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arXiv:2211.13107 · Physical review. D/Physical review. D.

All-mode renormalization for tensor network with stochastic noise

Erika Arai, Hiroshi Ohki, Shinji Takeda, Masaaki Tomii

Published Jun 20, 2023• 4 citations
arXivHTMLPDF

Abstract

In usual (nonstochastic) tensor network calculations, the truncated singular value decomposition is often used for approximating a tensor, and it causes systematic errors. By introducing stochastic noise in the approximation, however, one can avoid such systematic errors at the expense of statistical errors, which can be straightforwardly controlled. Therefore in principle, exact results can be obtained even at finite bond dimension up to the statistical errors. A previous study of the unbiased method implemented in tensor renormalization group algorithm, however, showed that the statistical errors for physical quantity are not negligible, and furthermore the computational cost is linearly proportional to a system volume. In this paper, we introduce a new way of stochastic noise such that the statistical error is suppressed, and moreover, in order to reduce the computational cost we propose common noise method whose cost is proportional to the logarithm of volume. We find that the method provides better accuracy for the free energy compared with the truncated singular value decomposition when applying to tensor renormalization group for Ising model on square lattice. Although the common noise method introduces systematic error originated from a correlation of noises, we show that the error can be described by a simple functional form in terms of the number of noises, thus the error can be straightforwardly controlled in an actual analysis. We also apply the method to the graph independent local truncation algorithm and show that the accuracy is further improved.

DOI
10.1103/physrevd.107.114515

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In usual (nonstochastic) tensor network calculations, the truncated singular value decomposition is often used for approximating a tensor, and it causes systematic errors. By introducing stochastic noise in the approximation, however, one can avoid such systematic errors at the expense of statistical errors, which can be straightforwardly controlled. Therefore in principle, exact results can be obtained even at finite bond dimension up to the statistical errors. A previous study of the unbiased method implemented in tensor renormalization group algorithm, however, showed that the statistical errors for physical quantity are not negligible, and furthermore the computational cost is linearly proportional to a system volume. In this paper, we introduce a new way of stochastic noise such that the statistical error is suppressed, and moreover, in order to reduce the computational cost we propose common noise method whose cost is proportional to the logarithm of volume. We find that the method provides better accuracy for the free energy compared with the truncated singular value decomposition when applying to tensor renormalization group for Ising model on square lattice. Although the common noise method introduces systematic error originated from a correlation of noises, we show that the error can be described by a simple functional form in terms of the number of noises, thus the error can be straightforwardly controlled in an actual analysis. We also apply the method to the graph independent local truncation algorithm and show that the accuracy is further improved.
We propose a hybrid stochastic method for the tensor renormalization group (TRG) approach. TRG is known as a powerful tool to study the many-body systems and quantum field theory on the lattice. It is based on a low-rank approximation of the tensor using the truncated singular value decomposition (SVD), whose computational cost increases as the bond dimension increases, so that efficient cost reduction techniques are highly demanded. We use noise vectors for the low-rank approximation with the truncated SVD, by which the truncation error is replaced with a statistical error due to noise, and an error estimation could be improved. We test this method in the classical Ising model and observe a better accuracy than TRG. We also discuss a cross contamination issue in a multiple use of the same noise vectors, and to remove this systematic error we consider position-dependent noise vectors.
Tensor renormalization group (TRG) is a coarse-graining algorithm for approximating the partition function using a tensor network in the field of elementary particle physics.Although the computational cost of TRG can be reduced using a randomized singular value decomposition, its computation time is still large.In this paper, we propose a cost-efficient cutoff method for calculating TRG by truncating small tensor elements.Numerical experiments showed that the proposed method is faster than the conventional one without degrading accuracy.
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.
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 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(D6)O(D^6)O(D6) computational cost in any ddd dimension (d≥2d \ge 2d≥2), where DDD is the truncated bond dimension of tensors. Although it is reduced to O(D5)O(D^5)O(D5) if a randomized singular value decomposition is employed, the total cost is O(Dd+3)O(D^{d+3})O(Dd+3) because the contraction part for creating a renormalized tensor with isometries has Dd+3D^{d+3}Dd+3 multiplications. We test our method in three dimensional Ising model and find that the numerical results are obtained for large DDDs with reasonable errors.
An algorithm of the tensor renormalization group is proposed based on a randomized algorithm for singular value decomposition. Our algorithm is applicable to a broad range of two-dimensional classical models. In the case of a square lattice, its computational complexity and memory usage are proportional to the fifth and the third power of the bond dimension, respectively, whereas those of the conventional implementation are of the sixth and the fourth power. The oversampling parameter larger than the bond dimension is sufficient to reproduce the same result as full singular value decomposition even at the critical point of the two-dimensional Ising model.