Abstract

As a solver for non-deterministic polynomial time (NP)-hard combinatorial optimization problems, the coherent Ising machine (CIM) is in the early stages of research, and the potential of this innovative physical system will be developed. Here, we propose a speed-up coherent Ising machine with a squeezed feedback system, which we call S-CIM. We couple squeezed feedback pulses generated by the squeezed feedback system into the degenerate optical parametric oscillator (DOPO) network. Simulations indicate that quantum inseparability of the coupled DOPO network is further enhanced during the whole optimization process, and quantum fluctuations are significantly smaller around the oscillation threshold. Computation experiments are performed on MAX-CUT problems of order between 4 and 20000. Numerical results demonstrate that S-CIM increases the optimal normalized output by 2.27% and significantly reduces the optimal computation time by 75.12%.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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    [Crossref]
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  5. I. H. Witten, E. Frank, M. A. Hall, and C. J. Pal, Data Mining: Practical Machine Learning Tools and Techniques (Morgan Kaufmann, 2016).
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    [Crossref]
  7. Z. Su, D. Xue, and Z. Ji, “Designing LED array for uniform illumination distribution by simulated annealing algorithm,” Opt. Express 20(S6), A843–855 (2012).
    [Crossref]
  8. P. Honzatko, J. Kanka, and B. Vrany, “Retrieval of the pulse amplitude and phase from cross-phase modulation spectrograms using the simulated annealing method,” Opt. Express 12(24), 6046–6052 (2004).
    [Crossref]
  9. T. Kadowaki and H. Nishimori, “Quantum annealing in the transverse Ising model,” Phys. Rev. E 58(5), 5355–5363 (1998).
    [Crossref]
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    [Crossref]
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    [Crossref]
  13. M. Gu and Á Perales, “Encoding universal computation in the ground states of Ising lattices,” Phys. Rev. E 86(1), 011116 (2012).
    [Crossref]
  14. M. R. Garey and D. S. Johnson, Computers and Intractability: a Guide to the Theory of NP-Completeness (Freeman, 2009).
  15. Y. Haribara, S. Utsunomiya, and Y. Yamamoto, “Computational principle and performance evaluation of coherent Ising machine based on degenerate optical parametric oscillator network,” Entropy 18(4), 151–166 (2016).
    [Crossref]
  16. T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 10(6), 415–419 (2016).
    [Crossref]
  17. H. Takesue and T. Inagaki, “10 GHz clock time-multiplexed degenerate optical parametric oscillators for a photonic Ising spin network,” Opt. Lett. 41(18), 4273–4276 (2016).
    [Crossref]
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    [Crossref]
  19. D. Maruo, S. Utsunomiya, and Y. Yamamoto, “Truncated Wigner theory of coherent Ising machines based on degenerate optical parametric oscillator network,” Phys. Scr. 91(8), 083010 (2016).
    [Crossref]
  20. K. Takata, A. Marandi, and Y. Yamamoto, “Quantum correlation in degenerate optical parametric oscillators with mutual injections,” Phys. Rev. A 92(4), 043821 (2015).
    [Crossref]
  21. D. F. Walls, “Squeezed states of light,” Nature 306(5939), 141–146 (1983).
    [Crossref]
  22. C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26(8), 1817–1839 (1982).
    [Crossref]
  23. H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13(6), 2226–2243 (1976).
    [Crossref]
  24. G. J. Milburn and D. F. Walls, “Squeezed states and intensity fluctuations in degenerate parametric oscillation,” Phys. Rev. A 27(1), 392–394 (1983).
    [Crossref]
  25. L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57(20), 2520–2523 (1986).
    [Crossref]
  26. Z. Dutton, J. H. Shapiro, and S. Guha, “LADAR resolution improvement using receivers enhanced with squeezed-vacuum injection and phase-sensitive amplification,” J. Opt. Soc. Am. B 27(6), A63–A72 (2010).
    [Crossref]
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    [Crossref]
  28. H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer-Verlag, 2002).
  29. D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 2007).
  30. L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
    [Crossref]
  31. C. Helmberg and F. A. Rendl, “spectral bundle method for semidefinite programming,” SIAM J. Control 10(3), 673–696 (2000).
    [Crossref]
  32. M. X. Goemans and D. P. Williamson, “Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming,” J. Assoc. Comput. Mach. 42(6), 1115–1145 (1995).
    [Crossref]

2017 (1)

