Hybrid stochastic method for the tensor renormalization group

TL;DR

This paper introduces a hybrid stochastic tensor renormalization group method that employs noise vectors to improve low-rank approximations, reducing computational costs and enhancing accuracy in lattice models.

Contribution

The paper presents a novel stochastic approach for TRG using noise vectors, addressing computational efficiency and error estimation in tensor approximations.

Findings

Achieved better accuracy than traditional TRG in the Ising model

Demonstrated the effectiveness of noise vectors in error reduction

Proposed position-dependent noise vectors to eliminate systematic errors

Abstract

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.