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 γ. The qubit energy levels are driven nonlinearly in time, ∝sgn(t)|t|~v, and we derive the transition probability in the limit of sufficiently fast noise, for arbitrary exponent v. The level occupation after the transition depends strongly on v, and there exists a critical v_c with qualitative difference between v< v_c and v> v_c. When v< v_c, the final state is always fully incoherent with equal population of both quantum levels, even for arbitrarily weak noise. For v>v_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 v_c=1/2, while for slow noise v_c<1/2 and it depends on γ. We also discuss phase coherence, which is relevant when the qubit has a nonzero minimum energy gap. The qualitative dependency on v is the same for the phase coherence and level occupation. The state after the transition does, in general, depend on γ. For fixed v, increasing γ decreases the final state coherence when v<1 and increases the final state coherence when v> 1. Only the conventional linear driving is independent of γ.
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