  The goal of this article is to contribute to a better understanding of the
relations between the exact controllability of nonlinear PDEs and the control
theory for ODEs based on Lie brackets, through a study of the Schr\"odinger PDE
with bilinear control. We focus on the small-time local controllability (STLC)
around an equilibrium, when the linearized system is not controllable. We study
the second order order term in the Taylor expansion of the state, with respect
to the control. For scalar input ODEs, quadratic terms never recover
controllability: they induce signed drifts in the dynamics. Thus proving STLC
requires to go at least to the third order. Similar results were proved for the
bilinear Schr\"odinger PDE with scalar input controls. In this article, we
study the case of multi-input systems. We clarify among the quadratic Lie
brackets, those that allow to recover STLC: they are bilinear with respect to 2
different controls. For ODEs, our result is a consequence of Sussman's
sufficient condition $S(\theta)$ (when focused on quadratic terms), but we
propose a new proof, designed to prepare an easier transfer to PDEs. This proof
relies on a representation formula of the state inspired by the Magnus formula.
By adapting it, we prove a new STLC result for the bilinear Schr\"odinger PDE.
