Abstract

Conventional computing architectures have no known efficient algorithms for combinatorial optimization tasks such as the Ising problem, which requires finding the ground state spin configuration of an arbitrary Ising graph. Physical Ising machines have recently been developed as an alternative to conventional exact and heuristic solvers; however, these machines typically suffer from decreased ground state convergence probability or universality for high edge-density graphs or arbitrary graph weights, respectively. We experimentally demonstrate a proof-of-principle integrated nanophotonic recurrent Ising sampler (INPRIS), using a hybrid scheme combining electronics and silicon-on-insulator photonics, that is capable of converging to the ground state of various four-spin graphs with high probability. The INPRIS results indicate that noise may be used as a resource to speed up the ground state search and to explore larger regions of the phase space, thus allowing one to probe noise-dependent physical observables. Since the recurrent photonic transformation that our machine imparts is a fixed function of the graph problem and therefore compatible with optoelectronic architectures that support GHz clock rates (such as passive or non-volatile photonic circuits that do not require reprogramming at each iteration), this work suggests the potential for future systems that could achieve orders-of-magnitude speedups in exploring the solution space of combinatorially hard problems.

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

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2020 (1)

C. Roques-Carmes, Y. Shen, C. Zanoci, M. Prabhu, F. Atieh, L. Jing, T. Dubček, C. Mao, M. R. Johnson, V. Čeperić, and J. D. Joannopoulos, “Heuristic recurrent algorithms for photonic Ising machines,” Nat. Commun. 11, 1–8 (2020).
[Crossref]

2019 (5)

M. Babaeian, D. T. Nguyen, V. Demir, M. Akbulut, P.-A. Blanche, Y. Kaneda, S. Guha, M. A. Neifeld, and N. Peyghambarian, “A single shot coherent Ising machine based on a network of injection-locked multicore fiber lasers,” Nat. Commun. 10, 3516 (2019).
[Crossref]

D. Pierangeli, G. Marcucci, and C. Conti, “Large-scale photonic Ising machine by spatial light modulation,” Phys. Rev. Lett. 122, 213902 (2019).
[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,” Sci. Adv. 5, eaau0823 (2019).
[Crossref]

R. Hamerly, L. Bernstein, A. Sludds, M. Soljačić, and D. Englund, “Large-scale optical neural networks based on photoelectric multiplication,” Phys. Rev. X 9, 021032 (2019).
[Crossref]

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]

2018 (6)

A. H. Atabaki, S. Moazeni, F. Pavanello, H. Gevorgyan, J. Notaros, L. Alloatti, M. T. Wade, C. Sun, S. A. Kruger, H. Meng, K. Al Qubaisi, I. Wang, B. Zhang, A. Khilo, C. V. Baiocco, M. A. Popović, V. M. Stojanović, and R. J. Ram, “Integrating photonics with silicon nanoelectronics for the next generation of systems on a chip,” Nature 556, 349–354 (2018).
[Crossref]

N. C. Harris, J. Carolan, D. Bunandar, M. Prabhu, M. Hochberg, T. Baehr-Jones, M. L. Fanto, A. M. Smith, C. C. Tison, P. M. Alsing, and D. Englund, “Linear programmable nanophotonic processors,” Optica 5, 1623 (2018).
[Crossref]

M. R. Vázquez, V. Bharadwaj, B. Sotillo, S.-Z. A. Lo, R. Ramponi, N. I. Zheludev, G. Lanzani, S. M. Eaton, and C. Soci, “Optical NP problem solver on laser-written waveguide platform,” Opt. Express 26, 702–710 (2018).
[Crossref]

F. Böhm, T. Inagaki, K. Inaba, T. Honjo, K. Enbutsu, T. Umeki, R. Kasahara, and H. Takesue, “Understanding dynamics of coherent Ising machines through simulation of large-scale 2D Ising models,” Nat. Commun. 9, 5020 (2018).
[Crossref]

