In each optimization cycle, the FM model is trained to predict the total energy after the relaxation calculation from the initial proton configuration. By solving the FM model using an Ising machine, an initial proton configuration that leads to stable proton configurations after the relaxation calculation is suggested. For the suggested configuration, the total energy is calculated by relaxation calculations using VASP, and the amount of training data for the FM model is increased. Subsequently, in each cycle, a relaxation calculation must be performed using VASP to acquire the total energy with the required calculation accuracy. The optimization task is therefore to determine the stable proton configurations using as few cycles as possible. For the purpose of this study, scandium-doped barium zirconate, which is a representative electrolyte candidate for PCFCs, is employed as a model system to perform the stable proton configuration searches by combining FMQA and VASP. In addition, a graphics processing unit-based Ising machine is used to solve the FM model.