  Superconducting qubits typically use a dispersive readout scheme, where a
resonator is coupled to a qubit such that its frequency is qubit-state
dependent. Measurement is performed by driving the resonator, where the
transmitted resonator field yields information about the resonator frequency
and thus the qubit state. Ideally, we could use arbitrarily strong resonator
drives to achieve a target signal-to-noise ratio in the shortest possible time.
However, experiments have shown that when the average resonator photon number
exceeds a certain threshold, the qubit is excited out of its computational
subspace in a process we refer to as a measurement-induced state transition
(MIST). These transitions degrade readout fidelity, and constitute leakage
which precludes further operation of the qubit in, for example, error
correction. Here we study these transitions experimentally with a transmon
qubit by measuring their dependence on qubit frequency, average resonator
photon number, and qubit state, in the regime where the resonator frequency is
lower than the qubit frequency. We observe signatures of resonant transitions
between levels in the coupled qubit-resonator system that exhibit noisy
behavior when measured repeatedly in time. We provide a semi-classical model of
these transitions based on the rotating wave approximation and use it to
predict the onset of state transitions in our experiments. Our results suggest
the transmon is excited to levels near the top of its cosine potential
following a state transition, where the charge dispersion of higher transmon
levels explains the observed noisy behavior of state transitions. Moreover, we
show that occupation in these higher energy levels poses a major challenge for
fast qubit reset.
