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Explore the geometric foundations of parameterized thermal states and their applications in quantum machine learning through this 23-minute conference talk from QTML 2025. Delve into the mathematical formulations underlying Fisher–Bures and Kubo–Mori information matrices for thermal states, and discover quantum algorithms that estimate these matrix elements using classical sampling, Hamiltonian simulation, and the Hadamard test. Learn how these theoretical developments enable a natural gradient descent algorithm specifically designed for quantum Boltzmann machine learning that accounts for thermal state geometry. Examine the fundamental limitations of Hamiltonian parameter estimation when working with thermal-state samples, including an asymptotically optimal measurement algorithm for single-parameter estimation. Understand how the natural gradient approach can be applied to any machine learning problem utilizing the quantum Boltzmann machine ansatz, bridging quantum information theory with practical machine learning applications.
