  We study how heralded qubit losses during the preparation of a
two-dimensional cluster state, a universal resource state for one-way quantum
computation, affect its computational power. Above the percolation threshold we
present a polynomial-time algorithm that concentrates a universal cluster
state, using resources that scale optimally in the size of the original
lattice. On the other hand, below the percolation threshold, we show that
single qubit measurements on the faulty lattice can be efficiently simulated
classically. We observe a phase transition at the threshold when the amount of
entanglement in the faulty lattice directly relevant to the computational power
changes exponentially.
