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Large-scale Ising spin network based on degenerate optical parametric oscillators

Abstract

Solving combinatorial optimization problems is becoming increasingly important in modern society, where the analysis and optimization of unprecedentedly complex systems are required. Many such problems can be mapped onto the ground-state-search problem of the Ising Hamiltonian, and simulating the Ising spins with physical systems is now emerging as a promising approach for tackling such problems. Here, we report a large-scale network of artificial spins based on degenerate optical parametric oscillators (DOPOs), paving the way towards a photonic Ising machine capable of solving difficult combinatorial optimization problems. We generate >10,000 time-division-multiplexed DOPOs using dual-pump four-wave mixing in a highly nonlinear fibre placed in a cavity. Using those DOPOs, a one-dimensional Ising model is simulated by introducing nearest-neighbour optical coupling. We observe the formation of spin domains and find that the domain size diverges near the DOPO threshold, which suggests that the DOPO network can simulate the behaviour of low-temperature Ising spins.

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Figure 1: Ising model and set-up for generating artificial Ising spins based on DOPOs.
Figure 2: DOPO measurement results (without optical coupling).
Figure 3: Results observed with a >10,000-spin 1D Ising machine.

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References

  1. Papadimitriou, C. H. & Steiglits, K. Combinatorial Optimization: Algorithms and Complexity (Dover Publications, 1998).

    Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  4. Kim, K. et al. Quantum simulation of frustrated Ising spins with trapped ions. Nature 465, 590–593 (2010).

    Article  ADS  Google Scholar 

  5. Yamaoka, M. et al. 20k-spin Ising chip for combinatorial optimization problem with CMOS annealing. Proc. Inter. Solid-State Circ. Conf. 24.3 (2015).

  6. Mahboob, I. & Yamaguchi, H. An electromechanical Ising machine. Preprint at http://arxiv.org/abs/1505.02467 (2015).

  7. Utsunomiya, S., Takata, K. & Yamamoto, Y. Mapping of Ising models onto injection-locked laser systems. Opt. Express 19, 18091–18108 (2011).

    Article  ADS  Google Scholar 

  8. Takata, K., Utsunomiya, S. & Yamamoto, Y. Transient time of an Ising machine based on injection-locked laser network. New J. Phys. 14, 013052 (2012).

    Article  ADS  Google Scholar 

  9. Wang, Z., Marandi, A., Wen, K., Byer, R. L. & Yamamoto, Y. Coherent Ising machine based on degenerated optical parametric oscillators. Phys. Rev. A 88, 063853 (2013).

    Article  ADS  Google Scholar 

  10. Haribara, Y., Yamamoto, Y., Kawarabayashi, K. I. & Utsunomiya, S. A coherent Ising machine for MAX-CUT problems: performance evaluation against semidefinite programming relaxation and simulated annealing. Preprint at http://arxiv.org/abs/1501.07030 (2015).

  11. Marandi, A., Wang, Z., Takata, K., Byer, R. L. & Yamamoto, Y. Network of time-multiplexed optical parametric oscillators as a coherent Ising machine. Nature Photon. 8, 937–942 (2014).

    Article  ADS  Google Scholar 

  12. Nabors, C. D., Yang, S. T., Day, T. & Byer, R. L. Coherence properties of a doubly-resonant monolithic optical parametric oscillator. J. Opt. Soc. Am. B 7, 815–820 (1990).

    Article  ADS  Google Scholar 

  13. Marandi, A., Leindecker, N. C., Pervak, V., Byer, R. L. & Vodopyanov, K. L. Coherence properties of a broadband femtosecond mid-IR optical parametric oscillator operating at degeneracy. Opt. Express 20, 7255–7262 (2012).

    Article  ADS  Google Scholar 

  14. Serkland, D. K., Bartolini, G. D., Agarwal, A., Kumar, P. & Kath, W. L. Pulsed degenerate optical parametric oscillator based on a nonlinear-fiber Sagnac interferometer. Opt. Lett. 23, 795–797 (1998).

