Automatic Differentiation for Second Renormalization of Tensor Networks

TL;DR

This paper introduces a differentiable tensor renormalization group framework using automatic differentiation, enabling automatic gradient computation to improve tensor network simulations of lattice models and quantum systems.

Contribution

It presents a novel $ ext{∂}$TRG framework that automates second renormalization of tensor networks via backpropagation, enhancing efficiency and accuracy in many-body simulations.

Findings

Effective in solving the square-lattice Ising model

Successfully simulates quantum systems at finite temperature

Achieves high efficiency with GPU acceleration

Abstract

Tensor renormalization group (TRG) constitutes an important methodology for accurate simulations of strongly correlated lattice models. Facilitated by the automatic differentiation technique widely used in deep learning, we propose a uniform framework of differentiable TRG ($\partial$TRG) that can be applied to improve various TRG methods, in an automatic fashion. Essentially, $\partial$TRG systematically extends the concept of second renormalization [PRL 103, 160601 (2009)] where the tensor environment is computed recursively in the backward iteration, in the sense that given the forward process of TRG, $\partial$TRG automatically finds the gradient through backpropagation, with which one can deeply "train" the tensor networks. We benchmark $\partial$TRG in solving the square-lattice Ising model, and demonstrate its power by simulating one- and two-dimensional quantum systems at finite temperature. The deep optimization as well as GPU acceleration renders $\partial$TRG manybody simulations with high efficiency and accuracy.