# Quantum Maximum Entropy Inference and Hamiltonian Learning

arXiv: https://arxiv.org/abs/2407.11473 · HTML: https://mlsanity.com/paper/2407.11473/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 51 references that are in the map.
- First posted on arXiv 16 Jul 2024.

## Builds on (most cited first, 9 of 9)

- [Efficiently learning Ising models on arbitrary graphs](https://mlsanity.com/paper/1411.6156/) (2014)
- [Interaction Screening: Efficient and Sample-Optimal Learning of Ising Models](https://mlsanity.com/paper/1605.07252/) (2016)
- [Learning Graphical Models Using Multiplicative Weights](https://mlsanity.com/paper/1706.06274/) (2017)
- [Sample-efficient learning of quantum many-body systems](https://mlsanity.com/paper/2004.07266/) (2020)
- [Learning quantum Hamiltonians at any temperature in polynomial time](https://mlsanity.com/paper/2310.02243/) (2023)
- [Information Theoretic Properties of Markov Random Fields, and their Algorithmic Applications](https://mlsanity.com/paper/1705.11107/) (2017)
- [Inferring the quantum density matrix with machine learning](https://mlsanity.com/paper/1904.05903/) (2019)
- [Damped Anderson Mixing for Deep Reinforcement Learning: Acceleration, Convergence, and Stabilization](https://mlsanity.com/paper/2110.08896/) (2021)
- [Classical and Quantum Iterative Optimization Algorithms Based on Matrix Legendre-Bregman Projections](https://mlsanity.com/paper/2209.14185/) (2022)

## Abstract (arXiv)

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.