Perturbative Contrastive Physical Learning in Classical and Quantum Systems

Responses to perturbations are key to understanding physical systems. The ability to contrast such responses—comparing how a system reacts under slightly different conditions—provides a mechanism for learning. Here, we introduce Perturbative Contrastive Physical Learning (PCPL), a general framework in which learning emerges from measurable contrasts between pairs of physical states produced by changes to inputs, boundary conditions, parameters, or interpreter functions, for example. PCPL unifies and extends prior approaches: Equilibrium Propagation appears when contrasts arise from differences between free and nudged equilibria in energy-based systems, while Frequency Propagation corresponds to contrasts extracted from sinusoidally driven, frequency-demodulated responses. We demonstrate PCPL in both classical and quantum platforms: (i) spring networks that update bond stiffness using measured displacements and forces, and (ii) continuous-variable photonic circuits trained via homodyne measurements and finite-difference Jacobians. In both settings, PCPL learns classifiers and continuous mappings. Finally, we discuss how pre- and post-processing of measurement data that can improve classical–quantum interfacing and contribute to more autonomous physical learning systems.