QR factorization decomposes a matrix A into an orthogonal matrix Q and an upper triangular matrix R. It is the workhorse of numerical linear algebra — least squares, eigenvalue computation, rank determination all reduce to it. Digital implementations run in O(N³) operations for an N×N matrix. The cubic scaling is fundamental to the algorithm, not the hardware.
De Pasquale and colleagues (arXiv:2602.20701) implement QR factorization on a programmable photonic mesh — a triangular array of tunable two-mode interferometers. Light enters one side, propagates through the mesh, and exits the other. Each interferometer is a 2×2 unitary transformation controlled by a phase shifter. The mesh as a whole implements an N×N unitary transformation. The key: configuring the mesh through sequential local power routing steps — nulling specific output ports one by one — naturally performs the Givens rotations that constitute the QR algorithm.
The result: O(N²) physical operations, not O(N³). The speedup comes from the fact that each Givens rotation, which requires O(N) digital operations to apply, is a single physical adjustment of one interferometer. The mesh performs the rotation by propagating light through it. The propagation IS the matrix multiplication.
The upper triangular factor R is directly readable from the optical outputs. The unitary factor Q is encoded in the mesh configuration itself — the physical state of the interferometers is the matrix. To compute eigenvalues, the configured mesh is reused in a mirrored arrangement that implements the iterative QR step, with each iteration taking O(N²) rather than O(N³).
The general observation: when a computation has a natural physical representation, the hardware can absorb complexity that would be explicit in software. The cubic cost of QR is a property of sequential digital execution, not of the mathematical structure. A parallel physical system that natively represents the operations pays only for the information content of the input, not for the sequential enumeration of steps.