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Adaptive Ising Machine Based on Phase Locking of an Auto-Oscillator to a Biharmonic External Driving with Noise

Eleonora Raimondo1, Andrea Grimaldi2, Vasyl Tyberkevych3, Riccardo Tomasello2, Anna Giordano4, Mario Carpentieri2, Andrei Slavin3, Massimo Chiappini1, and Giovanni Finocchio1,5,*

  • *Contact author: gfinocchio@unime.it
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Phys. Rev. Lett. 136, 177201 – Published 29 April, 2026

DOI: https://doi.org/10.1103/jv9m-4mny

Abstract

An auto-oscillator driven by a harmonic signal at about twice its free-running frequency is characterized by a bistable phase dynamics where the two states are separated by π radians. This phase bistability enables an oscillator to emulate a single Ising spin, providing a fundamental building block for the oscillator-based Ising machines (OIM). At the same time, a driving signal close to the oscillator free-running frequency locks the oscillator’s phase at a single value, playing the role of a magnetic field bias in ensembles of real spins. We introduce a universal theory of phase auto-oscillators driven by a biharmonic signal (having frequency components close to single and double of the free-running oscillator frequency) with noise; with it, we show how deterministic phase locking and stochastic phase slips can be continuously tuned by varying the relative amplitudes and frequencies of the driving components. Using, as an example, a spin-torque nano-oscillator, we numerically validate this theory by implementing a deterministic Ising machine paradigm, a probabilistic one, and dual-mode operation of the two. This demonstration introduces the concept of adaptive Ising machines (AIM), a unified oscillator-based architecture that dynamically combines both regimes within the same hardware platform by properly tuning the amplitudes of the biharmonic driving relative to the noise strength. Benchmarking on different classes of combinatorial optimization problems, the AIM exhibits complementary performance compared to OIMs and probabilistic Ising machines, with adaptability to the specific problem class. This Letter introduces the first OIM capable of transitioning between deterministic and probabilistic computation taking advantage of a proper design of the trade-off between the strength of phase-locking of an auto-oscillator to a biharmonic external driving and noise, opening a path toward scalable, CMOS-compatible hardware for hybrid optimization and inference.

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References (37)

