**5. Conclusions**

We evaluated the COS for Ising spin problems based on eigenmode characterization. After a square lattice (N × N) Ising problem was converted to a coupled oscillator model that includes positive and negative coupling, the equation of motion, which was reduced to an eigenmode problem, was solved. For smaller problems (N = 10), the oscillation amplitude (eigenvector) was delocalized (i.e., it covered the entire oscillator lattice) and a fairly good solution was obtained. For larger problems, the oscillation became localized, forming oscillator clusters with low frustration density (unfrustrated clusters). From a map of unfrustrated clusters for lower eigenmodes, we found that frustration tends to localize at the boundary between clusters. Frustration localization, where three of the four couplings are frustrated, is useful for reducing frustration by flipping the sign of the amplitude. Localization and the flipping method were applied to problems with N = 40. Good solutions with an accuracy of 97.2% in average (with respect to the exact ground state) were obtained simply by checking the lowest 13 (≤N/2) candidate eigenmodes. A benchmark study demonstrated that the computation time required to reach a fairly good solution (*n*COmax) for the COS is three times shorter than that for SA.

**Author Contributions:** Conceptualization, T.S.; software, S.M., O.I. and Y.H.; formal analysis, S.M., O.I. and Y.H.; investigation, S.M., O.I. and Y.H.; writing—original draft preparation, S.M.; writing— review and editing, T.S.; funding acquisition, T.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was partially supported by JSPS KAKENHI (grant number 18H04490 and 20H04645) and partially by MEXT Quantum Leap Flagship Program Grant No. JPMXS0118067246.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Conflicts of Interest:** The authors declare no conflict of interest.
