Quantum Information Geometry and Boltzmann Machines

The Stochastic Reconfiguration algorithm has been recently proposed to efficiently train Neural-Network Quantum States, i.e., Restricted Boltzmann Machines (RBMs) with complex parameters built to simulate the ground state of a quantum many-body problem. The SR algorithm is not only a convenient algorithm for the training these RBMs, but it is also theoretically justified once a non-Euclidean manifold structure, based on Quantum Information Geometry, is defined over the search space associated to a RBM. I will show that the gradient descent optimization algorithm used for the many-body problem is the quantum generalization of the Riemannian natural gradient introduced by Amari. Moreover, I will compare the geometry of the Neural-Network Quantum States with the one of the Quantum Boltzmann Machine.