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Scaling out Ising machines using a multi-chip architecture for simulated bifurcation

Nature Electronics volume 4, pages 208–217 (2021)Cite this article

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An Author Correction to this article was published on 27 July 2021

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Abstract

Ising machines are hardware devices that can solve ground-state search problems of Ising spin models and could be of use in solving various practical combinatorial optimization problems. However, large-scale systems have to be implemented by partitioning into subsystems that are hard to synchronize and where communication between them is difficult. Here, we report a scale-out architecture for Ising machines that provides enlarged machine sizes and enhanced processing speeds by using multiple connected chips. The architecture is based on the partitioned version of a quantum-inspired algorithm called simulated bifurcation. To maintain time consistency between multiple chips and a sufficiently small stall rate for every time-evolution step in simulated bifurcation, the architecture relies on an autonomous synchronization mechanism that is implemented in the information exchange processes between neighbouring chips and leads to scalability of computational throughput. Our eight-FPGA (field-programmable gate array) simulated bifurcation machine can obtain high-quality solutions to a 16,384-node MAX-CUT problem in 1.2 ms, which is 828 times faster than an optimized implementation of simulated annealing.

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Fig. 1: Proposed cluster architecture for the partitioned SB.
Fig. 2: Communication modules.
Fig. 3: Computation modules.
Fig. 4: Scalability of the cluster architecture for SB.
Fig. 5: Comparison of SB and simulated annealing for an all-to-all connected 16,384-node MAX-CUT problem.

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Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Code availability

The cluster simulator used in this work is available from the corresponding author upon reasonable request.

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References

  1. Soon, W. & Ye, H. Q. Currency arbitrage detection using a binary integer programming model. Int. J. Math. Educ. Sci. Technol. 42, 369–376 (2011).

    Article  MathSciNet  Google Scholar 

  2. Tatsumura, K., Hidaka, R., Yamasaki. M., Sakai, Y. & Goto, H. A currency arbitrage machine based on the simulated bifurcation algorithm for ultrafast detection of optimal opportunity. In Proc. IEEE International Symposium on Circuits and Systems (ISCAS) 1–5 (IEEE, 2020).

  3. Hernandez, M. & Aramon, M. Enhancing quantum annealing performance for the molecular similarity problem. Quantum Inf. Process. 16, 133 (2017).

    Article  MathSciNet  Google Scholar 

  4. Li, R. Y., Di Felice, R., Rohs, R. & Lidar, D. A. Quantum annealing versus classical machine learning applied to a simplified computational biology problem. npj Quantum Inf. 4, 14 (2018).

    Article  Google Scholar 

  5. Perdomo-Ortiz, A., Dickson, N., Drew-Brook, M., Rose, G. & Aspuru-Guzik, A. Finding low-energy conformations of lattice protein models by quantum annealing. Sci. Rep. 2, 571 (2012).

    Article  Google Scholar 

  6. Sakaguchi, H. et al. Boltzmann sampling by degenerate optical parametric oscillator network for structure-based virtual screening. Entropy 18, 365 (2016).

    Article  Google Scholar 

  7. Kanamaru, S. et al. Efficient ising model mapping to solving slot placement problem. In Proc. IEEE International Conference on Consumer Electronics (ICCE) 1–6 (IEEE, 2019).

  8. Boyda, E. et al. Deploying a quantum annealing processor to detect tree cover in aerial imagery of California. PLoS ONE 12, e0172505 (2017).

    Article  Google Scholar 

  9. Kirkpatrick, S., Gelatt, C. D. & Vecchi, M. P. Optimization by simulated annealing. Science 220, 671–680 (1983).

    Article  MathSciNet  Google Scholar 

  10. Barahona, F. On the computational complexity of Ising spin glass models. J. Phys. A 15, 3241–3253 (1982).

    Article  MathSciNet  Google Scholar 

  11. Lucas, A. Ising formulations of many NP problems. Front. Phys. 2, 5 (2014).

    Article  Google Scholar 

  12. Johnson, M. W. et al. Quantum annealing with manufactured spins. Nature 473, 194–198 (2011).

    Article  Google Scholar 

  13. Maezawa, M. et al. Toward practical-scale quantum annealing machine for prime factoring. J. Phys. Soc. Jpn 88, 061012 (2019).

    Article  Google Scholar 

  14. Yoshimura, C., Hayashi, M., Okuyama, T. & Yamaoka, M. Implementation and evaluation of FPGA-based annealing processor for Ising model by use of resource sharing. Int. J. Netw. Comput. 7, 154–172 (2017).

    Google Scholar 

  15. Takemoto, T., Hayashi, M., Yoshimura, C. & Yamaoka, M. A 2 × 30k-spin multi-chip scalable CMOS annealing processor based on a processing-in-memory approach for solving large-scale combinatorial optimization problems. IEEE J. Solid State Circuits 55, 145–156 (2019).

    Article  Google Scholar 

  16. Yamamoto, K., Ikebe, M., Asai, T., Motomura, M. & Takamaeda-Yamazaki, S. FPGA-based annealing processor with time-division multiplexing. IEICE Trans. Inf. Syst. E102.D, 2295–2305 (2019).

    Article  Google Scholar 

  17. McMahon, P. L. et al. A fully programmable 100-spin coherent Ising machine with all-to-all connections. Science 354, 614–617 (2016).

    Article  MathSciNet  Google Scholar 

  18. Inagaki, T. et al. A coherent Ising machine for 2,000-node optimization problems. Science 354, 603–606 (2016).

    Article  Google Scholar 

  19. Pierangeli, D., Marcucci, G. & Conti, C. Large-scale photonic Ising machine by spatial light modulation. Phys. Rev. Lett. 122, 213902 (2019).

