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Complex vector gain-based annealer for minimizing XY Hamiltonians

James S. Cummins and Natalia G. Berloff*

  • *Contact author: n.g.berloff@damtp.cam.ac.uk
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Phys. Rev. Applied 25, 034080 – Published 25 March, 2026

DOI: https://doi.org/10.1103/lpwn-t7sv

Abstract

This paper is a contribution to the Physical Review Applied collection titled Physics-Inspired Computing.

We present the complex vector gain-based annealer (CVGA), an analog computing platform designed to overcome energy barriers in XY Hamiltonians through a higher-dimensional representation. Traditional gain-based solvers using optical or photonic hardware typically represent each XY spin with a single complex field. These solvers often struggle with large energy barriers in complex landscapes, leading to relaxation into excited states. CVGA addresses these limitations by using two complex fields to represent each XY spin and dynamically evolving the energy landscape through time-dependent annealing. Operating in a higher-dimensional space, CVGA bridges energy barriers in this expanded space during the continuous phase evolution, thus avoiding entrapment in local minima. We introduce several graph structures that pose challenges for XY minimization and use them to benchmark CVGA against single-dimension XY solvers, highlighting the benefits of higher-dimensional operation.

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This article appears in the following collection:

Collection on Physics-Inspired Computing

Physical Review Applied is pleased to present a Collection on Physics-Inspired Computing, highlighting the rapidly evolving field of energy-efficient computing techniques, from hardware technologies to algorithms, where physics inspiration serves as the crucial link. Contributions to this collection will be published throughout 2025. This Collection is being curated by Guest Editors Kerem Camsari and Supriyo Datta.

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References (55)

