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.

翻译：机器学习和人工智能在科学技术各分支中的广泛应用，催生了对高能效替代硬件平台的需求。尽管针对多种平台已提出并实现了此类神经形态方法，但物理提取训练所需梯度仍具挑战性，因为通用方法仅存在于特定情况。平衡传播（EP）是一种针对弛豫至平衡态的经典能量模型提出并应用的方法。本文揭示了EP与昂萨格互易性之间的直接关联，并利用该关联推导出EP的量子版本。该方法可用于优化依赖于任意量子系统可观测量期望值的损失函数。具体而言，我们通过监督学习和无监督学习示例阐释这一新概念，其中输入或可求解任务具有量子力学特性，例如量子多体基态识别、量子相探索、传感及相边界探测。我们提出未来量子EP可用于解决诸如量子相发现等任务，即使对于数值模拟困难甚至部分未知的哈密顿量，也可通过量子模拟器实现。本方案适用于多种量子模拟平台，包括离子链、超导量子比特阵列、中性原子里德伯光镊阵列以及光学晶格中的强相互作用原子体系。