  Ising machines have attracted attention as efficient solvers for
combinatorial optimization problems, which are formulated as ground-state
(lowest-energy) search problems of the Ising model. Due to the limited
bit-width of coefficients on Ising machines, the Ising model must be
transformed into a bit-width reduced (BWR) Ising model. According to previous
research, the bit-width reduction method, which adds auxiliary spins, ensures
that the ground state of the BWR Ising model is theoretically the same as the
Ising model before bit-width reduction (original Ising model). However, while
the dynamical process is closely related to solution accuracy, how the BWR
Ising model progresses towards the ground state remains to be elucidated.
Therefore, we compared the dynamical processes of these models using simulated
annealing (SA). Our findings reveal significant differences in the dynamical
process across models. Analysis from the viewpoint of statistical mechanics
found that the BWR Ising model has two characteristic properties: an effective
temperature and a slow relaxation. These properties alter the temperature
schedule and spin flip probability in the BWR Ising model, leading to
differences in the dynamical process. Therefore, to obtain the same dynamical
process as the original Ising model, we proposed SA parameters for the BWR
Ising model. We demonstrated the proposed SA parameters using a square lattice
Ising model, in which all coefficients were set uniformly to the same positive
values or randomly. Our experimental evaluations demonstrated that the
dynamical process of the BWR and original Ising model became closer.
