Quantum-thermal annealing with a cluster-flip algorithm

A quantum-thermal annealing method using a cluster-flip algorithm is studied in the two-dimensional spin-glass model. The temperature $\left(T\right)$ and the transverse field $\left(\Gamma \right)$ are decreased simultaneously with the same rate along a linear path on the $T\text{−}\Gamma$ plane. We found that the additional pulse of the transverse field to the frozen local spins produces a good approximate solution with a low computational cost.

Figure 1

Paths for three types of annealing method in the plane of the temperature and the transverse field. Note that when the quantum Monte Carlo method is applied to QA in practice, the QA path is lifted from the $\Gamma$ axis because a small but nonzero temperature is introduced.

Figure 2

(Color online) Annealing time $\tau$ dependence of the minimum residual energy per site for a $100\times 100$ two-dimensional Ising spin glass. The cluster QTA is performed with $\beta \Gamma =20$ and 100. Correspondingly, the cluster QA is done with $\beta =20$ and 100. The broken curve with upper triangles is the data obtained by the cluster QTA with the local pulse process performed with $\beta \Gamma =10$. The initial value of the transverse field is chosen to be ${\Gamma }_0=1$ in all cases. Each annealing method is tried at least 100 times.

Figure 3

(Color online) Comparison of the minimum residual energies per site obtained by the cluster QTA, single QA, and CA. The abscissa is the computational cost (total CPU time) which is defined by the annealing time $\tau$ multiplied by $\beta \Gamma$ for the cluster QTA, $\tau$ multiplied by $M$ for the single QA, and $\tau$ for CA. The initial temperature for CA is $T_0=3$ and the initial transverse field for the other methods is set ${\Gamma }_0=1$.