X. Lin, Y. Rivenson, N. T. Yardimci, M. Veli, Y. Luo, M. Jarrahi, and A. Ozcan, “All-optical machine learning using diffractive deep neural networks,” Science 361, 1004–1008 (2018).
[Crossref]

C. R. de Galarreta, A. M. Alexeev, Y.-Y. Au, M. Lopez-Garcia, M. Klemm, M. Cryan, J. Bertolotti, and C. D. Wright, “Nonvolatile reconfigurable phase-change metadevices for beam steering in the near infrared,” Adv. Funct. Mater. 28, 1704993 (2018).
[Crossref]

2017 (4)

R. Burgwal, W. R. Clements, D. H. Smith, J. C. Gates, W. S. Kolthammer, J. J. Renema, and I. A. Walmsley, “Using an imperfect photonic network to implement random unitaries,” Opt. Express 25, 28236 (2017).
[Crossref]

S. Tsukamoto, M. Takatsu, S. Matsubara, and H. Tamura, “An accelerator architecture for combinatorial optimization problems,” Fujitsu Sci. Tech. J. 53, 8–13 (2017).

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljačić, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. C. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447–452 (2017).
[Crossref]

2016 (4)

P. L. McMahon, A. Marandi, Y. Haribara, R. Hamerly, C. Langrock, S. Tamate, T. Inagaki, H. Takesue, S. Utsunomiya, K. Aihara, R. L. Byer, M. M. Fejer, H. Mabuchi, and Y. Yamamoto, “A fully programmable 100-spin coherent Ising machine with all-to-all connections,” Science 354, 614–617 (2016).
[Crossref]

W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walsmley, “Optimal design for universal multiport interferometers,” Optica 3, 1460 (2016).
[Crossref]

T. Inagaki, Y. Haribara, K. Igarashi, T. Sonobe, S. Tamate, T. Honjo, A. Marandi, P. L. McMahon, T. Umeki, K. Enbutsu, O. Tadanaga, H. Takenouchi, K. Aihara, K.-I. Kawarabayashi, K. Inoue, S. Utsunomiya, and H. Takesue, “A coherent Ising machine for 2000-node optimization problems,” Science 354, 603–606 (2016).
[Crossref]

Q. Wang, E. T. F. Rogers, B. Gholipour, C.-M. Wang, G. Yuan, J. Teng, and N. I. Zheludev, “Optically reconfigurable metasurfaces and photonic devices based on phase change materials,” Nat. Photonics 10, 60–65 (2016).
[Crossref]

2015 (5)

D. A. B. Miller, “Perfect optics with imperfect components,” Optica 2, 747 (2015).
[Crossref]

Y. Okawachi, M. Yu, K. Luke, D. O. Carvalho, S. Ramelow, A. Farsi, M. Lipson, and A. L. Gaeta, “Dual-pumped degenerate Kerr oscillator in a silicon nitride microresonator,” Opt. Lett. 40, 5267–5270 (2015).
[Crossref]

S. Utsunomiya, N. Namekata, K. Takata, D. Akamatsu, S. Inoue, and Y. Yamamoto, “Binary phase oscillation of two mutually coupled semiconductor lasers,” Opt. Express 23, 6029–6040 (2015).
[Crossref]

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref]

N. Ghofraniha, I. Viola, F. Di Maria, G. Barbarella, G. Gigli, L. Leuzzi, and C. Conti, “Experimental evidence of replica symmetry breaking in random lasers,” Nat. Commun. 6, 6058 (2015).
[Crossref]

2014 (6)

A. Lucas, “Ising formulations of many NP problems,” Front. Phys. 2, 5 (2014).
[Crossref]

K. Wu, J. García de Abajo, C. Soci, P. Ping Shum, and N. I. Zheludev, “An optical fiber network oracle for NP-complete problems,” Light. Sci. Appl. 3, 147 (2014).
[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, 937–942 (2014).
[Crossref]

N. C. Harris, Y. Ma, J. Mower, T. Baehr-Jones, D. Englund, M. Hochberg, and C. Galland, “Efficient, compact and low loss thermo-optic phase shifter in silicon,” Opt. Express 22, 10487 (2014).
[Crossref]

N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: a simple way to prevent neural networks from overfitting,” J. Mach. Learn. Res. 15, 1929–1958 (2014).

