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We address the question of how many maximally entangled photon pairs areneeded in order to build up cluster states for quantum computing using thetoolbox of linear optics. As the needed gates in dual-rail encoding arenecessarily probabilistic with known optimal success probability, this questionamounts to finding the optimal strategy for building up cluster states, fromthe perspective of classical control. We develop a notion of classicalstrategies, and present rigorous statements on the ultimate maximal and minimaluse of resources of the globally optimal strategy. We find that this strategy -being also the most robust with respect to decoherence - gives rise to anadvantage of already more than an order of magnitude in the number of maximallyentangled pairs when building chains with an expected length of L=40, comparedto other legitimate strategies. For two-dimensional cluster states, we presenta first scheme achieving the optimal quadratic asymptotic scaling. Thisanalysis shows that the choice of appropriate classical control leads to a verysignificant reduction in resource consumption.Comment: 5 pages, 2 figures, title changed, presentation improved, bounds  improved, minor errors corrected, references update

Minimal resources for linear optical one-way computing