Minimally entangled typical thermal states algorithm with Trotter gates

We improve the efficiency of the minimally entangled typical thermal states (METTS) algorithm without breaking the Abelian symmetries. By adding the operation of Trotter gates that respects the Abelian symmetries to the METTS algorithm, we find that a correlation between successive states in Markov-chain Monte Carlo sampling decreases by orders of magnitude. We measure the performance of the improved METTS algorithm through the simulations of the canonical ensemble of the Bose-Hubbard model and confirm that the reduction of the autocorrelation leads to the reduction of computation time. We show that our protocol using the operation of Trotter gates is effective also for the simulations of the grand canonical ensemble.

Figure 1

The alignment of physical sites (blue circles) and ancilla sites (orange squares) which does not require any additional changes of operators for enlarging local Hilbert spaces of edge sites.

Figure 2

The $\tau$ dependence of the upper bound for autocorrelation time $-1/ln|{\lambda }_2|$ in the 1D unit-filled Bose-Hubbard model with $U/J=1.0$ at inverse temperature $\beta J=0.25$. The system size $L$ is set to six. For the parameters of the unitary operator ${\left[{\widehat{U}}_{\mathrm{T}}(\tau /n)\right]}^n$, we use $(n,U^{\prime })=(1,U)$ (blue solid line), $(n,U^{\prime })=(2,U)$ (orange dashed line), and $(n,U^{\prime })=(2,0)$ (green dashed-dotted line).

Figure 3

The $\tau$ dependence of the upper bound for autocorrelation time $-1/ln|{\lambda }_2|$ in the 1D unit-filled Bose-Hubbard model with $U/J=20.0$. For the parameters of the unitary operator ${\left[{\widehat{U}}_{\mathrm{T}}(\tau /n)\right]}^n$, we use $(n,U^{\prime })=(1,U)$ (blue solid line), $(n,U^{\prime })=(2,U)$ (orange dashed line), $(n,U^{\prime })=(2,J)$ (green dashed-dotted line), and $(n,U^{\prime })=(2,0)$ (red dotted line). Except the Hubbard interaction, any other parameters are taken to be the same with those in Fig. 2.

Figure 4

The ratio $R$ as a function of the block size $N_{\mathrm{b}}$ obtained from the six-site unit-filled 1D Bose-Hubbard model with $U/J=1.0$ at inverse temperature $\beta J=0.25$. For the parameters of unitary operators, we use $(\tau ,n,U^{\prime })=(0.25J^{-1},1,U)$ (orange dashed line) and $(\tau ,n,U^{\prime })=(1.0J^{-1},2,U)$ (green dashed-dotted line). The solid blue line corresponds to a simulation without the Trotter gate.

Figure 5

The chemical potential $\mu$–dependencies of the filling factor $\nu$ (top), the internal energy per site $\langle \widehat{H}\rangle /\left(JL\right)$ (middle), and the compressibility $\kappa$ (bottom) of the 1D Bose-Hubbard model in the large $U$ limit calculated by using the METTS algorithms with and without the Trotter gates. The inverse temperature $\beta$ is $5.0J^{-1}$. For the parameters characterizing the Trotter gates, we use $\tau J=3.6$ and $n=2$. The expectation values and the one-sigma error bars of the METTS results are estimated from successive 8192 samples.