Low-rank representation of tensor network operators with long-range pairwise interactions

TL;DR

This paper introduces novel low-rank tensor network representations for long-range interactions like Coulomb forces, using matrix completion and hierarchical low-rank formats to improve efficiency and applicability.

Contribution

It develops a modified incremental SVD method for MPOs and extends hierarchical low-rank techniques to general tensor network operators, enabling efficient representation of long-range interactions.

Findings

Efficient MPO representation of Coulomb interactions with O(log(N) log(N/ε)) terms.

Hierarchical low-rank formats applicable to both MPOs and PEPOs.

Modified ISVD yields equivalent MPOs to prior methods.

Abstract

Tensor network operators, such as the matrix product operator (MPO) and the projected entangled-pair operator (PEPO), can provide efficient representation of certain linear operators in high dimensional spaces. This paper focuses on the efficient representation of tensor network operators with long-range pairwise interactions such as the Coulomb interaction. For MPOs, we find that all existing efficient methods exploit a peculiar "upper-triangular low-rank" (UTLR) property, i.e. the upper-triangular part of the matrix can be well approximated by a low-rank matrix, while the matrix itself can be full-rank. This allows us to convert the problem of finding the efficient MPO representation into a matrix completion problem. We develop a modified incremental singular value decomposition method (ISVD) to solve this ill-conditioned matrix completion problem. This algorithm yields equivalent MPO representation to that developed in [Stoudenmire and White, Phys. Rev. Lett. 2017]. In order to efficiently treat more general tensor network operators, we develop another strategy for compressing tensor network operators based on hierarchical low-rank matrix formats, such as the hierarchical off-diagonal low-rank (HODLR) format, and the $\mathcal{H}$-matrix format. Though the pre-constant in the complexity is larger, the advantage of using the hierarchical low-rank matrix format is that it is applicable to both MPOs and PEPOs. For the Coulomb interaction, the operator can be represented by a linear combination of $\mathcal{O}(\log(N)\log(N/\epsilon))$ MPOs/PEPOs, each with a constant bond dimension, where $N$ is the system size and $\epsilon$ is the accuracy of the low-rank truncation. Neither the modified ISVD nor the hierarchical low-rank algorithm assumes that the long-range interaction takes a translation-invariant form.