  A fundamental and open question is whether cross-Kerr nonlinearities can be
used to construct a controlled-phase (CPHASE) gate. Here we propose a gate
constructed from a discrete set of atom-mediated cross-Kerr interaction sites
with counter-propagating photons. We show that the average gate fidelity $F$
between a CPHASE and our proposed gate increases as the number of interaction
sites increases and the spectral width of the photon decreases, e.g. with 12
sites we find $F > 99\%$.
