  An exact solution is presented for the time-dependent wavefunction of a
Kramers doublet which propagates around a quantum ring with tuneable Rashba
spin-orbit interaction. By propagating in segments it is shown that
Kramers-doublet qubits may be defined for which transformations on the Bloch
sphere may be performed for an integral number of revolutions around the ring.
The conditions for full coverage of the Bloch sphere are determined and
explained in terms of sequential qubit rotations due to electron motion along
the segments, with change of rotation axes between segments due to adiabatic
changes in the Rashba spin-orbit interaction. Prospects and challenges for
possible realizations are discussed for which rings based on InAs quantum wires
are promising candidates.
