Natural gradient and parameter estimation for quantum Boltzmann machines

The study on natural gradient and parameter estimation for quantum Boltzmann machines introduces essential formulas for the Fisher-Bures and Kubo-Mori information matrices of parameterized thermal states. These advancements are crucial in the realm of quantum information science, where thermal states are increasingly recognized for their fundamental role. The proposed quantum algorithms leverage classical sampling and Hamiltonian simulation, facilitating improved estimation of matrix elements of thermal states. This research culminates in a natural gradient descent algorithm tailored for quantum Boltzmann machine learning, which optimally considers the geometry of thermal states. Furthermore, it establishes significant limitations on the capacity to estimate Hamiltonian parameters when only thermal-state samples are accessible, thus paving the way for more efficient quantum algorithms in machine learning applications.