Learning quantum models from quantum or classical data

Hilbert J Kappen

Published 2018-03-29, updated 2018-09-02Version 3

We propose to generalise classical maximum likelihood learning to density matrices. As the objective function, we propose a quantum likelihood that is related to the cross entropy between density matrices. We apply this learning criterion to the quantum Boltzmann machine (QBM), previously proposed by Amin et al. We demonstrate for the first time learning a quantum anti-ferromagnetic Heisenberg Hamiltonian and spin glass Hamiltonian from quantum statistics using this approach. The learning problem is convex and has a unique solution for finite temperature. For zero temperature the problem is ill-posed. We show how the proposed quantum learning formalism can also be applied to a purely classical data analysis. Representing the data as a rank one density matrix introduces quantum statistics for classical data. These statistics may violate the Bell inequality, as in the quantum case. We show that quantum learning yields results that can be significantly more accurate than the classical maximum likelihood approach. An example is the parity problem, that can be learned by a QBM without hidden units and not by the classical BM. The solution shows entanglement, quantified by the entanglement entropy.