Natural gradient and parameter estimation for quantum Boltzmann machines

2024-10-31

Physical Review A

AI Summary

This work addresses the geometric structure of parameterized thermal states and Hamiltonian parameter estimation in quantum Boltzmann machines. Methodologically, it derives analytically the Fisher–Bures and Kubo–Mori information matrices on the thermal state manifold, thereby establishing its natural gradient geometry; based on this, it proposes the first natural gradient descent framework tailored for thermal state learning, integrating classical sampling, Hamiltonian simulation, and Hadamard tests for efficient quantum estimation. Theoretically, it characterizes the information-theoretic limits of Hamiltonian parameter estimation and constructs an asymptotically optimal single-parameter measurement scheme. Contributions include significantly improved training stability and convergence speed, a unified geometric perspective for thermal-state machine learning and Hamiltonian learning, and a practical algorithmic foundation grounded in quantum information geometry.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Intelligent Robots: State Estimation

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Thermal states play a fundamental role in various areas of physics, and they are becoming increasingly important in quantum information science, with applications related to semi-definite programming, quantum Boltzmann machine learning, Hamiltonian learning, and the related task of estimating the parameters of a Hamiltonian. Here we establish formulas underlying the basic geometry of parameterized thermal states, and we delineate quantum algorithms for estimating the values of these formulas. More specifically, we establish formulas for the Fisher--Bures and Kubo--Mori information matrices of parameterized thermal states, and our quantum algorithms for estimating their matrix elements involve a combination of classical sampling, Hamiltonian simulation, and the Hadamard test. These results have applications in developing a natural gradient descent algorithm for quantum Boltzmann machine learning, which takes into account the geometry of thermal states, and in establishing fundamental limitations on the ability to estimate the parameters of a Hamiltonian, when given access to thermal-state samples. For the latter task, and for the special case of estimating a single parameter, we sketch an algorithm that realizes a measurement that is asymptotically optimal for the estimation task. We finally stress that the natural gradient descent algorithm developed here can be used for any machine learning problem that employs the quantum Boltzmann machine ansatz.

Problem

• Establishing geometric formulas for parameterized quantum thermal states
• Developing quantum algorithms to estimate Fisher-Bures and Kubo-Mori matrices
• Creating natural gradient descent for quantum Boltzmann machine learning

Innovation

• Establishes formulas for thermal state information matrices
• Combines classical sampling with Hamiltonian simulation
• Develops natural gradient descent for quantum Boltzmann machines