  In dispersive readout schemes, qubit-induced nonlinearity typically limits
the measurement fidelity by reducing the signal-to-noise ratio (SNR) when the
measurement power is increased. Contrary to seeing the nonlinearity as a
problem, here we propose to use it to our advantage in a regime where it can
increase the SNR. We show analytically that such a regime exists if the qubit
has a many-level structure. We also show how this physics can account for the
high-fidelity avalanchelike measurement recently reported by Reed {\it et al.}
[arXiv:1004.4323v1].
