  We study the structure of the ground states of local stoquastic Hamiltonians
and show that under mild assumptions the following distributions can
efficiently approximate one another: (a) distributions arising from ground
states of stoquastic Hamiltonians, (b) distributions arising from ground states
of stoquastic frustration-free Hamiltonians, (c) Gibbs distributions of local
classical Hamiltonian, and (d) distributions represented by real-valued Deep
Boltzmann machines. In addition, we highlight regimes where it is possible to
efficiently classically sample from the above distributions.
