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

Performance of quantum registers in diamond in the presence of spin impurities