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We investigate the use of tensor-train approaches to the solution of thetime-dependent Schr\"odinger equation for chain-like quantum systems withon-site and nearest-neighbor interactions only. Using efficient low-rank tensortrain representations, we aim at reducing the memory consumption as well as thecomputation costs. As an example, coupled excitons and phonons modeled in termsof Fr\"ohlich-Holstein type Hamiltonians are studied here. By comparing ourtensor-train based results with semi-analytical results, we demonstrate the keyrole of the ranks of the quantum state vectors. Typically, an excellent qualityof the solutions is found only when the maximum number of ranks exceed acertain value. One class of propagation schemes builds on splitting theHamiltonian into two groups of interleaved nearest-neighbor interactions whichcommutate within each of the groups. In particular, the 4-th order Yoshida-Neriand the 8-th order Kahan-Li symplectic compositions are demonstrated to yieldvery accurate results, close to machine precision. However, due to thecomputational costs, currently their use is restricted to rather short chains.That also applies to propagations based on the time-dependent variationalprinciple, typically used in the context of matrix product states. Yet anotherclass of propagators involves explicit, time-symmetrized Euler integrators.Especially the 4-th order variant is recommended for quantum simulations oflonger chains, even though the high precision of the splitting schemes cannotbe reached. Moreover, the scaling of the computational effort with thedimensions of the local Hilbert spaces is much more favorable for thedifferencing than for the splitting or variational schemes.

Quantum dynamics of coupled excitons and phonons in chain-like systems: tensor train approaches and higher-order propagators