Abstract

The coherent Ising machine (CIM) implemented by degenerate optical parametric oscillator (DOPO) networks is a novel optical platform to accelerate computation of hard combinatorial optimization problems. Nevertheless, with the increase of the problem size, the probability of the machine being trapped by local minima increases exponentially. According to the quantum adiabatic theorem, a physical system will remain in its instantaneous ground state if the time-dependent Hamiltonian varies slowly enough. Here, we propose a method to help the machine partially avoid getting stuck in local minima by introducing quantum adiabatic evolution to the ground-state-search process of the CIM, which we call A-CIM. Numerical simulation results demonstrate that A-CIM can obtain improved solution accuracy in solving MAXCUT problems of vertices ranging from 10 to 2000 than CIM. The proposed machine that is based on quantum adiabatic theorem is expected to solve optimization problems more correctly.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Finding exact solutions or sampling good approximate solutions of combinatorial optimization problems is of paramount importance for various application fields, including finance [1], machine learning [2], circuit design [3] and drug discovery [4]. Numerous such problems belong to the nondeterministic polynomial time (NP)-hard or NP complete complexity classes, which are considered difficult to solve with modern digital computers because the computation time grows exponentially with the problem size [5]. It is well-known that many combinatorial problems can be mapped onto Ising models. Then finding the optimal solution of a combinatorial optimization problem is equivalent to finding the lowest energy state of the Ising Hamiltonian [6]. The energy of an N-spin Ising model without external magnetic fields is given by [7]

$$H ={-} \sum\limits_{i < j}^N {{J_{ij}}{\sigma _i}{\sigma _j},}$$
where σi denotes the i-th spin that takes a value of +1 or -1, Jij represents the coupling constant between the i-th and the j-th spins. Recently, extensive research focuses on optical Ising machines to take advantage of the high speed and parallelization of light. Optical solvers have been realized with opto-electronic oscillators [8], injection-locked lasers [911], spatial optical modulation [1214] and integrated nanophotonic circuits [1517].

In addition to the above schemes, Stanford University and the NTT group proposed a new physical system using time-multiplexed degenerate optical parametric oscillators (DOPOs) and measurement-feedback approach [18,19]. The schematic of CIM based on DOPOs is shown in Fig. 1. Above the oscillation threshold, the phase of each DOPO pulse takes only 0 or π relative to the pump phase. Therefore, each of DOPO pulses can represent an Ising spin and the sign of in-phase amplitude ci of the signal DOPOs denotes the spin up or down state [2022]. The feedback pulse calculated by a field-programmable gate array (FPGA) for the i-th pulse is fi. A coupling pulse conveying the feedback signal fi is injected into the i-th DOPO to achieve spin-spin interaction. The overall photon decay rate of the DOPO networks is proportional to the energy landscape of the Ising Hamiltonian. According to the minimum-gain principle, the network is likely to oscillate in one of the ground states [23].

 figure: Fig. 1.

Fig. 1. Experimental set up of CIM with measurement-feedback control . SHG: second harmonic generation, PPLN: periodically poled lithium niobate, PSA: phase-sensitive amplifier, BHD: balanced homodyne detection, FPGA: field-programmable gate array, IM: intensity modulator, PM: phase modulator. r is a constant and denotes the coupling strength.

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Adiabatic quantum computing (AQC) is a method to solve combinatorial optimization problems and help the system escape the local minima using quantum adiabatic theorem [24]. In the adiabatic evolution, the system is evolved from the ground state of a simple Hamiltonian towards the desired complicated Hamiltonian encoding an optimization problem. The performance of AQC depends on the energy gap between the ground state and higher energy states of the Hamiltonian. If we change the Hamiltonian sufficiently slowly to maintain the minimum energy gap greater than zero, the final state of the system is equal to the optimal solution of the problem of interest [25]. Quantum annealers based on AQC have been implemented by superconducting circuit [26,27], Kerr-nonlinear resonators [28], ion trap system [29] and nuclear magnetic resonance [30]. Realizing adiabatic evolution on optical platforms which possess high-speed and parallelization is also a promising computing architecture. The CIM shares many similarities with the quantum annealer. Both systems are designed to solve the Ising problem and implemented by mapping the ground-state-search process to the underlying physical dynamics [31]. As a form of annealing machines, the CIM is intrinsically able to take advantage of adiabatic evolution to improve its computation performance.

