  The term Tensor Network States (TNS) refers to a number of families of states
that represent different ans\"atze for the efficient description of the state
of a quantum many-body system. Matrix Product States (MPS) are one particular
case of TNS, and have become the most precise tool for the numerical study of
one dimensional quantum many-body systems, as the basis of the Density Matrix
Renormalization Group method. Lattice Gauge Theories (LGT), in their
Hamiltonian version, offer a challenging scenario for these techniques. While
the dimensions and sizes of the systems amenable to TNS studies are still far
from those achievable by 4-dimensional LGT tools, Tensor Networks can be
readily used for problems which more standard techniques, such as Markov chain
Monte Carlo simulations, cannot easily tackle. Examples of such problems are
the presence of a chemical potential or out-of-equilibrium dynamics. We have
explored the performance of Matrix Product States in the case of the Schwinger
model, as a widely used testbench for lattice techniques. Using finite-size,
open boundary MPS, we are able to determine the low energy states of the model
in a fully non-perturbative manner and to accurately extract the mass spectrum.
In this work we extend the analysis to the determination of the chiral
condensate, both for massless and massive fermions. The method allows for
accurate finite size and continuum limit extrapolations and produces remarkably
precise results, thus showing the feasibility of these techniques for gauge
theory problems.
