Numerically efficient unitary evolution for Hamiltonians beyond nearest-neighbors

Matrix product states (MPSs) and matrix product operators (MPOs) are fundamental tools in the study of quantum many-body systems, particularly in the context of tensor network methods such as Time-Evolving Block Decimation (TEBD). However, constructing compact MPO representations for Hamiltonians with interactions beyond nearest-neighbors, such as those arising in atomic, molecular, and optical (AMO) systems or in systems with ring geometry, remains a challenge. In this paper, we propose a novel approach for the direct construction of compact MPOs tailored specifically for the exponential of spin Hamiltonians. This approach allows for a more efficient time evolution, using TEBD, of spin systems with interactions beyond nearest-neighbors, such as long-range spin-chains, periodic systems and more complex cluster model, with interactions involving more than two spins.