Quantum Maximum Entropy Inference and Hamiltonian Learning

TL;DR

This work extends classical maximum entropy inference to quantum Hamiltonian learning by formulating Quantum Iterative Scaling (QIS) and gradient-descent (GD) on the dual problem. It provides a precise Jacobian for QIS, $J_{QIS} = \mathds{1} - P^{-1} L$, and contrasts it with $J_{GD} = \mathds{1} - \eta L$, enabling Ostrowski-based convergence analysis; by proving $L \preceq P$ and leveraging strong convexity, the authors establish a polynomial convergence rate $1 - \Omega(1/m^2)$ for local Hamiltonians. The paper also introduces two quasi-Newton accelerations—Anderson mixing for QIS and L-BFGS for GD—achieving orders-of-magnitude speedups in Hamiltonian inference experiments. These advances support efficient quantum Hamiltonian learning and highlight practical routes for accelerating fixed-point quantum optimization routines, with implications for quantum machine learning and state reconstruction.

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.

Figures (7)

• Figure 1: The maximum entropy problem and its dual problem. Here, $S(\rho) = -\mathop{\mathrm{tr}}\nolimits(\rho \ln \rho)$ is the von Neumann entropy of $\rho$.
• Figure 2: Comparison of QIS and GD algorithms. The loss in measured by the error in the objective function of the maximum entropy problem.
• Figure 3: Comparisons of the AM-QIS and L-BFGS-GD algorithms. The \ref{['fig:amqis-lbfgs-a']} on the left uses the Barzilai-Borwein method to choose the step size and \ref{['fig:amqis-lbfgs-b']} on the right uses fixed step size. The dotted (red) and dashed (green) lines represent the performance of the algorithms when the input Hamiltonian terms are complete satisfying $\sum_{j} F_{j} = \mathds{1}$.
• Figure 4: Two optimization problems of the Kullback-Leibler divergence that are dual to each other.
• Figure 5: Results of the Ising Hamiltonian for $7$ qubits.

Theorems & Definitions (18)

• Theorem 4.1
• Theorem 4.2
• Theorem 4.3
• Theorem 4.4
• proof : Proof of \ref{['thm:qis-convergence']}
• proof : Proof of \ref{['thm:jacobian-qis']}
• proof : Proof of \ref{['thm:jacobian-gd']}
• Theorem B.1: Ostrowski's theorem Ost66
• Lemma C.1: Quantum Belief Propagation Has07
• Lemma C.2: Bochner's Theorem