A. Yamamura, K. Aihara, and Y. Yamamoto, “Quantum model for coherent Ising machines: Discrete-time measurement feedback formulation,” Phys. Rev. A 96(5), 053834 (2017).
[Crossref]

2016 (4)

D. Maruo, S. Utsunomiya, and Y. Yamamoto, “Truncated Wigner theory of coherent Ising machines based on degenerate optical parametric oscillator network,” Phys. Scr. 91(8), 083010 (2016).
[Crossref]

Y. Haribara, S. Utsunomiya, and Y. Yamamoto, “Computational principle and performance evaluation of coherent Ising machine based on degenerate optical parametric oscillator network,” Entropy 18(4), 151–166 (2016).
[Crossref]

T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 10(6), 415–419 (2016).
[Crossref]

H. Takesue and T. Inagaki, “10 GHz clock time-multiplexed degenerate optical parametric oscillators for a photonic Ising spin network,” Opt. Lett. 41(18), 4273–4276 (2016).
[Crossref]

2015 (2)

K. Takata, A. Marandi, and Y. Yamamoto, “Quantum correlation in degenerate optical parametric oscillators with mutual injections,” Phys. Rev. A 92(4), 043821 (2015).
[Crossref]

E. G. Rieffel, D. Venturelli, B. O’Gorman, M. B. Do, E. M. Prystay, and V. N. Smelyanskiy, “A case study in programming a quantum annealer for hard operational planning problems,” Quantum Inf. Process. 14(1), 1–36 (2015).
[Crossref]

2014 (1)

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

2013 (1)

Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “Coherent Ising machine based on degenerate optical parametric oscillators,” Phys. Rev. A 88(6), 063853 (2013).
[Crossref]

2012 (2)

M. Gu and Á Perales, “Encoding universal computation in the ground states of Ising lattices,” Phys. Rev. E 86(1), 011116 (2012).
[Crossref]

Z. Su, D. Xue, and Z. Ji, “Designing LED array for uniform illumination distribution by simulated annealing algorithm,” Opt. Express 20(S6), A843–855 (2012).
[Crossref]

2010 (1)

2004 (2)

P. Honzatko, J. Kanka, and B. Vrany, “Retrieval of the pulse amplitude and phase from cross-phase modulation spectrograms using the simulated annealing method,” Opt. Express 12(24), 6046–6052 (2004).
[Crossref]

D. B. Kitchen, H. Decornez, J. R. Furr, and J. Bajorath, “Docking and scoring in virtual screening for drug discovery: methods and applications,” Nat. Rev. Drug Discovery 3(11), 935–949 (2004).
[Crossref]

2000 (2)

L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
[Crossref]

C. Helmberg and F. A. Rendl, “spectral bundle method for semidefinite programming,” SIAM J. Control 10(3), 673–696 (2000).
[Crossref]

1998 (1)

T. Kadowaki and H. Nishimori, “Quantum annealing in the transverse Ising model,” Phys. Rev. E 58(5), 5355–5363 (1998).
[Crossref]

1995 (1)

M. X. Goemans and D. P. Williamson, “Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming,” J. Assoc. Comput. Mach. 42(6), 1115–1145 (1995).
[Crossref]

1987 (1)

J. D. Bryngelson and P. G. Wolynes, “Spin glasses and the statistical mechanics of protein folding,” Proc. Natl. Acad. Sci. 84(21), 7524–7528 (1987).
[Crossref]

1986 (1)

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57(20), 2520–2523 (1986).
[Crossref]

1983 (3)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
[Crossref]

D. F. Walls, “Squeezed states of light,” Nature 306(5939), 141–146 (1983).
[Crossref]

G. J. Milburn and D. F. Walls, “Squeezed states and intensity fluctuations in degenerate parametric oscillation,” Phys. Rev. A 27(1), 392–394 (1983).
[Crossref]

1982 (2)

F. Barahona, “On the computational complexity of Ising spin glass models,” J. Phys. A: Math. Gen. 15(10), 3241–3253 (1982).
[Crossref]

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26(8), 1817–1839 (1982).
[Crossref]

1981 (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23(8), 1693–1708 (1981).
[Crossref]

1976 (1)

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13(6), 2226–2243 (1976).
[Crossref]

Aihara, K.

A. Yamamura, K. Aihara, and Y. Yamamoto, “Quantum model for coherent Ising machines: Discrete-time measurement feedback formulation,” Phys. Rev. A 96(5), 053834 (2017).
[Crossref]

Bajorath, J.

