When the number of Sc ions (i.e., 𝑥) is equal to seven or eight, only one configuration exists wherein the scandium ions can be substituted for symmetry. However, the number of initial proton configurations for the relaxation calculation drastically increases under these conditions, reaching 10 11 and 10 12 for 𝑥 = 7 and 8, respectively, without symmetry. In these cases, owing to the large number of candidates, conventional BBO methods such as Bayesian optimization cannot be performed. Thus, stable proton configuration searches were performed by FMQA optimization using 𝐸 " . More specifically, three independent FMQA optimizations were carried out, and further relaxation calculations were performed for the top 15 configurations. The results of the optimization cycle using 𝐸 " are shown in Figures ) and S2(d), and the most stable configurations are summarized in Figure , wherein the numbers of scandium ions (𝑥) are 2, 5,