  We present a matrix-product state (MPS) based band-Lanczos method as solver
for quantum cluster methods such as the variational cluster approximation
(VCA). While a na\"ive implementation of MPS as cluster solver would barely
improve its range of applicability, we show that our approach makes it possible
to treat cluster geometries well beyond the reach of exact diagonalization
methods. The key modifications we introduce are a continuous energy truncation
combined with a convergence criterion that is more robust against approximation
errors introduced by the MPS representation and provides a bound to deviations
in the resulting Green's function. The potential of the resulting cluster
solver is demonstrated by computing the self-energy functional for the
single-band Hubbard model at half filling in the strongly correlated regime, on
different cluster geometries. Here, we find that only when treating large
cluster sizes, observables can be extrapolated to the thermodynamic limit,
which we demonstrate at the example of the staggered magnetization. Treating
clusters sizes with up to $6\times 6$ sites we obtain excellent agreement with
quantum Monte-Carlo results.
