Symplectic Reck: In-Situ Learning of Gaussian Quantum Operations

Programmable photonic circuits are a natural platform for physical learning, where the learned computation is performed by the hardware itself rather than by a digital model. We propose a physical learning algorithm and architecture for continuous-variable quantum systems where a photonic circuit is configured in situ to invert an unknown lossless Gaussian quantum operation. Our contributions are threefold. (i) Theory: we prove that generic $S \in \mathrm{Sp}(2 N, \mathbb{R})$ admits a triangular decomposition into two-mode and single-mode symplectic gates. (ii) Architecture: we introduce the symplectic Reck mesh, a triangular array of two-mode symplectic gates that physically realizes this factorization, generalizing the unitary Reck mesh widely used in optical neural networks. (iii) Physical learning: when placed after an unknown Gaussian unitary, the mesh is trained in situ by nulling local response blocks one at a time using simple optical probes and measurements. We identify a measure that predicts when training becomes hard and numerically demonstrate finite-shot training and robustness to gate errors. Our work extends self-configuring optical networks from unitary to symplectic transformations, bringing physical learning to continuous-variable quantum photonic hardware.