Abstract

We propose a mapping protocol to implement Ising models in injection-locked laser systems. The proposed scheme is based on optical coherent feedback and can be potentially applied for large-scale Ising problems.

© 2011 OSA

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  1. M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (W. H. Freeman and Company, 1979).
  2. K. Binder and A. A. Young, “Spin glasses: experimental facts, theoretical concepts, and open questions,” Rev. Mod. Phys. 58, 801–976 (1986).
    [CrossRef]
  3. V. Dotsenko, Introduction to the Replica Theory of Disordered Statistical Systems (Cambridge University Press, 2000).
    [CrossRef]
  4. H. Nishimori, Statistical Physics of Spin Glasses and Information Processing: An Introduction (Oxford University Press, 2001).
    [CrossRef]
  5. A. Das and B. K. Chakrabarti, “Colloquium: quantum annealing and analog quantum computation,” Rev. Mod. Phys. 80, 1061–1081 (2008).
    [CrossRef]
  6. P. Ray, B. K. Chakrabarti, and A. Chakrabarti, “Sherrington–Kirkpatrick model in a transverse field: absence of replica symmetry breaking due to quantum fluctuations,” Phys. Rev. B 39, 11828–11832 (1989).
    [CrossRef]
  7. B. Appoloni, C. Carvalho, and D. de Falco, “Quantum stochastic optimization,” Stochastic Proc. Appl. 33, 233–244 (1989).
    [CrossRef]
  8. R. Martonak, G. E. Santoro, and E. Tosatti, “Quantum annealing by the path-integral Monte Carlo method: the two-dimensional random Ising model,” Phys. Rev. B 66, 094203 (2002).
    [CrossRef]
  9. G. E. Santoro, R. Martonak, E. Tosatti, and R. Car, “Theory of quantum annealing of an Ising spin glass,” Science 295(5564), 2427–2430 (2002).
    [CrossRef] [PubMed]
  10. G. E. Santoro and E. Tosatti, “Quantum to classical and back,” Nat. Phys. 3, 593–594 (2007).
    [CrossRef]
  11. R. D. Somma and C. D. Batista, “Quantum approach to classical statistical mechanics,” Phys. Rev. Lett. 99, 030603 (2007).
    [CrossRef] [PubMed]
  12. J. Brooke, D. Bitko, T. F. Rosenbau, and G. Aeppli, “Quantum annealing of a disordered magnet,” Science 284(5415), 779–781 (1999).
    [CrossRef] [PubMed]
  13. G. Aeppli and T. F. Rosenbaum, in Quantum Annealing and Related Optimization Methods, A. Das and B. K. Chakrabarti, eds. (Springer Verlag, 2005).
  14. M. Steffen, W. van Dam, T. Hogg, G. Breyta, and I. Chuang, “Experimental implementation of an adiabatic quantum optimization algorithm,” Phys. Rev. Lett. 90, 067903 (2003).
    [CrossRef] [PubMed]
  15. T. Byrnes, K. Yan, and Y. Yamamoto, “Optimization using Bose-Einstein condensation and measurement-feedback circuits,” arXiv:0909.2530v2.
  16. K. Yan, T. Byrnes, and Y. Yamamoto, “Kinetic Monte Carlo study of accelerated optimization problem search using Bose-Einstein condensates,” Prog. Inform. 8, 1–9 (2011).
  17. F. Barahona, “On the computational complexity of Ising spin glass models,” J. Phys. A: Math. Gen. 15, 3241–3253 (1982).
    [CrossRef]
  18. S. Kobayashi and T. Kimura, “Injection locking in AIGaAs semiconductor laser,” IEEE J. Quantum Electron. 17(5), 681–689 (1981).
    [CrossRef]
  19. S. Kobayashi, Y. Yamamoto, and T. Kimura, “Optical FM signal amplification and FM noise reduction in an injection locked AlGaAs semiconductor laser,” Electron. Lett. 17(22), 849–851 (1981).
    [CrossRef]
  20. L. Gillner, G. Bjork, and Y. Yamamoto, “Quantum noise properties of an injection-locked laser oscillator with pump-noise suppression and squeezed injection,” Phys. Rev A 41(9), 5053–5065 (1990).
    [CrossRef] [PubMed]
  21. H. A. Haus and Y. Yamamoto, “Quantum noise of an injection-locked laser oscillator,” Phys. Rev. A 29, 1261–1274 (1984).
    [CrossRef]
  22. M. Surgent, M. O. Scully, and W. E. Lamb, Laser Physics (Westview Press, 1978), Chap. 20, pp. 331–335.
  23. H. Haug, “Quantum-mechanical rate equations for semiconductor lasers,” Phys. Rev.184, 338–348 (1969).
    [CrossRef]
  24. S. F. Yu, Analysis and Design of Vertical Cavity Surface Emitting Laser (Wiley-Interscience, 2003), Chap. 8.
    [CrossRef]
  25. Y. Yamamoto, S. Machida, and O. Neilsson, “Amplitude squeezing in a pump-noise-suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
    [CrossRef] [PubMed]
  26. C. Weisbuch and B. Vinter, Quantum Semiconductor Structures: Fundamentals and Applications (Academic Press, 1991), Chap. 3, pp. 65–69.
  27. K. Takata, S. Utsunomiya, and Y. Yamamoto, “Transient time of an Ising machine based on injection-locked lasers: contribution of locking bandwidth and Zeeman component,” (to be submitted).
  28. S. D. Personick, “Receiver design for digital fiber optic communication systems, I,” Bell Syst. Tech. J. 52(6), 843–874 (1973).
  29. S. Inoue, H. Ohzu, S. Machida, and Y. Yamamoto, “Quantum correlation between longitudinal mode intensities in a multimode squeezed semiconductor laser,” Phys. Rev. A 46, 2757–2765 (1992).
    [CrossRef] [PubMed]
  30. A. P. Young, S. Knysh, and V. N. Smelyanskiy, “First-order phase transition in the quantum adiabatic algorithm,” Phys. Rev. Lett. 104, 020502 (2010).
    [CrossRef] [PubMed]
  31. Y. Zhang, Y. Xia, Z. Man, and G. Guo, “Simulation of the Ising model, memory for Bell states and generation of four-atom entangled states in cavity QED,” Sci. China, Ser. G 52, 700–707 (2009).
    [CrossRef]

