  We analyze the generation of linear optical cluster states (LOCS) via
addition of one and two qubits. Existing approaches employ the stochastic
linear optical two-qubit CZ gate with success rate of 1/9 per fusion operation.
The question of optimality of the CZ gate with respect to LOCS generation
remains open. We report that there are alternative schemes to the CZ gate that
are exponentially more efficient and show that sequential LOCS growth is
globally optimal. We find that the optimal cluster growth operation is a state
transformation on a subspace of the full Hilbert space. We show that the
maximal success rate of fusing n photonic qubits or m Bell pairs is 1/2^n-1 and
1/4^m-1 respectively and give an explicit optical design.