Z. Cheng, H. K. Tsang, X. Wang, K. Xu, and J.-B. Xu, “In-plane optical absorption and free carrier absorption in graphene-on-silicon waveguides,” IEEE J. Sel. Top. Quantum Electron. 20, 43–48 (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, 063853 (2013).
[Crossref]

2011 (2)

Q. Bao, H. Zhang, Z. Ni, Y. Wang, L. Polavarapu, Z. Shen, Q.-H. Xu, D. Tang, and K. P. Loh, “Monolayer graphene as a saturable absorber in a mode-locked laser,” Nano Res. 4, 297–307 (2011).
[Crossref]

B. J. Eggleton, B. Luther-Davies, and K. Richardson, “Chalcogenide photonics,” Nat. Photonics 5, 141 (2011).
[Crossref]

2009 (1)

A. R. Honerkamp-Smith, S. L. Veatch, and S. L. Keller, “An introduction to critical points for biophysicists; observations of compositional heterogeneity in lipid membranes,” Biochimica et Biophys. Acta (BBA) – Biomembr. 1788, 53–63 (2009).
[Crossref]

2005 (2)

D. J. Earl and M. W. Deem, “Parallel tempering: theory, applications, and new perspectives,” Phys. Chem. Chem. Phys. 7, 3910 (2005).
[Crossref]

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
[Crossref]

1996 (1)

W. M. Macready, A. G. Siapas, and S. A. Kauffman, “Criticality and parallelism in combinatorial optimization,” Science 271, 56–59 (1996).
[Crossref]

1994 (1)

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

1986 (1)

A. Vergis, K. Steiglitz, and B. Dickinson, “The complexity of analog computation,” Math. Comput. Simul. 28, 91–113 (1986).
[Crossref]

1984 (1)

P. Peretto, “Collective properties of neural networks: a statistical physics approach,” Biol. Cybern. 50, 51–62 (1984).
[Crossref]

1983 (1)

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

1982 (1)

J. J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proc. Natl. Acad. Sci. USA 79, 2554–2558 (1982).
[Crossref]

1974 (1)

W. A. Little, “The existence of persistent states in the brain,” Math. Biosci. 19, 101–120 (1974).
[Crossref]

Aarts, E.

E. Aarts and J. Korst, Simulated Annealing and Boltzmann Machines: A Stochastic Approach to Combinatorial Optimization and Neural Computing (Wiley, 1989).

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]

T. Inagaki, Y. Haribara, K. Igarashi, T. Sonobe, S. Tamate, T. Honjo, A. Marandi, P. L. McMahon, T. Umeki, K. Enbutsu, O. Tadanaga, H. Takenouchi, K. Aihara, K.-I. Kawarabayashi, K. Inoue, S. Utsunomiya, and H. Takesue, “A coherent Ising machine for 2000-node optimization problems,” Science 354, 603–606 (2016).
[Crossref]

P. L. McMahon, A. Marandi, Y. Haribara, R. Hamerly, C. Langrock, S. Tamate, T. Inagaki, H. Takesue, S. Utsunomiya, K. Aihara, R. L. Byer, M. M. Fejer, H. Mabuchi, and Y. Yamamoto, “A fully programmable 100-spin coherent Ising machine with all-to-all connections,” Science 354, 614–617 (2016).
[Crossref]

Akamatsu, D.

Akbulut, M.

