  While relatively easy to engineer, static transverse coupling between a qubit
and a cavity mode satisfies the criteria for a quantum non-demolition (QND)
measurement only if the coupling between the qubit and cavity is much less than
their mutual detuning. This can put significant limits on the speed of the
measurement, requiring trade-offs in the circuit design between coupling,
detuning, and decoherence introduced by the cavity mode. Here, we study a
circuit in which the qubit-cavity and the cavity-feedline coupling can be
turned on and off, which helps to isolate the qubit. We do not rely on the
rotating-wave or dispersive approximations, but solve the full transverse
interaction between the qubit and the cavity mode. We show that by carefully
choosing the detuning and interaction time, we can exploit a recurrence in the
qubit-cavity dynamics in a way that makes it possible to perform very fast,
high fidelity, QND measurements. Here, the qubit measurement is performed more
like a gate operation between the qubit and the cavity, where the cavity state
can be amplified, squeezed, and released in a time-sequenced fashion. In
addition, we also show that the non-demolition property of the off-resonant
approximation breaks down much faster than its dispersive property, suggesting
that many of the dispersive measurements to date have been implemented outside
the QND regime.