In the present article, we demonstrate quantum adiabatic evolution on the CIM. The machine starts with the ground state of an Ising Hamiltonian in homogeneous coupling coefficient (Jij = 1). Then we linearly vary the spin couplings with sufficiently long intervals to adiabatically change instantaneous Hamiltonian. The spin configuration of the system in the final is the optimal solution of the target problem. By varying Ising Hamiltonian adiabatically in the ground state search process, the convergence to the Ising ground state can be enhanced. In our computational experiments, to understand the performance of the A-CIM, we first solve MAX-CUT problems of N = 10 ∼ 100 vertices. Furthermore, the performance of the A-CIM is also tested on G-set instances and fully connected complete graphs of 800 ∼ 2000 nodes. Our numerical results demonstrate that applying quantum adiabatic theory on CIM can improve solution quality without additional devices.

2. Methods

To implement adiabatic evolution on networks of DOPOs, we discretize the continuous time-dependent Ising Hamiltonian into M equal intervals. A fully connected complete graph with N vertices and N(N - 1)/2 edges of weight 1 is encoded into the beginning Hamiltonian H[0]= Hb. The optimal cut value of this graph is known to be N2/4. The final Hamiltonian H[M]= Hp encodes the target MAX-CUT problem which is computationally difficult to find its ground state straightforward. The schematic diagram of the adiabatic evolution process is shown in Fig. 2. The time-dependent Ising Hamiltonian H[t] is a linear interpolation from H[0] to H[M] [14,30]:

$$H(t) ={-} \sum\limits_{i < j}^N {{J_{ij}}(t){\sigma _i}{\sigma _j} = (t/kM){H_p} + (1 - (t/kM)){H_b},}$$
where annealing time t∈[0,M, …, (k – 1)M, kM] and k is the roundtrips in each step.

 figure: Fig. 2.

Fig. 2. (a) Simplified illustration of the adiabatic computing principle. The energy landscapes with several mountains and valleys correspond to the Hamiltonian that starts from Hb and evolves towards the problem Hp. When the evolution is performed slowly enough, the system remains in the ground state throughout the computing process and obtains the optimal solution in the final. (b)The evolution of spin configurations in ground state search process versus normalized annealing time for a fully-connected complete graph with 10 vertices. Each small rectangle represents an Ising spin (orange: spin down, blue: spin up). Insets show the MAX-CUT graphs of Hb and Hp (blue/magenta line represents the edge weight that takes +1 or -1).

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Theoretically, given long enough annealing time, the system will succeed in reaching the ground state of the target problem [24]. However, we observe freeze-out effects which make the system stuck in excited states in many cases [32]. As shown in Fig. 3(a), once the system reaches the freeze point, the phase of each DOPO pulse will remain unchanged for the rest of the computation and the spin configuration will remain in an excited state. Freeze-out effects hinder the system from evolving to a lower energy state and have a negative impact on computing performance. We propose an approach by randomly flipping a few spins at the end of each adiabatic step to decrease the Ising energy, so the machine can continue evolving towards the Ising ground state, as shown in Fig. 3(b). It is worth mentioning that random flipping spins can improve the computation performance of the CIM to a certain extent, for the case where the system state is stuck in a high excited energy state caused by the freeze-out effects.

 figure: Fig. 3.

