Cost reduction of the bottleneck contraction in the higher-order tensor renormalization group

Hideaki Oba

Published Jan 1, 2021• 0 citations

Abstract

We reduce the computational cost of the bottleneck contraction in the higher-order tensor renormalization group (HOTRG) by using the singular value decomposition. To conﬁrm the effectiveness of the proposed technique, we examine the HOTRG with or without the proposed technique by using the square lattice Ising model at criticality. Then, we ﬁnd that the proposed method provides the consistent results with the original HOTRG while the computational cost and the elapsed time are reduced. Therefore, we conclude that the performance of the proposed method is better than that of the original one.

DOI: 10.24517/0002000259

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Publisher

Abstract

DOI: 10.24517/0002000259

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d

with the computational cost

O(D^{3d})

depending on the truncated bond dimension

D

. We also introduce the minimally-decomposed TRG (MDTRG) as the R-HOTRG on the tensor of order

d+1

with

O(D^{2d + 1})

and a triad representation of the MDTRG (Triad-MDTRG) with

O(D^{d+3})

. The results from these formulations are consistent with the HOTRG result with the same truncated bond dimension

D

O(χ^{2d+1})

, where

χ

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(χ^{\max(d+3,7)})

, though the total cost of ATRG with the method remains

O(χ^{2d+1})

. Moreover, the memory cost of the whole algorithm can be reduced from

O(χ^{2d})

O(χ^{\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.

O(\chi^{2d+1})

, where

\chi

is the maximum bond dimension and

d

is the dimensionality of the system. We propose an alternative method for the bond-swapping 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})

O(\chi^{\max(d+1,6)})

. We examine ATRG with or without the proposed method in the 4D 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 a higher-order tensor renormalization 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.