Natural gradient and parameter estimation for quantum Boltzmann machines

TL;DR

This paper develops quantum algorithms for estimating geometric properties of thermal states, enabling improved parameter estimation and natural gradient methods in quantum Boltzmann machine learning.

Contribution

It derives formulas for quantum Fisher information matrices of thermal states and proposes algorithms for their estimation, advancing quantum machine learning techniques.

Findings

Formulas for Fisher--Bures and Kubo--Mori information matrices of thermal states

Quantum algorithms combining sampling, Hamiltonian simulation, and Hadamard test

An asymptotically optimal measurement for single-parameter Hamiltonian estimation

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.