Adiabatic geometric phase gate with a quantized control field

We study the performance of an adiabatic, geometric phase gate with a quantized driving field numerically, and develop an analytical approximation that shows how the qubit becomes entangled with the driving field. This results in a scaling of the gate error probability versus the energy in the control field that has the same form as that found for dynamic, nonadiabatic gates, but with a prefactor that would typically be several orders of magnitude larger, because of the adiabaticity constraint. In the approximation we have used, which should be valid for sufficiently large control fields, the main source of decoherence (and hence error) is the “which path” information carried by the photons that would be radiated by the driven qubit.