Randomized higher-order tensor renormalization group

TL;DR

This paper introduces a randomized higher-order tensor renormalization group (HOTRG) method that reduces computational costs using randomized SVD, with consistent results compared to traditional HOTRG.

Contribution

The paper develops a novel randomized HOTRG approach and variants like MDTRG and Triad-MDTRG, improving efficiency for tensor network calculations.

Findings

The randomized HOTRG achieves similar accuracy to standard HOTRG.

The computational cost is significantly reduced with the new methods.

Results are consistent across different tensor network formulations.

Abstract

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 $d$ with the computational cost $O(D^{3d})$ depending on the truncated bond dimension $D$. We also introduce the minimally-decomposed TRG (MDTRG) as the R-HOTRG on the tensor of order $d+1$ with $O(D^{2d + 1})$ and a triad representation of the MDTRG (Triad-MDTRG) with $O(D^{d+3})$. The results from these formulations are consistent with the HOTRG result with the same truncated bond dimension $D$.