Fig. 3. The evolution of spin configurations as a function of normalized annealing time for an all-to-all connected complete graph with 100 vertices (orange: spin down, blue: spin up). (a) Time evolution to the ground state abruptly stops after the freeze-out point (red dashed line) and the obtained cut value is 335. (b)By randomly flipping several spins at the end of each adiabatic step, the machine can obtain the optimal solution (cut value = 338). For the rest of the annealing time(M>17), the spin configuration remains unchanged due to the phase space of the instantaneous Hamiltonians is very similar with that of Hp

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3. Results

3.1 For small size graphs

From the advantages and properties of quantum adiabatic algorithm, we expect that the A-CIM will enhance the convergence to the Ising ground state. To verify our conjecture, we first perform computational experiments on fully-connected complete graphs with vertices ranging from 10 to 100 and the weights of edges belong to {±1}. We conducted the numerical simulation of the c-number Langevin expressed by Eq. (3) to evaluate the performance of the CIM and the A-CIM [33].

$$d{c_i} = \left[ {( - 1 + p - c_i^2 - s_i^2){c_i} + r\sum\limits_j {{J_{ij}}{c_j}} } \right]dt + \frac{1}{{{A_s}}}\sqrt {c_i^2 + s_i^2 + \frac{1}{2}} d{W_i},$$
here, t is a unitless time normalized to twice of the signal photon lifetime in the fiber cavity; p is the normalized pump rate; ci and si are the normalized in-phase and quadrature-phase amplitude components of the $i$th DOPO; r denotes the coupling strength; As is the steady-state amplitude at p = 2; Wi is the Gaussian noise term. Figure 4 shows the evolution of the in-phase components ci as a function of the computation time for the CIM and the A-CIM in one trial.

 figure: Fig. 4.

Fig. 4. Normalized DOPO signal amplitudes vary with normalized time when solving a N = 100 MAX-CUT problem on (a) the CIM and (b) the A-CIM. Each line represents an OPO.

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Considering that different random graphs of the same size may have different computational difficulties, we generate 20 random instances for each size by a machine independent graph generator [34]. Each instance is solved 500 times to estimate the success probability. Figure 5(a) shows the success probabilities of A-CIM and CIM vary as the graph size. The normalized annealing time M is chosen for 30. The pump rate p is linearly increased from p = 0.5 at t = 0 to p = 1.0 at t = 100 for CIM [35]; p is linearly increased from 0.7 to 1.0 for each adiabatic step of A-CIM. The solutions obtained by BiqMac MAX-CUT solver are used as a benchmark to evaluate the success probability [36]. As shown in Fig. 5(a), the performance of the A-CIM is superior to the CIM. The probability of obtaining ground states (100% accuracy) is increased from 4% to 23.61% compared with the CIM; for 99% accuracy, the probability is increased from 4% to 25.02%. The solution accuracy is the probability of finding a solution within x% of the best solution.

 figure: Fig. 5.

Fig. 5. (a) Observed probabilities of obtaining a solution within 100% and 99% accuracy of the optimal (maximum cut) solution as a function of graph size N, for A-CIM and CIM respectively. (b) The success probabilities versus normalized annealing time M for MAX-CUT problems with 30, 60 and 90 vertices. The dotted line represents the computation results of the CIM.

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We also investigate how the machine performance depends on the annealing time. Figure 5(b) shows the success probabilities vary as the normalized annealing time M. The total run time is T = niter × M and the number of iterations in each step is kept constant to niter = 1000, so the speed of varying the Hamiltonian is depended on the number of evolution steps M. For the relatively small M, the energy gap between the instantaneous ground state and the first excited state may not always remain greater than zero, so the system may transition to an excited state during the adiabatic evolution process. The adiabatic condition is determined by success probabilities reaching a plateau [37,38]. The probabilities of obtaining optimal solution grow as the increase of M from 1 to about 20 and can be barely improved for longer annealing time. Therefore, we should vary the time-dependent Hamiltonian during at least 20 evolution steps to ensure the effectiveness of the adiabatic evolution.

