  The quantum Boltzmann machine (QBM) is a generative machine learning model
for both classical data and quantum states. Training the QBM consists of
minimizing the relative entropy from the model to the target state. This
requires QBM expectation values which are computationally intractable for large
models in general. It is therefore important to develop heuristic training
methods that work well in practice. In this work, we study a heuristic method
characterized by a nested loop: the inner loop trains the $\beta$-variational
quantum eigensolver ($\beta$-VQE) by Liu et al (2021 Mach. Learn.: Sci.
Technol.2 025011) to approximate the QBM expectation values; the outer loop
trains the QBM to minimize the relative entropy to the target. We show that
low-rank representations obtained by $\beta$-VQE provide an efficient way to
learn low-rank target states, such as classical data and low-temperature
quantum tomography. We test the method on both classical and quantum target
data with numerical simulations of up to 10 qubits. For the cases considered
here, the obtained QBMs can model the target to high fidelity. We implement a
trained model on a physical quantum device. The approach offers a valuable
route towards variationally training QBMs on near-term quantum devices.
