A closed-form free-probability law for loss-induced conditioning in deep photonic mesh networks

Abstract

Deep optical neural networks built from cascaded Mach–Zehnder-interferometer (MZI) meshes promise fast, low-energy linear algebra, but optical loss limits how many layers can be cascaded and no closed-form theory has described that limit. We show that for a product of independent Haar-random meshes the squared-singular-value distribution is the L-fold free multiplicative convolution of the single-layer loss spectrum, and from it we derive a closed-form law for the dynamical-isometry order parameter χ(L) = Var[log s_i(W)]: χ(L) = σ_loss² L (1 + σ_loss² L/3) + O(σ⁶). The correction coefficient γ = 1/3 is shape-independent to leading order and exact in the free multiplicative Brownian scaling limit. It is fixed by the second-order coefficient (1/12) of the log-variance under free multiplicative convolution—a probabilistic fact, the decomposition of log into a semicircle plus a uniform law, established by Auer and Voit; our contribution is not this coefficient but its identification as the law governing loss-limited depth in photonic meshes. Large-scale simulations show that real, structured, noisy Clements meshes follow the same law as the idealized Haar product to about 1–2% up to N = 2048 modes, the free correction (a ≈13% departure from a naive additive model at the depths tested) being required to fit them; because asymptotic freeness of deterministic meshes is an open problem, the real-mesh law is supported empirically rather than proven. The same conditioning controls optimization: in a deep linear network a lossless control isolates loss as the cause of a depth-dependent training collapse, which we show to be ill-conditioning rather than vanishing gradients. It also ranks mesh architectures by loss resilience through a single layout quantity, the variance of the per-mode path length—rectangular (Clements) meshes are predicted to tolerate about 85× more depth than triangular (Reck) meshes at N = 128—and carries its own regime of validity, crossing over to a classical additive law when the meshes mix weakly. At the moderate depths of present devices the measured coefficient is slightly smaller, ≈0.30, well described by the empirical finite-depth fit γ_eff = 1/3 + α/L with α ≈ −0.34 (a finite-size fit, not a derived identity). Files: the preprint (PDF); the simulation and derivation-verification code with result tables, notebooks and a README mapping each script to the results in the preprint (code-and-data ZIP; code under the MIT License); and the LaTeX source of the preprint (source ZIP). The use of AI tools in this work is disclosed in the preprint.