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Abstract

The widespread adoption of machine learning and artificial intelligence in all branches of science and tech- nology has created a need for energy-efficient, alternative hardware platforms. While such neuromorphic ap- proaches have been proposed and realised for a wide range of platforms, physically extracting the gradients required for training remains challenging as generic approaches only exist in certain cases. Equilibrium prop- agation (EP) is such a procedure that has been introduced and applied to classical energy-based models which relax to an equilibrium. Here, we show a direct connection between EP and Onsager reciprocity and exploit this to derive a quantum version of EP. This can be used to optimize loss functions that depend on the expec- tation values of observables of an arbitrary quantum system. Specifically, we illustrate this new concept with supervised and unsupervised learning examples in which the input or the solvable task is of quantum mechani- cal nature, e.g., the recognition of quantum many-body ground states, quantum phase exploration, sensing and phase boundary exploration. We propose that in the future quantum EP may be used to solve tasks such as quan- tum phase discovery with a quantum simulator even for Hamiltonians which are numerically hard to simulate or even partially unknown. Our scheme is relevant for a variety of quantum simulation platforms such as ion chains, superconducting qubit arrays, neutral atom Rydberg tweezer arrays and strongly interacting atoms in optical lattices.