  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.
