  The bottleneck part of anisotropic tensor renormalization group (ATRG) is a
swapping bonds part which consists of a contraction of two tensors and a
partial singular value decomposition of a matrix, and their computational costs
are $O(\chi^{2d+1})$, where $\chi$ is the maximum bond dimension and $d$ is the
dimensionality of a system. We propose an alternative method for the swapping
bonds part and it scales with $O(\chi^{\max(d+3,7)})$, though the total cost of
ATRG with the method remains $O(\chi^{2d+1})$. Moreover, the memory cost of the
whole algorithm can be reduced from $O(\chi^{2d})$ to $O(\chi^{\max(d+1,6)})$.
We examine ATRG with or without the proposed method in the four-dimensional
Ising model and find that the free energy density of the proposed algorithm is
consistent with that of the original ATRG while the elapsed time is
significantly reduced. We also compare the proposed algorithm with higher-order
tensor renromalization group (HOTRG) and find that the value of the free energy
density of the proposed algorithm is lower than that of HOTRG in the fixed
elapsed time.
