  A quantum system constrained to a degenerate energy eigenspace can undergo a
nontrival time evolution upon adiabatic driving, described by a non-Abelian
Berry phase. This type of dynamics may provide logical gates in quantum
computing that are robust against timing errors. A strong candidate to realize
such holonomic quantum gates is an electron or hole spin qubit trapped in a
spin-orbit-coupled semiconductor, whose twofold Kramers degeneracy is protected
by time-reversal symmetry. Here, we propose and quantitatively analyze
protocols to measure the non-Abelian Berry phase by pumping a spin qubit
through a loop of quantum dots. One of these protocols allows to characterize
the local internal Zeeman field directions in the dots of the loop. We expect a
near-term realisation of these protocols, as all key elements have been already
demonstrated in spin-qubit experiments. These experiments would be important to
assess the potential of holonomic quantum gates for spin-based quantum
information processing.
