  We demonstrate a dispersive measurement pulse shaping technique that allows
for arbitrarily fast quantum non-demolition, single-quadrature measurements of
non-linear systems and unconditionally leaves the measurement resonator empty.
For single-qubit measurements, current measurements are limited to the 99%
fidelity range due to relaxation during the process. However, trying to go to
shorter times to circumvent this with square or composite digital pulses leads
to leftover cavity population after measurement of the same order of error.
These effects can be suppressed using simple smooth pulse shapes from a similar
family of pulses as DRAG shaping, used in the context of leakage removal in
superconducting qubits; here, it can be derived exactly for arbitrarily many
measured modes. Beyond single qubits, the measurement pulses are fully general
to dispersive measurement systems. This includes multi-qubit and multi-state
(leakage) measurements where the measurement can be done in a single shot and
with a single homodyne phase. Another major challenge for fast measurement is
depopulating Purcell filter cavities, which we show can readily be achieved
using derivative shaping. Finally, we show how to apply the technique to
cascaded cavity systems, e.g~for fast remote entanglement generation.
