Scaling of success probabilities for linear optics gates

By using the abstract linear-optical network derived in [S. Scheel and N. L\"utkenhaus, New J. Phys. \textbf{6}, 51 (2004)] we show that for the lowest possible ancilla photon numbers the probability of success of realizing a (single-shot) generalized nonlinear sign shift gate on an ($N+1$)-dimensional signal state scales as $1/N^2$. We limit ourselves to single-shot gates without conditional feed-forward. We derive our results by using determinants of Vandermonde-type over a polynomial basis which is closely related to the well-known Jacobi polynomials.