The nitrogen vacancy center in diamond has attracted much attention in recent years. As a stable and optically addressable spin qubit it has many potential applications in Quantum information processing tasks.

Many proof of principle experiments demonstrating importing important milestones like initialization, single shot readout and even three-qubit error correction have been performed to date. However, in all these cases fidelity was limited.

We investigate the question, how well one can hope to control an effective three qubit-system consisting of an NV and a nearby, strongly coupled carbon 13C nuclear spin. We use a simple but realistic model of the system including a non-Markovian decoherence for the vacancy spin.

First, we investigate the controllability in principle by checking that the set of reachable operations is indeed the full unitary group. Then we go on to present the result of numerical simulations of single and multi-qubit gates obtained by using single-frequency microwave driving pulses to control the system. Lastly, we give a brief overview over optimal control techniques and present preliminary results for multi-frequency driving using the CRAB method.