To verify the BBO performance using the FMQA algorithm for exploring stable proton configurations, optimization was performed for cases where the total energies of 𝐸 ! , 𝐸 " , and 𝐸 # the number of cycles required to obtain the most stable proton configuration for Sc@(1,2). Here, the most stable configurations are defined as those with total energies ranging from the lowest total energy up to 0.005 eV. For comparison, five random samplings were conducted, starting from each initial training dataset used in the FMQA optimizations; thus, 50 runs were performed. Figure summarizes the results for obtained Sc@(1,6) and Sc@(1,8), wherein the most stable proton configurations were identified within 150 BBO cycles using the FMQA algorithm. Consequently, it was found that the FMQA outperformed the random sampling approach. Notably, when random sampling is employed, the number of cycles required to find stable proton configurations for 𝐸 ! is larger than those required for 𝐸 " and 𝐸 # . This can be attributed to the fact that the number of most stable configurations for 𝐸 ! is smaller than those for 𝐸 " and 𝐸 # . Furthermore, Figures ) and 3(d)