  In this work we develop several new simulation algorithms for 1D many-body
quantum mechanical systems combining the Matrix Product State variational
ansatz with Taylor, Pad\'e and Arnoldi approximations to the evolution
operator. By comparing with previous techniques based on MPS and DMRG we
demonstrate that the Arnoldi method is the best one, reaching extremely good
accuracy with moderate resources. Finally we apply this algorithm to studying
how correlations are transferred from the atomic to the molecular cloud when
crossing a Feschbach resonance with two-species hard-core bosons in a 1D
optical lattice.