2011 (1)

K. Yan, T. Byrnes, and Y. Yamamoto, “Kinetic Monte Carlo study of accelerated optimization problem search using Bose-Einstein condensates,” Prog. Inform. 8, 1–9 (2011).

2010 (1)

A. P. Young, S. Knysh, and V. N. Smelyanskiy, “First-order phase transition in the quantum adiabatic algorithm,” Phys. Rev. Lett. 104, 020502 (2010).
[CrossRef] [PubMed]

2009 (1)

Y. Zhang, Y. Xia, Z. Man, and G. Guo, “Simulation of the Ising model, memory for Bell states and generation of four-atom entangled states in cavity QED,” Sci. China, Ser. G 52, 700–707 (2009).
[CrossRef]

2008 (1)

A. Das and B. K. Chakrabarti, “Colloquium: quantum annealing and analog quantum computation,” Rev. Mod. Phys. 80, 1061–1081 (2008).
[CrossRef]

2007 (2)

G. E. Santoro and E. Tosatti, “Quantum to classical and back,” Nat. Phys. 3, 593–594 (2007).
[CrossRef]

R. D. Somma and C. D. Batista, “Quantum approach to classical statistical mechanics,” Phys. Rev. Lett. 99, 030603 (2007).
[CrossRef] [PubMed]

2003 (1)

M. Steffen, W. van Dam, T. Hogg, G. Breyta, and I. Chuang, “Experimental implementation of an adiabatic quantum optimization algorithm,” Phys. Rev. Lett. 90, 067903 (2003).
[CrossRef] [PubMed]

2002 (2)

R. Martonak, G. E. Santoro, and E. Tosatti, “Quantum annealing by the path-integral Monte Carlo method: the two-dimensional random Ising model,” Phys. Rev. B 66, 094203 (2002).
[CrossRef]

G. E. Santoro, R. Martonak, E. Tosatti, and R. Car, “Theory of quantum annealing of an Ising spin glass,” Science 295(5564), 2427–2430 (2002).
[CrossRef] [PubMed]

1999 (1)

J. Brooke, D. Bitko, T. F. Rosenbau, and G. Aeppli, “Quantum annealing of a disordered magnet,” Science 284(5415), 779–781 (1999).
[CrossRef] [PubMed]

1992 (1)

S. Inoue, H. Ohzu, S. Machida, and Y. Yamamoto, “Quantum correlation between longitudinal mode intensities in a multimode squeezed semiconductor laser,” Phys. Rev. A 46, 2757–2765 (1992).
[CrossRef] [PubMed]

1990 (1)

L. Gillner, G. Bjork, and Y. Yamamoto, “Quantum noise properties of an injection-locked laser oscillator with pump-noise suppression and squeezed injection,” Phys. Rev A 41(9), 5053–5065 (1990).
[CrossRef] [PubMed]

1989 (2)

P. Ray, B. K. Chakrabarti, and A. Chakrabarti, “Sherrington–Kirkpatrick model in a transverse field: absence of replica symmetry breaking due to quantum fluctuations,” Phys. Rev. B 39, 11828–11832 (1989).
[CrossRef]

B. Appoloni, C. Carvalho, and D. de Falco, “Quantum stochastic optimization,” Stochastic Proc. Appl. 33, 233–244 (1989).
[CrossRef]

1986 (2)

K. Binder and A. A. Young, “Spin glasses: experimental facts, theoretical concepts, and open questions,” Rev. Mod. Phys. 58, 801–976 (1986).
[CrossRef]

Y. Yamamoto, S. Machida, and O. Neilsson, “Amplitude squeezing in a pump-noise-suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
[CrossRef] [PubMed]

1984 (1)

H. A. Haus and Y. Yamamoto, “Quantum noise of an injection-locked laser oscillator,” Phys. Rev. A 29, 1261–1274 (1984).
[CrossRef]

1982 (1)

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

1981 (2)

S. Kobayashi and T. Kimura, “Injection locking in AIGaAs semiconductor laser,” IEEE J. Quantum Electron. 17(5), 681–689 (1981).
[CrossRef]

S. Kobayashi, Y. Yamamoto, and T. Kimura, “Optical FM signal amplification and FM noise reduction in an injection locked AlGaAs semiconductor laser,” Electron. Lett. 17(22), 849–851 (1981).
[CrossRef]

1973 (1)

S. D. Personick, “Receiver design for digital fiber optic communication systems, I,” Bell Syst. Tech. J. 52(6), 843–874 (1973).

