Stoquastic ground states are classical thermal distributions

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.