Native QR Factorization on Programmable Photonic Meshes

We propose a photonic native procedure for computing the QR factorization of a matrix using a programmable unitary interferometer mesh. The method configures the mesh through a sequence of local power routing steps within tunable two mode interferometric elements, while reading out the resulting upper triangular factor directly from the optical outputs. The number of physical operations grows as $O(N\log_2N)$ with matrix size $N$, reducing the runtime relative to standard digital QR routines, which scale cubically ($O(N^3)$). Beyond single factorizations, the same architecture supports iterative spectral computations by reusing the configured interferometer in a mirrored arrangement that implements the core update step of the QR eigenvalue algorithm. We also describe related optical procedures for Hessenberg reduction and bidiagonalization, serving as compatible preprocessors for QR and SVD workflows. A comparison with the systolic array computational architecture is provided. Our approach exhibits comparable asymptotic complexity for blocked QR decomposition and is more efficient for Hessenberg reduction and bidiagonalization.