Low-rank Tree Tensor Network Operators for Long-Range Pairwise Interactions

TL;DR

This paper introduces a new method for efficiently representing long-range pairwise interactions in quantum many-body systems using tree tensor network operators, which are more compact than previous approaches.

Contribution

It establishes a direct link between hierarchical low-rank matrices and TTNOs, enabling more compact representations of long-range interactions in tensor network methods.

Findings

Numerical experiments validate the effectiveness of TTNOs.

TTNOs outperform matrix product operators in representing long-range interactions.

The method achieves significant compression of interaction matrices.

Abstract

Compactly representing and efficently applying linear operators are fundamental ingredients in tensor network methods for simulating quantum many-body problems and solving high-dimensional problems in scientific computing. In this work, we study such representations for tree tensor networks, the so called tree tensor network operators (TTNOs), paying particular attention to Hamiltonian operators that involve long-range pairwise interactions between particles. Generalizing the work by Lin, Tong, and others on matrix product operators, we establish a direct connection between the hierarchical low-rank structure of the interaction matrix and the TTNO property. This connection allows us to arrive at very compact TTNO representations by compressing the interaction matrix into a hierarchically semi-separable matrix. Numerical experiments for different quantum spin systems validate our results and highlight the potential advantages of TTNOs over matrix product operators.