In the case where 𝑥 = 2, the doping and proton configurations were simultaneously optimized by BBO using FMQA, and the results are presented in Figure for searching with the total energies 𝐸 ! , 𝐸 " , and 𝐸 # . The results of random sampling are also shown for comparison, and the optimization performance of FMQA was observed to outperform that of random sampling. As a model system for proton conductors, scandium-doped barium zirconate was used, wherein the number of scandium ions (𝑥) was set as 2, 5, 7, or 8 in the 2 × 2 × 2 supercell model. For 𝑥 = 2, the number of possible proton configurations was 52,992, and so it was possible to calculate the total energies for all cases using VASP, and the most stable configuration was identified. In this case, the optimization performance of FMQA algorithm always outperformed that of random sampling for finding the most stable configurations. Subsequently, using the FMQA algorithm, the proton configurations were optimized for 𝑥 = 5, 7, and 8, wherein the numbers of initial configurations were in the order of 10 9 , 10 11 , and 10 12 , respectively. Notably, such vast number of possible configurations are difficult to treat using conventional BBO methods, such as Bayesian optimization, because of the computation times required. Moreover, the optimization results identified the formation of hydrogen bond pairs in the stable proton configurations, regardless of the number of scandium ions. Notably, simultaneous optimization of the doped scandium ions and proton configurations was achieved. It is therefore expected that this optimization strategy based on a combination of Ising machines and density functional theory calculations will be able to solve various issues related to configuration optimization in solid-state materials, and promote novel scientific discoveries.