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We present a Lie-algebraic classification and detailed construction of thedynamical invariants, also known as Lewis-Riesenfeld invariants, of thefour-level systems including two-qubit systems which are most relevant andsufficiently general for quantum control and computation. These invariants notonly solve the time-dependent Schr\"odinger equation of four-level systemsexactly but also enable the control, and hence quantum computation based onwhich, of four-level systems fast and beyond adiabatic regimes.

Dynamical invariants for quantum control of four-level systems