A simplified and improved approach to tensor network operators in two dimensions

TL;DR

This paper introduces a simplified method to reformulate 2D tensor network operators as MPO-like structures, enabling more efficient and accurate evaluation of 2D Hamiltonians with finite-range and long-range interactions.

Contribution

It presents a novel reformulation of PEPOs into tensor network operators resembling MPOs, reducing complexity and computational cost in 2D tensor network calculations.

Findings

Enables on-the-fly evaluation of PEPOs using MPOs

Improves energy evaluation efficiency for 2D Hamiltonians

Offers a more accurate and efficient encoding of long-range interactions

Abstract

Matrix product states (MPS) and matrix product operators (MPOs) are one dimensional tensor networks that underlie the modern density matrix renormalization group (DMRG) algorithm. The use of MPOs accounts for the high level of generality and wide range of applicability of DMRG. However, current algorithms for two dimensional (2D) tensor network states, known as projected entangled-pair states (PEPS), rarely employ the associated 2D tensor network operators, projected entangled-pair operators (PEPOs), due to their computational cost and conceptual complexity. To lower these two barriers, we describe how to reformulate a PEPO into a set of tensor network operators that resemble MPOs by considering the different sets of local operators that are generated from sequential bipartitions of the 2D system. The expectation value of a PEPO can then be evaluated on-the-fly using only the action of MPOs and generalized MPOs at each step of the approximate contraction of the 2D tensor network. This technique allows for the simpler construction and more efficient energy evaluation of 2D Hamiltonians that contain finite-range interactions, and provides an improved strategy to encode long-range interactions that is orders of magnitude more accurate and efficient than existing schemes.