3.2 For large size graphs

To see how the performance of the A-CIM scales to larger problem size, we solved the sparsely connected G-set instances and fully connected complete graphs with the number of vertices ranging from 800 to 2000. Each problem instance is solved 100 times. The cut values obtained by GW-SDP algorithm are used as a benchmark to evaluate the computational accuracy [39]. The numerical simulation results of CIM and A-CIM for G-set graphs are summarized in Table 1. V denotes the number of vertices, E denotes the number of edges, USDP is the optimal solution obtained by GW-SDP algorithm. CCIM and 〈CCIM〉 are the best and average cut values solved by CIM in 1000 roundtrips, respectively. CA-CIM and 〈CA-CIM〉 are the best and average cut values obtained by A-CIM of 30 discretization steps, respectively. To compare computed results more effectively, we normalize the cut values C generated from CIM and A-CIM according to (C + Eneg)/(USDP + Eneg), where Eneg ≥ 0 is the number of negative edges of a graph [40,41]. The outcomes show the performance improvement of the A-CIM with respect to the CIM. The computational accuracy of the A-CIM is increased from 0.124% to 0.754% compared with the CIM.

Tables Icon

Table 1. Normalized cut values of A-CIM and CIM for G-set graphs.

The A-CIM has a more obvious advantage for denser graphs. The cut value histograms obtained by A-CIM and CIM in 100 runs for fully-connected graphs (maximally dense) of 800, 1000 and 2000 vertices are shown in Fig. 6. The results indicate that the A-CIM achieves higher cut values than those obtained by CIM.

 figure: Fig. 6.

Fig. 6. Cut value histograms for (a) 800, (b) 1000, (c) 2000-node complete graphs solved by the CIM (blue bars) and the A-CIM (orange bars), respectively.

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4. Analysis of computation time as a function of the problem size

In the CIM experimental architecture, the roundtrip time depends on the cavity length of the fiber. Assuming the pulse repetition frequency is 1 GHz, the number of pulses accommodated in the fiber should be increased in proportional to the fiber length. The computation time of a single run for the A-CIM is estimated by (the number of roundtrips) ${\times} $ (cavity roundtrip time) ${\times} $ (discretization steps). For each problem size, the evolution steps M remain constant at 30 and the number of roundtrips in each adiabatic step is 1000. For MAXCUT problem graphs of N ≤ 2000, the fiber cavity is set at 400 m, the computation time of the A-CIM is approximately equal to 60 ms and exhibits a problem-size independent character. In practice, the discretization steps of changing the Hamiltonian is generally longer than the annealing time required for the machine to obtain the best result. The computation time can be reduced by increasing the pulse repetition frequency and using shortcuts to adiabaticity [42].

5. Conclusion

We have realized adiabatic computing on networks of DOPOs for the improvement of the CIM, in which the spin coupling coefficient is changed sufficiently slowly to perform adiabatic evolution. This scheme can partly help the machine avoid getting trapped in local minimum. The performance of the A-CIM is superior compared with the CIM on small and large-scale problems especially for dense graphs. The computation time of the A-CIM is on the order of tens of milliseconds and independent on the size of the problem. Further improvement to the A-CIM can be achieved by combining it with reinforcement quantum annealing [4346]. The quantum reinforcement algorithm is able to increase the minimal energy gap and reduce the adiabatic computing time. Another interesting potential research direction is how the proposed method can be extended to solve more complex Ising Hamiltonian and physical models [4749].

Funding

National Natural Science Foundation of China (61775234, 61975232).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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References