Aeppli, G.

J. Brooke, D. Bitko, T. F. Rosenbau, and G. Aeppli, “Quantum annealing of a disordered magnet,” Science 284(5415), 779–781 (1999).
[CrossRef] [PubMed]

G. Aeppli and T. F. Rosenbaum, in Quantum Annealing and Related Optimization Methods, A. Das and B. K. Chakrabarti, eds. (Springer Verlag, 2005).

Appoloni, B.

B. Appoloni, C. Carvalho, and D. de Falco, “Quantum stochastic optimization,” Stochastic Proc. Appl. 33, 233–244 (1989).
[CrossRef]

Barahona, F.

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

Batista, C. D.

R. D. Somma and C. D. Batista, “Quantum approach to classical statistical mechanics,” Phys. Rev. Lett. 99, 030603 (2007).
[CrossRef] [PubMed]

Binder, K.

K. Binder and A. A. Young, “Spin glasses: experimental facts, theoretical concepts, and open questions,” Rev. Mod. Phys. 58, 801–976 (1986).
[CrossRef]

Bitko, D.

J. Brooke, D. Bitko, T. F. Rosenbau, and G. Aeppli, “Quantum annealing of a disordered magnet,” Science 284(5415), 779–781 (1999).
[CrossRef] [PubMed]

Bjork, G.

L. Gillner, G. Bjork, and Y. Yamamoto, “Quantum noise properties of an injection-locked laser oscillator with pump-noise suppression and squeezed injection,” Phys. Rev A 41(9), 5053–5065 (1990).
[CrossRef] [PubMed]

Breyta, G.

M. Steffen, W. van Dam, T. Hogg, G. Breyta, and I. Chuang, “Experimental implementation of an adiabatic quantum optimization algorithm,” Phys. Rev. Lett. 90, 067903 (2003).
[CrossRef] [PubMed]

Brooke, J.

J. Brooke, D. Bitko, T. F. Rosenbau, and G. Aeppli, “Quantum annealing of a disordered magnet,” Science 284(5415), 779–781 (1999).
[CrossRef] [PubMed]

Byrnes, T.

K. Yan, T. Byrnes, and Y. Yamamoto, “Kinetic Monte Carlo study of accelerated optimization problem search using Bose-Einstein condensates,” Prog. Inform. 8, 1–9 (2011).

T. Byrnes, K. Yan, and Y. Yamamoto, “Optimization using Bose-Einstein condensation and measurement-feedback circuits,” arXiv:0909.2530v2.

Car, R.

G. E. Santoro, R. Martonak, E. Tosatti, and R. Car, “Theory of quantum annealing of an Ising spin glass,” Science 295(5564), 2427–2430 (2002).
[CrossRef] [PubMed]

Carvalho, C.

B. Appoloni, C. Carvalho, and D. de Falco, “Quantum stochastic optimization,” Stochastic Proc. Appl. 33, 233–244 (1989).
[CrossRef]

Chakrabarti, A.

P. Ray, B. K. Chakrabarti, and A. Chakrabarti, “Sherrington–Kirkpatrick model in a transverse field: absence of replica symmetry breaking due to quantum fluctuations,” Phys. Rev. B 39, 11828–11832 (1989).
[CrossRef]

Chakrabarti, B. K.

A. Das and B. K. Chakrabarti, “Colloquium: quantum annealing and analog quantum computation,” Rev. Mod. Phys. 80, 1061–1081 (2008).
[CrossRef]

P. Ray, B. K. Chakrabarti, and A. Chakrabarti, “Sherrington–Kirkpatrick model in a transverse field: absence of replica symmetry breaking due to quantum fluctuations,” Phys. Rev. B 39, 11828–11832 (1989).
[CrossRef]

Chuang, I.

M. Steffen, W. van Dam, T. Hogg, G. Breyta, and I. Chuang, “Experimental implementation of an adiabatic quantum optimization algorithm,” Phys. Rev. Lett. 90, 067903 (2003).
[CrossRef] [PubMed]

Das, A.

A. Das and B. K. Chakrabarti, “Colloquium: quantum annealing and analog quantum computation,” Rev. Mod. Phys. 80, 1061–1081 (2008).
[CrossRef]

de Falco, D.

B. Appoloni, C. Carvalho, and D. de Falco, “Quantum stochastic optimization,” Stochastic Proc. Appl. 33, 233–244 (1989).
[CrossRef]

Dotsenko, V.

V. Dotsenko, Introduction to the Replica Theory of Disordered Statistical Systems (Cambridge University Press, 2000).
[CrossRef]

Garey, M. R.

M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (W. H. Freeman and Company, 1979).

Gillner, L.

L. Gillner, G. Bjork, and Y. Yamamoto, “Quantum noise properties of an injection-locked laser oscillator with pump-noise suppression and squeezed injection,” Phys. Rev A 41(9), 5053–5065 (1990).
[CrossRef] [PubMed]

Guo, G.

