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

Abstract

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