Quantum Equilibrium Propagation: gradient-descent training of quantum systems

Equilibrium propagation (EP) is a training framework for physical systems that minimize an energy function. A key feature of EP is that it uses the system's intrinsic physics during both inference and training, making it a candidate for the development of energy-efficient processors for machine learning. EP has been explored in various classical physical systems, including classical Ising networks and elastic networks. We extend EP to quantum systems, where the energy function that is minimized is the mean energy functional (expectation value of the Hamiltonian), whose minimum is the Hamiltonian's ground state. As examples, we study the settings of the transverse-field Ising network and the quantum harmonic oscillator network -- quantum analogues of the Ising network and elastic network.