Y. Zhang, Y. Xia, Z. Man, and G. Guo, “Simulation of the Ising model, memory for Bell states and generation of four-atom entangled states in cavity QED,” Sci. China, Ser. G 52, 700–707 (2009).
[CrossRef]

Haug, H.

H. Haug, “Quantum-mechanical rate equations for semiconductor lasers,” Phys. Rev.184, 338–348 (1969).
[CrossRef]

Haus, H. A.

H. A. Haus and Y. Yamamoto, “Quantum noise of an injection-locked laser oscillator,” Phys. Rev. A 29, 1261–1274 (1984).
[CrossRef]

Hogg, T.

M. Steffen, W. van Dam, T. Hogg, G. Breyta, and I. Chuang, “Experimental implementation of an adiabatic quantum optimization algorithm,” Phys. Rev. Lett. 90, 067903 (2003).
[CrossRef] [PubMed]

Inoue, S.

S. Inoue, H. Ohzu, S. Machida, and Y. Yamamoto, “Quantum correlation between longitudinal mode intensities in a multimode squeezed semiconductor laser,” Phys. Rev. A 46, 2757–2765 (1992).
[CrossRef] [PubMed]

Johnson, D. S.

M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (W. H. Freeman and Company, 1979).

Kimura, T.

S. Kobayashi, Y. Yamamoto, and T. Kimura, “Optical FM signal amplification and FM noise reduction in an injection locked AlGaAs semiconductor laser,” Electron. Lett. 17(22), 849–851 (1981).
[CrossRef]

S. Kobayashi and T. Kimura, “Injection locking in AIGaAs semiconductor laser,” IEEE J. Quantum Electron. 17(5), 681–689 (1981).
[CrossRef]

Knysh, S.

A. P. Young, S. Knysh, and V. N. Smelyanskiy, “First-order phase transition in the quantum adiabatic algorithm,” Phys. Rev. Lett. 104, 020502 (2010).
[CrossRef] [PubMed]

Kobayashi, S.

S. Kobayashi, Y. Yamamoto, and T. Kimura, “Optical FM signal amplification and FM noise reduction in an injection locked AlGaAs semiconductor laser,” Electron. Lett. 17(22), 849–851 (1981).
[CrossRef]

S. Kobayashi and T. Kimura, “Injection locking in AIGaAs semiconductor laser,” IEEE J. Quantum Electron. 17(5), 681–689 (1981).
[CrossRef]

Lamb, W. E.

M. Surgent, M. O. Scully, and W. E. Lamb, Laser Physics (Westview Press, 1978), Chap. 20, pp. 331–335.

Machida, S.

S. Inoue, H. Ohzu, S. Machida, and Y. Yamamoto, “Quantum correlation between longitudinal mode intensities in a multimode squeezed semiconductor laser,” Phys. Rev. A 46, 2757–2765 (1992).
[CrossRef] [PubMed]

Y. Yamamoto, S. Machida, and O. Neilsson, “Amplitude squeezing in a pump-noise-suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
[CrossRef] [PubMed]

Man, Z.

Y. Zhang, Y. Xia, Z. Man, and G. Guo, “Simulation of the Ising model, memory for Bell states and generation of four-atom entangled states in cavity QED,” Sci. China, Ser. G 52, 700–707 (2009).
[CrossRef]

Martonak, R.

R. Martonak, G. E. Santoro, and E. Tosatti, “Quantum annealing by the path-integral Monte Carlo method: the two-dimensional random Ising model,” Phys. Rev. B 66, 094203 (2002).
[CrossRef]

G. E. Santoro, R. Martonak, E. Tosatti, and R. Car, “Theory of quantum annealing of an Ising spin glass,” Science 295(5564), 2427–2430 (2002).
[CrossRef] [PubMed]

Neilsson, O.

Y. Yamamoto, S. Machida, and O. Neilsson, “Amplitude squeezing in a pump-noise-suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
[CrossRef] [PubMed]

Nishimori, H.

H. Nishimori, Statistical Physics of Spin Glasses and Information Processing: An Introduction (Oxford University Press, 2001).
[CrossRef]

Ohzu, H.

S. Inoue, H. Ohzu, S. Machida, and Y. Yamamoto, “Quantum correlation between longitudinal mode intensities in a multimode squeezed semiconductor laser,” Phys. Rev. A 46, 2757–2765 (1992).
[CrossRef] [PubMed]

Personick, S. D.

S. D. Personick, “Receiver design for digital fiber optic communication systems, I,” Bell Syst. Tech. J. 52(6), 843–874 (1973).

Ray, P.

P. Ray, B. K. Chakrabarti, and A. Chakrabarti, “Sherrington–Kirkpatrick model in a transverse field: absence of replica symmetry breaking due to quantum fluctuations,” Phys. Rev. B 39, 11828–11832 (1989).
[CrossRef]

Rosenbau, T. F.

J. Brooke, D. Bitko, T. F. Rosenbau, and G. Aeppli, “Quantum annealing of a disordered magnet,” Science 284(5415), 779–781 (1999).
[CrossRef] [PubMed]

Rosenbaum, T. F.

G. Aeppli and T. F. Rosenbaum, in Quantum Annealing and Related Optimization Methods, A. Das and B. K. Chakrabarti, eds. (Springer Verlag, 2005).

