Skip to main content
SpringerLink
Log in
Book cover

International Conference on Unconventional Computation and Natural Computation

UCNC 2019: Unconventional Computation and Natural Computation pp 232–256Cite as

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

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNTCS,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.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   49.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   64.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    Moreover, as we show in Sect. 3.4, our scheme is inherently resistant to variability even without such calibration.

  2. 2.

    More generally, \(c_{ij}\)s can be any \(2\pi \)-periodic odd functions, which are better suited to practical oscillators.

  3. 3.

    In the Ising Hamiltonian (1), \(J_{ij}\) is only defined when \(i<j\); here we assume that \(J_{ij} = J_{ji}\) for all i,  j.

  4. 4.

    More generally, we can use \(\{2k\pi ~|~k\in \mathbf {Z}\}\) and \(\{2k\pi +\pi ~|~k\in \mathbf {Z}\}\) to represent the two states for each oscillator’s phase.

  5. 5.

    The G-set problems are available for download as set1 at http://www.optsicom.es/maxcut.

  6. 6.

    G1\(\sim \)21 are of size 800; G22\(\sim \)42 are of size 2000; G43\(\sim \)47, G51\(\sim \)54 are of size 1000; G48\(\sim \)50 are of size 3000.

  7. 7.

    Their results and runtime are available for download at http://www.optsicom.es/maxcut in the “Computational Experiences” section.

  8. 8.

    Ising machines can be used on general graph colouring problems, and this four-colouring problem is chosen here for illustrative purposes. Four-colouring a planar graph is actually not NP-hard and there exist polynomial-time algorithms for it [46].

  9. 9.

    Hawaii and Alaska are considered adjacent such that their colours will be different in the map.

References

  1. Ising, E.: Beitrag zur theorie des ferromagnetismus. Zeitschrift für Physik A Hadrons and Nuclei 31(1), 253–258 (1925)

    Google Scholar 

  2. Brush, S.G.: History of the Lenz-Ising Model. Rev. Mod. Phys. 39, 883–893 (1967)

    Article  Google Scholar 

  3. Barahona, F.: On the computational complexity of Ising spin glass models. J. Phys. A: Math. Gen. 15(10), 3241 (1982)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  5. McMahon, P.L., et al.: A fully-programmable 100-spin coherent Ising machine with all-to-all connections. Science 354, 5178 (2016)

    Article  MathSciNet  Google Scholar 

  6. Inagaki, T., et al.: A coherent Ising machine for 2000-node optimization problems. Science 354(6312), 603–606 (2016)

    Article  Google Scholar 

  7. Johnson, M.W., et al.: Quantum annealing with manufactured Spins. Nature 473(7346), 194 (2011)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Chapter  Google Scholar 

  11. Lucas, A.: Ising formulations of many NP problems. arXiv preprint arXiv:1302.5843 (2013)

  12. Wang, T., Roychowdhury, J.: Oscillator-based Ising Machine. arXiv preprint arXiv:1709.08102 (2017)

  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)

    Article  MathSciNet  Google Scholar 

  14. Jensen, T.R., Toft, B.: Graph Coloring Problems. Wiley, New York (2011)

    MATH  Google Scholar 

  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. 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. 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)

    Chapter  Google Scholar 

  18. Kuramoto, Y.: Chemical Oscillations, Waves and Turbulence. Dover, New York (2003)

    MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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_29

    Chapter  Google Scholar 

  21. Wang, T.: Sub-harmonic Injection Locking in Metronomes. arXiv preprint arXiv:1709.03886 (2017)

  22. Wang, T.: Achieving Phase-based Logic Bit Storage in Mechanical Metronomes. arXiv preprint arXiv:1710.01056 (2017)

  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. 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)

    Article  Google Scholar 

  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. 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. Harris, R., et al.: Experimental demonstration of a robust and scalable flux qubit. Phys. Rev. B 81(13), 134–510 (2010)

    Google Scholar 

  28. Rønnow, T.F., et al.: Defining and detecting quantum speedup. Science 345(6195), 420–424 (2014)

    Article  Google Scholar 

  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. Mahboob, I., Okamoto, H., Yamaguchi, H.: An electromechanical Ising Hamiltonian. Sci. Adv. 2(6), e1600236 (2016)

    Article  Google Scholar 

  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. 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. Winfree, A.: Biological rhythms and the behavior of populations of coupled oscillators. Theor. Biol. 16, 15–42 (1967)

    Article  Google Scholar 

  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/10.1109/81.847872

    Article  Google Scholar 

  35. Wang, T., Roychowdhury, J.: OIM: Oscillator-based Ising Machines for Solving Combinatorial Optimisation Problems. arXiv preprint arXiv:1903.07163 (2019)

    Chapter  Google Scholar 

  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. Shinomoto, S., Kuramoto, Y.: Phase transitions in active rotator systems. Progress Theoret. Phys. 75(5), 1105–1110 (1986)

    Article  Google Scholar 

  38. Lyapunov, A.M.: The general problem of the stability of motion. Int. J. Control 55(3), 531–534 (1992)

    Article  MathSciNet  Google Scholar 

  39. Hopfield, J.J., Tank, D.W.: “Neural” computation of decisions in optimization problems. Biol. Cybern. 52(3), 141–152 (1985)

    MATH  Google Scholar 

  40. Ercsey-Ravasz, M., Toroczkai, Z.: Optimization hardness as transient chaos in an analog approach to constraint satisfaction. Nat. Phys. 7(12), 966 (2011)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  42. Myklebust, T.: Solving maximum cut problems by simulated annealing. arXiv preprint arXiv:1505.03068 (2015)

  43. Helmberg, C., Rendl, F.: A spectral bundle method for semidefinite programming. SIAM J. Optim. 10(3), 673–696 (2000)

    Article  MathSciNet  Google Scholar 

  44. Martí, R., Duarte, A., Laguna, M.: Advanced scatter search for the max-cut problem. INFORMS J. Comput. 21(1), 26–38 (2009)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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 

Download references

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.

Author information

Authors and Affiliations

  1. Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, USA

    Tianshi Wang & Jaijeet Roychowdhury

Authors
  1. Tianshi Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jaijeet Roychowdhury
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tianshi Wang .

Editor information

Editors and Affiliations

  1. University of Saskatchewan, Saskatoon, SK, Canada

    Ian McQuillan

  2. University of Electro-Communications, Tokyo, Japan

    Shinnosuke Seki

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Wang, T., Roychowdhury, J. (2019). OIM: Oscillator-Based Ising Machines for Solving Combinatorial Optimisation Problems. In: McQuillan, I., Seki, S. (eds) Unconventional Computation and Natural Computation. UCNC 2019. Lecture Notes in Computer Science(), vol 11493. Springer, Cham. https://doi.org/10.1007/978-3-030-19311-9_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-19311-9_19

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-19310-2

  • Online ISBN: 978-3-030-19311-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation