  The widespread adoption of machine learning and artificial intelligence in
all branches of science and technology has created a need for energy-efficient,
alternative hardware platforms. While such neuromorphic approaches 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 propagation (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 expectation 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 mechanical 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 quantum 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.
