Symplectic Reck: In-Situ Learning of Gaussian Quantum Operations

arXiv:2611.10206v1 Announce Type: new Abstract: Programmable photonic circuits are a natural platform for physical learning, where hardware parameters are trained directly from measured feedback. We propose a physical learning algorithm and architecture for continuous-variable quantum systems where a photonic circuit is trained in situ to invert an unknown lossless Gaussian quantum operation. We prove that a generic symplectic matrix $S\in\mathrm{Sp}(2N,\mathbb{R})$ admits a decomposition into a triangular array of two-mode symplectic gates on neighboring modes, followed by single-mode symplectic gates. We then introduce the symplectic Reck mesh, which physically realizes this factorization, generalizing the unitary Reck mesh widely used in optical neural networks.