Abstract

The coherent Ising machine (CIM) enables efficient sampling of low-lying energy states of the Ising Hamiltonian with all-to-all connectivity by encoding the spins in the amplitudes of pulsed modes in an optical parametric oscillator (OPO). The interaction between the pulses is realized by means of measurement-based optoelectronic feedforward, which enhances the gain for lower-energy spin configurations. We present an efficient method of simulating the CIM on a classical computer that outperforms the CIM itself, as well as the noisy mean-field annealer in terms of both the quality of the samples and the computational speed. It is furthermore advantageous with respect to the CIM in that it can handle Ising Hamiltonians with arbitrary real-valued node coupling strengths. These results illuminate the nature of the faster performance exhibited by the CIM and may give rise to a new class of quantum-inspired algorithms of classical annealing that can successfully compete with existing methods.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (1)

T. Leleu, Y. Yamamoto, P. L. McMahon, and K. Aihara, “Destabilization of local minima in analog spin systems by correction of amplitude heterogeneity,” Phys. Rev. Lett. 122, 040607 (2019).
[Crossref] [PubMed]

2018 (1)

G. Torlai, G. Mazzola, J. Carrasquilla, M. Troyer, R. Melko, and G. Carleo, “Neural-network quantum state tomography,” Nat. Phys. 14, 447 (2018).
[Crossref]

2017 (6)

G. Carleo and M. Troyer, “Solving the quantum many-body problem with artificial neural networks,” Science 355, 602–606 (2017).
[Crossref] [PubMed]

W. R. Clements, J. J. Renema, Y. H. Wen, H. M. Chrzanowski, W. S. Kolthammer, and I. A. Walmsley, “Gaussian optical ising machines,” Phys. Rev. A 96, 043850 (2017).
[Crossref]

F. Neukart, G. Compostella, C. Seidel, D. von Dollen, S. Yarkoni, and B. Parney, “Traffic flow optimization using a quantum annealer,” Front. ICT 4, 29 (2017).
[Crossref]

Y. Haribara, H. Ishikawa, S. Utsunomiya, K. Aihara, and Y. Yamamoto, “Performance evaluation of coherent ising machines against classical neural networks,” Quantum Sci. Technol. 2, 044002 (2017).
[Crossref]

Y. Yamamoto, K. Aihara, T. Leleu, K.-i. Kawarabayashi, S. Kako, M. Fejer, K. Inoue, and H. Takesue, “Coherent ising machines—optical neural networks operating at the quantum limit,” npj Quantum Inf. 3, 49 (2017).
[Crossref]

J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, “Quantum machine learning,” Nature 549, 195 (2017).
[Crossref] [PubMed]

2016 (3)

Takahiro Inagaki, Yoshitaka Haribara, Koji Igarashi, Tomohiro Sonobe, Shuhei Tamate, Toshimori Honjo, Alireza Marandi, Peter L. McMahon, Takeshi Umeki, Koji Enbutsu, Osamu Tadanaga, Hirokazu Takenouchi, Kazuyuki Aihara, Ken-ichi Kawarabayashi, Kyo Inoue, Shoko Utsunomiya, and Hiroki Takesue, “A coherent ising machine for 2000-node optimization problems,” Science,  354(6312):603–606, 2016.
[Crossref] [PubMed]

Peter L. McMahon, Alireza Marandi, Yoshitaka Haribara, Ryan Hamerly, Carsten Langrock, Shuhei Tamate, Takahiro Inagaki, Hiroki Takesue, Shoko Utsunomiya, Kazuyuki Aihara, Robert L. Byer, M. M. Fejer, Hideo Mabuchi, and Yoshihisa Yamamoto, “A fully programmable 100-spin coherent ising machine with all-to-all connections,” Science,  354(6312):614–617, 2016.
[Crossref] [PubMed]

G. Rosenberg, P. Haghnegahdar, P. Goddard, P. Carr, K. Wu, and M. L. De Prado, “Solving the optimal trading trajectory problem using a quantum annealer,” IEEE J. Sel. Top. Signal Process. 10, 1053–1060 (2016).
[Crossref]

2014 (3)

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, 937 (2014).
[Crossref]

P. I. Bunyk, E. M. Hoskinson, M. W. Johnson, E. Tolkacheva, F. Altomare, A. J. Berkley, R. Harris, J. P. Hilton, T. Lanting, A. J. Przybysz, and J. Whittaker, “Architectural considerations in the design of a superconducting quantum annealing processor,” IEEE Transactions on Appl. Supercond. 24, 1–10 (2014).
[Crossref]

