  A large body of work has demonstrated that parameterized artificial neural
networks (ANNs) can efficiently describe ground states of numerous interesting
quantum many-body Hamiltonians. However, the standard variational algorithms
used to update or train the ANN parameters can get trapped in local minima,
especially for frustrated systems and even if the representation is
sufficiently expressive. We propose a parallel tempering method that
facilitates escape from such local minima. This methods involves training
multiple ANNs independently, with each simulation governed by a Hamiltonian
with a different "driver" strength, in analogy to quantum parallel tempering,
and it incorporates an update step into the training that allows for the
exchange of neighboring ANN configurations. We study instances from two classes
of Hamiltonians to demonstrate the utility of our approach using Restricted
Boltzmann Machines as our parameterized ANN. The first instance is based on a
permutation-invariant Hamiltonian whose landscape stymies the standard training
algorithm by drawing it increasingly to a false local minimum. The second
instance is four hydrogen atoms arranged in a rectangle, which is an instance
of the second quantized electronic structure Hamiltonian discretized using
Gaussian basis functions. We study this problem in a minimal basis set, which
exhibits false minima that can trap the standard variational algorithm despite
the problem's small size. We show that augmenting the training with quantum
parallel tempering becomes useful to finding good approximations to the ground
states of these problem instances.