  1. M. W. Johnson, M. H. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A. J. Berkley, J. Johansson, P. Bunyk et al., Quantum annealing with manufactured spins, Nature 473, 194 (2011).
  2. V. S. Denchev, S. Boixo, S. V. Isakov, N. Ding, R. Babbush, V. Smelyanskiy, J. Martinis, and H. Neven, What is the computational value of finite range tunneling? Phys. Rev. X 6, 031015 (2016).
  3. F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. Brandao, D. A. Buell et al., Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019).
  4. P. L. McMahon, A. Marandi, Y. Haribara, R. Hamerly, C. Langrock, S. Tamate, T. Inagaki, H. Takesue, S. Utsunomiya, K. Aihara et al., A fully programmable 100-spin coherent Ising machine with all-to-all connections, Science 354, 614 (2016).
  5. T. Inagaki, Y. Haribara, K. Igarashi, T. Sonobe, S. Tamate, T. Honjo, A. Marandi, P. L. McMahon, T. Umeki, K. Enbutsu et al., A coherent Ising machine for 2000-node optimization problems, Science 354, 603 (2016).
  6. Y. Yamamoto, K. Aihara, T. Leleu, K.-I. Kawarabayashi, S. Kako, M. Fejer, K. Inoue, and H. Takesue, Coherent Ising machines—Optical neural networks operating at the quantum limit, npj Quantum Inf. 3, 1 (2017).
  7. T. Honjo, T. Sonobe, K. Inaba, T. Inagaki, T. Ikuta, Y. Yamada, T. Kazama, K. Enbutsu, T. Umeki, and R. Kasahara, 100,000-spin coherent Ising machine, Sci. Adv. 7, eabh0952 (2021).
  8. T. Honjo, K. Inaba, T. Inagaki, T. Ikuta, Y. Yamada, and H. Takesue, in 2022 IEEE 22nd International Conference on Nanotechnology (NANO) (IEEE, Palma de Mallorca, Spain, 2022), pp. 405–408.
  9. F. Cai, S. Kumar, T. Van Vaerenbergh, X. Sheng, R. Liu, C. Li, Z. Liu, M. Foltin, S. Yu, Q. Xia et al., Power-efficient combinatorial optimization using intrinsic noise in memristor Hopfield neural networks, Nat. Electron. 3, 409 (2020).
  10. M. Babaeian, D. T. Nguyen, V. Demir, M. Akbulut, P.-A. Blanche, Y. Kaneda, S. Guha, M. A. Neifeld, and N. Peyghambarian, A single shot coherent Ising machine based on a network of injection-locked multicore fiber lasers, Nat. Commun. 10, 3516 (2019).
  11. V. Pal, S. Mahler, C. Tradonsky, A. A. Friesem, and N. Davidson, Rapid fair sampling of the XY spin Hamiltonian with a laser simulator, Phys. Rev. Res. 2, 033008 (2020).
  12. M. Parto, W. Hayenga, A. Marandi, D. N. Christodoulides, and M. Khajavikhan, Realizing spin Hamiltonians in nanoscale active photonic lattices, Nat. Mater. 19, 725 (2020).
  13. D. Pierangeli, G. Marcucci, and C. Conti, Large-scale photonic Ising machine by spatial light modulation, Phys. Rev. Lett. 122, 213902 (2019).
  14. C. Roques-Carmes, Y. Shen, C. Zanoci, M. Prabhu, F. Atieh, L. Jing, T. Dubček, C. Mao, M. R. Johnson, V. Čeperić et al., Heuristic recurrent algorithms for photonic Ising machines, Nat. Commun. 11, 249 (2020).
  15. K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, Quantum simulation of frustrated Ising spins with trapped ions, Nature 465, 590 (2010).
  16. N. G. Berloff, M. Silva, K. Kalinin, A. Askitopoulos, J. D. Töpfer, P. Cilibrizzi, W. Langbein, and P. G. Lagoudakis, Realizing the classical XY Hamiltonian in polariton simulators, Nat. Mater. 16, 1120 (2017).
  17. K. P. Kalinin, A. Amo, J. Bloch, and N. G. Berloff, Polaritonic XY-Ising machine, Nanophotonics 9, 4127 (2020).
  18. M. Vretenar, B. Kassenberg, S. Bissesar, C. Toebes, and J. Klaers, Controllable Josephson junction for photon Bose-Einstein condensates, Phys. Rev. Res. 3, 023167 (2021).
  19. A. Litvinenko, R. Khymyn, R. Ovcharov, and J. Åkerman, A 50-spin surface acoustic wave Ising machine, Commun. Phys. 8, 58 (2025).
  20. A. Lucas, Ising formulations of many NP problems, Front. Phys. 2, 5 (2014).
  21. M. Yamaoka, C. Yoshimura, M. Hayashi, T. Okuyama, H. Aoki, and H. Mizuno, A 20k-spin Ising chip to solve combinatorial optimization problems with CMOS annealing, IEEE J. Solid-State Circuits 51, 303 (2016).
  22. K. Tanahashi, S. Takayanagi, T. Motohashi, and S. Tanaka, Application of Ising machines and a software development for Ising machines, J. Phys. Soc. Jpn. 88, 061010 (2019).
  23. A. Momeni, B. Rahmani, B. Scellier, L. G. Wright, P. L. McMahon, C. C. Wanjura, Y. Li, A. Skalli, N. G. Berloff, T. Onodera et al., Training of physical neural networks, Nature 645, 53 (2025).
  24. X. Guo, T. D. Barrett, Z. M. Wang, and A. Lvovsky, Backpropagation through nonlinear units for the all-optical training of neural networks, Photonics Res. 9, B71 (2021).
  25. J. Spall, X. Guo, and A. Lvovsky, Training neural networks with end-to-end optical backpropagation, Adv. Photonics 7, 016004 (2025).
  26. P. Freire, E. Manuylovich, J. E. Prilepsky, and S. K. Turitsyn, Artificial neural networks for photonic applications—From algorithms to implementation: Tutorial, Adv. Opt. Photonics 15, 739 (2023).
  27. M. Gilli, D. Maringer, and E. Schumann, Numerical Methods and Optimization in Finance (Academic Press, Amsterdam, 2019).
  28. R. Z. Wang, J. S. Cummins, M. Syed, N. Stroev, G. Pastras, J. Sakellariou, S. Tsintzos, A. Askitopoulos, D. Veraldi, M. C. Strinati et al., Efficient computation using spatial-photonic Ising machines with low-rank and circulant matrix constraints, Commun. Phys. 8, 86 (2025).
