  A possible two-qubit gate for optical quantum computing is the parity gate
based on the weak Kerr effect. Two photonic qubits modulate the phase of a
coherent state, and a quadrature measurement of the coherent state reveals the
parity of the two qubits without destroying the photons. This can be used to
create so-called cluster states, a universal resource for quantum computing.
Here, the effect of self-phase modulation on the parity gate is studied,
introducing generating functions for the Wigner function of a modulated
coherent state. For materials with non-EIT-based Kerr nonlinearities, there is
typically a self-phase modulation that is half the magnitude of the cross-phase
modulation. Therefore, this effect cannot be ignored. It is shown that for a
large class of physical implementations of the phase modulation, the quadrature
measurement cannot distinguish between odd and even parity. Consequently, weak
nonlinear parity gates must be implemented with physical systems where the
self-phase modulation is negligable.