D. Sornette, “Physics and financial economics (1776–2014): puzzles, ising and agent-based models,” Reports on Prog. Phys. 77, 062001 (2014).
[Crossref]

2013 (2)

U. Benlic and J.-K. Hao, “Breakout local search for the max-cut problem,” Eng. Appl. Artif. Intell. 26, 1162 (2013).
[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, 063853 (2013).
[Crossref]

2012 (1)

A. Perdomo-Ortiz, N. Dickson, M. Drew-Brook, G. Rose, and A. Aspuru-Guzik, “Finding low-energy conformations of lattice protein models by quantum annealing,” Sci. Reports 2, 571 (2012).
[Crossref]

2010 (1)

H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4, 261 (2010).
[Crossref]

2009 (1)

J. Ambjorn, K. N. Anagnostopoulos, R. Loll, and I. Pushkina, “Shaken, but not stirred: Potts model coupled to quantum gravity,” Nucl. Phys. B807, 251–264 (2009).
[Crossref]

1999 (2)

K. A. Smith, “Neural networks for combinatorial optimization: a review of more than a decade of research,” INFORMS J. on Comput. 11, 15–34 (1999).
[Crossref]

N. Qian, “On the momentum term in gradient descent learning algorithms,” Neural networks 12, 145–151 (1999).
[Crossref]

1986 (1)

J. J. Hopfield and D. W. Tank, “Computing with neural circuits: A model,” Science 233, 625–633 (1986).
[Crossref] [PubMed]

1983 (1)

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

1982 (1)

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

1977 (1)

J. B. Zuber and C. Itzykson, “Quantum field theory and the two-dimensional ising model,” Phys. Rev. D 15, 2875–2884 (1977).
[Crossref]

Aihara, K.

T. Leleu, Y. Yamamoto, P. L. McMahon, and K. Aihara, “Destabilization of local minima in analog spin systems by correction of amplitude heterogeneity,” Phys. Rev. Lett. 122, 040607 (2019).
[Crossref] [PubMed]

Y. Haribara, H. Ishikawa, S. Utsunomiya, K. Aihara, and Y. Yamamoto, “Performance evaluation of coherent ising machines against classical neural networks,” Quantum Sci. Technol. 2, 044002 (2017).
[Crossref]

Y. Yamamoto, K. Aihara, T. Leleu, K.-i. Kawarabayashi, S. Kako, M. Fejer, K. Inoue, and H. Takesue, “Coherent ising machines—optical neural networks operating at the quantum limit,” npj Quantum Inf. 3, 49 (2017).
[Crossref]

Aihara, Kazuyuki

Takahiro Inagaki, Yoshitaka Haribara, Koji Igarashi, Tomohiro Sonobe, Shuhei Tamate, Toshimori Honjo, Alireza Marandi, Peter L. McMahon, Takeshi Umeki, Koji Enbutsu, Osamu Tadanaga, Hirokazu Takenouchi, Kazuyuki Aihara, Ken-ichi Kawarabayashi, Kyo Inoue, Shoko Utsunomiya, and Hiroki Takesue, “A coherent ising machine for 2000-node optimization problems,” Science,  354(6312):603–606, 2016.
[Crossref] [PubMed]

Peter L. McMahon, Alireza Marandi, Yoshitaka Haribara, Ryan Hamerly, Carsten Langrock, Shuhei Tamate, Takahiro Inagaki, Hiroki Takesue, Shoko Utsunomiya, Kazuyuki Aihara, Robert L. Byer, M. M. Fejer, Hideo Mabuchi, and Yoshihisa Yamamoto, “A fully programmable 100-spin coherent ising machine with all-to-all connections,” Science,  354(6312):614–617, 2016.
[Crossref] [PubMed]

Altomare, F.

P. I. Bunyk, E. M. Hoskinson, M. W. Johnson, E. Tolkacheva, F. Altomare, A. J. Berkley, R. Harris, J. P. Hilton, T. Lanting, A. J. Przybysz, and J. Whittaker, “Architectural considerations in the design of a superconducting quantum annealing processor,” IEEE Transactions on Appl. Supercond. 24, 1–10 (2014).
[Crossref]

Ambjorn, J.

J. Ambjorn, K. N. Anagnostopoulos, R. Loll, and I. Pushkina, “Shaken, but not stirred: Potts model coupled to quantum gravity,” Nucl. Phys. B807, 251–264 (2009).
[Crossref]

Anagnostopoulos, K. N.

