Percolation thresholds for photonic quantum computing

Despite linear-optical fusion (Bell measurement) being probabilistic, photonic cluster states for universal quantum computation can be prepared without feed-forward by fusing small n-photon entangled clusters, if the success probability of each fusion attempt is above a threshold, $${\mathrm{\lambda }}_{\mathrm{c}}^{(n)}$$ λ c ( n ) . We prove a general bound $${\mathrm{\lambda }}_{\mathrm{c}}^{(n)} \ge 1/(n - 1)$$ λ c ( n ) ≥ 1 ∕ ( n - 1 ) , and develop a conceptual method to construct long-range-connected clusters where $${\mathrm{\lambda }}_{\mathrm{c}}^{(n)}$$ λ c ( n ) becomes the bond percolation threshold of a logical graph. This mapping lets us find constructions that require lower fusion success probabilities than currently known, and settle a heretofore open question by showing that a universal cluster state can be created by fusing 3-photon clusters over a 2D lattice with a fusion success probability that is achievable with linear optics and single photons, making this attractive for integrated-photonic realizations.