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.

该论文介绍了均衡传播（EP）与 Onsager 互易性之间的直接关系，并利用此关系推导出 EP 的量子版本。该方法可用于优化依赖于任意量子系统的可观测期望值的损失函数，且适用于识别量子多体基态、量子相探索、感知和相界探索等与量子力学有关的学习任务。该方案对于离子链、超导量子位阵列、中性原子 Rydberg 钳子阵列和强相互作用原子光晶格等各种量子模拟平台都具有重要意义。