  We theoretically analyze a system where two electrons are trapped separately
in two quantum dots on a suspended carbon nanotube (CNT), subject to external
ac electric driving. An indirect mechanically-induced coupling of two distant
single electron spins is induced by the interaction between the spins and the
mechanical motion of the CNT. We show that a two-qubit iSWAP gate and arbitrary
single-qubit gates can be obtained from the intrinsic spin-orbit coupling.
Combining the iSWAP gate and single-qubit gates, maximally entangled states of
two spins can be generated in a single step by varying the frequency and the
strength of the external electric driving field. The spin-phonon coupling can
be turned off by electrostatically shifting the electron wave function on the
nanotube.
