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arXiv:2103.05629 · Physical Review Research

Efficient sampling of ground and low-energy Ising spin configurations with a coherent Ising machine

Edwin Ng, Tatsuhiro Onodera, Satoshi Kako, Peter L. McMahon, Hideo Mabuchi, Y. Yamamoto

Published Jan 3, 2022• 32 citations
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Abstract

We show that the nonlinear stochastic dynamics of a measurement-feedback-based coherent Ising machine (MFB-CIM) in the presence of quantum noise can be exploited to sample degenerate ground and low-energy spin configurations of the Ising model. We formulate a general discrete-time Gaussian-state model of the MFB-CIM which faithfully captures the nonlinear dynamics present at and above system threshold. This model overcomes the limitations of both mean-field models, which neglect quantum noise, and continuous-time models, which assume long photon lifetimes. Numerical simulations of our model show that when the MFB-CIM is operated in a quantum-noise-dominated regime with short photon lifetimes (i.e., low cavity finesse), homodyne monitoring of the system can efficiently produce samples of low-energy Ising spin configurations, requiring many fewer roundtrips to sample than suggested by established high-finesse, continuous-time models. We find that sampling performance is robust to, or even improved by, turning off or altogether reversing the sign of the parametric drive, but performance is critically reduced in the absence of optical nonlinearity. For the class of MAX-CUT problems with binary-signed edge weights, the number of roundtrips sufficient to fully sample all spin configurations up to the first-excited Ising energy, including all degeneracies, scales as $1.08^N$. At a problem size of $N = 100$ with a few dozen (median of 20) such desired configurations per instance, we have found median sufficient sampling times of $6\times10^6$ roundtrips; in an experimental implementation of an MFB-CIM with a 10 GHz repetition rate, this corresponds to a wall-clock sampling time of 60 ms.

DOI
10.1103/physrevresearch.4.013009

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We show that the nonlinear stochastic dynamics of a measurement-feedback-based coherent Ising machine (MFB-CIM) in the presence of quantum noise can be exploited to sample degenerate ground and low-energy spin configurations of the Ising model. We formulate a general discrete-time Gaussian-state model of the MFB-CIM which faithfully captures the nonlinear dynamics present at and above system threshold. This model overcomes the limitations of both mean-field models, which neglect quantum noise, and continuous-time models, which assume long photon lifetimes. Numerical simulations of our model show that when the MFB-CIM is operated in a quantum-noise-dominated regime with short photon lifetimes (i.e., low cavity finesse), homodyne monitoring of the system can efficiently produce samples of low-energy Ising spin configurations, requiring many fewer roundtrips to sample than suggested by established high-finesse, continuous-time models. We find that sampling performance is robust to, or even improved by, turning off or altogether reversing the sign of the parametric drive, but performance is critically reduced in the absence of optical nonlinearity. For the class of MAX-CUT problems with binary-signed edge weights, the number of roundtrips sufficient to fully sample all spin configurations up to the first-excited Ising energy, including all degeneracies, scales as 1.08N1.08^N1.08N. At a problem size of N=100N = 100N=100 with a few dozen (median of 20) such desired configurations per instance, we have found median sufficient sampling times of 6×1066\times10^66×106 roundtrips; in an experimental implementation of an MFB-CIM with a 10 GHz repetition rate, this corresponds to a wall-clock sampling time of 60 ms.
Abstract A Coherent Ising machine (CIM) is an oscillator-network-based analog computing system to circumvent the bottleneck in von Neumann digital computing architectures. The CIM consists of a network of degenerate optical parametric oscillators (DOPOs) and is designed to find a ground state or perform Boltzmann sampling for all degenerate ground states and low-energy excited states in combinatorial optimization problems. A nonlinear measurement feedback scheme, called chaotic amplitude control (CAC), has recently been proposed to correct pulse amplitude inhomogeneity and thereby faithfully map the Ising Hamiltonian to the loss landscape of the DOPO network. However, the quantum limit of the CIM-CAC performance is not fully explored yet. This work clarifies how the quantum noise squeezing and the measurement-induced state shift in repeated indirect quantum measurements improve the system performance. From the numerical simulation on the Ising model with the Zeeman terms, obtained from combinatorial clustering problems formulated as constrained optimization problems, it is revealed that the CIM-CAC operating in a single photon per pulse regime dramatically outperforms the standard CIM-CAC with a large photon number per pulse. This is because the standard CIM-CAC is often trapped in a periodic trajectory and cannot escape from there. On the other hand, the significant improvement is brought by the noise-induced amplitude jump in the single photon CIM-CAC.
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.
Coherent Ising machines (CIMs), leveraging the bistable physical properties of coherent light to emulate Ising spins, exhibit great potential as hardware accelerators for tackling complex combinatorial optimization problems. Recent advances have demonstrated that the performance of CIMs can be enhanced either by incorporating large random noise or higher-order nonlinearities, yet their combined effects on CIM performance remain mainly unexplored. In this work, we develop a numerical CIM model that utilizes a tunable fifth-order polynomial nonlinear dynamic function under large noise levels, which has the potential to be implemented in all-optical platforms. We propose a normal form of a CIM model that allows for both supercritical and subcritical pitchfork bifurcation operational regimes, with fifth-order nonlinearity and tunable hyperparameters to control the Ising spin dynamics. In the benchmark studies, we simulate various sets of MaxCut problems using our fifth-order polynomial CIM model. The results show a significant performance improvement, achieving an average of 59.5% improvement in median time-to-solution (TTS) and an average of 6 times improvement in median success rate (SR) for dense Maxcut problems in the BiqMac library, compared to the commonly used third-order polynomial CIM model with low noise. The fifth-order polynomial CIM model in the large-noise regime also shows better performance trends as the problem size scales up. These findings reveal the enhancements on the computational performance of Ising machines in the large-nose regime from fifth-order nonlinearity, showing important implications for both simulation and hardware perspectives.
Combinatorial optimization problems are computationally hard in general, but they are ubiquitous in our modern life. A coherent Ising machine (CIM) based on a multiple-pulse degenerate optical parametric oscillator (DOPO) is an alternative approach to solve these problems by a specialized physical computing system. To evaluate its potential performance, computational experiments are performed on maximum cut (MAX-CUT) problems against traditional algorithms such as semidefinite programming relaxation of Goemans-Williamson and simulated annealing by Kirkpatrick, et al. The numerical results empirically suggest that the almost constant computation time is required to obtain the reasonably accurate solutions of MAX-CUT problems on a CIM with the number of vertices up to 2×1042 \times 10^42×104 and the number of edges up to 10810^8108.
Recently, the coherent Ising machine (CIM) as a degenerate optical parametric oscillator (DOPO) network has been researched to solve Ising combinatorial optimization problems. We formulate a theoretical model for the CIM with discrete-time measurement feedback processes, and perform numerical simulations for the simplest network, composed of two degenerate optical parametric oscillator pulses with the antiferromagnetic mutual coupling. We evaluate the extent to which quantum coherence exists during the optimization process.
Computers based on physical systems are increasingly anticipated to overcome the impending limitations on digital computer performance. One such computer is a coherent Ising machine (CIM) for solving combinatorial optimization problems. Here, we report a CIM with 100,512 degenerate optical parametric oscillator pulses working as the Ising spins. We show that the CIM delivers fine solutions to maximum cut problems of 100,000-node graphs drastically faster than standard simulated annealing. Moreover, the CIM, when operated near the phase transition point, provides some extremely good solutions and a very broad distribution. This characteristic will be useful for applications that require fast random sampling such as machine learning.