  Non-abelian geometric phases have been proposed as an essential ingredient in
logical gate implementation -- their geometric nature guarantees their
invariance under reparametrizations of the associated cyclic path in parameter
space. However, they are still dependent on deformations of that path, due to,
e.g., noise. The first question that we tackle in this work is how to quantify
in a meaningful way this effect of noise, focusing, for concreteness, on the
nuclear quadrupole resonance hamiltonian -- other systems of this nature can
clearly be treated analogously. We consider a precessing magnetic field that
drives adiabatically a degenerate doublet, and is subjected to noise, the
effects of which on the Wilczek-Zee holonomy are computed analytically. A
critical review of previous related works reveals a series of assumptions, like
sudden jumps in the field, or the presence of white noise, that might violate
adiabaticity. We propose a state-independent measure of the effect, and then
consider sinusoidal noise in the field, of random amplitude and phase. We find
that all integer noise frequencies $m \neq 2$ behave similarly, in a manner
reminiscent of the abelian case, but that noise of frequency $m = 2$ has a very
different, and, at the same time, very pronounced effect, that might well
affect robustness estimations.