J. Ambjorn, K. N. Anagnostopoulos, R. Loll, and I. Pushkina, “Shaken, but not stirred: Potts model coupled to quantum gravity,” Nucl. Phys. B807, 251–264 (2009).
[Crossref]

Anschuetz, E. R.

E. R. Anschuetz, J. P. Olson, A. Aspuru-Guzik, and Y. Cao, “Variational quantum factoring,” arXiv:1808.08927 (2018).

Aspuru-Guzik, A.

A. Perdomo-Ortiz, N. Dickson, M. Drew-Brook, G. Rose, and A. Aspuru-Guzik, “Finding low-energy conformations of lattice protein models by quantum annealing,” Sci. Reports 2, 571 (2012).
[Crossref]

E. R. Anschuetz, J. P. Olson, A. Aspuru-Guzik, and Y. Cao, “Variational quantum factoring,” arXiv:1808.08927 (2018).

Atieh, Fadi

Charles Roques-Carmes, Yichen Shen, Cristian Zanoci, Mihika Prabhu, Fadi Atieh, Li Jing, Tena Dubcek, Vladimir Ceperic, John D Joannopoulos, Dirk Englund, and Marin Soljacic. “Photonic recurrent Ising sampler,” arXiv:1811.02705 (2018).

Banderier, C.

C. Banderier, H.-K. Hwang, V. Ravelomana, and V. Zacharovas, “Average case analysis of np-complete problems: Maximum independent set and exhaustive search algorithms,” Proc. AofA2009.

Barahona, F.

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

Baxter, R. J.

R. J. Baxter, Exactly solved models in statistical mechanics (Dover Publications, 2014).

Benlic, U.

U. Benlic and J.-K. Hao, “Breakout local search for the max-cut problem,” Eng. Appl. Artif. Intell. 26, 1162 (2013).
[Crossref]

Berkley, A. J.

P. I. Bunyk, E. M. Hoskinson, M. W. Johnson, E. Tolkacheva, F. Altomare, A. J. Berkley, R. Harris, J. P. Hilton, T. Lanting, A. J. Przybysz, and J. Whittaker, “Architectural considerations in the design of a superconducting quantum annealing processor,” IEEE Transactions on Appl. Supercond. 24, 1–10 (2014).
[Crossref]

A. D. King, W. Bernoudy, J. King, A. J. Berkley, and T. Lanting, “Emulating the coherent ising machine with a mean-field algorithm,” arXiv:1806.08422 (2018).

Berloff, N. G.

K. P. Kalinin and N. G. Berloff, “Global optimization of spin hamiltonians with gain-dissipative systems,” arXiv:1807.00699 (2018).

Bernoudy, W.

A. D. King, W. Bernoudy, J. King, A. J. Berkley, and T. Lanting, “Emulating the coherent ising machine with a mean-field algorithm,” arXiv:1806.08422 (2018).

C. C. McGeoch, W. Bernoudy, and J. King, “Comment on “scaling advantages of all-to-all connectivity in physical annealers: the coherent Ising machine vs d-wave 2000q”,” arXiv:1807.00089 (2018).

Biamonte, J.

J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, “Quantum machine learning,” Nature 549, 195 (2017).
[Crossref] [PubMed]

Bian, Z.

Z. Bian, F. Chudak, W. Macready, A. Roy, R. Sebastiani, and S. Varotti, “Solving sat and maxsat with a quantum annealer: Foundations, encodings, and preliminary results,” arXiv:1811.02524 (2018).

Bilbro, G.

G. Bilbro, R. Mann, T. K. Miller, W. E. Snyder, D. E. Van den Bout, and M. White, “Optimization by mean field annealing,” in Advances in Neural Information Processing Systems (Neural Information Processing Systems Foundation, Inc., 1989) 91–98.

Blanzieri, E.

E. Blanzieri and D. Pastorello, “Quantum annealing tabu search for qubo optimization,” arXiv:1810.09342 (2018).

Bunyk, P. I.

P. I. Bunyk, E. M. Hoskinson, M. W. Johnson, E. Tolkacheva, F. Altomare, A. J. Berkley, R. Harris, J. P. Hilton, T. Lanting, A. J. Przybysz, and J. Whittaker, “Architectural considerations in the design of a superconducting quantum annealing processor,” IEEE Transactions on Appl. Supercond. 24, 1–10 (2014).
[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, 937 (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, 063853 (2013).
[Crossref]

R. Hamerly, T. Inagaki, P. L. McMahon, D. Venturelli, A. Marandi, T. Onodera, E. Ng, C. Langrock, K. Inaba, T. Honjo, K. Enbutsu, T. Umeki, R. Kasahara, S. Utsunomiya, S. Kako, K.-i. Kawarabayashi, R. L. Byer, M. M. Fejer, H. Mabuchi, D. Englund, E. Rieffel, H. Takesue, and Y. Yamamoto, “Experimental investigation of performance differences between Coherent Ising Machines and a quantum annealer,” arXiv:1805.05217 (2018).

