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OIM: Oscillator-Based Ising Machines for Solving Combinatorial Optimisation Problems

  • Tianshi WangEmail author
  • Jaijeet Roychowdhury
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11493)

Abstract

We present a new way to make Ising machines, i.e., using networks of coupled self-sustaining nonlinear oscillators. Our scheme is theoretically rooted in a novel result that establishes that the phase dynamics of coupled oscillator systems, under the influence of subharmonic injection locking, are governed by a Lyapunov function that is closely related to the Ising Hamiltonian of the coupling graph. As a result, the dynamics of such oscillator networks evolve naturally to local minima of the Lyapunov function. Two simple additional steps (i.e., adding noise, and turning subharmonic locking on and off smoothly) enable the network to find excellent solutions of Ising problems. We demonstrate our method on Ising versions of the MAX-CUT and graph colouring problems, showing that it improves on previously published results on several problems in the G benchmark set. Our scheme, which is amenable to realisation using many kinds of oscillators from different physical domains, is particularly well suited for CMOS IC implementation, offering significant practical advantages over previous techniques for making Ising machines. We present working hardware prototypes using CMOS electronic oscillators.

Notes

Acknowledgements

The authors would like to thank the reviewers for the useful comments and in particular anonymous reviewer No. 2 for pointing us to Ercsey-Ravasz/Toroczkai and Yin’s work on designing dynamical systems to solve NP-complete problems.

