Efficient Graph State Construction Under the Barrett and Kok Scheme

Recently Barrett and Kok (BK) proposed an elegant method for entangling separated matter qubits. They outlined a strategy for using their entangling operation (EO) to build graph states, the resource for one-way quantum computing. However by viewing their EO as a graph fusion event, one perceives that each successful event introduces an ideal redundant graph edge, which growth strategies should exploit. For example, if each EO succeeds with probability p=0.4 then a highly connected graph can be formed with an overhead of only about ten EO attempts per graph edge. The BK scheme then becomes competitive with the more elaborate entanglement procedures designed to permit p to approach unity.