  The negatively charged nitrogen-vacancy (NV) center spin in diamond can be
used to realize quantum computation and to sense magnetic fields. Its spin
triplet consists of three levels labeled with its spin z-components of +1, 0,
and -1. Without external field, the +1 and -1 states are degenerate and higher
than the 0 state due to the zero-field splitting. By taking the symmetrical and
anti-symmetrical superpositions of the +1 and -1 states as our qubit basis, we
obtain exact evolution operator of the NV center spin under time-dependent
magnetic field by mapping the three-level system on time-dependent quantum
two-level systems with exact analytical solutions. With our exact evolution
operator of the NV center spin including three levels, we show that arbitrary
qubits can be prepared from the starting 0 state and arbitrary rapid quantum
logic gates of these qubits can be realized with magnetic fields. In addition,
it is made clear that the typical quantum logic gates can be accomplished
within a few nanoseconds and the fidelity can be very high because only
magnetic field strength needs to be controlled in this approach. These results
should be useful to realizing quantum computing with the NV center spin systems
in diamond and exploring other effects and applications.
