Quantum Maximum Entropy Inference and Hamiltonian Learning

TL;DR

This paper extends maximum entropy inference algorithms to quantum systems, analyzing their convergence and introducing quasi-Newton enhancements, with applications to Hamiltonian learning.

Contribution

It provides the first rigorous convergence analysis of quantum iterative scaling and explores quasi-Newton methods for improved quantum inference algorithms.

Findings

Quantum iterative scaling converges with established spectral bounds.

Quasi-Newton methods significantly accelerate convergence.

Algorithms enable practical Hamiltonian learning in quantum systems.

Abstract

Maximum entropy inference and learning of graphical models are pivotal tasks in learning theory and optimization. This work extends algorithms for these problems, including generalized iterative scaling (GIS) and gradient descent (GD), to the quantum realm. While the generalization, known as quantum iterative scaling (QIS), is straightforward, the key challenge lies in the non-commutative nature of quantum problem instances, rendering the convergence rate analysis significantly more challenging than the classical case. Our principal technical contribution centers on a rigorous analysis of the convergence rates, involving the establishment of both lower and upper bounds on the spectral radius of the Jacobian matrix for each iteration of these algorithms. Furthermore, we explore quasi-Newton methods to enhance the performance of QIS and GD. Specifically, we propose using Anderson mixing and the L-BFGS method for QIS and GD, respectively. These quasi-Newton techniques exhibit remarkable efficiency gains, resulting in orders of magnitude improvements in performance. As an application, our algorithms provide a viable approach to designing Hamiltonian learning algorithms.