  29. P. W. Anderson, in Basic Notions of Condensed Matter Physics (CRC Press, Boca Raton, 2018).
  30. P. Minnhagen, The two-dimensional Coulomb gas, vortex unbinding, and superfluid-superconducting films, Rev. Mod. Phys. 59, 1001 (1987).
  31. W. H. Zurek, in Formation and Interactions of Topological Defects: Proceedings of a NATO Advanced Study Institute on Formation and Interactions of Topological Defects (Springer, New York, NY, 1995), pp. 349–378.
  32. P.-G. De Gennes and J. Prost, The Physics of Liquid Crystals, 83 (Oxford University Press, Oxford, 1993).
  33. D. Gallina and G. Pastor, Disorder-induced transformation of the energy landscapes and magnetization dynamics in two-dimensional ensembles of dipole-coupled magnetic nanoparticles, Phys. Rev. X 10, 021068 (2020).
  34. K. A. Dill, S. B. Ozkan, M. S. Shell, and T. R. Weikl, The protein folding problem, Annu. Rev. Biophys. 37, 289 (2008).
  35. E. J. Candes, Y. C. Eldar, T. Strohmer, and V. Voroninski, Phase retrieval via matrix completion, SIAM Rev. 57, 225 (2015).
  36. M. Syed and N. G. Berloff, Physics-enhanced bifurcation optimisers: All you need is a canonical complex network, IEEE J. Sel. Top. Quantum Electron. 29, 1 (2023).
  37. J. S. Cummins, H. Salman, and N. G. Berloff, Ising Hamiltonian minimization: Gain-based computing with manifold reduction of soft spins vs quantum annealing, Phys. Rev. Res. 7, 013150 (2025).
  38. J. S. Cummins and N. G. Berloff, Vector Ising spin annealer for minimizing Ising Hamiltonians, Commun. Phys. 8, 225 (2025).
  39. K. P. Kalinin and N. G. Berloff, Global optimization of spin Hamiltonians with gain-dissipative systems, Sci. Rep. 8, 17791 (2018).
  40. S. Abreu, I. Boikov, M. Goldmann, T. Jonuzi, A. Lupo, S. Masaad, L. Nguyen, E. Picco, G. Pourcel, A. Skalli et al., A photonics perspective on computing with physical substrates, Rev. Phys., 12100093 (2024).
  41. N. Stroev and N. G. Berloff, Discrete polynomial optimization with coherent networks of condensates and complex coupling switching, Phys. Rev. Lett. 126, 050504 (2021).
  42. K. Kalinin, G. Mourgias-Alexandris, H. Ballani, N. G. Berloff, J. H. Clegg, D. Cletheroe, C. Gkantsidis, I. Haller, V. Lyutsarev, F. Parmigiani et al., Analog Iterative Machine (AIM): using light to solve quadratic optimization problems with mixed variables, arXiv preprint arXiv:2304.12594.
  43. K. P. Kalinin, J. Gladrow, J. Chu, J. H. Clegg, D. Cletheroe, D. J. Kelly, B. Rahmani, G. Brennan, B. Canakci, F. Falck et al., Analog optical computer for AI inference and combinatorial optimization, Nature 645, 354 (2025).
  44. H. Ohadi, R. Gregory, T. Freegarde, Y. Rubo, A. Kavokin, N. Berloff, and P. Lagoudakis, Nontrivial phase coupling in polariton multiplets, Phys. Rev. X 6, 031032 (2016).
  45. N. Stroev and N. G. Berloff, Programmable k-local Ising machines and all-optical Kolmogorov-Arnold networks on photonic platforms, arXiv preprint arXiv:2508.17440.
  46. M. C. Strinati, D. Pierangeli, and C. Conti, All-optical scalable spatial coherent Ising machine, Phys. Rev. Appl. 16, 054022 (2021).
  47. A. B. Ayoub and D. Psaltis, High speed, complex wavefront shaping using the digital micro-mirror device, Sci. Rep. 11, 18837 (2021).
  48. K. P. Kalinin and N. G. Berloff, Computational complexity continuum within Ising formulation of NP problems, Commun. Phys. 5, 20 (2020).
  49. S. Istrail, Statistical mechanics, three-dimensionality and NP-completeness: I. Universality of intractability for the partition function of the Ising model across non-planar lattices, in Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, New York, NY, 2000), pp. 87–96.
  50. T. F. Rønnow, Z. Wang, J. Job, S. Boixo, S. V. Isakov, D. Wecker, J. M. Martinis, D. A. Lidar, and M. Troyer, Defining and detecting quantum speedup, Science 345, 420 (2014).
  51. P. B. Wigley, P. J. Everitt, A. van den Hengel, J. W. Bastian, M. A. Sooriyabandara, G. D. McDonald, K. S. Hardman, C. D. Quinlivan, P. Manju, C. C. Kuhn et al., Fast machine-learning online optimization of ultra-cold-atom experiments, Sci. Rep. 6, 25890 (2016).
  52. D. Subires, F. J. Gómez-Ruiz, A. Ruiz-García, D. Alonso, and A. Del Campo, Benchmarking quantum annealing dynamics: The spin-vector Langevin model, Phys. Rev. Res. 4, 023104 (2022).
  53. R. Hamerly, T. Inagaki, P. L. McMahon, D. Venturelli, A. Marandi, T. Onodera, E. Ng, C. Langrock, K. Inaba, T. Honjo et al., Experimental investigation of performance differences between coherent Ising machines and a quantum annealer, Sci. Adv. 5, eaau0823 (2019).
  54. M. P. Harrigan, K. J. Sung, M. Neeley, K. J. Satzinger, F. Arute, K. Arya, J. Atalaya, J. C. Bardin, R. Barends, S. Boixo et al., Quantum approximate optimization of non-planar graph problems on a planar superconducting processor, Nat. Phys. 17, 332 (2021).
  55. F. Böhm, G. Verschaffelt, and G. Van der Sande, A poor man’s coherent Ising machine based on opto-electronic feedback systems for solving optimization problems, Nat. Commun. 10, 3538 (2019).

Outline

Information

Phys. Rev. Applied 25, 034080– Published 25 March, 2026
  • Received 25 December 2024
  • Revised 24 November 2025
  • Accepted 20 January 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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