  The concept of off-diagonal geometric phase (GP) has been introduced in order
to recover interference information about the geometry of quantal evolution
where the standard GPs are not well-defined. In this Letter, we propose a
physical setting for realizing non-Abelian off-diagonal GPs. The proposed
non-Abelian off-diagonal GPs can be implemented in a cyclic chain of four
qubits with controllable nearest-neighbor interactions. Our proposal seems to
be within reach in various nano-engineered systems and therefore opens up for
first experimental test of the non-Abelian off-diagonal GP.
