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The Potts model is a generalization of the Ising model with $Q>2$ components.In the fully connected ferromagnetic Potts model, a first-order phasetransition is induced by varying thermal fluctuations. Therefore, thecomputational time required to obtain the ground states by simulated annealingexponentially increases with the system size. This study analytically confirmsthat the transverse magnetic-field quantum annealing induces a first-orderphase transition. This result implies that quantum annealing does notexponentially accelerate the ground-state search of the ferromagnetic Pottsmodel. To avoid the first-order phase transition, we propose an iterativeoptimization method using a half-hot constraint that is applicable to bothquantum and simulated annealing. In the limit of $Q \to \infty$, a saddle pointequation under the half-hot constraint is identical to the equation describingthe behavior of the fully connected ferromagnetic Ising model, thus confirminga second-order phase transition. Furthermore, we verify the same relationbetween the fully connected Potts glass model and the Sherrington--Kirkpatrickmodel under assumptions of static approximation and replica symmetric solution.The proposed method is expected to obtain low-energy states of the Potts modelswith high efficiency using Ising-type computers such as the D-Wave quantumannealer and the Fujitsu Digital Annealer.

Efficient quantum and simulated annealing of Potts models using a half-hot constraint