  We introduce a new class of generative quantum-neural-network-based models
called Quantum Hamiltonian-Based Models (QHBMs). In doing so, we establish a
paradigmatic approach for quantum-probabilistic hybrid variational learning,
where we efficiently decompose the tasks of learning classical and quantum
correlations in a way which maximizes the utility of both classical and quantum
processors. In addition, we introduce the Variational Quantum Thermalizer (VQT)
for generating the thermal state of a given Hamiltonian and target temperature,
a task for which QHBMs are naturally well-suited. The VQT can be seen as a
generalization of the Variational Quantum Eigensolver (VQE) to thermal states:
we show that the VQT converges to the VQE in the zero temperature limit. We
provide numerical results demonstrating the efficacy of these techniques in
illustrative examples. We use QHBMs and the VQT on Heisenberg spin systems, we
apply QHBMs to learn entanglement Hamiltonians and compression codes in
simulated free Bosonic systems, and finally we use the VQT to prepare thermal
Fermionic Gaussian states for quantum simulation.
