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 prove 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.

翻译：热态在物理学的多个领域中扮演着基础性角色，并且在量子信息科学中正变得日益重要，其应用涉及半定规划、量子玻尔兹曼机器学习、哈密顿量学习以及相关的哈密顿量参数估计任务。本文建立了参数化热态基本几何结构的基础公式，并阐述了用于估计这些公式值的量子算法。具体而言，我们证明了参数化热态的Fisher-Bures与Kubo-Mori信息矩阵的公式，而用于估计其矩阵元素的量子算法结合了经典采样、哈密顿量模拟和Hadamard测试。这些结果可用于开发一种考虑热态几何结构的自然梯度下降算法以用于量子玻尔兹曼机器学习，并可用于确立在给定热态样本条件下估计哈密顿量参数能力的基本限制。对于后一任务（特别是单参数估计情形），我们概述了一种实现渐近最优估计测量的算法。最后需要强调的是，本文开发的自然梯度下降算法可适用于任何采用量子玻尔兹曼机拟设的机器学习问题。