  Previous analyses of conditional \phi-phase gates for photonic qubits that
treat cross-phase modulation (XPM) in a causal, multimode, quantum field
setting suggest that a large (~\pi rad) nonlinear phase shift is always
accompanied by fidelity-degrading noise [J. H. Shapiro, Phys. Rev. A 73, 062305
(2006); J. Gea-Banacloche, Phys. Rev. A 81, 043823 (2010)]. Using an atomic
V-system to model an XPM medium, we present a conditional phase gate that, for
sufficiently small nonzero \phi, has high fidelity. The gate is made cascadable
by using using a special measurement, principal mode projection, to exploit the
quantum Zeno effect and preclude the accumulation of fidelity-degrading
departures from the principal-mode Hilbert space when both control and target
photons illuminate the gate.
