  We establish bounds to the necessary resource consumption when building up
cluster states for one-way computing using probabilistic gates. Emphasis is put
on state preparation with linear optical gates, as the probabilistic character
is unavoidable here. We identify rigorous general bounds to the necessary
consumption of initially available maximally entangled pairs when building up
one-dimensional cluster states with individually acting linear optical quantum
gates, entangled pairs and vacuum modes. As the known linear optics gates have
a limited maximum success probability, as we show, this amounts to finding the
optimal classical strategy of fusing pieces of linear cluster states. A formal
notion of classical configurations and strategies is introduced for
probabilistic non-faulty gates. We study the asymptotic performance of
strategies that can be simply described, and prove ultimate bounds to the
performance of the globally optimal strategy. The arguments employ methods of
random walks and convex optimization. This optimal strategy is also the one
that requires the shortest storage time, and necessitates the fewest
invocations of probabilistic gates. For two-dimensional cluster states, we
find, for any elementary success probability, an essentially deterministic
preparation of a cluster state with quadratic, hence optimal, asymptotic
scaling in the use of entangled pairs. We also identify a percolation effect in
state preparation, in that from a threshold probability on, almost all
preparations will be either successful or fail. We outline the implications on
linear optical architectures and fault-tolerant computations.
