosen to sample
hidden-spins configurations, in particular at the critical point, is analyzed.
We discuss the implications of these findings for what concerns the problem of
simulating adiabatic quantum optimization using stochastic algorithms on
classical computers.
