  The quantum many-body problem lies at the center of the most important open
challenges in condensed matter, quantum chemistry, atomic, nuclear, and
high-energy physics. While quantum Monte Carlo, when applicable, remains the
most powerful numerical technique capable of treating dozens or hundreds of
degrees of freedom with high accuracy, it is restricted to models that are not
afflicted by the infamous sign problem. A powerful alternative that has emerged
in recent years is the use of neural networks as variational estimators for
quantum states. In this work, we propose a symmetry-projected variational
solution in the form of linear combinations of simple restricted Boltzmann
machines. This construction allows one to explore states outside of the
original variational manifold and increase the representation power with
moderate computational effort. Besides allowing one to restore spatial
symmetries, an expansion in terms of Krylov states using a Lanczos recursion
offers a solution that can further improve the quantum state accuracy. We
illustrate these ideas with an application to the Heisenberg $J_1-J_2$ model on
the square lattice, a paradigmatic problem under debate in condensed matter
physics, and achieve start-of-the-art accuracy in the representation of the
ground state.