Byer, Robert L.

Peter L. McMahon, Alireza Marandi, Yoshitaka Haribara, Ryan Hamerly, Carsten Langrock, Shuhei Tamate, Takahiro Inagaki, Hiroki Takesue, Shoko Utsunomiya, Kazuyuki Aihara, Robert L. Byer, M. M. Fejer, Hideo Mabuchi, and Yoshihisa Yamamoto, “A fully programmable 100-spin coherent ising machine with all-to-all connections,” Science,  354(6312):614–617, 2016.
[Crossref] [PubMed]

Cao, Y.

E. R. Anschuetz, J. P. Olson, A. Aspuru-Guzik, and Y. Cao, “Variational quantum factoring,” arXiv:1808.08927 (2018).

Carleo, G.

G. Torlai, G. Mazzola, J. Carrasquilla, M. Troyer, R. Melko, and G. Carleo, “Neural-network quantum state tomography,” Nat. Phys. 14, 447 (2018).
[Crossref]

G. Carleo and M. Troyer, “Solving the quantum many-body problem with artificial neural networks,” Science 355, 602–606 (2017).
[Crossref] [PubMed]

Carr, P.

G. Rosenberg, P. Haghnegahdar, P. Goddard, P. Carr, K. Wu, and M. L. De Prado, “Solving the optimal trading trajectory problem using a quantum annealer,” IEEE J. Sel. Top. Signal Process. 10, 1053–1060 (2016).
[Crossref]

Carrasquilla, J.

G. Torlai, G. Mazzola, J. Carrasquilla, M. Troyer, R. Melko, and G. Carleo, “Neural-network quantum state tomography,” Nat. Phys. 14, 447 (2018).
[Crossref]

Caulfield, H. J.

H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4, 261 (2010).
[Crossref]

Ceperic, Vladimir

Charles Roques-Carmes, Yichen Shen, Cristian Zanoci, Mihika Prabhu, Fadi Atieh, Li Jing, Tena Dubcek, Vladimir Ceperic, John D Joannopoulos, Dirk Englund, and Marin Soljacic. “Photonic recurrent Ising sampler,” arXiv:1811.02705 (2018).

Chrzanowski, H. M.

W. R. Clements, J. J. Renema, Y. H. Wen, H. M. Chrzanowski, W. S. Kolthammer, and I. A. Walmsley, “Gaussian optical ising machines,” Phys. Rev. A 96, 043850 (2017).
[Crossref]

Chudak, F.

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Clements, W. R.

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Figures (4)

Fig. 1
Fig. 1 CIM setup. Each pulse undergoes optical squeezing, linear and non-linear loss, as well as displacement. FPGA: field-programmable gate array.
Fig. 2
Fig. 2 a) Evolution of the components of the “spin” vector x⃗ = {xi}. b) Time dependence of the pump-loss factor v and the quantity ‖��(x⃗) − ��(Ĵx⃗)‖ [the symbol ��(·) denoting normalization] which shows the proximity of x⃗ to the eigenvector of Ĵ with the highest eigenvalue.
Fig. 3
Fig. 3 Histograms for the graphs G22, G39 [26] as well as the fully connected K2000 graph. For each graph the dependence between the number of runs and cut value is presented for the NMFA and SimCim algorithms and for CIM experimental results [13]. For G22 and G39 the results of CIM and algorithms also compared to BLS algorithm which gives the best known cuts.
Fig. 4
Fig. 4 Results for a random graph of dimension 800 with real-valued couplings Jij that are normally distributed with zero mean and variance equal to one.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

H = 1 2 i , j J i j σ i σ j
j J i j X j ,
Δ a i = w a i * γ a i s | a i | 2 a i + ζ j J i j x j + f i / 2 .
Δ x i = w x i γ x i s ( x i 2 + p i 2 ) x i + ζ j J i j x j + Re f i ; Δ p i = w p i γ p i s ( x i 2 + p i 2 ) p i + Im f i .
Δ x i = v x i + ζ j J i j x j + f i ,
ϕ ( x ) = { x for | x | x sat ; x sat otherwise
{ Δ e i = ( v + ζ Λ i i ) e i + f i ; | j R j i e j | x sat ,
cut = 1 2 i < j J i j ( 1 σ i σ j ) ,

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