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

## TL;DR

This paper investigates the limitations of quantum and simulated annealing for Potts models, revealing first-order phase transitions, and proposes a half-hot constraint method to improve ground state search efficiency.

## Contribution

It introduces a half-hot constraint approach that avoids first-order phase transitions in Potts models, enhancing annealing methods' effectiveness.

## Key findings

- Quantum annealing induces first-order phase transitions in Potts models.
- The half-hot constraint method transforms the transition to second-order.
- The method enables efficient low-energy state search using Ising-type computers.

## Abstract

The Potts model is a generalization of the Ising model with $Q>2$ components. In the fully connected ferromagnetic Potts model, a first-order phase transition is induced by varying thermal fluctuations. Therefore, the computational time required to obtain the ground states by simulated annealing exponentially increases with the system size. This study analytically confirms that the transverse magnetic-field quantum annealing induces a first-order phase transition. This result implies that quantum annealing does not exponentially accelerate the ground-state search of the ferromagnetic Potts model. To avoid the first-order phase transition, we propose an iterative optimization method using a half-hot constraint that is applicable to both quantum and simulated annealing. In the limit of $Q \to \infty$, a saddle point equation under the half-hot constraint is identical to the equation describing the behavior of the fully connected ferromagnetic Ising model, thus confirming a second-order phase transition. Furthermore, we verify the same relation between the fully connected Potts glass model and the Sherrington--Kirkpatrick model under assumptions of static approximation and replica symmetric solution. The proposed method is expected to obtain low-energy states of the Potts models with high efficiency using Ising-type computers such as the D-Wave quantum annealer and the Fujitsu Digital Annealer.