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arXiv:2602.08987 · Physical review. D/Physical review. D.

Forward-mode automatic differentiation for the tensor renormalization group and its relation to the impurity method

Yoshiyuki Sugimoto

Published Apr 10, 2026• 1 citation
arXivHTMLPDF

Abstract

We propose a forward-mode automatic differentiation (AD) framework for tensor renormalization group methods. In this approach, evaluating the derivatives of the partition function up to the order of k increases the matrix-multiplication cost by a factor of ( k + 1 ) ( k + 2 ) / 2 compared to computing the free energy alone, and the memory footprint is only k + 1 times that of the original calculation. In the limit where the derivatives of the singular value decomposition are neglected, we establish a theoretical correspondence between our forward-mode AD and conventional impurity methods. Numerically, we find that the proposed AD algorithm can calculate internal energy and specific heat significantly higher accuracy than the impurity method at comparable computational cost. We also provide a practical procedure to extract critical exponents from derivatives of the renormalized tensor in tensor renormalization group calculations in both two and three dimensions. In addition, we discuss how to efficiently differentiate an arbitrary tensor network.

DOI
10.1103/9wb1-rrcn

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We propose a forward-mode automatic differentiation (AD) framework for tensor renormalization group methods. In this approach, evaluating the derivatives of the partition function up to the order of k increases the matrix-multiplication cost by a factor of ( k + 1 ) ( k + 2 ) / 2 compared to computing the free energy alone, and the memory footprint is only k + 1 times that of the original calculation. In the limit where the derivatives of the singular value decomposition are neglected, we establish a theoretical correspondence between our forward-mode AD and conventional impurity methods. Numerically, we find that the proposed AD algorithm can calculate internal energy and specific heat significantly higher accuracy than the impurity method at comparable computational cost. We also provide a practical procedure to extract critical exponents from derivatives of the renormalized tensor in tensor renormalization group calculations in both two and three dimensions. In addition, we discuss how to efficiently differentiate an arbitrary tensor network.
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.
Machine learning techniques have recently gained prominence in physics, yielding a host of new results and insights. One key concept is that of backpropagation, which computes the exact gradient of any output of a program with respect to any input. This is achieved efficiently within the differentiable programming paradigm, which utilizes automatic differentiation (AD) of each step of a computer program and the chain rule. A classic application is in training neural networks. Here, we apply this methodology instead to the numerical renormalization group (NRG), a powerful technique in computational quantum many-body physics. We demonstrate how derivatives of NRG outputs with respect to Hamiltonian parameters can be accurately and efficiently obtained. Physical properties can be calculated using this differentiable NRG scheme---for example, thermodynamic observables from derivatives of the free energy. Susceptibilities can be computed by adding source terms to the Hamiltonian, but still evaluated with AD at precisely zero field. As an outlook, we briefly discuss the derivatives of dynamical quantities and a possible route to the vertex.
We propose a multi-impurity method for the bond-weighted tensor renormalization group (BWTRG) to compute the higher-order moment of physical quantities in a two-dimensional system. The replacement of the bond weight with an impurity matrix in a bond-weighted triad tensor network represents a physical quantity such as the magnetization and the energy. We demonstrate that the accuracy of the proposed method is much higher than the conventional tensor renormalization group for the Ising model and the five-state Potts model. Furthermore, we perform the finite-size scaling analysis and observe that the dimensionless quantity characterizing the structure of the fixed point tensor satisfies the same scaling relation in the critical region as the Binder parameter. The estimated critical temperature dependence on the bond dimension indicates that the exponent relating the correlation length to the bond dimension varies continuously with respect to the BWTRG hyperparameter. We find that BWTRG with the optimal hyperparameter is more efficient in terms of computational time than alternative approaches based on the matrix product state in estimating the critical temperature.
We demonstrate how to compute the correlation function by using the tensor renormalization group for 2D Ising model. From the correlation function around the critical temperature, we extract the correlation length and the spontaneous magnetization and then determine some critical exponents. Attractive feature of our method to extract the critical exponents is that numerical derivative, that causes a loss of significant digits, and additional simulation parameters, that demand more computational cost and complicated analysis, are not required. On the other hand, our approach requires an additional treatment of impurity tensors, but the additional cost is still tolerable.
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.
Gradient-based optimization methods are the established state-of-the-art paradigm to study strongly entangled quantum systems in two dimensions with projected entangled pair states. However, the key ingredient, the gradient itself, has proven challenging to calculate accurately and reliably in the case of a corner transfer matrix-based approach. Automatic differentiation (AD), which is the best known tool for calculating the gradient, still suffers some crucial shortcomings. Some of these are known, like the problem of excessive memory usage and the divergences which may arise when differentiating a singular value decomposition (SVD). Importantly, we find that there is also a fundamental inaccuracy in the currently used backpropagation of SVD that had not been noted before. In this paper, we describe all these problems and provide them with compact and easy to implement solutions. We analyze the impact of these changes and find that the last problem—the use of the correct gradient—is by far the dominant one and thus should be considered a crucial patch to any AD application that makes use of an SVD for truncation.