  A conventional realization of quantum logic gates and control is based on
resonant Rabi oscillations of the occupation probability of the system. This
approach has certain limitations and complications, like counter-rotating
terms. We study an alternative paradigm for implementing quantum logic gates
based on Landau-Zener-St\"{u}ckelberg-Majorana (LZSM) interferometry with
non-resonant driving and the alternation of adiabatic evolution and
non-adiabatic transitions. Compared to Rabi oscillations, the main differences
are a non-resonant driving frequency and a small number of periods in the
external driving. We explore the dynamics of a multilevel quantum system under
LZSM drives and optimize the parameters for increasing single- and two-qubit
gates speed. We define the parameters of the external driving required for
implementing some specific gates using the adiabatic-impulse model. The LZSM
approach can be applied to a large variety of multi-level quantum systems and
external driving, providing a method for implementing quantum logic gates on
them.
