  The large deviation (LD) statistics of dynamical observables is encoded in
the spectral properties of deformed Markov generators. Recent works have shown
that tensor network methods are well suited to compute the relevant leading
eigenvalues and eigenvectors accurately. However, the efficient generation of
the corresponding rare trajectories is a harder task. Here we show how to
exploit the MPS approximation of the dominant eigenvector to implement an
efficient sampling scheme which closely resembles the optimal (so-called
"Doob") dynamics that realises the rare events. We demonstrate our approach on
three well-studied lattice models, the Fredrickson-Andersen and East
kinetically constrained models (KCMs), and the symmetric simple exclusion
process (SSEP). We discuss how to generalise our approach to higher dimensions.
