Pre-Asymptotic Trainability in Photonic Variational Circuits under Postselection

TL;DR

This paper investigates how passive photonic variational circuits with postselection can avoid barren plateaus, showing that certain regimes maintain polynomial gradient variance decay, unlike others with exponential concentration.

Contribution

It demonstrates that the interplay of passive linear optics, postselection, and observables influences trainability, providing insights for designing effective photonic quantum algorithms.

Findings

Allow-bunching and collision-free regimes maintain polynomial gradient variance decay.

Dual-rail postselection induces exponential concentration of gradients.

Gradient behavior depends on the type of postselection and the task observables.

Abstract

Barren plateaus in variational quantum circuits are commonly attributed to strong mixing dynamics that cause gradient variance to vanish exponentially with system size. Passive photonic circuits, central to linear optical quantum computing, challenge this picture: although their Hilbert space can be exponentially large, their dynamics are constrained to a Lie algebra whose dimension scales as the square of the number of modes. In photonic systems, postselection also plays a central role, with gradient concentration governed not by the Hilbert-space dimension but by how it reshapes the effective observable. Through exact statevector simulations, we compare allow-bunching evolution, collision-free filtering, and dual-rail postselection. In the allow-bunching and collision-free regimes, gradient variance remains consistent with polynomial rather than exponential decay over the tested system sizes. By contrast, dual-rail postselection induces exponential concentration beyond moderate system sizes, robustly across three initialization ensembles. These results indicate that photonic barren plateaus are governed by the interplay between passive linear-optical dynamics, postselection geometry, and task observables, offering practical guidance for designing near-term photonic variational architectures.