    Article  Google Scholar 

  20. Pierangeli, D., Rafayelyan, M., Conti, C. & Gigan, S. Scalable spin-glass optical simulator. Preprint at https://arxiv.org/abs/2006.00828 (2020).

  21. Pierangeli, D., Marcucci, G. & Conti, C. Adiabatic evolution on a spatial-photonic Ising machine. Optica 7, 1535–1543 (2020).

    Article  Google Scholar 

  22. Tsukamoto, S., Takatsu, M., Matsubara, S. & Tamura, H. An accelerator architecture for combinatorial optimization problems. Fujitsu Sci. Technol. J. 53, 8–13 (2017).

    Google Scholar 

  23. Isakov, S. V., Zintchenko, I. N., Rønnow, T. F. & Troyer, M. Optimised simulated annealing for Ising spin glasses. Comput. Phys. Commun. 192, 265–271 (2015).

    Article  MathSciNet  Google Scholar 

  24. Hamerly, R. et al. Experimental investigation of performance differences between coherent Ising machines and a quantum annealer. Sci. Adv. 5, eaau0823 (2019).

    Article  Google Scholar 

  25. Goto, H., Tatsumura, K. & Dixon, A. R. Combinatorial optimization by simulating adiabatic bifurcations in nonlinear Hamiltonian systems. Sci. Adv. 5, eaav2372 (2019).

    Article  Google Scholar 

  26. Goto, H. Bifurcation-based adiabatic quantum computation with a nonlinear oscillator network. Sci. Rep. 6, 21686 (2016).

    Article  Google Scholar 

  27. Goto, H. Quantum computation based on quantum adiabatic bifurcations of Kerr-nonlinear parametric oscillators. J. Phys. Soc. Jpn 88, 061015 (2019).

    Article  Google Scholar 

  28. Bello, L., Strinati, M. C., DallaTorre, E. G. & Pe’er, A. Persistent coherent beating in coupled parametric oscillators. Phys. Rev. Let. 123, 083901 (2019).

    Article  Google Scholar 

  29. Strinati, M. C., Bello, L., Pe’er, A. & DallaTorre, E. G. Theory of coupled parametric oscillators beyond coupled Ising spins. Phys. Rev. A 100, 023835 (2019).

    Article  Google Scholar 

  30. Strinati, M. C. et al. Coherent dynamics in frustrated coupled parametric oscillators. New J. Phys. 22, 085005 (2020).

    Article  MathSciNet  Google Scholar 

  31. Tatsumura, K., Dixon, A. R. & Goto, H. FPGA-based simulated bifurcation machine. In Proc. IEEE International Conference on Field-Programmable Logic and Applications (FPL) 59–66 (IEEE, 2019).

  32. Hennessy, J. L. & Patterson, D. A. in Computer Architecture: A Quantitative Approach 6th edn, Appendix I (Elsevier, 2017).

  33. Del Sozzo, E., Rabozzi, M., Di Tucci, L., Sciuto, D. & Santambrogio, M. D. A scalable FPGA design for cloud n-body simulation. In Proc. IEEE International Conference on Application-Specific Systems, Architectures and Processors (ASAP) 1–8 (IEEE, 2018).

  34. Huthmann, J., Shin, A. Podobas, A., Sano, K. & Takizawa, H. Scaling performance for N-body stream computation with a ring of FPGAs. In Proc. 10th International Symposium on Highly-Efficient Accelerators and Reconfigurable Technologies (HEART) 1–6 (2019).

  35. Goemans, M. X. & Williamson, D. P. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145 (1995).

    Article  MathSciNet  Google Scholar 

  36. Hopfield, J. J. Neural networks and physical systems with emergent collective computational abilities. Proc. Natl Acad. Sci. USA 79, 2554–2558 (1982).

    Article  MathSciNet  Google Scholar 

  37. Bianchi, F. M., Maiorino, E., Kampffmeyer, M. C., Rizzi, A. & Jenssen, R. Recurrent Neural Networks for Short-Term Load Forecasting: an Overview and Comparative Analysis (Springer, 2017).

  38. Hopfield, J. J. & Tank, D. W. Computing with neural circuits: a model. Science 233, 625–633 (1986).

    Article  Google Scholar 

  39. Arora, S. & Barak, B. Computational Complexity: A Modern Approach (Cambridge Univ. Press, 2009).

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Acknowledgements

We thank Y. Kaneko, Y. Hikichi and Y. Kawamata for their support.

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Authors and Affiliations

  1. Corporate Research and Development Center, Toshiba Corporation, Kawasaki, Kanagawa, Japan

    Kosuke Tatsumura, Masaya Yamasaki & Hayato Goto

Authors
  1. Kosuke Tatsumura
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  2. Masaya Yamasaki
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  3. Hayato Goto
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Contributions

K.T. and H.G. conceived the project. K.T. designed the cluster architecture. K.T. and M.Y. implemented the design with FPGAs and measured the performance. H.G. implemented and evaluated the optimized simulated annealing. K.T. analysed and evaluated the results. K.T. wrote the manuscript. All authors reviewed and edited the manuscript.

Corresponding author

Correspondence to Kosuke Tatsumura.

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Competing interests

K.T. and H.G. are inventors on two US patent applications related to this work filed by the Toshiba Corporation (no. 16/123146, filed 6 September 2018; no. 16/118646, filed 31 August 2018).

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Peer review information Nature Electronics thanks Claudio Conti, Kentaro Sano and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary Figs. 1–9 and Discussion.

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Tatsumura, K., Yamasaki, M. & Goto, H. Scaling out Ising machines using a multi-chip architecture for simulated bifurcation. Nat Electron 4, 208–217 (2021). https://doi.org/10.1038/s41928-021-00546-4

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