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  1. A. Soler-Dominguez, A. A. Juan, and R. Kizys, “A survey on financial applications of metaheuristics,” ACM Comput. Surv. 50(1), 1–23 (2017).
    [Crossref]
  2. M. Kommenda, J. Karder, A. Beham, B. Burlacu, G. Kronberger, S. Wagner, and M. Affenzeller, “Optimization Networks for Integrated Machine Learning,” in Computer Aided Systems Theory – EUROCAST 2017, R. Moreno-Díaz, F. Pichler, and A. Quesada-Arencibia, eds. (Springer International Publishing, 2018), pp. 392–399.
  3. S. M. Sait and M. M. Zahra, “Tabu search based circuit optimization,” Eng. Appl. Artif. Intell. 15(3-4), 357–368 (2002).
    [Crossref]
  4. 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]
  5. F. Barahona, “On the computational complexity of Ising spin glass models,” J. Phys. A: Math. Gen. 15(10), 3241–3253 (1982).
    [Crossref]
  6. A. Lucas, “Ising formulations of many NP problems,” Front. Phys. 2, 5 (2014).
    [Crossref]
  7. E. Ising, “Beitrag zur theorie des ferromagnetismus,” Z. Phys. 31(1), 253–258 (1925).
    [Crossref]
  8. F. Böhm, G. Verschaffelt, and G. Van der Sande, “A poor man’s coherent Ising machine based on opto-electronic feedback systems for solving optimization problems,” Nat. Commun. 10(1), 3538 (2019).
    [Crossref]
  9. S. Utsunomiya, K. Takata, and Y. Yamamoto, “Mapping of Ising models onto injection-locked laser systems,” Opt. Express 19(19), 18091–18108 (2011).
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2021 (2)

D. Inoue, A. Okada, T. Matsumori, K. Aihara, and H. Yoshida, “Traffic signal optimization on a square lattice with quantum annealing,” Sci. Rep. 11(1), 3303 (2021).
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N. Yoshimura, M. Tawada, S. Tanaka, J. Arai, S. Yagi, H. Uchiyama, and N. Togawa, “Mapping Induced Subgraph Isomorphism Problems to Ising Models and Its Evaluations by an Ising Machine,” IEICE Trans. Inf. Syst. E104.D(4), 481–489 (2021).
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2020 (9)

L. Luo, H. Liu, N. Huang, and Z. Wang, “Speed-up coherent Ising machine with a squeezed feedback system,” Opt. Express 28(2), 1914 (2020).
[Crossref]

K. Mills, P. Ronagh, and I. Tamblyn, “Finding the ground state of spin Hamiltonians with reinforcement learning,” Nat. Mach. Intell. 2(9), 509–517 (2020).
[Crossref]

R. Ayanzadeh, M. Halem, and T. Finin, “Reinforcement Quantum Annealing: A Hybrid Quantum Learning Automata,” Sci. Rep. 10(1), 7952 (2020).
[Crossref]

D. Pierangeli, G. Marcucci, D. Brunner, and C. Conti, “Noise-enhanced spatial-photonic Ising machine,” Nanophotonics 9(13), 4109–4116 (2020).
[Crossref]

D. Pierangeli, G. Marcucci, and C. Conti, “Adiabatic evolution on a spatial-photonic Ising machine,” Optica 7(11), 1535–1543 (2020).
[Crossref]

C. Roques-Carmes, Y. Shen, C. Zanoci, M. Prabhu, F. Atieh, L. Jing, T. Dubček, C. Mao, M. R. Johnson, V. Čeperić, J. D. Joannopoulos, D. Englund, and M. Soljačić, “Heuristic recurrent algorithms for photonic Ising machines,” Nat. Commun. 11(1), 249 (2020).
[Crossref]

M. Parto, W. Hayenga, A. Marandi, D. N. Christodoulides, and M. Khajavikhan, “Realizing spin Hamiltonians in nanoscale active photonic lattices,” Nat. Mater. 19(7), 725–731 (2020).
[Crossref]

M.-C. Chen, M. Gong, X. Xu, X. Yuan, J.-W. Wang, C. Wang, C. Ying, J. Lin, Y. Xu, Y. Wu, S. Wang, H. Deng, F. Liang, C.-Z. Peng, S. C. Benjamin, X. Zhu, C.-Y. Lu, and J.-W. Pan, “Demonstration of Adiabatic Variational Quantum Computing with a Superconducting Quantum Coprocessor,” Phys. Rev. Lett. 125(18), 180501 (2020).
[Crossref]

S. Kako, T. Leleu, Y. Inui, F. Khoyratee, S. Reifenstein, and Y. Yamamoto, “Coherent Ising Machines with Error Correction Feedback,” Adv. Quantum Technol. 3(11), 2000045 (2020).
[Crossref]

2019 (4)

