Quantum Maximum Entropy Inference and Hamiltonian Learning
This work addresses the challenge of quantum inference and Hamiltonian learning for researchers in quantum machine learning, but it is incremental as it builds on classical algorithms.
The paper tackles the problem of extending maximum entropy inference and graphical model learning algorithms to quantum settings, where non-commutativity complicates convergence analysis, and shows that quasi-Newton methods like Anderson mixing and L-BFGS achieve orders of magnitude performance improvements.
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.