It is also possible to optimize the doping configuration of scandium ions simultaneously with the proton configuration. The doping configuration should also be treated as a bit string, and so the bits for the scandium/zirconium sites were prepared as 1 for scandium ions and 0 for zirconium ions. In the 2 × 2 × 2 supercell model, eight bits were used to represent the doping configuration, which was defined by {𝑞 0 } 0&",…,1 . In other words, when simultaneously considering the doping and proton configurations, in addition to the 216-bit, {𝑞 $% } $&",…,#),%&",…,* was also employed for doping.