On the Sample Complexity of Quantum Boltzmann Machine Learning

TL;DR

This paper establishes that Quantum Boltzmann Machines can be trained efficiently with polynomial sample complexity, avoiding barren plateaus, and introduces pre-training strategies that further reduce data requirements.

Contribution

It provides an operational framework for QBM learning, proves polynomial sample complexity bounds, and explores effective pre-training methods.

Findings

QBM learning can be achieved with polynomially many Gibbs states.

Pre-training on subsets of parameters reduces sample complexity.

Numerical verification confirms theoretical results.

Abstract

Quantum Boltzmann machines (QBMs) are machine-learning models for both classical and quantum data. We give an operational definition of QBM learning in terms of the difference in expectation values between the model and target, taking into account the polynomial size of the data set. By using the relative entropy as a loss function this problem can be solved without encountering barren plateaus. We prove that a solution can be obtained with stochastic gradient descent using at most a polynomial number of Gibbs states. We also prove that pre-training on a subset of the QBM parameters can only lower the sample complexity bounds. In particular, we give pre-training strategies based on mean-field, Gaussian Fermionic, and geometrically local Hamiltonians. We verify these models and our theoretical findings numerically on a quantum and a classical data set. Our results establish that QBMs are promising machine learning models.