  We study geometric quantum phases corresponding to analogues of the Anandan
quantum phase [J. Anandan, Phys. Lett. A {\bf138}, 347 (1989)] based on a
possible scenario of the Lorentz symmetry violation background in a tensor
background. We also show that quantum holonomies associated with the analogue
of the Anandan quantum phase can be determined, and discuss a way of performing
one-qubit quantum gates by analogy with the holonomic quantum computation [P.
Zanardi and M. Rasetti, Phys. Lett. A {\bf264}, 94 (1999)].
