arXiv:2307.14191

Randomized higher-order tensor renormalization group

Katsumasa Nakayama

Published Jul 26, 2023• 2 citations

Abstract

The higher-order tensor renormalization group (HOTRG) is a fundamental method to calculate the physical quantities by using a tensor network representation. This method is based on the singular value decomposition (SVD) to take the contraction of all indices in the network with an approximation. For the SVD, randomized singular value decomposition (R-SVD) is a powerful method to reduce computational costs of SVD. However, HOTRG with the randomized method is not established. We propose a randomized HOTRG method in a dimension $d$ 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 $D$.

DOI: 10.48550/arxiv.2307.14191

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arXiv:2307.14191

Randomized higher-order tensor renormalization group

Katsumasa Nakayama

Published Jul 26, 2023• 2 citations

arXiv HTML PDF

Abstract

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

DOI: 10.48550/arxiv.2307.14191

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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 $O(χ^{2d+1})$ , 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 $O(χ^{2d})$ to $O(χ^{\max(d+1,6)})$ . 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.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 $O(χ^{2d+1})$, 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 $O(χ^{2d})$ to $O(χ^{\max(d+1,6)})$. 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.

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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

We propose a new renormalization scheme of tensor networks made only of third order tensors. The isometry used for coarse-graining the network can be prepared at an

O(D^6)

computational cost in any

d

dimension (

d \ge 2

), where

D

is the truncated bond dimension of tensors. Although it is reduced to

O(D^5)

if a randomized singular value decomposition is employed, the total cost is

O(D^{d+3})

because the contraction part for creating a renormalized tensor with isometries has

D^{d+3}

multiplications. We test our method in three dimensional Ising model and find that the numerical results are obtained for large

D

s with reasonable errors.

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.