How local is the information in MPS/PEPS tensor networks?

TL;DR

This paper introduces a local approximation method for calculating expectation values in 2D PEPS tensor networks representing gapped ground states, leveraging local patches and semi-definite programming to achieve high precision.

Contribution

The authors propose a novel local patch-based approach combined with semi-definite programming to efficiently estimate expectation values in PEPS, especially in frustrated systems.

Findings

Accurate expectation values up to 3-4 digits with small patches

Method works well even when patch size is smaller than correlation length

Approach provides rigorous bounds on expectation values

Abstract

Two dimensional tensor networks such as projected entangled pairs states (PEPS) are generally hard to contract. This is arguably the main reason why variational tensor network methods in 2D are still not as successful as in 1D. However, this is not necessarily the case if the tensor network represents a gapped ground state of a local Hamiltonian; such states are subject to many constraints and contain much more structure. In this paper we introduce a new approach for approximating the expectation value of a local observable in ground states of local Hamiltonians that are represented as PEPS tensor-networks. Instead of contracting the full tensor-network, we try to estimate the expectation value using only a local patch of the tensor-network around the observable. Surprisingly, we demonstrate that this is often easier to do when the system is frustrated. In such case, the spanning vectors of the local patch are subject to non-trivial constraints that can be utilized via a semi-definite program to calculate rigorous lower- and upper-bounds on the expectation value. We test our approach in 1D systems, where we show how the expectation value can be calculated up to at least 3 or 4 digits of precision, even when the patch radius is smaller than the correlation length.