Transmon qubits arise from the quantization of nonlinear resonators, systems that are prone to the buildup of strong, possibly even chaotic, fluctuations. One may wonder to what extent fast gate operations, which involve the transient population of states outside the computational subspace, can be affected by such instabilities. Wehere consider the eigenphases and -states of the time evolution operators describing a universal gate set, and analyze them by methodology otherwise applied in the context of many-body physics. Specifically, we discuss their spectral statistic, the distribution of time dependent level curvatures, and state occupations in- and outside the computational subspace. We observe 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. We find that for these gates even small variations of Hamiltonian or control parameters lead to large gate errors and speculate on the consequences for the practical implementation of quantum control.