Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

IEEE International Symposium on Parallel & Distributed Processing, Workshops and Phd Forum

Optimization of Reordering Procedures in HOTRG for Distributed Parallel Computing

Haruka Yamada, Akira Imakura, Toshiyuki Imamura, Tetsuya Sakurai

Published May 1, 2018• 2 citations
Publisher

Abstract

Higher-order tensor renormalization group (HOTRG) is a coarse-graining algorithm for approximating the partition function in the field of elementary particle physics using a tensor network. Coarse-graining in HOTRG comprises an approximation step and a contraction step, and the contraction step is performed with tensor reorderings and matrix products. In this paper, we introduce a naive parallel implementation of HOTRG and propose optimal reordering procedures for a three-dimensional (3D) classical cubic lattice Ising model. Numerical experiments on the K computer show that the elapsed time of the proposed procedure is 6.88 times faster than the naive one for the reorderings.

DOI
10.1109/ipdpsw.2018.00150

Related papers

Coarse-Graining Tensor Renormalization

Tao Xiang

Coarse-graining renormalization aims to reformulate a tensor network model with a coarse-grained one at a larger scale. It has attracted particular attention in recent years because it opens a new avenue to unveil the entanglement structure of a tensor network model under the scaling transformation. This chapter reviews and compares the tensor renormalization group (TRG) and other coarse-graining methods developed in the past two decades. The methods can be divided into two groups according to whether or not the renormalization effect of the environment tensors is incorporated in the optimization of local tensors. The local optimization methods include TRG, HOTRG (a variant of TRG based on the higher-order singular value decomposition), tensor network renormalization (TNR), and loop-TNR. The global optimization methods include the second renormalized TRG and HOTRG, referred to as SRG and HOSRG, respectively. Among all these coarse-graining methods, HOTRG and HOSRG are the only two that can be readily extended and efficiently applied to three-dimensional classical or two-dimensional quantum lattice models.

Cambridge University Press eBooks•Aug 31, 2023•0 citations

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

Hideaki Oba

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.

•Jan 1, 2021•0 citations

Global optimization of tensor renormalization group using the corner transfer matrix

Satoshi Morita, Naoki Kawashima

A tensor network renormalization algorithm with global optimization based on the corner transfer matrix is proposed. Since the environment is updated by the corner transfer matrix renormalization group method, the forward-backward iteration is unnecessary, which is a time-consuming part of other methods with global optimization. In addition, a further approximation reducing the order of the computational cost of contraction for the calculation of the coarse-grained tensor is proposed. The computational time of our algorithm in two dimensions scales as the sixth power of the bond dimension, while the higher-order tensor renormalization group and the higher-order second renormalization group methods have the seventh power. We perform benchmark calculations in the Ising model on the square lattice and show that the time-to-solution of the proposed algorithm is faster than that of other methods.

arXiv:2009.01997•Physical review. B./Physical review. B•Jan 22, 2021•12 citations

Tensor network algorithm by coarse-graining tensor renormalization on finite periodic lattices

Hui‐Hai Zhao, Z. Y. Xie, Tao Xiang +1 more

We develop coarse-graining tensor renormalization group algorithms to compute physical properties of two-dimensional lattice models on finite periodic lattices. Two different coarse-graining strategies, one based on the tensor renormalization group and the other based on the higher-order tensor renormalization group, are introduced. In order to optimize the tensor network model globally, a sweeping scheme is proposed to account for the renormalization effect from the environment tensors under the framework of second renormalization group. We demonstrate the algorithms by the classical Ising model on the square lattice and the Kitaev model on the honeycomb lattice, and show that the finite-size algorithms achieve substantially more accurate results than the corresponding infinite-size ones.

arXiv:1510.03333•Physical review. B./Physical review. B•Mar 9, 2016•40 citations

Anisotropic tensor renormalization group

Daiki Adachi, Tsuyoshi Okubo, Synge Todo

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.

arXiv:1906.02007•Physical review. B./Physical review. B•Aug 24, 2020•83 citations