Santoro, G. E.

G. E. Santoro and E. Tosatti, “Quantum to classical and back,” Nat. Phys. 3, 593–594 (2007).
[CrossRef]

R. Martonak, G. E. Santoro, and E. Tosatti, “Quantum annealing by the path-integral Monte Carlo method: the two-dimensional random Ising model,” Phys. Rev. B 66, 094203 (2002).
[CrossRef]

G. E. Santoro, R. Martonak, E. Tosatti, and R. Car, “Theory of quantum annealing of an Ising spin glass,” Science 295(5564), 2427–2430 (2002).
[CrossRef] [PubMed]

Scully, M. O.

M. Surgent, M. O. Scully, and W. E. Lamb, Laser Physics (Westview Press, 1978), Chap. 20, pp. 331–335.

Smelyanskiy, V. N.

A. P. Young, S. Knysh, and V. N. Smelyanskiy, “First-order phase transition in the quantum adiabatic algorithm,” Phys. Rev. Lett. 104, 020502 (2010).
[CrossRef] [PubMed]

Somma, R. D.

R. D. Somma and C. D. Batista, “Quantum approach to classical statistical mechanics,” Phys. Rev. Lett. 99, 030603 (2007).
[CrossRef] [PubMed]

Steffen, M.

M. Steffen, W. van Dam, T. Hogg, G. Breyta, and I. Chuang, “Experimental implementation of an adiabatic quantum optimization algorithm,” Phys. Rev. Lett. 90, 067903 (2003).
[CrossRef] [PubMed]

Surgent, M.

M. Surgent, M. O. Scully, and W. E. Lamb, Laser Physics (Westview Press, 1978), Chap. 20, pp. 331–335.

Takata, K.

K. Takata, S. Utsunomiya, and Y. Yamamoto, “Transient time of an Ising machine based on injection-locked lasers: contribution of locking bandwidth and Zeeman component,” (to be submitted).

Tosatti, E.

G. E. Santoro and E. Tosatti, “Quantum to classical and back,” Nat. Phys. 3, 593–594 (2007).
[CrossRef]

G. E. Santoro, R. Martonak, E. Tosatti, and R. Car, “Theory of quantum annealing of an Ising spin glass,” Science 295(5564), 2427–2430 (2002).
[CrossRef] [PubMed]

R. Martonak, G. E. Santoro, and E. Tosatti, “Quantum annealing by the path-integral Monte Carlo method: the two-dimensional random Ising model,” Phys. Rev. B 66, 094203 (2002).
[CrossRef]

Utsunomiya, S.

K. Takata, S. Utsunomiya, and Y. Yamamoto, “Transient time of an Ising machine based on injection-locked lasers: contribution of locking bandwidth and Zeeman component,” (to be submitted).

van Dam, W.

M. Steffen, W. van Dam, T. Hogg, G. Breyta, and I. Chuang, “Experimental implementation of an adiabatic quantum optimization algorithm,” Phys. Rev. Lett. 90, 067903 (2003).
[CrossRef] [PubMed]

Vinter, B.

C. Weisbuch and B. Vinter, Quantum Semiconductor Structures: Fundamentals and Applications (Academic Press, 1991), Chap. 3, pp. 65–69.

Weisbuch, C.

C. Weisbuch and B. Vinter, Quantum Semiconductor Structures: Fundamentals and Applications (Academic Press, 1991), Chap. 3, pp. 65–69.

Xia, Y.

Y. Zhang, Y. Xia, Z. Man, and G. Guo, “Simulation of the Ising model, memory for Bell states and generation of four-atom entangled states in cavity QED,” Sci. China, Ser. G 52, 700–707 (2009).
[CrossRef]

Yamamoto, Y.

K. Yan, T. Byrnes, and Y. Yamamoto, “Kinetic Monte Carlo study of accelerated optimization problem search using Bose-Einstein condensates,” Prog. Inform. 8, 1–9 (2011).

S. Inoue, H. Ohzu, S. Machida, and Y. Yamamoto, “Quantum correlation between longitudinal mode intensities in a multimode squeezed semiconductor laser,” Phys. Rev. A 46, 2757–2765 (1992).
[CrossRef] [PubMed]

L. Gillner, G. Bjork, and Y. Yamamoto, “Quantum noise properties of an injection-locked laser oscillator with pump-noise suppression and squeezed injection,” Phys. Rev A 41(9), 5053–5065 (1990).
[CrossRef] [PubMed]

Y. Yamamoto, S. Machida, and O. Neilsson, “Amplitude squeezing in a pump-noise-suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
[CrossRef] [PubMed]

H. A. Haus and Y. Yamamoto, “Quantum noise of an injection-locked laser oscillator,” Phys. Rev. A 29, 1261–1274 (1984).
[CrossRef]

S. Kobayashi, Y. Yamamoto, and T. Kimura, “Optical FM signal amplification and FM noise reduction in an injection locked AlGaAs semiconductor laser,” Electron. Lett. 17(22), 849–851 (1981).
[CrossRef]

T. Byrnes, K. Yan, and Y. Yamamoto, “Optimization using Bose-Einstein condensation and measurement-feedback circuits,” arXiv:0909.2530v2.

