  Hamiltonian learning is crucial to the certification of quantum devices and
quantum simulators. In this paper, we propose a hybrid quantum-classical
Hamiltonian learning algorithm to find the coefficients of the Pauli operator
components of the Hamiltonian. Its main subroutine is the practical
log-partition function estimation algorithm, which is based on the minimization
of the free energy of the system. Concretely, we devise a stochastic
variational quantum eigensolver (SVQE) to diagonalize the Hamiltonians and then
exploit the obtained eigenvalues to compute the free energy's global minimum
using convex optimization. Our approach not only avoids the challenge of
estimating von Neumann entropy in free energy minimization, but also reduces
the quantum resources via importance sampling in Hamiltonian diagonalization,
facilitating the implementation of our method on near-term quantum devices.
Finally, we demonstrate our approach's validity by conducting numerical
experiments with Hamiltonians of interest in quantum many-body physics.
