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arXiv:2507.21909 · Physical review. D/Physical review. D.

Triad representation for the anisotropic tensor renormalization group in four dimensions

Yoshiyuki Sugimoto, Shoichi Sasaki

Published Nov 25, 2025• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/c3qc-tn48

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Progress of Theoretical and Experimental Physics•Oct 23, 2019•9 citations
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.
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.
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.
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 tensor renormalization group (TRG) is a highly effective method for calculating the free energy of lattice models and has demonstrated exceptional accuracy for several low-dimensional systems. However, its application to higher-dimensional models is often hampered by a large memory footprint and high computational cost. In this paper, we propose a new algorithm, “tt-lattice”, which approximates the tensor at each lattice vertex using a tensor train (tt) decomposition. The algorithm is designed to maintain the tt-format throughout the TRG computation, which significantly reduces both memory usage and computational cost. For a uniform lattice model, the computational cost of tt-lattice scales as O(d2log⁡(L)χ6)\mathcal {O}(d^2\log (L)\chi ^6)O(d2log(L)χ6), with the largest intermediate tensor size of O(χ4)\mathcal {O}(\chi ^4)O(χ4), where d is the lattice dimension, L is the linear size of each lattice axis, and χ\chiχ is the cutoff of the number of lattice edge states. To validate our algorithm, we conducted numerical experiments on a Gaussian field model, for which the exact free energy is computable in any lattice dimensions. The results demonstrate that the tt-lattice method can calculate the free energy with reasonable accuracy even for a 10D lattice model with modern computational architectures.
Abstract We propose a second renormalization group (SRG) in the triad representation of tensor networks. The SRG method improves two parts of the triad tensor renormalization group, which are the decomposition of intermediate tensors and the preparation of isometries, taking the influence of environment tensors into account. Every fundamental tensor including environment tensor is given as a rank-3 tensor, and the computational cost of the proposed algorithm scales with O \mathcal{O} O O (χ5) employing the randomized SVD where χ is the bond dimension of tensors. We test this method in the classical Ising model on the two dimensional square lattice, and find that numerical results are obtained in good accuracy for a fixed computational 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.