D. B. Kitchen, H. Decornez, J. R. Furr, and J. Bajorath, “Docking and scoring in virtual screening for drug discovery: methods and applications,” Nat. Rev. Drug Discovery 3(11), 935–949 (2004).
[Crossref]

Barahona, F.

F. Barahona, “On the computational complexity of Ising spin glass models,” J. Phys. A: Math. Gen. 15(10), 3241–3253 (1982).
[Crossref]

Bryngelson, J. D.

J. D. Bryngelson and P. G. Wolynes, “Spin glasses and the statistical mechanics of protein folding,” Proc. Natl. Acad. Sci. 84(21), 7524–7528 (1987).
[Crossref]

Byer, R. L.

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “Coherent Ising machine based on degenerate optical parametric oscillators,” Phys. Rev. A 88(6), 063853 (2013).
[Crossref]

Carmichael, H. J.

H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer-Verlag, 2002).

Caves, C. M.

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26(8), 1817–1839 (1982).
[Crossref]

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23(8), 1693–1708 (1981).
[Crossref]

Cirac, J. I.

L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
[Crossref]

Decornez, H.

D. B. Kitchen, H. Decornez, J. R. Furr, and J. Bajorath, “Docking and scoring in virtual screening for drug discovery: methods and applications,” Nat. Rev. Drug Discovery 3(11), 935–949 (2004).
[Crossref]

Do, M. B.

E. G. Rieffel, D. Venturelli, B. O’Gorman, M. B. Do, E. M. Prystay, and V. N. Smelyanskiy, “A case study in programming a quantum annealer for hard operational planning problems,” Quantum Inf. Process. 14(1), 1–36 (2015).
[Crossref]

Duan, L. M.

L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
[Crossref]

Dutton, Z.

Frank, E.

I. H. Witten, E. Frank, M. A. Hall, and C. J. Pal, Data Mining: Practical Machine Learning Tools and Techniques (Morgan Kaufmann, 2016).

Furr, J. R.

D. B. Kitchen, H. Decornez, J. R. Furr, and J. Bajorath, “Docking and scoring in virtual screening for drug discovery: methods and applications,” Nat. Rev. Drug Discovery 3(11), 935–949 (2004).
[Crossref]

Garey, M. R.

M. R. Garey and D. S. Johnson, Computers and Intractability: a Guide to the Theory of NP-Completeness (Freeman, 2009).

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
[Crossref]

Giedke, G.

L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
[Crossref]

Goemans, M. X.

M. X. Goemans and D. P. Williamson, “Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming,” J. Assoc. Comput. Mach. 42(6), 1115–1145 (1995).
[Crossref]

Gu, M.

M. Gu and Á Perales, “Encoding universal computation in the ground states of Ising lattices,” Phys. Rev. E 86(1), 011116 (2012).
[Crossref]

Guha, S.

Hall, J. L.

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57(20), 2520–2523 (1986).
[Crossref]

Hall, M. A.

I. H. Witten, E. Frank, M. A. Hall, and C. J. Pal, Data Mining: Practical Machine Learning Tools and Techniques (Morgan Kaufmann, 2016).

Hamerly, R.

T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 10(6), 415–419 (2016).
[Crossref]

Haribara, Y.

Y. Haribara, S. Utsunomiya, and Y. Yamamoto, “Computational principle and performance evaluation of coherent Ising machine based on degenerate optical parametric oscillator network,” Entropy 18(4), 151–166 (2016).
[Crossref]

Helmberg, C.

C. Helmberg and F. A. Rendl, “spectral bundle method for semidefinite programming,” SIAM J. Control 10(3), 673–696 (2000).
[Crossref]

Honzatko, P.

Inaba, K.

T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 10(6), 415–419 (2016).
[Crossref]

Inagaki, T.

T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 10(6), 415–419 (2016).
[Crossref]

H. Takesue and T. Inagaki, “10 GHz clock time-multiplexed degenerate optical parametric oscillators for a photonic Ising spin network,” Opt. Lett. 41(18), 4273–4276 (2016).
[Crossref]

Inoue, K.

T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 10(6), 415–419 (2016).
[Crossref]

Ji, Z.

Johnson, D. S.

M. R. Garey and D. S. Johnson, Computers and Intractability: a Guide to the Theory of NP-Completeness (Freeman, 2009).

Kadowaki, T.

