  A non-equilibrium open-dissipative neural network, such as a coherent Ising
machine based on mutually coupled optical parametric oscillators, has been
proposed and demonstrated as a novel computing machine for hard combinatorial
optimization problems. However, there are two challenges in the previously
proposed approach: (1) The machine can be trapped by local minima which
increases exponentially with problem size and (2) the machine fails to map a
target Hamiltonian correctly on the loss landscape of a neural network due to
oscillator amplitude heterogeneity. Both of them lead to erroneous solutions
rather than correct answers. In this paper, we show that it is possible to
overcome these two problems partially but simultaneously by introducing error
detection and correction feedback mechanism. The proposed machine achieves
efficient sampling of degenerate ground states and low-energy excited states
via its inherent migration property during a solution search process.
