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

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.Comment: 16 pages, 10 figure

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Investigation of the Superconducting Gap Structure in SrFe2_2(As0.7_{0.7}P0.3_{0.3})2_2 by Magnetic Penetration Depth and Flux Flow Resistivity Analysis

We measured the microwave surface impedances and obtained the superfluid density and flux flow resistivity in single crystals of a phosphor-doped iron-based superconductor SrFe$_2$(As$_{1-x}$P$_{x}$)$_2$ single crystals ($x=0.30$, $T_c=25 \mathrm{K}$). At low temperatures, the superfluid density, $n_s (T)/n_s(0)$, obeys a power law, $n_s (T)/n_s (0)=1-C(T/T_c)^n$, with a fractional exponent of $n=1.5$-1.6. The flux flow resistivity was significantly enhanced at low magnetic fields. These features are consistent with the presences of both a gap with line nodes and nodeless gaps with a deep minimum. The remarkable difference observed in the superconducting gap structure between SrFe$_2$(As$_{1-x}$P$_{x}$)$_2$ and BaFe$_2$(As$_{1-x}$P$_{x}$)$_2$ in our experiments is important for clarifying the mechanism of iron-based superconductivity