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.
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The authors declare that all data supporting the findings of this study are available within the supplementary information files of this paper.
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The code that supports the findings of this study is available from the first author (zw321@cam.ac.uk) upon request.
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Acknowledgements
The authors acknowledge the support from HORIZON EIC-2022-PATHFINDERCHALLENGES-01 HEISINGBERG Project 101114978. R.Z.W. and N.G.B. acknowledge the support from the Julian Schwinger Foundation Grant No. JSF-19-02-0005. N.G.B. also acknowledges support from Weizmann-UK Make Connection Grant 142568 and the EPSRC UK Multidisciplinary Centre for Neuromorphic Computing (grant UKRI982).
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R.Z.W. developed the simulation code and performed numerical simulations. G.L. performed large-scale numerical simulations, produced Fig. 4, and provided comments during manuscript preparation. S.G., D.P., M.C.S., and C.C. contributed implementation considerations for physical hardware, including expected operating timescales, and provided input on experimental feasibility. R.Z.W. wrote the initial manuscript draft. N.G.B. substantially revised and edited the manuscript. N.G.B. supervised the study.
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Communications Physics thanks Mostafa Honari, Roman Khymyn and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
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Wang, R.Z., Li, G., Gentilini, S. et al. Phase retrieval via gain-based photonic XY-Hamiltonian optimization. Commun Phys (2026). https://doi.org/10.1038/s42005-026-02525-7
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DOI: https://doi.org/10.1038/s42005-026-02525-7
