  We study superadiabatic quantum control of a three-level quantum system whose
energy spectrum exhibits multiple avoided crossings. In particular, we
investigate the possibility of treating the full control task in terms of
independent two-level Landau-Zener problems. We first show that the time
profiles of the elements of the full control Hamiltonian are characterized by
peaks centered around the crossing times. These peaks decay algebraically for
large times. In principle, such a power-law scaling invalidates the hypothesis
of perfect separability. Nonetheless, we address the problem from a pragmatic
point of view by studying the fidelity obtained through separate control as a
function of the intercrossing separation. This procedure may be a good approach
to achieve approximate adiabatic driving of a specific instantaneous eigenstate
in realistic implementations.
