  According to idealized models, a strong Kerr non-linearity may be used to
build optical quantum gates for optical quantum information processing by
inducing conditional phase shifts on quantum states. Recently, Shapiro (PRA 73,
062305 (2006)) argued that for a Kerr medium with non-instantaneous but fast
response, essentially no phase shift is induced on two-single-photon input
states, and thus a quantum gate build from such a medium cannot work. Here we
show that a fast response Kerr medium induces some but very little phase shifts
on a two-single-photon input state, and it is insufficient for high fidelity
quantum computation. We point out that this is caused by the medium imparting
spectral entanglement to the input photons. We further show that a way to
circumvent this problem and achieve a high fidelity gate, is to engineer the
dispersion properties of the medium to give a dominant spectral effect over the
non-instantaneous response, in addition to satisfying a phase matching
condition.