T. Kadowaki and H. Nishimori, “Quantum annealing in the transverse Ising model,” Phys. Rev. E 58(5), 5355–5363 (1998).
[Crossref]

Kanka, J.

Kimble, H. J.

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57(20), 2520–2523 (1986).
[Crossref]

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
[Crossref]

Kitchen, D. B.

D. B. Kitchen, H. Decornez, J. R. Furr, and J. Bajorath, “Docking and scoring in virtual screening for drug discovery: methods and applications,” Nat. Rev. Drug Discovery 3(11), 935–949 (2004).
[Crossref]

Marandi, A.

K. Takata, A. Marandi, and Y. Yamamoto, “Quantum correlation in degenerate optical parametric oscillators with mutual injections,” Phys. Rev. A 92(4), 043821 (2015).
[Crossref]

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “Coherent Ising machine based on degenerate optical parametric oscillators,” Phys. Rev. A 88(6), 063853 (2013).
[Crossref]

Marc, M.

M. Marc, G. Parisi, and M. Virasoro, Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications (World Scientific Publishing Company, 1987).

Maruo, D.

D. Maruo, S. Utsunomiya, and Y. Yamamoto, “Truncated Wigner theory of coherent Ising machines based on degenerate optical parametric oscillator network,” Phys. Scr. 91(8), 083010 (2016).
[Crossref]

Milburn, G. J.

G. J. Milburn and D. F. Walls, “Squeezed states and intensity fluctuations in degenerate parametric oscillation,” Phys. Rev. A 27(1), 392–394 (1983).
[Crossref]

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 2007).

Nishimori, H.

T. Kadowaki and H. Nishimori, “Quantum annealing in the transverse Ising model,” Phys. Rev. E 58(5), 5355–5363 (1998).
[Crossref]

O’Gorman, B.

E. G. Rieffel, D. Venturelli, B. O’Gorman, M. B. Do, E. M. Prystay, and V. N. Smelyanskiy, “A case study in programming a quantum annealer for hard operational planning problems,” Quantum Inf. Process. 14(1), 1–36 (2015).
[Crossref]

Pal, C. J.

I. H. Witten, E. Frank, M. A. Hall, and C. J. Pal, Data Mining: Practical Machine Learning Tools and Techniques (Morgan Kaufmann, 2016).

Parisi, G.

M. Marc, G. Parisi, and M. Virasoro, Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications (World Scientific Publishing Company, 1987).

Perales, Á

M. Gu and Á Perales, “Encoding universal computation in the ground states of Ising lattices,” Phys. Rev. E 86(1), 011116 (2012).
[Crossref]

Prystay, E. M.

E. G. Rieffel, D. Venturelli, B. O’Gorman, M. B. Do, E. M. Prystay, and V. N. Smelyanskiy, “A case study in programming a quantum annealer for hard operational planning problems,” Quantum Inf. Process. 14(1), 1–36 (2015).
[Crossref]

Rendl, F. A.

C. Helmberg and F. A. Rendl, “spectral bundle method for semidefinite programming,” SIAM J. Control 10(3), 673–696 (2000).
[Crossref]

Rieffel, E. G.

E. G. Rieffel, D. Venturelli, B. O’Gorman, M. B. Do, E. M. Prystay, and V. N. Smelyanskiy, “A case study in programming a quantum annealer for hard operational planning problems,” Quantum Inf. Process. 14(1), 1–36 (2015).
[Crossref]

Shapiro, J. H.

Smelyanskiy, V. N.

E. G. Rieffel, D. Venturelli, B. O’Gorman, M. B. Do, E. M. Prystay, and V. N. Smelyanskiy, “A case study in programming a quantum annealer for hard operational planning problems,” Quantum Inf. Process. 14(1), 1–36 (2015).
[Crossref]

Su, Z.

Takata, K.

K. Takata, A. Marandi, and Y. Yamamoto, “Quantum correlation in degenerate optical parametric oscillators with mutual injections,” Phys. Rev. A 92(4), 043821 (2015).
[Crossref]

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

Takesue, H.

T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nat. Photonics 10(6), 415–419 (2016).
[Crossref]

H. Takesue and T. Inagaki, “10 GHz clock time-multiplexed degenerate optical parametric oscillators for a photonic Ising spin network,” Opt. Lett. 41(18), 4273–4276 (2016).
[Crossref]

Utsunomiya, S.

