Training Quantum Boltzmann Machines with the $\beta$-Variational Quantum Eigensolver

TL;DR

This paper introduces a heuristic method for training quantum Boltzmann machines using the $eta$-variational quantum eigensolver, enabling efficient learning of low-rank quantum and classical data with potential for near-term quantum device implementation.

Contribution

The paper proposes a nested-loop training approach combining $eta$-VQE with QBM training, facilitating practical training of QBMs on near-term quantum hardware.

Findings

High-fidelity modeling of classical and quantum target data.

Efficient learning of low-rank states using $eta$-VQE.

Successful implementation on a 10-qubit quantum device.

Abstract

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.