(Figure (a)), Sc@(1,6), and Sc@(1,8), respectively. For each case, the number of possible proton configurations as the initial states for the relaxation calculations was defined as 24Cx × 8 x = 17,664, without considering symmetry. For all configurations, DFT relaxation calculations were performed in advance using a Fugaku supercomputer, and the total energies of 𝐸 ! , 𝐸 " , and 𝐸 # were obtained with different accuracies. Thus, for 𝑥 = 2, the optimal solutions for the stable proton configurations were known for the various relaxation steps, and these solutions were used to estimate the efficiency of the FMQA algorithm in finding stable proton configurations. In addition, during each optimization cycle, the total energy of the selected proton configuration can be accessed from the previously calculated total energies. Consequently, it is not necessary to perform an additional DFT calculation to estimate the efficiency.