Tensor renormalization group with randomized singular value decomposition

Abstract

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.

SuSMoST: Surface Science Modeling and Simulation Toolkit

“…[47] The second one uses decomposition of the T tensor (Equation (4)) on 4 three-legged tensors as recently proposed by Morita, Igarashi, Zhao, and Kawashima. [61] The second F I G U R E 4 Computational cost of transfer-matrix calculations for two hard-disks models [60] [Color figure can be viewed at wileyonlinelibrary.com] F I G U R E 5 Computational cost of TRG calculations for 1NN harddisks model [60] using Levin and Nave [47] and Morita et al [61] algorithms [Color figure can be viewed at wileyonlinelibrary.com] implementation uses randomized SVD algorithm [62] and can be leveraged by graphical processing unit via CUDA [63] interface. The second implementation greatly reduces memory and computation time requirements, but in some cases of low-temperature and strong attractive interactions its benefits fade out due to dramatic accuracy loss.…”

Cost-efficient cutoff method for tensor renormalization group with randomized singular value decomposition

“…Using RSVD [8], Morita et al reduced computational cost for both of the approximation step and the contraction step [4]. In this subsection, we present an outline of the algorithm for TRG with RSVD.…”

“…In this section, we evaluate the effect on accuracy and the performance of the cutoff method proposed in Section 3 for the same model in Section 3. We set the bond dimension D to 100 and set q = 0 in RSVD as in [4]. The singular values are normalized so that the largest singular value equals to 1.0.…”

“…In this approach, the partition function is represented by a network of tensors that is then approximated by coarse-graining the tensor network. There are many coarse-graining methods for different target physical models such as DMRG [1,2], tensor renormalization group (TRG) [3,4], TNR [5] and HOTRG [6]. For the details of the tensor network methods, we refer [7] and the references therein.…”

“…The computational cost of both steps is O(D 6 ), where a parameter D is related to the accuracy and is called the bond dimension. Morita et al reduced the computational cost of these two steps to O(D 5 ) [4] using a randomized SVD (RSVD) [8]. In general, the partition function should be calculated for various temperatures to obtain the critical temperature.…”