  Readout of the state of a superconducting qubit by homodyne detection of the
output signal from a dispersively coupled microwave resonator is a common
technique in circuit quantum electrodynamics, and is often claimed to be
quantum non-demolition (QND) up to the same order of approximation as in the
dispersive approximation. However, in this work we show that only in the limit
of infinite measurement time is this protocol QND, as the formation of a
dressed coherent state in the qubit-cavity system applies an effective rotation
to the qubit state. We show how this rotation can be corrected by a coherent
operation, leading to improved qubit initialization by measurement and coherent
feedback.