K. Takata, S. Utsunomiya, and Y. Yamamoto, “Transient time of an Ising machine based on injection-locked lasers: contribution of locking bandwidth and Zeeman component,” (to be submitted).

Yan, K.

K. Yan, T. Byrnes, and Y. Yamamoto, “Kinetic Monte Carlo study of accelerated optimization problem search using Bose-Einstein condensates,” Prog. Inform. 8, 1–9 (2011).

T. Byrnes, K. Yan, and Y. Yamamoto, “Optimization using Bose-Einstein condensation and measurement-feedback circuits,” arXiv:0909.2530v2.

Young, A. A.

K. Binder and A. A. Young, “Spin glasses: experimental facts, theoretical concepts, and open questions,” Rev. Mod. Phys. 58, 801–976 (1986).
[CrossRef]

Young, A. P.

A. P. Young, S. Knysh, and V. N. Smelyanskiy, “First-order phase transition in the quantum adiabatic algorithm,” Phys. Rev. Lett. 104, 020502 (2010).
[CrossRef] [PubMed]

Yu, S. F.

S. F. Yu, Analysis and Design of Vertical Cavity Surface Emitting Laser (Wiley-Interscience, 2003), Chap. 8.
[CrossRef]

Zhang, Y.

Y. Zhang, Y. Xia, Z. Man, and G. Guo, “Simulation of the Ising model, memory for Bell states and generation of four-atom entangled states in cavity QED,” Sci. China, Ser. G 52, 700–707 (2009).
[CrossRef]

Bell Syst. Tech. J. (1)

S. D. Personick, “Receiver design for digital fiber optic communication systems, I,” Bell Syst. Tech. J. 52(6), 843–874 (1973).

Electron. Lett. (1)

S. Kobayashi, Y. Yamamoto, and T. Kimura, “Optical FM signal amplification and FM noise reduction in an injection locked AlGaAs semiconductor laser,” Electron. Lett. 17(22), 849–851 (1981).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. Kobayashi and T. Kimura, “Injection locking in AIGaAs semiconductor laser,” IEEE J. Quantum Electron. 17(5), 681–689 (1981).
[CrossRef]

J. Phys. A: Math. Gen. (1)

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

Nat. Phys. (1)

G. E. Santoro and E. Tosatti, “Quantum to classical and back,” Nat. Phys. 3, 593–594 (2007).
[CrossRef]

Phys. Rev A (1)

L. Gillner, G. Bjork, and Y. Yamamoto, “Quantum noise properties of an injection-locked laser oscillator with pump-noise suppression and squeezed injection,” Phys. Rev A 41(9), 5053–5065 (1990).
[CrossRef] [PubMed]

Phys. Rev. A (3)

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Figures (7)

Fig. 1
Fig. 1

A proposed injection-locked laser system for finding the ground state of an Ising model Eq. (2). A master laser output is equally split into M paths and injected into M slave lasers via an optical isolator. At a time t < 0 (initialization), the injection signal from the master laser has a vertical linear polarization so that all slave lasers are initialized in vertical linear polarization states |V1|V2 ... |V M . At a time t = 0, the combined attenuator, HWP and QWP can implement the Zeeman term λi . Also at a time t = 0, each slave laser output is injected to other slave lasers via a horizontal linear polarizer, phase shifter, and attenuator but without an isolator. This mutual injection-locking can implement the Ising interaction term Jij . After a steady state condition is reached, the two polarization components of each slave laser are detected by a polarization beam splitter (PBS) and two photodetectors.

Fig. 2
Fig. 2

Each single photon occupies M slave lasers simultaneously as its partial waves. Each slave laser starts from the state, | V = 1 2 ( | R + | L ) ), and ends in the state |R〉 or |L〉 with a high probability, which is the computational result. This computational process is described by the mode competition or amplitude modulation in the |R〉-|L〉 basis and by the mutual interference or phase modulation in the |D〉-|〉 basis.