D. Maruo, S. Utsunomiya, and Y. Yamamoto, “Truncated Wigner theory of coherent Ising machines based on degenerate optical parametric oscillator network,” Phys. Scr. 91(8), 083010 (2016).
[Crossref]

Y. Haribara, S. Utsunomiya, and Y. Yamamoto, “Computational principle and performance evaluation of coherent Ising machine based on degenerate optical parametric oscillator network,” Entropy 18(4), 151–166 (2016).
[Crossref]

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
[Crossref]

Venturelli, D.

E. G. Rieffel, D. Venturelli, B. O’Gorman, M. B. Do, E. M. Prystay, and V. N. Smelyanskiy, “A case study in programming a quantum annealer for hard operational planning problems,” Quantum Inf. Process. 14(1), 1–36 (2015).
[Crossref]

Virasoro, M.

M. Marc, G. Parisi, and M. Virasoro, Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications (World Scientific Publishing Company, 1987).

Vrany, B.

Walls, D. F.

D. F. Walls, “Squeezed states of light,” Nature 306(5939), 141–146 (1983).
[Crossref]

G. J. Milburn and D. F. Walls, “Squeezed states and intensity fluctuations in degenerate parametric oscillation,” Phys. Rev. A 27(1), 392–394 (1983).
[Crossref]

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 2007).

Wang, Z.

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “Coherent Ising machine based on degenerate optical parametric oscillators,” Phys. Rev. A 88(6), 063853 (2013).
[Crossref]

Wen, K.

Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “Coherent Ising machine based on degenerate optical parametric oscillators,” Phys. Rev. A 88(6), 063853 (2013).
[Crossref]

Williamson, D. P.

M. X. Goemans and D. P. Williamson, “Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming,” J. Assoc. Comput. Mach. 42(6), 1115–1145 (1995).
[Crossref]

Witten, I. H.

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A. Yamamura, K. Aihara, and Y. Yamamoto, “Quantum model for coherent Ising machines: Discrete-time measurement feedback formulation,” Phys. Rev. A 96(5), 053834 (2017).
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Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “Coherent Ising machine based on degenerate optical parametric oscillators,” Phys. Rev. A 88(6), 063853 (2013).
[Crossref]

K. Takata, A. Marandi, and Y. Yamamoto, “Quantum correlation in degenerate optical parametric oscillators with mutual injections,” Phys. Rev. A 92(4), 043821 (2015).
[Crossref]

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13(6), 2226–2243 (1976).
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Phys. Scr. (1)

D. Maruo, S. Utsunomiya, and Y. Yamamoto, “Truncated Wigner theory of coherent Ising machines based on degenerate optical parametric oscillator network,” Phys. Scr. 91(8), 083010 (2016).
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Figures (8)

Fig. 1.
Fig. 1. (a). The schematic of the speed-up coherent Ising machine with a squeezed feedback system (S-CIM). The PSA connected to BS1 amplifies the quadrature amplitude component of each extracted DOPO pulse, while the BS2 couples the modulated squeezed feedback pulse with the target DOPO pulse to implement the given Ising Hamiltonian. A: the network of N degenerate optical parametric oscillators (DOPOs). B: the squeezed feedback system. C: the injection coupler. (b) The corresponding calculation model of the coupled DOPO network based on squeezed feedback process.
Fig. 2.
Fig. 2. Total variances of the EPR operators ${\Delta }\hat{u}_ + ^2 + {\Delta }\hat{v}_ - ^2$ versus normalized pump rate E for three different CIM schemes. The numerical values are computed from two feedback process models: the valid coupling modes approach (Fig. 2(a)) and the unitary displacement operator approach (Fig. 2(b)).
Fig. 3.
Fig. 3. Variances of the quadrature amplitude component ${\hat{x}_{s12}}$ versus normalized pump rate E for CIM and S-CIM with gradually increasing pump rate, respectively. The numerical values are computed from two feedback process models: the valid coupling modes approach (Fig. 3(a)) and the unitary displacement operator approach (Fig. 3(b)). Since the curves of ${\hat{x}_{s12}}$ and ${\hat{x}_{s21}}$ are almost identical, only the former is given here.
Fig. 4.
Fig. 4. In the 2-coupled DOPO network, the probabilities of finding |↑↓>, |↑↑>, |↓↓> and |↓↑> states at four different stages of the optimization process when the final selected ground state is |↑↓>. The gain of the coupled DOPO network (yellow dotted line) gradually increases from below the coupled oscillation threshold to reach the minimum global loss (blue curve) corresponding to the ground state of the Ising Hamiltonian.
Fig. 5.
Fig. 5. In the 2-coupled DOPO network, the probability evolutions of two ground-state spin configurations versus normalized time τ for CIM and S-CIM. The green, red, blue and pink portions in each curve represent the four procedures of the optimization process of quantum parallel search, quantum filtering, spontaneous symmetry breaking, and quantum-to-classical crossover, respectively. The optimization process of CIM is divided by yellow dotted lines, and the optimization process of S-CIM is divided by yellow solid lines.
Fig. 6.
Fig. 6. In the 2-coupled DOPO network, the average photon number n versus normalized pump rate $p = \frac{E}{{{E_d}}}$ for CIM and S-CIM, respectively.
Fig. 7.
Fig. 7. In the 4-coupled DOPO network, the comparison of spin configuration distributions in 1000 trials of numerical simulations for CIM and S-CIM, respectively.
Fig. 8.
Fig. 8. Quadrature amplitude components of DOPOs versus normalized time τ for CIM and S-CIM against the MAX-CUT problem on the G1 graph.

