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A quantum system constrained to a degenerate energy eigenspace can undergo anontrival time evolution upon adiabatic driving, described by a non-AbelianBerry phase. This type of dynamics may provide logical gates in quantumcomputing that are robust against timing errors. A strong candidate to realizesuch holonomic quantum gates is an electron or hole spin qubit trapped in aspin-orbit-coupled semiconductor, whose twofold Kramers degeneracy is protectedby time-reversal symmetry. Here, we propose and quantitatively analyzeprotocols to measure the non-Abelian Berry phase by pumping a spin qubitthrough a loop of quantum dots. One of these protocols allows to characterizethe local internal Zeeman field directions in the dots of the loop. We expect anear-term realisation of these protocols, as all key elements have been alreadydemonstrated in spin-qubit experiments. These experiments would be important toassess the potential of holonomic quantum gates for spin-based quantuminformation processing.

Protocols to measure the non-Abelian Berry phase by pumping a spin qubit through a quantum-dot loop