  Transmon qubits arise from the quantization of nonlinear resonators, systems
that are prone to the buildup of strong, possibly chaotic, fluctuations. Such
instabilities will likely affect fast gate operations which involve the
transient population of higher excited states outside the computational
subspace. Here we show that a statistical analysis of the instantaneous
eigenphases of the time evolution operator, in particular of their curvatures,
allows for identifying the subspace affected by chaotic fluctuations. Our
analysis shows that fast entangling gates, operating at speeds close to the
so-called quantum speed limit, contain transient regimes where the dynamics
indeed becomes partially chaotic for just two transmons.
