  We investigate the use of tensor-train approaches to the solution of the
time-dependent Schr\"odinger equation for chain-like quantum systems with
on-site and nearest-neighbor interactions only. Using efficient low-rank tensor
train representations, we aim at reducing the memory consumption as well as the
computation costs. As an example, coupled excitons and phonons modeled in terms
of Fr\"ohlich-Holstein type Hamiltonians are studied here. By comparing our
tensor-train based results with semi-analytical results, we demonstrate the key
role of the ranks of the quantum state vectors. Typically, an excellent quality
of the solutions is found only when the maximum number of ranks exceed a
certain value. One class of propagation schemes builds on splitting the
Hamiltonian into two groups of interleaved nearest-neighbor interactions which
commutate within each of the groups. In particular, the 4-th order Yoshida-Neri
and the 8-th order Kahan-Li symplectic compositions are demonstrated to yield
very accurate results, close to machine precision. However, due to the
computational costs, currently their use is restricted to rather short chains.
That also applies to propagations based on the time-dependent variational
principle, typically used in the context of matrix product states. Yet another
class of propagators involves explicit, time-symmetrized Euler integrators.
Especially the 4-th order variant is recommended for quantum simulations of
longer chains, even though the high precision of the splitting schemes cannot
be reached. Moreover, the scaling of the computational effort with the
dimensions of the local Hilbert spaces is much more favorable for the
differencing than for the splitting or variational schemes.
