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arXiv:1908.07295

Cost Reduction of Swapping Bonds Part in Anisotropic Tensor Renormalization Group

Hideaki Oba

Published Aug 20, 2019• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.1908.07295

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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(χ2d+1)O(χ^{2d+1})O(χ2d+1), where χχχ is the maximum bond dimension and ddd is the dimensionality of a system. We propose an alternative method for the swapping bonds part and it scales with O(χmax⁡(d+3,7))O(χ^{\max(d+3,7)})O(χmax(d+3,7)), though the total cost of ATRG with the method remains O(χ2d+1)O(χ^{2d+1})O(χ2d+1). Moreover, the memory cost of the whole algorithm can be reduced from O(χ2d)O(χ^{2d})O(χ2d) to O(χmax⁡(d+1,6))O(χ^{\max(d+1,6)})O(χmax(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.
Abstract The bottleneck part of an anisotropic tensor renormalization group (ATRG) is a bond-swapping part that consists of a contraction of two tensors and a partial singular value decomposition of a matrix, and their computational costs are O(χ2d+1)O(\chi^{2d+1})O(χ2d+1), where χ\chiχ is the maximum bond dimension and ddd is the dimensionality of the system. We propose an alternative method for the bond-swapping part and it scales with O(χmax⁡(d+3,7))O(\chi^{\max(d+3,7)})O(χmax(d+3,7)), though the total cost of ATRG with the method remains O(χ2d+1)O(\chi^{2d+1})O(χ2d+1). Moreover, the memory cost of the whole algorithm can be reduced from O(χ2d)O(\chi^{2d})O(χ2d) to O(χmax⁡(d+1,6))O(\chi^{\max(d+1,6)})O(χmax(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.
We propose a different tensor renormalization group algorithm, anisotropic tensor renormalization group (ATRG), for lattice models in arbitrary dimensions. The proposed method shares the same versatility with the higher-order tensor renormalization group (HOTRG) algorithm, i.e., it preserves the lattice topology after the renormalization. In comparison with HOTRG, both the computation cost and the memory footprint of our method are drastically reduced, especially in higher dimensions, by renormalizing tensors in an anisotropic way after the singular value decomposition. We demonstrate the ability of ATRG for the square lattice and the simple cubic lattice Ising models. Although the accuracy of the present method degrades when compared with HOTRG of the same bond dimension, the accuracy with fixed computation time is improved greatly due to the drastic reduction of the computation cost.
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.
Anisotropic Tensor Renormalization Group (ATRG) is a powerful algorithm for four-dimensional tensor network calculations. However, the larger bond dimensions are known to be difficult to achieve in practice due to the higher computational cost. Adopting the methods of the minimally decomposed TRG and its triad prescriptions, we construct a triad representation of the four- dimensional ATRG by decomposing the unit-cell tensor. We observe that this combining approach can significantly improve the computational cost even with maintaining the convergence accuracy of the free energy in the four-dimensional Ising model. In addition, we also show that a further improvement can be achieved in terms of the computational cost when our proposed approach is implemented in parallel on GPUs.
The development of tensor renormalization group (TRG) algorithm in higher dimensions is an important and urgent task, as the TRG is expected to provide a way to overcome the sign problem in lattice quantum chromodynamics (QCD) calculations at finite density. One possible approach that enables faster computations in four-dimensional lattice theories is the anisotropic tensor renormalization group (ATRG). However, the computational cost remains substantial and requires significant computational resources. In this paper, we propose a novel algorithm, called the triad-ATRG, which is based on the ATRG and other improved TRG variants with triad network representation. This method achieves lower scaling with respect to the bond dimension, while minimizing the loss of accuracy in the free energy and other physical quantities. We also present parallel implementations of both the ATRG and triad-ATRG on multiple GPUs, which significantly improve performance compared to CPU-based calculations for the four-dimensional system.
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 ddd with the computational cost O(D3d)O(D^{3d})O(D3d) depending on the truncated bond dimension DDD. We also introduce the minimally-decomposed TRG (MDTRG) as the R-HOTRG on the tensor of order d+1d+1d+1 with O(D2d+1)O(D^{2d + 1})O(D2d+1) and a triad representation of the MDTRG (Triad-MDTRG) with O(Dd+3)O(D^{d+3})O(Dd+3). The results from these formulations are consistent with the HOTRG result with the same truncated bond dimension DDD.