Coarse-graining renormalization by higher-order singular value decomposition

Z. Y. Xie, Jing Chen, Mingpu Qin +3 more

We propose a novel coarse-graining tensor renormalization group method based on the higher-order singular value decomposition. This method provides an accurate but low computational cost technique for studying both classical and quantum lattice models in two or three dimensions. We have demonstrated this method using the Ising model on the square and cubic lattices. By keeping up to 16 bond basis states, we obtain by far the most accurate numerical renormalization group results for the three-dimensional Ising model. We have also applied the method to study the ground state as well as finite temperature properties for the two-dimensional quantum transverse Ising model and obtain the results which are consistent with published data.

arXiv:1201.1144•Physical Review B•Jul 30, 2012•392 citations
Higher-order tensor renormalization group (HOTRG) is a coarse-graining algorithm for approximating the partition function in the field of elementary particle physics using a tensor network. Coarse-graining in HOTRG comprises an approximation step and a contraction step, and the contraction step is performed with tensor reorderings and matrix products. In this paper, we introduce a naive parallel implementation of HOTRG and propose optimal reordering procedures for a three-dimensional (3D) classical cubic lattice Ising model. Numerical experiments on the K computer show that the elapsed time of the proposed procedure is 6.88 times faster than the naive one for the reorderings.
Coarse-graining renormalization aims to reformulate a tensor network model with a coarse-grained one at a larger scale. It has attracted particular attention in recent years because it opens a new avenue to unveil the entanglement structure of a tensor network model under the scaling transformation. This chapter reviews and compares the tensor renormalization group (TRG) and other coarse-graining methods developed in the past two decades. The methods can be divided into two groups according to whether or not the renormalization effect of the environment tensors is incorporated in the optimization of local tensors. The local optimization methods include TRG, HOTRG (a variant of TRG based on the higher-order singular value decomposition), tensor network renormalization (TNR), and loop-TNR. The global optimization methods include the second renormalized TRG and HOTRG, referred to as SRG and HOSRG, respectively. Among all these coarse-graining methods, HOTRG and HOSRG are the only two that can be readily extended and efficiently applied to three-dimensional classical or two-dimensional quantum lattice models.
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.
A tensor network renormalization algorithm with global optimization based on the corner transfer matrix is proposed. Since the environment is updated by the corner transfer matrix renormalization group method, the forward-backward iteration is unnecessary, which is a time-consuming part of other methods with global optimization. In addition, a further approximation reducing the order of the computational cost of contraction for the calculation of the coarse-grained tensor is proposed. The computational time of our algorithm in two dimensions scales as the sixth power of the bond dimension, while the higher-order tensor renormalization group and the higher-order second renormalization group methods have the seventh power. We perform benchmark calculations in the Ising model on the square lattice and show that the time-to-solution of the proposed algorithm is faster than that of other methods.
We develop coarse-graining tensor renormalization group algorithms to compute physical properties of two-dimensional lattice models on finite periodic lattices. Two different coarse-graining strategies, one based on the tensor renormalization group and the other based on the higher-order tensor renormalization group, are introduced. In order to optimize the tensor network model globally, a sweeping scheme is proposed to account for the renormalization effect from the environment tensors under the framework of second renormalization group. We demonstrate the algorithms by the classical Ising model on the square lattice and the Kitaev model on the honeycomb lattice, and show that the finite-size algorithms achieve substantially more accurate results than the corresponding infinite-size ones.
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 propose a novel coarse-graining tensor renormalization group method based on the higher-order singular value decomposition. This method provides an accurate but low computational cost technique for studying both classical and quantum lattice models in two or three dimensions. We have demonstrated this method using the Ising model on the square and cubic lattices. By keeping up to 16 bond basis states, we obtain by far the most accurate numerical renormalization group results for the three-dimensional Ising model. We have also applied the method to study the ground state as well as finite temperature properties for the two-dimensional quantum transverse Ising model and obtain the results which are consistent with published data.