  A key challenge in quantum computing is speeding up measurement and
initialization. Here, we experimentally demonstrate a dispersive measurement
method for superconducting qubits that simultaneously measures the qubit and
returns the readout resonator to its initial state. The approach is based on
universal analytical pulses and requires knowledge of the qubit and resonator
parameters, but needs no direct optimization of the pulse shape, even when
accounting for the nonlinearity of the system. Moreover, the method generalizes
to measuring an arbitrary number of modes and states. For the qubit readout, we
can drive the resonator to $\sim 10^2$ photons and back to $\sim 10^{-3}$
photons in less than $3 \kappa^{-1}$, while still achieving a $T_1$-limited
assignment error below 1\%. We also present universal pulse shapes and
experimental results for qutrit readout.
