Machina Ex Quanta: Rise of the Quantum Boltzmann Machines

Boltzmann machines are a well established model for classical machine learning, based on ideas rooted in physics like thermal distributions. Roughly eight years ago now, quantum Boltzmann machines were proposed as a generalization of their classical counterparts and can be understood as parameterized thermal states of local Hamiltonians. While this generalization allows for non-commuting Hamiltonians inaccessible in the classical case, training them for various optimization tasks had remained an obstacle. In this talk, I'll discuss how my coauthors and I have overcome this obstacle by deriving analytical expressions for the gradient and providing quantum algorithms that can estimate it. The algorithms make use of standard primitives such as the Hadamard test, classical sampling, and Hamiltonian simulation. I'll also show how to train quantum Boltzmann machines using a metric-aware gradient descent algorithm, which makes use of analytical expressions we have derived for quantum generalizations of the Fisher information matrix and quantum algorithms we have constructed for estimating information matrix elements. Finally, I'll introduce a new model called evolved quantum Boltzmann machines, which uses parameterized time-evolved thermal states as an ansatz, extending the conventional model of quantum Boltzmann machines. Evolved quantum Boltzmann machines incorporate real and imaginary time evolution according to two different non-commuting parameterized Hamiltonians, and they can also be trained using metric-aware gradient descent. If time permits, I'll mention applications of our findings to estimating time-evolved thermal states. Based on joint work with Daniel Koch, Michele Minervini, Dhrumil Patel, Saahil Patel, and available at arXiv:2410.12935, arXiv:2410.24058, and arXiv:2501.03367.