Fig. 3
Fig. 3

The time evolution of the average photon numbers nRi and nLi . A fractional spontaneous emission coupling efficiency is β = 10−5, an injection current level is I = 32mA ( I I t h 2 ) and attenuation parameters are α = 1 200 and ζ = 1 500 . (a) M = 2, (J 12, λ 1, λ 2)=6, 1, 9 10 , (b) M = 5, (J 12, J 13, J 14, J 15, J 23, J 24, J 25, J 34, J 35, J 45, λ 1, λ 2, λ 3, λ 4, λ 5)=(6, 11 2 , 5, 9 2 , 4, 7 2 , 3, 5 2 , 2, 3 2 , 1, 9 10 , 9 10 , 9 10 , 9 10 ), (c) M = 7, (J 12, J 13, J 14, J 15, J 16, J 17, J 23, J 24, J 25, J 26, J 27, J 34, J 35, J 36, J 37, J 45, J 46, J 47, J 56, J 57, J 67, λ 1, λ 2, λ 3, λ 4, λ 5, λ 6, λ 7) = (6, 52 9 , 50 9 , 16 3 , 46 9 , 44 9 , 14 3 , 40 9 , 38 9 , 4, 34 9 , 32 9 , 10 3 , 28 9 , 26 9 , 8 3 , 22 9 , 20 9 , 2, 16 9 , 14 9 , 1, 9 10 , 9 10 , 9 10 , 9 10 , 9 10 , 9 10 ) and (d) M = 10, (J 12, J 13, J 14, J 15, J 16, J 17, J 18, J 19, J 110, J 23, J 24, J 25, J 26, J 27, J 28, J 29, J 210, J 34, J 35, J 36, J 37, J 38, J 39, J 310, J 45, J 46, J 47, J 48, J 49, J 410, J 56, J 57, J 58, J 59, J 510, J 67, J 68, J 69, J 610, J 78, J 79, J 710, J 89, J 810, J 910, λ 1, λ 2, λ 3, λ 4, λ 5, λ 6, λ 7, λ 8, λ 9, λ 10) = (6, 53 9 , 52 9 , 17 3 , 50 9 , 49 9 , 16 3 , 47 9 , 46 9 , 5, 44 9 , 43 9 , 14 3 , 41 9 , 40 9 , 13 3 , 38 9 , 37 9 , 4, 35 9 , 34 9 , 11 3 , 32 9 , 31 9 , 10 3 , 29 9 , 28 9 , 3, 26 9 , 25 9 , 8 3 , 23 9 , 22 9 , 7 3 , 20 9 , 19 9 , 2, 17 9 , 16 9 , 5 3 , 14 9 , 13 9 , 4 3 , 11 9 , 10 9 , 1, 9 10 , 9 10 , 9 10 , 9 10 , 9 10 , 9 10 , 9 10 , 9 10 , 9 10 ), where ξ i j = α J i j max i j M [ | J i j | , | λ i | ] and η i = α λ 1 max i j M [ | J i j | , | λ i | ] for each M.

Fig. 4
Fig. 4

The time evolution of the average electron number Ni for (a) M = 2 to (d) M = 10. The numerical parameters are identical to those in Fig. 3.

Fig. 5
Fig. 5

The time evolution of the photon numbers nRi and nLi in a three site problem. Numerical parameters are β = 10−5, Ith = 16mA, I I t h 2 , α = 1 350 and ζ = 1 200 . In (a) and (c) the Ising Hamiltonian is given with (J 12, J 13, J 23, λ 1, λ 2, λ 3) = (1, 1, 0, 1 10 , 1 30 , 1 10 ), for which the ground state is (σ 1, σ 2, σ 3) = (1, −1, −1). In (b) and (d) the Ising Hamiltonian is given with (J 12, J 13, J 23, λ 1, λ 2, λ 3) = (1, 1, 0, 1 5 , 1 10 , 1 20 ), for which the ground state is (σ 1, σ 2, σ 3) = (−1, 1, 1). (a)(b) The numerical results without photon number noise. (c)(d) The numerical results with the Poissonian photon number noise. A Gaussian distributed noise term with a zero mean and standard distribution of n R i or n L i is added at every time step Δt = 10−12 sec in the Range-Kutta numerical integration.

Fig. 6
Fig. 6

(a)The time evolution of the slowly varying amplitudes ADi 0 and AD̄i 0. (b)The time evolution of the slowly varying phases ϕDi and ϕD̄i . The phases of ± π 4 correspond to complete circular polarizations. (c)The time evolution of the photon number nRi and nLi calculated in the |D〉 and |〉 basis. (d)The time evolution of the photon number nRi and nLi calculated in the |R〉 and |L〉 basis. The numerical parameters are M = 3, (J 12, J 13, J 23, λ 1, λ 2, λ 3) = (6, 3, 2, 3 10 , 1 10 , 1 10 ), α = 1 250 and η = 1 200 .

Fig. 7
Fig. 7

(a) Implementation of the Zeeman term λi by an attenuator, HWP and QWP. When ηi > 0, the horizontal polarization component of the injection signal couples to the right and left circular polarization modes of the slave laser with π and zero phases, respectively. (b) Implementation of the Ising interaction term Jij by a horizontal linear polarizer, phase shifter and attenuator. The magnitude of the Ising interaction term |Jij | can be implemented by an attenuator. The horizontal linear polarizer output of the slave laser i is proportional to n R i n L i . Therefore, the phase of the injection signal from the slave laser i to the slave laser j can implement a sign of the Ising interaction term Jij /|Jij |.

Equations (38)

