  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 train the network without having to clamp
the output layer during training. We also study the low temperature limit of
equilibrium propagation.