M. Babaeian, D. T. Nguyen, V. Demir, M. Akbulut, P.-A. Blanche, Y. Kaneda, S. Guha, M. A. Neifeld, and N. Peyghambarian, “A single shot coherent Ising machine based on a network of injection-locked multicore fiber lasers,” Nat. Commun. 10, 3516 (2019).
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Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1. Photonic recurrent Ising sampler. (a) An arbitrary Ising graph described by a coupling matrix ${\bf K}$ is processed to produce the fixed transformation for the recurrent sampler, shown in (b). (b) Single algorithm iteration of the photonic recurrent Ising sampler. An in-phase optical signal encoding a spin state is fed to an optical matrix multiplication unit encoding ${\bf C} = 2\sqrt {\bf K}$, where ${\bf K}$ is the coupling matrix of the desired Ising graph. The output signal is noisy, with a distribution that is Gaussian with standard deviation $\phi$, and goes through an analog nonlinear unit before being fed back to the chip input. Considering as an example a nine-spin 2D antiferromagnetic graph, with coupling and ground state shown in (c), the simulated energy evolution as a function of time is shown in (d). (e) Simulated energy distribution of the optical output, which converges in probability to the Gibbs distribution of the associated Ising problem [Eq. (3)], for which the ground state (c) is exponentially more likely than higher-energy states at low temperatures.
Fig. 2.
Fig. 2. Experimental realization of integrated nanophotonic coherent Ising sampler. (a) A 1550 nm laser diode is coupled to a single input port of a silicon-on-insulator programmable nanophotonic processor (PNP). Output mode intensities are measured with an InGaAs photodiode array, then processed in the electronic domain to determine PNP phase settings for the next iteration. (b) The PNP comprises 88 Mach–Zehnder interferometers with 176 individually controlled thermal phase shifters [15,16] and encodes a circuit consisting of input routing (red) and a U(5) unitary matrix (green). (c) Each algorithm step, shown conceptually in Fig. 1(b), requires four passes through the PNP chip. Each use of the chip performs a unitary matrix product of a state-preparation rotation matrix ${\bf R}$, the desired unitary (${\bf U}$ or ${{\bf U}^\dagger}$), and one of two homodyne detection matrices (${{\bf h}_{\bf 1}}$ or ${{\bf h}_{\bf 2}}$). Phase-intensity reconstruction, a diagonal matrix multiplication, Gaussian noise addition, and a nonlinear threshold unit are applied in the electronic domain.
Fig. 3.
Fig. 3. Evaluating physical observables and finding ground states with the integrated nanophotonic coherent Ising sampler. (a) Experimental evaluation of the magnetization of a two-dimensional ferromagnet (red dots). The fit with theory allows us to evaluate intrinsic noise on the PNP ${\phi _{\text{int}}} = 0.6457$. (b)–(f) Left: ground state probability as a function of extrinsic noise ${\phi _{\text{ext}}}$ (orange), compared to simulated PNP with homodyne (blue) and ideal (green) phase-intensity reconstruction. Right: energy histograms for value of ${\phi _{\text{ext}}}$ showing best performance and comparison with random search. Inset: schematic representation of the Ising model being modeled by the PNP. Green (resp. orange) lines between nodes represent +1 (resp. −1) couplings.
Fig. 4.
Fig. 4. Phase space probing by extrinsic noise injection on ferromagnet problem. The area of each dot is proportional to the measured mean probability of observing the system in state $(y,x)$ at a given extrinsic noise level ${\phi _{\text{ext}}}$, where $y = ({\sigma _1},{\sigma _2})$ is the state of spins 1 and 2, and $x = ({\sigma _3},{\sigma _4})$ is the state of spins 3 and 4.

Equations (5)

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H ( K ) ( σ ) = 1 2 1 i , j N σ i K ij σ j ,
S ( t + 1 ) = f θ th ( N ( C S ( t ) , ϕ ) ) ,
lim t P ( σ ( t ) ) exp [ β H L ( σ ( t ) ) ] ,
H L ( σ ) = 1 β i log cosh ( β j J ij σ j )
β H ( J T J ) ( σ ) ,