Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Network of time-multiplexed optical parametric oscillators as a coherent Ising machine

Abstract

Finding the ground states of the Ising Hamiltonian1 maps to various combinatorial optimization problems in biology, medicine, wireless communications, artificial intelligence and social network. So far, no efficient classical and quantum algorithm is known for these problems and intensive research is focused on creating physical systems—Ising machines—capable of finding the absolute or approximate ground states of the Ising Hamiltonian2,3,4,5,6. Here, we report an Ising machine using a network of degenerate optical parametric oscillators (OPOs). Spins are represented with above-threshold binary phases of the OPOs and the Ising couplings are realized by mutual injections7. The network is implemented in a single OPO ring cavity with multiple trains of femtosecond pulses and configurable mutual couplings, and operates at room temperature. We programmed a small non-deterministic polynomial time-hard problem on a 4-OPO Ising machine and in 1,000 runs no computational error was detected.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Figure 1: Principle of operation of OPO Ising machine.
Figure 2: Experimental set-up.
Figure 3: Slow detector results.
Figure 4: Fast detector results.

References

  1. 1

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

    ADS  MathSciNet  Article  Google Scholar 

  2. 2

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

    ADS  MathSciNet  Article  Google Scholar 

  3. 3

    Kadowaki, T. & Nishimori, H. Quantum annealing in the transverse Ising model. Phys. Rev. E 58, 5355 (1998).

    ADS  Article  Google Scholar 

  4. 4

    Santoro, G. E., Martoak, R., Tosatti, E. & Car, R. Theory of quantum annealing of an Ising spin glass. Science 295, 2427–2430 (2002).

    ADS  Article  Google Scholar 

  5. 5

    Farhi, E. et al. A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science 292, 472–475 (2001).

    ADS  MathSciNet  Article  Google Scholar 

  6. 6

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

    ADS  Article  Google Scholar 

  7. 7

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

    ADS  Article  Google Scholar 

  8. 8

    Kitchen, D. B., Decornez, H., Furr, J. R. & Bajorath, J. Docking and scoring in virtual screening for drug discovery: methods and applications. Nature Rev. Drug Discov. 3, 935–949 (2004).

    Article  Google Scholar 

  9. 9

    Witten, I. H., Frank, E. & Hall, M. A. Data Mining: Practical Machine Learning Tools and Techniques (Morgan Kaufmann, 2005).

    MATH  Google Scholar 

  10. 10

    Papadimitriou, C. H. & Steiglitz, K. Combinatorial Optimization: Algorithms and Complexity (Courier Dover, 1998).

    MATH  Google Scholar 

  11. 11

    Hochbaum, D. S. Approximation Algorithms for NP-hard Problems (PWS, 1996).

    MATH  Google Scholar 

  12. 12

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

    MathSciNet  Article  Google Scholar 

  13. 13

    Adleman, L. M. Molecular computation of solutions to combinatorial problems. Science 266, 1021–1023 (1994).

    ADS  Article  Google Scholar 

  14. 14

    Wu, K., de Abajo, J. G., Soci, C., Shum, P. P. & Zheludev, N. I. An optical fiber network oracle for NP-complete problems. Light 3, e147 (2014).

    Article  Google Scholar 

  15. 15

    Dickson, N. G. et al. Thermally assisted quantum annealing of a 16-qubit problem. Nature Commun. 4, 1903 (2013).

    ADS  MathSciNet  Article  Google Scholar 

  16. 16

    Ronnow, T. F. et al. Defining and detecting quantum speedup. Science 345, 420–424 (2014).

    ADS  Article  Google Scholar 

  17. 17

    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).

    ADS  Article  Google Scholar 

  18. 18

    Roslund, J., De Araujo, R. M., Jiang, S., Fabre, C. & Treps, N. Wavelength-multiplexed quantum networks with ultrafast frequency combs. Nature Photon. 8, 109–113 (2014).

    ADS  Article  Google Scholar 

  19. 19

    Marandi, A., Leindecker, N. C., Vodopyanov, K. L. & Byer R. L. All-optical quantum random bit generation from intrinsically binary phase of parametric oscillators. Opt. Express 20, 19322–19330 (2012).

    ADS  Article  Google Scholar 

  20. 20

    Binder, K. & Young, A. P. Spin glasses: experimental facts, theoretical concepts, and open questions. Rev. Mod. Phys. 58, 801 (1986).

    ADS  Article  Google Scholar 

  21. 21

    Ising, E. Beitrag zur Theorie des Ferromagnetismus. Zeitschrift für Physik A, Hadrons and Nuclei 31, 253–258 (1925).

    Google Scholar 

  22. 22

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

    ADS  Article  Google Scholar 

  23. 23

    Simon, J. et al. Quantum simulation of antiferromagnetic spin chains in an optical lattice. Nature 472, 307–312 (2011).

    ADS  Article  Google Scholar 

  24. 24

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

    ADS  Article  Google Scholar 

  25. 25

    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).

    ADS  Article  Google Scholar 

  26. 26

    Wu, L., Kimble, H. J., Hall, J. L. & Wu, H. Generation of squeezed states by parametric down conversion. Phys. Rev. Lett. 57, 2520 (1986).

    ADS  Article  Google Scholar 

  27. 27

    Graham, R. in Coherence and Quantum Optics (eds Mandel, L. & Wolf, E.) 851–872 (Springer, 1973).

    Book  Google Scholar 

  28. 28

    Wolinsky, M. & Carmichael, H. J. Quantum noise in the parametric oscillator: from squeezed states to coherent-state superpositions. Phys. Rev. Lett. 60, 1836 (1988).

    ADS  Article  Google Scholar 

  29. 29

    Gottesman, D. The Heisenberg representation of quantum computers. Preprint at http://arxiv.org/abs/quant-ph/9807006 (1998).

  30. 30

    Galluccio, A., Loebl, M. & Vondrk, J. New algorithm for the Ising problem: partition function for finite lattice graphs. Phys. Rev. Lett. 84, 5924 (2000).

    ADS  Article  Google Scholar 

Download references

Acknowledgements

The authors thank S.E. Harris, H. Mabuchi, M. Armen, S. Utsunomiya, S. Tamate, K. Yan and Y. Haribara for discussions and K. Ingold, C.W. Rudy, C. Langrock and K. Urbanek for experimental support. The work is supported by the FIRST Quantum Information Processing project.

Author information

Affiliations

Authors

Contributions

A.M. and Y.Y. conceived the idea and designed the experiment. A.M. and K.T. carried out the experiment. Z.W. performed the numerical simulations. Y.Y. and R.L.B. guided the work. A.M. wrote the manuscript, with input from all authors.

Corresponding author

Correspondence to Alireza Marandi.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary information

Supplementary information (PDF 2667 kb)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Marandi, A., Wang, Z., Takata, K. et al. Network of time-multiplexed optical parametric oscillators as a coherent Ising machine. Nature Photon 8, 937–942 (2014). https://doi.org/10.1038/nphoton.2014.249

Download citation

Further reading

Search

Quick links