  In recent decades there has been a rapid development of methods to
experimentally control individual quantum systems. A broad range of quantum
control methods has been developed for two-level systems, however the
complexity of multi-level quantum systems make the development of analogous
control methods extremely challenging. Here, we exploit the equivalence between
multi-level systems with SU(2) symmetry and spin-1/2 systems to develop a
technique for generating new robust, high-fidelity, multi-level control
methods. As a demonstration of this technique, we develop new adiabatic and
composite multi-level quantum control methods and experimentally realise these
methods using an $^{171}$Yb$^+$ ion system. We measure the average infidelity
of the process in both cases to be around $10^{-4}$, demonstrating that this
technique can be used to develop high-fidelity multi-level quantum control
methods and can, for example, be applied to a wide range of quantum computing
protocols including implementations below the fault-tolerant threshold in
trapped ions.