    Article  ADS  Google Scholar 

  15. Okawachi, Y. et al. Dual-pumped degenerate Kerr oscillator in a silicon nitride microresonator. Opt. Lett. 40, 5267–5270 (2015).

    Article  ADS  Google Scholar 

  16. McKinstrie, C. & Radic, S. Phase-sensitive amplification in fibre. Opt. Express 12, 4973–4979 (2004).

    Article  ADS  Google Scholar 

  17. Fan, J. & Migdall, A. Phase-sensitive four-wave mixing and Raman suppression in a microstructure fibre with dual laser pumps. Opt. Lett. 31, 2771–2773 (2006).

    Article  ADS  Google Scholar 

  18. Marhic, M. E., Hsia, C. H. & Jeong, J. M. Optical amplification in a nonlinear fibre interferometer. Electron. Lett. 27, 210–211 (1991).

    Article  ADS  Google Scholar 

  19. Levenson, J. A., Abram, I., Rivera, Th. & Grangier, Ph. Reduction of quantum noise in optical parametric amplification. J. Opt. Soc. Am. B 10, 2233–2238 (1993).

    Article  ADS  Google Scholar 

  20. Choi, S.-K., Vasilyev, M. & Kumar, P. Noiseless optical amplification of images. Phys. Rev. Lett. 83, 1938–1941 (1999).

    Article  ADS  Google Scholar 

  21. Imajuku, W., Takada, A. & Yamabayashi, Y. Low-noise amplifcation under the 3 dB noise figure in high-gain phase-sensitive fibre amplifier. Electron. Lett. 35, 1954–1955 (1999).

    Article  Google Scholar 

  22. Tong, Z. et al. Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers. Nature Photon. 5, 430–436 (2011).

    Article  ADS  Google Scholar 

  23. Ising, E. Beitrag zur theorie des ferromagnetismus. Zeitschrift fur Physik A 31, 253–258 (1925).

    Article  ADS  Google Scholar 

  24. Brown, W. F. Jr Micromagnetics (Wiley, 1963).

    MATH  Google Scholar 

  25. McCoy, B. M. & Wu, T. T. The Two-Dimensional Ising Model 2nd edn (Dover, 2014).

    MATH  Google Scholar 

  26. Agrawal, G. P. Nonlinear Fibre Optics (Academic, 1989).

    Google Scholar 

  27. Yamamoto, Y. & Inoue, K. Noise in amplifiers. J. Lightw. Technol. 21, 2895–2915 (2003).

    Article  ADS  Google Scholar 

  28. Drummond, P. D. & Corney, J. F. Quantum noise in optical fibers. I. Stochastic equations. J. Opt. Soc. Am. B 18, 139–152 (2001).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The authors thank S. Utsunomiya, A. Marandi, P. McMahon, K. Igarashi, S. Tamate, K. Takata, Y. Haribara, K. Shimizu, I. Mahboob and W. J. Munro for discussions, and H. Tamura for various types of support. This research was funded by the Impulsing Paradigm Change through Disruptive Technologies (ImPACT) Program of the Council of Science, Technology and Innovation (Cabinet Office, Government of Japan).

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Contributions

T.I. and H.T. constructed the DOPO set-up and performed the experiments. R.H. and K.Inaba developed the theoretical model. T.I., K.Inaba, R.H. and H.T. analysed the data. H.T., K.Inoue and Y.Y. conceived the concept for the experiment. All authors discussed the results and wrote the paper.

Corresponding author

Correspondence to Hiroki Takesue.

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The authors declare no competing financial interests.

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Inagaki, T., Inaba, K., Hamerly, R. et al. Large-scale Ising spin network based on degenerate optical parametric oscillators. Nature Photon 10, 415–419 (2016). https://doi.org/10.1038/nphoton.2016.68

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