Energy gap scaling of quantum annealing based on Wajnflasz--Pick model

The conventional quantum annealer, transverse field Ising model, is known to suffer from the energy gap shrinkage for the scalable system, which may not provide the solution of the optimization problem within the realistic time scale. An idea for possibly overcoming this problem proposed by Seki, Tanaka, and Kawabata is to employ the Wajnflasz-Pick model, i.e., a qudit. By analyzing the order of the phase transition within the mean-field approach with the uniform interaction, they have found that the performance of quantum annealing is controlled by the number of upper and lower states of the spin. In this study, in order to confirm their claim, we revisit the energy gap of the finite size Wajnflasz--Pick model with all-to-all random interaction.