  The self-Kerr interaction is an optical nonlinearity that produces a phase
shift proportional to the square of the number of photons in the field. At
present, many proposals use nonlinearities to generate photon-photon
interactions. For propagating fields these interactions result in undesirable
features such as spectral correlation between the photons. Here, we engineer a
discrete network composed of cross-Kerr interaction regions to simulate a
self-Kerr medium. The medium has effective long-range interactions implemented
in a physically local way. We compute the one- and two-photon S matrices for
fields propagating in this medium. From these scattering matrices we show that
our proposal leads to a high fidelity photon-photon gate. In the limit where
the number of nodes in the network tends to infinity, the medium approximates a
perfect self-Kerr interaction in the one- and two-photon regime.
