  We present a theoretical scheme that allows to perform a universal set of
holonomic gates on a two qubit register, formed by a $^{13}$C nuclear spin
coupled to the electron spin of a nitrogen-vacancy center in diamond. Strong
hyperfine interaction between the electron spin and the spins of the first
three shells of $^{13}$C atoms allows to operate the state of the register on
the submicrosecond timescale using microwave pulses only. We describe the
system and the operating regime analytically and numerically, as well as
simulate the initialization protocols.
