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.

翻译：紧凑表示并高效应用线性算子是张量网络方法在模拟量子多体问题和求解科学计算中高维问题时的基本要素。本文研究了树张量网络中的此类表示，即所谓的树张量网络算子，特别关注涉及粒子间长程对相互作用的哈密顿算子。通过推广Lin、Tong等人在矩阵乘积算子方面的工作，我们建立了交互矩阵的层次化低秩结构与树张量网络算子性质之间的直接联系。这种联系使我们能够通过将交互矩阵压缩为层次化半可分离矩阵，得到非常紧凑的树张量网络算子表示。针对不同量子自旋系统的数值实验验证了我们的结果，并凸显了树张量网络算子相对于矩阵乘积算子的潜在优势。