Quantum equilibrium propagation for efficient training of quantum systems based on Onsager reciprocity

The widespread adoption of machine learning and artificial intelligence in all branches of science and technology creates a need for energy-efficient, alternative hardware. While such neuromorphic systems have been demonstrated in a wide range of platforms, it remains an open challenge to find efficient and general physics-based training approaches. Equilibrium propagation (EP), the most widely studied approach, has been introduced for classical energy-based models relaxing to an equilibrium. Here, we show a direct connection between EP and Onsager reciprocity and exploit this to derive a quantum version of EP. For an arbitrary quantum system, this can now be used to extract training gradients with respect to all tuneable parameters via a single linear response experiment. We illustrate this new concept in examples in which the input or the task is of quantum-mechanical nature, e.g., the recognition of many-body ground states, phase discovery, sensing, and phase boundary exploration. Quantum EP may be used to solve challenges such as quantum phase discovery for Hamiltonians which are classically hard to simulate or even partially unknown. Our scheme is relevant for a variety of quantum simulation platforms such as ion chains, superconducting circuits, Rydberg atom tweezer arrays and ultracold atoms in optical lattices.