  We study Landau-Zener like dynamics of a qubit influenced by transverse
random telegraph noise. The telegraph noise is characterized by its coupling
strength, $v$ and switching rate, $\gamma$. The qubit energy levels are driven
nonlinearly in time, $\propto \sign(t)|t|^\nu$, and we derive the transition
probability in the limit of sufficiently fast noise, for arbitrary exponent
$\nu$. The longitudinal coherence after transition depends strongly on $\nu$,
and there exists a critical $\nu_c$ with qualitative difference between $\nu<
\nu_c$ and $\nu > \nu_c$. When $\nu<\nu_c$ the end state is always fully
incoherent with equal population of both quantum levels, even for arbitrarily
weak noise. For $\nu>\nu_c$ the system keeps some coherence depending on the
strength of the noise, and in the limit of weak noise no transition takes
place. For fast noise $\nu_c=1/2$, while for slow noise $\nu_c<1/2$ and it
depends on $\gamma$. We also discuss transverse coherence, which is relevant
when the qubit has a nonzero minimum energy gap. The qualitative dependency on
$\nu$ is the same for transverse as for longitudinal coherence. The state after
transition does in general depend on $\gamma$. For fixed $v$, increasing
$\gamma$ decreases the final state coherence when $\nu<1$ and increase the
final state coherence when $\nu>1$. Only the conventional linear driving is
independent of $\gamma$.