References

  1. 1.
    Ising, E.: Beitrag zur theorie des ferromagnetismus. Zeitschrift für Physik A Hadrons and Nuclei 31(1), 253–258 (1925)Google Scholar
  2. 2.
    Brush, S.G.: History of the Lenz-Ising Model. Rev. Mod. Phys. 39, 883–893 (1967)CrossRefGoogle Scholar
  3. 3.
    Barahona, F.: On the computational complexity of Ising spin glass models. J. Phys. A: Math. Gen. 15(10), 3241 (1982)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Marandi, A., Wang, Z., Takata, K., Byer, R.L., Yamamoto, Y.: Network of time-multiplexed optical parametric oscillators as a coherent Ising machine. Nat. Photonics 8(12), 937–942 (2014)CrossRefGoogle Scholar
  5. 5.
    McMahon, P.L., et al.: A fully-programmable 100-spin coherent Ising machine with all-to-all connections. Science 354, 5178 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Inagaki, T., et al.: A coherent Ising machine for 2000-node optimization problems. Science 354(6312), 603–606 (2016)CrossRefGoogle Scholar
  7. 7.
    Johnson, M.W., et al.: Quantum annealing with manufactured Spins. Nature 473(7346), 194 (2011)CrossRefGoogle Scholar
  8. 8.
    Bian, Z., Chudak, F., Israel, R., Lackey, B., Macready, W.G., Roy, A.: Discrete optimization using quantum annealing on sparse Ising models. Front. Phys. 2, 56 (2014)CrossRefGoogle Scholar
  9. 9.
    Yamaoka, M., Yoshimura, C., Hayashi, M., Okuyama, T., Aoki, H., Mizuno, H.: A 20k-spin Ising chip to solve combinatorial optimization problems with CMOS annealing. IEEE J. Solid-State Circuits 51(1), 303–309 (2016)CrossRefGoogle Scholar
  10. 10.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations, pp. 85–103. Springer, Boston (1972)CrossRefGoogle Scholar
  11. 11.
    Lucas, A.: Ising formulations of many NP problems. arXiv preprint arXiv:1302.5843 (2013)
  12. 12.
    Wang, T., Roychowdhury, J.: Oscillator-based Ising Machine. arXiv preprint arXiv:1709.08102 (2017)
  13. 13.
    Festa, P., Pardalos, P.M., Resende, M.G.C., Ribeiro, C.C.: Randomized heuristics for the MAX-CUT problem. Optim. Methods Softw. 17(6), 1033–1058 (2002)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Jensen, T.R., Toft, B.: Graph Coloring Problems. Wiley, New York (2011)zbMATHGoogle Scholar
  15. 15.
    Neogy, A., Roychowdhury, J.: Analysis and Design of Sub-harmonically Injection Locked Oscillators. In: Proceedings of the IEEE DATE, March 2012. http://dx.doi.org/10.1109/DATE.2012.6176677
  16. 16.
    Bhansali, P., Roychowdhury, J.: Gen-Adler: The generalized Adler’s equation for injection locking analysis in oscillators. In: Proceedings of the IEEE ASP-DAC, pp. 522–227, January 2009. http://dx.doi.org/10.1109/ASPDAC.2009.4796533
  17. 17.
    Kuramoto, Y.: Self-entrainment of a population of coupled non-linear oscillators. In: Araki, H. (ed.) International Symposium on Mathematical Problems in Theoretical Physics, pp. 420–422. Springer, Heidelberg (1975)CrossRefGoogle Scholar
  18. 18.
    Kuramoto, Y.: Chemical Oscillations, Waves and Turbulence. Dover, New York (2003)zbMATHGoogle Scholar
  19. 19.
    Acebrón, J.A., Bonilla, L.L., Vicente, C.J.P., Ritort, F., Spigler, R.: The kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77(1), 137 (2005)CrossRefGoogle Scholar
  20. 20.
    Wang, T., Roychowdhury, J.: PHLOGON: PHase-based LOGic using oscillatory nano-systems. In: Ibarra, O.H., Kari, L., Kopecki, S. (eds.) UCNC 2014. LNCS, vol. 8553, pp. 353–366. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-08123-6_29CrossRefGoogle Scholar
  21. 21.
    Wang, T.: Sub-harmonic Injection Locking in Metronomes. arXiv preprint arXiv:1709.03886 (2017)
  22. 22.
    Wang, T.: Achieving Phase-based Logic Bit Storage in Mechanical Metronomes. arXiv preprint arXiv:1710.01056 (2017)
  23. 23.
    Aramon, M., Rosenberg, G., Valiante, E., Miyazawa, T., Tamura, H., Katzgraber, H.G.: Physics-inspired optimization for quadratic unconstrained problems using a digital annealer. arXiv:1806.08815 [physics.comp-ph] August 2018
  24. 24.
    Gyoten, H., Hiromoto, M., Sato, T.: Area efficient annealing processor for ising model without random number generator. IEICE Trans. Inf. Syst. E101.D(2), 314–323 (2018)CrossRefGoogle Scholar
  25. 25.
    Gyoten, H., Hiromoto, M., Sato, T.: Enhancing the solution quality of hardware Ising-model solver via parallel tempering. In: Proceedings of the ICCAD, ICCAD 2018, pp. 70:1–70:8. ACM, New York (2018)Google Scholar
  26. 26.
    Bian, Z., Chudak, F., Macready, W.G., Rose, G.: The Ising model: teaching an old problem new tricks. D-Wave Syst. 2, 1–32 (2010)Google Scholar
  27. 27.
    Harris, R., et al.: Experimental demonstration of a robust and scalable flux qubit. Phys. Rev. B 81(13), 134–510 (2010)Google Scholar
  28. 28.
    Rønnow, T.F., et al.: Defining and detecting quantum speedup. Science 345(6195), 420–424 (2014)CrossRefGoogle Scholar
  29. 29.
    Denchev, V.S., et al.: What is the computational value of finite-range tunneling? Phys. Rev. X 6(3), 031015 (2016)Google Scholar
  30. 30.
    Mahboob, I., Okamoto, H., Yamaguchi, H.: An electromechanical Ising Hamiltonian. Sci. Adv. 2(6), e1600236 (2016)CrossRefGoogle Scholar
  31. 31.
    Camsari, K.Y., Faria, R., Sutton, B.M., Datta, S.: Stochastic p-bits for invertible logic. Phys. Rev. X 7(3), 031014 (2017)Google Scholar
  32. 32.
    Yamamoto, K., Huang, W., Takamaeda-Yamazaki, S., Ikebe, M., Asai, T., Motomura, M.: A time-division multiplexing Ising machine on FPGAs. In: Proceedings of the 8th International Symposium on Highly Efficient Accelerators and Reconfigurable Technologies, p. 3. ACM (2017)Google Scholar
  33. 33.
    Winfree, A.: Biological rhythms and the behavior of populations of coupled oscillators. Theor. Biol. 16, 15–42 (1967)CrossRefGoogle Scholar
  34. 34.
    Demir, A., Mehrotra, A., Roychowdhury, J.: Phase noise in oscillators: a unifying theory and numerical methods for characterization. IEEE Trans. Circuits Syst.- I: Fund. Th. Appl. 47, 655–674 (2000). http://dx.doi.org/10.1109/81.847872CrossRefGoogle Scholar
  35. 35.
    Wang, T., Roychowdhury, J.: OIM: Oscillator-based Ising Machines for Solving Combinatorial Optimisation Problems. arXiv preprint arXiv:1903.07163 (2019)CrossRefGoogle Scholar
  36. 36.
    Wang, T., Roychowdhury, J.: Design tools for oscillator-based computing systems. In: Proceedings IEEE DAC, pp. 188:1–188:6 (2015). http://dx.doi.org/10.1145/2744769.2744818
  37. 37.
    Shinomoto, S., Kuramoto, Y.: Phase transitions in active rotator systems. Progress Theoret. Phys. 75(5), 1105–1110 (1986)CrossRefGoogle Scholar
  38. 38.
    Lyapunov, A.M.: The general problem of the stability of motion. Int. J. Control 55(3), 531–534 (1992)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Hopfield, J.J., Tank, D.W.: “Neural” computation of decisions in optimization problems. Biol. Cybern. 52(3), 141–152 (1985)zbMATHGoogle Scholar
  40. 40.
    Ercsey-Ravasz, M., Toroczkai, Z.: Optimization hardness as transient chaos in an analog approach to constraint satisfaction. Nat. Phys. 7(12), 966 (2011)CrossRefGoogle Scholar
  41. 41.
    Yin, X., Sedighi, B., Varga, M., Ercsey-Ravasz, M., Toroczkai, Z., Hu, X.S.: Efficient analog circuits for boolean satisfiability. IEEE Trans. Very Large Scale Integr. (VLSI) Syst. 26(1), 155–167 (2018)CrossRefGoogle Scholar
  42. 42.
    Myklebust, T.: Solving maximum cut problems by simulated annealing. arXiv preprint arXiv:1505.03068 (2015)
  43. 43.
    Helmberg, C., Rendl, F.: A spectral bundle method for semidefinite programming. SIAM J. Optim. 10(3), 673–696 (2000)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Martí, R., Duarte, A., Laguna, M.: Advanced scatter search for the max-cut problem. INFORMS J. Comput. 21(1), 26–38 (2009)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Burer, S., Monteiro, R., Zhang, Y.: Rank-two relaxation heuristics for max-cut and other binary quadratic programs. SIAM J. Optim. 12(2), 503–521 (2002)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Robertson, N., Sanders, D.P., Seymour, P., Thomas, R.: Efficiently four-coloring planar graphs. In: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, pp. 571–575. ACM (1996)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Electrical Engineering and Computer SciencesUniversity of CaliforniaBerkeleyUSA

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