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. 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(5), eaau0823 (2019).
[Crossref]

D. Pierangeli, G. Marcucci, and C. Conti, “Large-Scale Photonic Ising Machine by Spatial Light Modulation,” Phys. Rev. Lett. 122(21), 213902 (2019).
[Crossref]

F. Böhm, G. Verschaffelt, and G. Van der Sande, “A poor man’s coherent Ising machine based on opto-electronic feedback systems for solving optimization problems,” Nat. Commun. 10(1), 3538 (2019).
[Crossref]

D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martínez-Garaot, and J. G. Muga, “Shortcuts to adiabaticity: Concepts, methods, and applications,” Rev. Mod. Phys. 91(4), 045001 (2019).
[Crossref]

2018 (4)

B.-X. Wang, M.-J. Tao, Q. Ai, T. Xin, N. Lambert, D. Ruan, Y.-C. Cheng, F. Nori, F.-G. Deng, and G.-L. Long, “Efficient quantum simulation of photosynthetic light harvesting,” npj Quantum Inf. 4(1), 52 (2018).
[Crossref]

A. Ramezanpour, “Enhancing the efficiency of quantum annealing via reinforcement: A path-integral Monte Carlo simulation of the quantum reinforcement algorithm,” Phys. Rev. A 98(6), 062309 (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(1), 5020 (2018).
[Crossref]

J. Zhang, F. Li, Y. Xie, C. Wu, B. Ou, W. Wu, and P. Chen, “Realizing an adiabatic quantum search algorithm with shortcuts to adiabaticity in an ion-trap system,” Phys. Rev. A 98(5), 052323 (2018).
[Crossref]

2017 (4)

S. Puri, C. K. Andersen, A. L. Grimsmo, and A. Blais, “Quantum annealing with all-to-all connected nonlinear oscillators,” Nat. Commun. 8(1), 15785 (2017).
[Crossref]

A. Soler-Dominguez, A. A. Juan, and R. Kizys, “A survey on financial applications of metaheuristics,” ACM Comput. Surv. 50(1), 1–23 (2017).
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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(7), 441–446 (2017).
[Crossref]

A. Ramezanpour, “Optimization by a quantum reinforcement algorithm,” Phys. Rev. A 96(5), 052307 (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(6312), 614–617 (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(6312), 603–606 (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 (2016).
[Crossref]

Y. Haribara, S. Utsunomiya, and Y. Yamamoto, “A coherent ising machine for MAX-CUT problems: Performance evaluation against semidefinite programming and simulated annealing,” Lect. Notes Phys. 911, 251–262 (2016).
[Crossref]

2014 (3)

S. Boixo, T. F. Rønnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, “Evidence for quantum annealing with more than one hundred qubits,” Nat. Phys. 10(3), 218–224 (2014).
[Crossref]

K. Takata and Y. Yamamoto, “Data search by a coherent Ising machine based on an injection-locked laser network with gradual pumping or coupling,” Phys. Rev. A 89(3), 032319 (2014).
[Crossref]

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

2011 (1)

2009 (1)

Q. Ai, W. Huo, G. L. Long, and C. P. Sun, “Nonadiabatic fluctuation in the measured geometric phase,” Phys. Rev. A 80(2), 024101 (2009).
[Crossref]

2006 (1)

G. E. Santoro and E. Tosatti, “Optimization using quantum mechanics: quantum annealing through adiabatic evolution,” J. Phys. A. Math. Gen. 39(36), R393–R431 (2006).
[Crossref]

2004 (1)

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]

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(6), 067903 (2003).
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2002 (1)

S. M. Sait and M. M. Zahra, “Tabu search based circuit optimization,” Eng. Appl. Artif. Intell. 15(3-4), 357–368 (2002).
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2001 (1)

E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, “A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem,” Science 292(5516), 472–475 (2001).
[Crossref]

1998 (1)

1995 (1)

M. X. Goemans and D. P. Williamson, “Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming,” J. ACM 42(6), 1115–1145 (1995).
[Crossref]

