  We report on theoretical research in photonic cluster-state computing.
Finding optimal schemes of generating non-classical photonic states is of
critical importance for this field as physically implementable photon-photon
entangling operations are currently limited to measurement-assisted stochastic
transformations. A critical parameter for assessing the efficiency of such
transformations is the success probability of a desired measurement outcome. At
present there are several experimental groups which are capable of generating
multi-photon cluster states carrying more than eight qubits. Separate photonic
qubits or small clusters can be fused into a single cluster state by a
probabilistic optical CZ gate conditioned on simultaneous detection of all
photons with 1/9 success probability of each gate. This design mechanically
follows the original theoretical scheme of cluster state generation proposed
more than a decade ago by Raussendorf, Browne and Briegel. The optimality of
the destructive CZ gate in application to linear optical cluster state
generation has not been analyzed previously. Our results reveal that this
method is far from the optimal one. Employing numerical optimization we have
identified that maximal success probability of fusing n unentangled dual-rail
optical qubits into a linear cluster state is equal to 1/2^(n-1); m-tuple of
photonic Bell pair states, commonly generated via spontaneous parametric
down-conversion, can be fused into a single cluster with the maximal success
probability of 1/4^(m-1).
