Tensor Network States with Low-Rank Tensors

TL;DR

This paper introduces low-rank constraints on tensor network tensors, significantly reducing computational costs while maintaining accuracy, with promising applications in quantum physics and machine learning.

Contribution

It proposes a novel approach of imposing low-rank constraints on tensor network tensors, improving efficiency without sacrificing accuracy.

Findings

Low-rank tensor networks can approximate quantum states accurately.

Choosing tensor rank around the bond dimension yields high accuracy.

Low-rank constraints outperform standard tensor networks in simulations.

Abstract

Tensor networks are used to efficiently approximate states of strongly-correlated quantum many-body systems. More generally, tensor network approximations may allow to reduce the costs for operating on an order-$N$ tensor from exponential to polynomial in $N$, and this has become a popular approach for machine learning. We introduce the idea of imposing low-rank constraints on the tensors that compose the tensor network. With this modification, the time and space complexities for the network optimization can be substantially reduced while maintaining high accuracy. We detail this idea for tree tensor network states (TTNS) and projected entangled-pair states. Simulations of spin models on Cayley trees with low-rank TTNS exemplify the effect of rank constraints on the expressive power. We find that choosing the tensor rank $r$ to be on the order of the bond dimension $m$, is sufficient to obtain high-accuracy groundstate approximations and to substantially outperform standard TTNS computations. Thus low-rank tensor networks are a promising route for the simulation of quantum matter and machine learning on large data sets.