1990 (2)

C.-P. Sun, “High-order adiabatic approximations related to non-Abelian Berry’s phase factors and nuclear quadrupole resonance,” Phys. Rev. D 41(4), 1318–1323 (1990).
[Crossref]

C. D. Nabors, S. T. Yang, T. Day, and R. L. Byer, “Coherence properties of a doubly resonant monolithic optical parametric oscillator,” J. Opt. Soc. Am. B 7(5), 815–820 (1990).
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1982 (1)

F. Barahona, “On the computational complexity of Ising spin glass models,” J. Phys. A: Math. Gen. 15(10), 3241–3253 (1982).
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1925 (1)

E. Ising, “Beitrag zur theorie des ferromagnetismus,” Z. Phys. 31(1), 253–258 (1925).
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Affenzeller, M.

M. Kommenda, J. Karder, A. Beham, B. Burlacu, G. Kronberger, S. Wagner, and M. Affenzeller, “Optimization Networks for Integrated Machine Learning,” in Computer Aided Systems Theory – EUROCAST 2017, R. Moreno-Díaz, F. Pichler, and A. Quesada-Arencibia, eds. (Springer International Publishing, 2018), pp. 392–399.

Agarwal, A.

Ai, Q.

B.-X. Wang, M.-J. Tao, Q. Ai, T. Xin, N. Lambert, D. Ruan, Y.-C. Cheng, F. Nori, F.-G. Deng, and G.-L. Long, “Efficient quantum simulation of photosynthetic light harvesting,” npj Quantum Inf. 4(1), 52 (2018).
[Crossref]

Q. Ai, W. Huo, G. L. Long, and C. P. Sun, “Nonadiabatic fluctuation in the measured geometric phase,” Phys. Rev. A 80(2), 024101 (2009).
[Crossref]

Aihara, K.

D. Inoue, A. Okada, T. Matsumori, K. Aihara, and H. Yoshida, “Traffic signal optimization on a square lattice with quantum annealing,” Sci. Rep. 11(1), 3303 (2021).
[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(6312), 614–617 (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(6312), 603–606 (2016).
[Crossref]

Andersen, C. K.

S. Puri, C. K. Andersen, A. L. Grimsmo, and A. Blais, “Quantum annealing with all-to-all connected nonlinear oscillators,” Nat. Commun. 8(1), 15785 (2017).
[Crossref]

Arai, J.

N. Yoshimura, M. Tawada, S. Tanaka, J. Arai, S. Yagi, H. Uchiyama, and N. Togawa, “Mapping Induced Subgraph Isomorphism Problems to Ising Models and Its Evaluations by an Ising Machine,” IEICE Trans. Inf. Syst. E104.D(4), 481–489 (2021).
[Crossref]

Atieh, F.

C. Roques-Carmes, Y. Shen, C. Zanoci, M. Prabhu, F. Atieh, L. Jing, T. Dubček, C. Mao, M. R. Johnson, V. Čeperić, J. D. Joannopoulos, D. Englund, and M. Soljačić, “Heuristic recurrent algorithms for photonic Ising machines,” Nat. Commun. 11(1), 249 (2020).
[Crossref]

Ayanzadeh, R.

R. Ayanzadeh, M. Halem, and T. Finin, “Reinforcement Quantum Annealing: A Hybrid Quantum Learning Automata,” Sci. Rep. 10(1), 7952 (2020).
[Crossref]

Baehr-Jones, T.

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(7), 441–446 (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]

Bartolini, G. D.

Beham, A.

M. Kommenda, J. Karder, A. Beham, B. Burlacu, G. Kronberger, S. Wagner, and M. Affenzeller, “Optimization Networks for Integrated Machine Learning,” in Computer Aided Systems Theory – EUROCAST 2017, R. Moreno-Díaz, F. Pichler, and A. Quesada-Arencibia, eds. (Springer International Publishing, 2018), pp. 392–399.

Benjamin, S. C.

