  The Nitrogen Vacancy Center in diamond coupled to addressable surrounding
nuclear spins forms a versatile building block for future quantum technologies.
While previous activities focused on sensing with only a single or very few
spins in operation, recently multi-qubit registers have been successfully
implemented for quantum information processing. Further progress requires a
detailed understanding of the performance of quantum protocols for consecutive
gate operations and thus, beyond established treatments for relaxation and
dephasing. Here, we provide such a theoretical analysis for a small spin
registers with up to four spins built out of NV and environmental constituents
in presence of ensembles of interacting impurity spins. Adapting a cluster
correlation expansion, we predict coherence properties as well as fidelities
for GHZ- and Bell-gate operations also in presence of decoupling pulses. The
influence of the volume density and the geometry of the spin-bath consisting of
the substitutional nitrogen- or $^{13}C$ atoms are also taken into account.
