*Article* **Investigation of Eigenmode-Based Coupled Oscillator Solver Applied to Ising Spin Problems**

**Shintaro Murakami, Okuto Ikeda, Yusuke Hirukawa and Toshiharu Saiki \***

Graduate School of Science and Technology, Keio University, Yokohama 223-8522, Japan; murashin1218@keio.jp (S.M.); okuto.ikeda@saiki.elec.keio.ac.jp (O.I.); yusuke.hirukawa@saiki.elec.keio.ac.jp (Y.H.)

**\***Correspondence: saiki@elec.keio.ac.jp

**Abstract:** We evaluate a coupled oscillator solver by applying it to square lattice (N × N) Ising spin problems for N values up to 50. The Ising problems are converted to a classical coupled oscillator model that includes both positive (ferromagnetic-like) and negative (antiferromagnetic-like) coupling between neighboring oscillators (i.e., they are reduced to eigenmode problems). A map of the oscillation amplitudes of lower-frequency eigenmodes enables us to visualize oscillator clusters with a low frustration density (unfrustrated clusters). We found that frustration tends to localize at the boundary between unfrustrated clusters due to the symmetric and asymmetric nature of the eigenmodes. This allows us to reduce frustration simply by flipping the sign of the amplitude of oscillators around which frustrated couplings are highly localized. For problems with N = 20 to 50, the best solutions with an accuracy of 96% (with respect to the exact ground state) can be obtained by simply checking the lowest ~N/2 candidate eigenmodes.

**Keywords:** combinatorial optimization; Ising spin glass; coupled oscillator; eigenmode; clustering
