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The nitrogen vacancy (NV) 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 multiqubit 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 small spin registers with up to four spins built out of NV and environmental constituents in the presence of ensembles of interacting impurity spins. Adapting a cluster correlation expansion, we predict coherence properties as well as fidelities for Greenberger-Horne-Zeilinger- and Bell-gate operations also in the presence of decoupling pulses. The influence of the volume density and the geometry of the spin-bath consisting of the substitutional nitrogen or ^13C atoms are also taken into account.
Maile et al. (2024) studied this question.