EPIC-CIM: Training Convolutional Neural Networks on a Coherent Ising Machine via Equilibrium Propagation

Quantum convolutional neural networks, due to the involvement of quantum measurements and discrete quantum state evolution, face inherent training challenges associated with non-differentiable operations and discrete optimization dynamics, which make conventional gradient-based learning difficult to apply effectively. In this context, energy-based learning provides a promising alternative by reformulating network training as an energy minimization process without explicit gradient backpropagation.In this framework, input data are processed through convolutional operations, followed by quantum sampling to generate intermediate binary representations, while the output layer also relies on quantum sampling to produce final predictions. The overall network energy is composed of convolutional feature matching terms, linear coupling terms at the output layer, and global output constraint terms, enabling both parameter updates and feature evolution to be described through physically interpretable energy dynamics. Furthermore, under the equilibrium propagation mechanism, the energy difference between the free phase and the weakly clamped phase is exploited to drive parameter updates without explicit gradient computation, thereby enabling stable and consistent learning in non-differentiable and discrete spaces. While remaining consistent with classical convolutional learning theory, the proposed framework enhances interpretability and observability through quantum energy modeling, offering a unified physical perspective for efficient QCNN training and the integration of quantum computing with artificial intelligence.