In quantum annealers and Ising machines, only 0/1 variables can be used; thus, it is necessary to encode atomistic configurations into a bit string composed of 0/1 variables. Oxygen ions can take the form of protonated (i.e., hydroxy) or non-protonated species. For protonated oxygen ions, a proton is placed at any one of eight possible positions in the initial state for structural relaxation (Figure ). Thus, nine bits were used to express the position of a proton at each oxygen atom site, and a one-hot constraint was considered to ensure that no more than two protons were attached to the same oxygen. In total, the proton configuration can be expressed as 24 × 9 = 216 bits, while the bit string can be expressed as 𝐪 = {𝑞 $% } $&",…,#),%&",…,* where 𝑞 $% = 0 or 1. The indices 𝑛 and 𝑖 represent the indices of the oxygen ions and their proton positions, respectively. For the proton position, 𝑖 = 1, … ,8 refers to the positions shown in Figure (b), while 𝑖 = 9 indicates the absence of a proton. The optimization cycle employed to obtain a stable proton configuration (i.e., the lowest total energy) using the FMQA algorithm therefore involved four key steps. Firstly, 𝑀 proton configurations were generated as the initial training dataset from 24Cx × 8 x configurations, which were used as the initial configurations for structural relaxation. Relaxation calculations were performed using VASP and the total energies were obtained. These data were used as the training data for the FM model, as shown in Figure . In step 2, the FM model was used as a surrogate model to predict the total energy from the initial proton configuration and was trained when bit string 𝐪 was input. The values of the total energy predicted by the FM are as follows: