Equilibrium propagation: the quantum and the thermal cases

Equilibrium propagation is a recently introduced method to use and train artificial neural networks in which the network is at the minimum (more generally extremum) of an energy functional. Equilibrium propagation has shown good performance on a number of benchmark tasks. Here, we extend equilibrium propagation in two directions. First, we show that there is a natural quantum generalization of equilibrium propagation in which a quantum neural network is taken to be in the ground state (more generally any eigenstate) of the network Hamiltonian, with a similar training mechanism that exploits the fact that the mean energy is extremal on eigenstates. Second, we extend the analysis of equilibrium propagation at finite temperature, showing that thermal fluctuations allow one to naturally optimize the network without having to clamp the output layer during training. We also study the low-temperature limit of equilibrium propagation and show how clamping can be avoided also in this limit.