# Phase retrieval via gain-based photonic XY-Hamiltonian optimization

## TL;DR

The paper presents a new method for solving phase retrieval problems using photonic networks, which outperforms existing methods in terms of speed and efficiency.

## Contribution

The novel contribution is reformulating phase retrieval as an XY Hamiltonian minimization problem solvable via gain-based photonic networks.

## Key findings

- The gain-based photonic solver outperforms the RRR algorithm in the medium-noise regime.
- The method retains performance advantage as problem size increases.
- The approach is suitable for optical parallelism and energy-efficient computation.

## Abstract

Phase-retrieval from coded diffraction patterns (CDP) is important to X-ray crystallography, diffraction tomography and astronomical imaging, yet remains a hard, non-convex inverse problem. We show that CDP recovery can be reformulated exactly as the minimization of a continuous-variable XY Hamiltonian and solved by gain-based photonic networks. The coupled-mode equations we exploit are the natural mean-field dynamics of exciton-polariton condensate lattices, coupled-laser arrays and driven photon Bose-Einstein condensates, while other hardware such as the spatial photonic Ising machine can implement the same update rule through high-speed digital feedback, preserving full optical parallelism. Numerical experiments on images, two- and three-dimensional vortices and unstructured complex data demonstrate that the gain-based solver consistently outperforms the state-of-the-art Relaxed-Reflect-Reflect (RRR) algorithm in the medium-noise regime (signal-to-noise ratios 10-40 dB) and retains this advantage as problem size scales. Because the physical platform performs the continuous optimisation, our approach promises fast, energy-efficient phase retrieval on readily available photonic hardware.

Recovering missing phase from measured intensity is central to X-ray crystallography and diffraction imaging but is a hard, non-convex inverse problem. The authors reformulate coded diffraction pattern phase retrieval exactly as minimizing a continuous XY energy and solve it with gain-based photonic oscillator networks with efficient scaling and noise tolerance.