  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 0.6 ms.
