  We analyse the use of entangled states to perform quantum computations non
locally among distant nodes in a quantum network. The complexity associated
with the generation of multiparticle entangled states is quantified in terms of
the concept of global cost. This parameter allows us to compare the use of
physical resources in different schemes. We show that for ideal channels and
for a sufficiently large number of nodes, the use of maximally entangled states
is advantageous over uncorrelated ones. For noisy channels, one has to use
entanglement purification procedures in order to create entangled states of
high fidelity. We show that under certain circumstances a quantum network
supplied with a maximally entangled input still yields a smaller global cost,
provided that $n$ belongs to a given interval $n\in [n_{min},n_{max}]$. The
values of $n_{min}$ and $n_{max}$ crucially depend on the purification
protocols used to establish the $n$-- processor entangled states, as well as on
the presence of decoherence processes during the computation. The phase
estimation problem has been used to illustrate this fact.
