  Fast discrimination between quantum states of superconducting artificial
atoms is an important ingredient for quantum information processing. In circuit
quantum electrodynamics, increasing the signal field amplitude in the readout
resonator, dispersively coupled to the artificial atom, improves the
signal-to-noise ratio and increases the measurement strength. Here we employ
this effect over two orders of magnitude in readout power, made possible by the
unique combination of a dimer Josephson junction array amplifier with a large
dynamic range, and the fact that the readout of our granular aluminum fluxonium
artificial atom remained quantum-non-demolition (QND) at relatively large
photon numbers in the readout resonator, up to $\overline{n} = 110$. Using
Bayesian inference, this allows us to detect quantum jumps faster than the
readout resonator response time $2/\kappa$, where $\kappa$ is the bandwidth of
the readout resonator.
