Native QR Factorization on Programmable Photonic Meshes

Abstract

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 quadratically ($O(N^2)$) with matrix size, matching the information content of a dense input and reducing 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. Finally, we analyze the impact of finite control resolution on the computed factors.

Figures (8)

• Figure 1: Optical architecture. (a) Fundamental tunable $2\times2$ interferometric block $T$. (b) $N=6$ Reck unitary mesh. (c) Structure of Givens array $Y_s$. (d) Compact representation using Givens arrays.
• Figure 2: Step by step physical QR decomposition for $N=6$. Columns of $A$ are injected sequentially while Givens arrays $Y_N$ through $Y_2$ zero subdiagonal elements. Red symbols mark the newly configured array $Y_{N+1-i}$ at step $(i)$. The final injection yields the last column of $R$.
• Figure 3: Optical iterative QR algorithm for $N=6$. (a) Two universal Reck interferometers with mirrored inverse Givens arrays ($Q^{(L)}$, $Q^{(R)}$). (b) Array $X_s$, the structural inverse of $Y_s$. (c) First three iterations, red symbols mark parameters determined at step $(i)$.
• Figure 4: Optical Hessenberg decomposition for $N=6$. (a) Two shortened (nonuniversal) Reck interferometers with mirrored inverse Givens arrays ($Q^{(L)}$, $Q^{(R)}$). (b) Decomposition workflow: red symbols mark the Givens arrays updated at step $(i)$.
• Figure 5: Optical bidiagonalization for $N=6$. (a) Setup with one shortened (nonuniversal) and one universal Reck multiport interferometer: $Q^{(L)}$ and $Q^{(R)}$ use mirrored inverse Givens arrays. (b) Decomposition workflow: red symbols mark the Givens arrays updated at step $(i)$.