  We present a novel semi-analytical methodology to construct optimal linear
optical circuits for heralded production of 3-photon GHZ and 2-photon Bell
states. We provide a detailed description and analysis of the resulting optical
schemes, which deliver success probabilities of 1/54 and 2/27 for dual-rail
encoded 3-GHZ and Bell states generation, respectively. Our results improve the
known constructive bounds on the success probabilities for 3-GHZ states and are
of particular importance for a ballistic quantum computing model, for which
these states provide an essential resource.