Tables (1)

Tables Icon

Table 1. Performance comparison of CIM and S-CIM in computing large-scale MAX-CUT problems on sample G-set graphs.

Equations (16)

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H = 1 j < l N J j l σ j σ l ,
H = H free + H int + H pump + H sr ,
H free = ω s j = 1 2 a ^ s j a ^ s j + ω s j = 1 2 a ^ p j a ^ p j ,
H int = i κ 2 j = 1 2 ( a ^ s j 2 a ^ p j a ^ p j 2 a ^ s j ) ,
H pump = i γ p j = 1 2 ( ϵ a ^ p j e i ω p t ϵ a ^ p j e i ω p t ) ,
H sr = i j = 1 2 a ^ s j Γ ^ R s j γ s + Γ ^ R s j a ^ s j γ s + a ^ p j Γ ^ R p j γ p + Γ ^ R p j a ^ p j γ p ,
a ^ s j = T 1 a ^ s j 1 T 1 a ^ v a c ,
a ^ s j , o u t = G ( T 1 a ^ s j 1 T 1 a ^ v a c ) + G 1 ( T 1 a ^ s j + 1 T 1 a ^ v a c + ) ,
ξ j l b ^ s l = ξ j l G T 1 ( a ^ s j , o u t cosh ( r ) a ^ s j , o u t e i θ sinh ( r ) ) ,
c ^ s j l = T 2 b ^ s l + 1 T 2 a ^ s j c ^ s j l = T 2 b ^ s l + 1 T 2 a ^ s j ,
ρ = e λ c ^ λ c ^ { e λ α λ α W ( α ) d α } d λ ,
d α s 12 = ( γ s α s 12 + κ α p 12 α s 12 ) d t + γ s d W s 12 ( t ) d α s 21 = ( γ s α s 21 + κ α p 21 α s 21 ) d t + γ s d W s 21 ( t ) d α p 12 = ( γ p α p 12 κ 2 α s 12 2 + ϵ ) d t + γ p d W p 12 ( t ) d α p 21 = ( γ p α p 21 κ 2 α s 21 2 + ϵ ) d t + γ p d W p 21 ( t ) ,
d A s 12 = { A s 12 + ( E A s 12 2 ) A s 12 ξ 21 A s 21 } d τ + g d W s 12 ( τ ) d A s 21 = { A s 21 + ( E A s 21 2 ) A s 21 ξ 12 A s 12 } d τ + g d W s 21 ( τ ) ,
d W s 12 = 1 4 e E γ S d W s 12 ( τ ) + A s 12 d W p 1 ( τ ) d W s 21 = 1 4 e E γ S d W s 21 ( τ ) + A s 21 d W p 2 ( τ ) ,
d A s j l = { A s j l + ( E A s j l 2 ) A s j l m = 1 N n = 1 N ξ m n A s m n } d τ + g d W s j l ( τ ) d W s j l ( τ ) = 1 4 e E γ S d W s j l ( τ ) + A s j l d W p j ( τ ) ( j l , m n ) .
c ^ s 12 j c ^ s 21 k c ^ s 12 l c ^ s 21 m s = α s 12 j α s 21 k α s 12 l α s 21 m W ( { α } ) d α ,

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