  We consider the relaxation of a spin qubit in a quantum dot propagating as a
whole in a one-dimensional semiconductor with hyperfine coupling. We show that
this motion leads to qualitatively new features in this process compared to
static quantum dots. For a fast straightforward motion, the initial spin
density decreases to zero with the relaxation rate independent of the spatial
spread of the electron wave function and inversely proportional to the electron
speed. However, for the oscillatory motion, the qubit acquires memory, and the
dephasing becomes Gaussian rather than exponential. After some time, one third
of the initial spin polarization is restored, as it happens for static qubits.
This revival can occur either through periodic peaks or through a monotonous
increase in the polarization, after a minimum, until a plateau has been
reached. Our results can be useful for the understanding of the spin dynamics
and decoherence in quantum wires.
