On Stability of Tensor Networks and Canonical Forms

TL;DR

This paper analyzes the stability of tensor networks like MPS and PEPS, providing bounds on how local errors can be amplified, and evaluates the benefits of canonical forms through theoretical and experimental approaches.

Contribution

It offers new bounds on error amplification in tensor networks and quantifies the advantages of canonical forms, addressing a key challenge in tensor network optimization.

Findings

Bounds on error amplification for tensor networks.

Canonical forms reduce error propagation.

Experimental validation on perturbed random MPS.

Abstract

Tensor networks such as matrix product states (MPS) and projected entangled pair states (PEPS) are commonly used to approximate quantum systems. These networks are optimized in methods such as DMRG or evolved by local operators. We provide bounds on the conditioning of tensor network representations to sitewise perturbations. These bounds characterize the extent to which local approximation error in the tensor sites of a tensor network can be amplified to error in the tensor it represents. In known tensor network methods, canonical forms of tensor network are used to minimize such error amplification. However, canonical forms are difficult to obtain for many tensor networks of interest. We quantify the extent to which error can be amplified in general tensor networks, yielding estimates of the benefit of the use of canonical forms. For the MPS and PEPS tensor networks, we provide simple forms on the worst-case error amplification. Beyond theoretical error bounds, we experimentally study the dependence of the error on the size of the network for perturbed random MPS tensor networks.