Nonlinear Unitary Circuits for Photonic Neural Networks

Photonics has unlocked the potential for the energy-efficient acceleration of deep learning. Conventional platforms for photonic deep learning have implemented linear matrix calculations and nonlinear activations separately following traditional deep learning architectures. However, critical challenges remain in achieving and evaluating nonlinear expressivity─the ability of neural networks to approximate nonlinear functions─including the substantial cost of electro-optical conversion or optical nonlinearity and the absence of a metric to quantify the expressivity of photonic platforms. Here, we propose the concept of nonlinear unitary (NU) circuits for norm-preserving photonic learning, which achieve both linear and nonlinear expressivity within a building block. We examine two-dimensional NU operations─characterized by norm-preserving mappings and nonconservative inner products─using this building block, which can be employed to construct high-dimensional circuits. Using deep nonlinear unitary circuits, we demonstrate exponential growth in trajectory length and near-uniform coverage of the output space by developing platform-transparent metrices to quantify the nonlinear expressivity of the circuits. Along with neuroevolutionary learning examples, our results pave the way to highly expressive photonic neural networks with rigorous quantification.