  1. J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort, and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys. 77, 137 (2005).
  2. T. Wang and J. Roychowdhury, OIM: Oscillator-based ising machines for solving combinatorial optimisation problems, in Unconventional Computation and Natural Computation (Springer, New York, 2019), pp. 232–256.
  3. T. Zhang et al., A review of Ising machines implemented in conventional and emerging technologies, IEEE Trans. Nanotechnol. 23, 704 (2024).
  4. D. I. Albertsson and A. Rusu, Highly reconfigurable oscillator-based ising machine through quasiperiodic modulation of coupling strength, Sci. Rep. 13, 4005 (2023).
  5. S. Chiavazzo, M. Calvanese Strinati, C. Conti, and D. Pierangeli, Ising machine by dimensional collapse of nonlinear polarization oscillators, Phys. Rev. Lett. 135, 063801 (2025).
  6. Z.-Y. Zhou, C. Gneiting, J. Q. You, and F. Nori, Frustration elimination and excited state search in coherent Ising machines, Phys. Rev. Lett. 134, 090401 (2025).
  7. T. Honjo et al., 100,000-spin coherent Ising machine, Sci. Adv. 7, 952 (2021).
  8. H. Yamashita, K. Okubo, S. Shimomura, Y. Ogura, J. Tanida, and H. Suzuki, Low-rank combinatorial optimization and statistical learning by spatial photonic Ising machine, Phys. Rev. Lett. 131, 063801 (2023).
  9. J. Vaidya, R. S. Surya Kanthi, and N. Shukla, Creating electronic oscillator-based Ising machines without external injection locking, Sci. Rep. 12, 981 (2022).
  10. M. K. Bashar, Z. Li, V. Narayanan, and N. Shukla, An FPGA-based Max-K-Cut accelerator exploiting oscillator synchronization model, in Proceedings of the 2024 25th International Symposium on Quality Electronic Design (ISQED) (IEEE, New York, 2024), pp. 1–8.
  11. A. Grimaldi, L. Sánchez-Tejerina, N. Anjum Aadit, S. Chiappini, M. Carpentieri, K. Camsari, and G. Finocchio, Spintronics-compatible approach to solving maximum-satisfiability problems with probabilistic computing, invertible logic, and parallel tempering, Phys. Rev. Appl. 17, 024052 (2022).
  12. Y.-T. Liu, C.-C. Peng, T.-Y. Hung, Y.-H. Huang, and C.-F. Pai, Advancing the problem-solving capabilities of ising machines based on spin Hall nano-oscillators, Phys. Rev. Appl. 22, 064009 (2024).
  13. K. Y. Camsari, R. Faria, B. M. Sutton, and S. Datta, Stochastic P-bits for invertible logic, Phys. Rev. X 7, 031014 (2017).
  14. N. A. Aadit, A. Grimaldi, M. Carpentieri, L. Theogarajan, J. M. Martinis, G. Finocchio, and K. Y. Camsari, Massively parallel probabilistic computing with sparse Ising machines, Natl. Electron. Rev. 5, 460 (2022).
  15. E. Raimondo et al., High-performance and reliable probabilistic Ising machine based on simulated quantum annealing, Phys. Rev. X 15, 041001 (2025).
  16. F. Böhm, D. Alonso-Urquijo, G. Verschaffelt, and G. Van der Sande, Noise-injected analog Ising machines enable ultrafast statistical sampling and machine learning, Nat. Commun. 13, 5847 (2022).
  17. G. Margiani, J. del Pino, T. L. Heugel, N. E. Bousse, S. Guerrero, T. W. Kenny, O. Zilberberg, D. Sabonis, and A. Eichler, Deterministic and stochastic sampling of two coupled Kerr parametric oscillators, Phys. Rev. Res. 5, L012029 (2023).
  18. J. Tosca, M. C. Strinati, C. Conti, and C. Ciuti, Emergent equilibrium in all-optical single quantum-trajectory Ising machines, Phys. Rev. Lett. 134, 230404 (2025).
  19. C. Delacour, Self-adaptive ising machines for constrained optimization, in 2025 Design, Automation & Test in Europe Conference (IEEE, New York, 2025), pp. 1–7.
  20. C. C. Rodrigues, C. M. Kersul, A. G. Primo, M. Lipson, T. P. M. Alegre, and G. S. Wiederhecker, Optomechanical synchronization across multi-octave frequency spans, Nat. Commun. 12, 5625 (2021).
  21. G. Beltramo, R. Horváth, G. Modica, and R. Braive, Injection-locking dynamics of two self-pulsing nanocavities optically coupled on photonic integrated circuit, Commun. Phys. 9, 12 (2025).
  22. D. Zou, Y. Song, O. Gat, M. Hu, and P. Grelu, Synchronization of the internal dynamics of optical soliton molecules, Optica 9, 1307 (2022).
  23. Y. Atobe, M. Tawada, and N. Togawa, A novel classical-Ising hybrid annealing method with QUBO model cutting, in Proceedings of the 2024 IEEE 67th International Midwest Symposium on Circuits and Systems (MWSCAS) (IEEE, New York, 2024), pp. 1154–1157.
  24. S. Kikuchi, N. Togawa, and S. Tanaka, Hybrid optimization method using simulated-annealing-based Ising machine and quantum annealer, J. Phys. Soc. Jpn. 92, 124002 (2023).
  25. See Supplemental Material at http://link.aps.org/supplemental/10.1103/jv9m-4mny for the full derivaton of the general theory of synchronization of a noisy phase oscillator, for spin-torque nano oscillator characterization and phase probability at different working points.
  26. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge, England, 2001).
  27. B. Fang et al., Giant spin-torque diode sensitivity in the absence of bias magnetic field, Nat. Commun. 7, 11259 (2016).
  28. Z. Zeng et al., Ultralow-current-density and bias-field-free spin-transfer nano-oscillator, Sci. Rep. 3, 1426 (2013).
  29. D. C. Ralph and M. D. Stiles, Spin transfer torques, J. Magn. Magn. Mater. 320, 1190 (2008).
  30. A. Meo, C. E. Cronshaw, S. Jenkins, A. Lees, and R. F. L. Evans, Spin-transfer and spin-orbit torques in the Landau–Lifshitz–Gilbert equation, J. Phys. Condens. Matter 35, 025801 (2023).
  31. N.-T. Phan et al., Unbiased random bitstream generation using injection-locked spin-torque nano-oscillators, Phys. Rev. Appl. 21, 034063 (2024).
  32. M. Tortarolo, B. Lacoste, J. Hem, C. Dieudonné, M.-C. Cyrille, J. A. Katine, D. Mauri, A. Zeltser, L. D. Buda-Prejbeanu, and U. Ebels, Injection locking at 2f of spin torque oscillators under influence of thermal noise, Sci. Rep. 8, 1728 (2018).
  33. J. F. Sierra, M. Quinsat, F. Garcia-Sanchez, U. Ebels, I. Joumard, A. S. Jenkins, B. Dieny, M.-C. Cyrille, A. Zeltser, and J. A. Katine, Influence of thermal fluctuations on the emission linewidth in MgO-based spin transfer oscillators, Appl. Phys. Lett. 101, 062407 (2012).
  34. N. Mohseni, P. L. McMahon, and T. Byrnes, Ising machines as hardware solvers of combinatorial optimization problems, Nat. Rev. Phys. 4, 363 (2022).
  35. T. Tran, M. Do, E. Rieffel, J. Frank, Z. Wang, B. O’Gorman, D. Venturelli, and J. Beck, A hybrid quantum-classical approach to solving scheduling problems, Proc. Int. Symp. Comb. Search 7, 98 (2021).
  36. MRAM-Info MRAM Industry Portal, https://www.mram-info.com/.
  37. www.petaspin.com.

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Phys. Rev. Lett. 136, 177201– Published 29 April, 2026
  • Received 12 October 2025
  • Revised 21 January 2026
  • Accepted 26 March 2026
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