M.-C. Chen, M. Gong, X. Xu, X. Yuan, J.-W. Wang, C. Wang, C. Ying, J. Lin, Y. Xu, Y. Wu, S. Wang, H. Deng, F. Liang, C.-Z. Peng, S. C. Benjamin, X. Zhu, C.-Y. Lu, and J.-W. Pan, “Demonstration of Adiabatic Variational Quantum Computing with a Superconducting Quantum Coprocessor,” Phys. Rev. Lett. 125(18), 180501 (2020).
[Crossref]

Blais, A.

S. Puri, C. K. Andersen, A. L. Grimsmo, and A. Blais, “Quantum annealing with all-to-all connected nonlinear oscillators,” Nat. Commun. 8(1), 15785 (2017).
[Crossref]

Böhm, F.

F. Böhm, G. Verschaffelt, and G. Van der Sande, “A poor man’s coherent Ising machine based on opto-electronic feedback systems for solving optimization problems,” Nat. Commun. 10(1), 3538 (2019).
[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(1), 5020 (2018).
[Crossref]

Boixo, S.

S. Boixo, T. F. Rønnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, “Evidence for quantum annealing with more than one hundred qubits,” Nat. Phys. 10(3), 218–224 (2014).
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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (6)

Fig. 1.
Fig. 1. Experimental set up of CIM with measurement-feedback control . SHG: second harmonic generation, PPLN: periodically poled lithium niobate, PSA: phase-sensitive amplifier, BHD: balanced homodyne detection, FPGA: field-programmable gate array, IM: intensity modulator, PM: phase modulator. r is a constant and denotes the coupling strength.
Fig. 2.
Fig. 2. (a) Simplified illustration of the adiabatic computing principle. The energy landscapes with several mountains and valleys correspond to the Hamiltonian that starts from Hb and evolves towards the problem Hp. When the evolution is performed slowly enough, the system remains in the ground state throughout the computing process and obtains the optimal solution in the final. (b)The evolution of spin configurations in ground state search process versus normalized annealing time for a fully-connected complete graph with 10 vertices. Each small rectangle represents an Ising spin (orange: spin down, blue: spin up). Insets show the MAX-CUT graphs of Hb and Hp (blue/magenta line represents the edge weight that takes +1 or -1).
Fig. 3.
Fig. 3. The evolution of spin configurations as a function of normalized annealing time for an all-to-all connected complete graph with 100 vertices (orange: spin down, blue: spin up). (a) Time evolution to the ground state abruptly stops after the freeze-out point (red dashed line) and the obtained cut value is 335. (b)By randomly flipping several spins at the end of each adiabatic step, the machine can obtain the optimal solution (cut value = 338). For the rest of the annealing time(M>17), the spin configuration remains unchanged due to the phase space of the instantaneous Hamiltonians is very similar with that of Hp
Fig. 4.
Fig. 4. Normalized DOPO signal amplitudes vary with normalized time when solving a N = 100 MAX-CUT problem on (a) the CIM and (b) the A-CIM. Each line represents an OPO.
Fig. 5.
Fig. 5. (a) Observed probabilities of obtaining a solution within 100% and 99% accuracy of the optimal (maximum cut) solution as a function of graph size N, for A-CIM and CIM respectively. (b) The success probabilities versus normalized annealing time M for MAX-CUT problems with 30, 60 and 90 vertices. The dotted line represents the computation results of the CIM.
Fig. 6.
Fig. 6. Cut value histograms for (a) 800, (b) 1000, (c) 2000-node complete graphs solved by the CIM (blue bars) and the A-CIM (orange bars), respectively.

Tables (1)

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Table 1. Normalized cut values of A-CIM and CIM for G-set graphs.

Equations (3)

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H = i < j N J i j σ i σ j ,
H ( t ) = i < j N J i j ( t ) σ i σ j = ( t / k M ) H p + ( 1 ( t / k M ) ) H b ,
d c i = [ ( 1 + p c i 2 s i 2 ) c i + r j J i j c j ] d t + 1 A s c i 2 + s i 2 + 1 2 d W i ,

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