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= i < j M J i j σ i z σ j z
= ( i < j J i j σ i z σ j z + i λ i σ i z ) .
σ i z = { 1 ( if n R i > n L i ) 1 ( if n R i < n L i ) ,
Δ ω L = ω Q P i n P o u t ,
| Ψ t = 0 = 1 2 ( | R + | L ) 1 1 2 ( | R + | L ) 2 1 2 ( | R + | L ) M = 1 2 M ( | R 1 | R 2 | R M + + | L 1 | L 2 | L M ) ,
d d t A ^ ( t ) = i ω r A ^ ( t ) 1 2 [ ω Q ω μ 2 ( χ ˜ i i χ ˜ r ) ] A ^ ( t ) + ω Q F 0 e i ω t + f ˜ G + f ^ L .
d d t n ^ ( t ) = ω Q n ^ ( t ) + E ˜ C V n ^ ( t ) + E ˜ C V + ω Q ( F 0 * A ^ ( t ) + A ^ ( t ) F 0 ) + F ^ n ( t ) .
F ^ n ( t ) F ^ n ( s ) = δ ( t s ) [ ω Q n ^ + E ˜ C V ( n ^ + 1 ) ]
d d t N ˜ ( t ) = P N ˜ ( t ) τ s p E ˜ C V n ^ ( t ) E ˜ C V + F ˜ c ( t ) ,
F ˜ c ( t ) F ˜ c ( s ) = δ ( t s ) [ P + N ˜ τ s p + E ˜ C V ( n ^ + 1 ) ] .
F ^ n ( t ) F ˜ c ( t ) = δ ( t s ) [ E ˜ C V ( n ^ + 1 ) ] .
A ( t ) = A 0 ( t ) e i [ ω t + ϕ 0 ( t ) ] ,
d d t A 0 ( t ) = 1 2 [ ( ω Q ) E C V ] A 0 ( t ) + ω Q F 0 cos [ ϕ 0 ( t ) ] ,
d d t n ( t ) = ( ω Q E C V ) n ( t ) + E C V + 2 ω Q F 0 A 0 ( t ) cos [ ϕ 0 ( t ) ] ,
d d t ϕ 0 ( t ) = ( ω ω 0 ) ω Q F 0 A 0 ( t ) sin [ ϕ 0 ( t ) ] ,
d d t n ( t ) = ( ω Q ) n ( t ) + E C V n ( t ) + E C V + 2 ω Q n ( t ) ζ n M .
d d t n R i = ( ω Q E C V R i ) n R i + E C V R i + 2 ω Q n R i ζ n M ,
d d t n L i = ( ω Q E C V L i ) n L i + E C V L i + 2 ω Q n L i ζ n M ,
d d t N R i = P 2 N R i ( t ) τ s p E C V R i n R i ( t ) E C V R i N R i ( t ) N L i ( t ) τ s p i n ,
d d t N L i = P 2 N L i ( t ) τ s p E C V L i n L i ( t ) E C V L i N L i ( t ) N R i ( t ) τ s p i n .
d d t N i ( t ) = P N i ( t ) τ s p E C V i [ n R i ( t ) + n L i ( t ) + 2 ] .
d d t n R i = ( ω Q E C V i ) n R i + E C V i + 2 ω Q n R i [ ( ζ η i ) n M j i 1 2 ξ i j ( n R j n L j ) ]
d d t n L i = ( ω Q E C V i ) n L i + E C V i + 2 ω Q n L i [ ( ζ + η i ) n M + j i 1 2 ξ i j ( n R j n L j ) ] .
E C V i = ω Q 2 ω Q ζ n M ( n R i + n L i ) n R i + n L i + 2 ω Q n R i n L i n R i + n L i [ η i n M n R i + n L i + j i 1 2 ξ i j n R j n L j n R i + n L i ] .
i n R i n L i n T i [ η i n M n T i + j i 1 2 ξ i j n R j n L j n T i ] ,
j η i n M n T i σ i z + i < j ξ i j σ i z σ j z ,
η i = α λ i max [ | J i j | , | λ i | ] ,
ξ i j = α J i j max [ | J i j | , | λ i | ] ,
C ( σ 1 , σ 2 , σ M ) = { 1 ( for the ground state ) 0 ( for all the other states ) .
A D i ( t ) = A D i 0 ( t ) exp { ı [ ω t + ϕ D i ( t ) ] } ,
A D ¯ i ( t ) = A D ¯ i 0 ( t ) exp { ı [ ω t + ϕ D ¯ i ( t ) ] }
d d t A D i 0 ( t ) = 1 2 ( ω Q E C V i ) A D i 0 ( t ) + ω Q n M ζ 2 + η i 2 cos [ δ ϕ D i ( t ) ] j i 1 2 ξ i j ω Q { n D j cos [ ϕ D j ( t ) ϕ D i ( t ) ] n D ¯ j cos [ ϕ D ¯ j ( t ) ϕ D i ( t ) ] } ,
d d t ϕ D i ( t ) = ( ω ω 0 ) + ω Q 1 Q D i 0 ( t ) { n M ζ 2 + η i 2 sin [ δ ϕ D i ( t ) ] j i 1 2 ξ i j [ n D j sin ( ϕ D j ( t ) ϕ D i ( t ) ) n D ¯ j sin ( ϕ D ¯ j ( t ) ϕ D i ( t ) ) ] } ,
d d t N i ( t ) = P N i ( t ) τ s p E C V i [ n D i ( t ) + n D ¯ i ( t ) + 2 ] ,
n R i ( t ) = | ( 1 + ı ) 2 A D i 0 ( t ) exp [ ı ϕ D i ( t ) ] + ( 1 ı ) 2 A D ¯ i 0 ( t ) exp [ ı ϕ D ¯ i ( t ) ] | 2 ,
n L i ( t ) = | ( 1 ı ) 2 A D i 0 ( t ) exp [ ı ϕ D i ( t ) ] + ( 1 + ı ) 2 A D ¯ i 0 ( t ) exp [ ı ϕ D ¯ i ( t ) ] | 2 .
P e = 1 2 erfc ( Q 2 ) ,
4 Q 2 = m R i m L i 2 Δ m R i 2 + Δ m L i 2 = η D ( ω Q ) n R i T ( 1 R ) 2 1 + R 144 ,

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