  With the help of the spin-orbit interaction, we propose a scheme to perform
holonomic single qubit gates on the electron spin confined to a quantum dot.
The manipulation is done in the absence (or presence) of an applied magnetic
field. By adiabatic changing the position of the confinement potential, one can
rotate the spin state of the electron around the Bloch sphere in semiconductor
heterostructures. The dynamics of the system is equivalent to employing an
effective non-Abelian gauge potential whose structure depends on the type of
the spin-orbit interaction. As an example, we find an analytic expression for
the electron spin dynamics when the dot is moved around a circular path (with
radius R) on the two dimensional electron gas (2DEG), and show that all single
qubit gates can be realized by tuning the radius and orientation of the
circular paths. Moreover, using the Heisenberg exchange interaction, we
demonstrate how one can generate two-qubit gates by bringing two quantum dots
near each other, yielding a scalable scheme to perform quantum computing on
arbitrary N qubits.
  This proposal shows a way of realizing holonomic quantum computers in
solid-state systems.
