  A negatively charged nitrogen vacancy (NV) center in diamond has been
recognized as a good solid-state qubit. A system consisting of the electronic
spin of the NV center and hyperfine-coupled nitrogen and additionally nearby
carbon nuclear spins can form a quantum register of several qubits for quantum
information processing or as a node in a quantum repeater. Several impressive
experiments on the hybrid electron and nuclear spin register have been
reported, but fidelities achieved so far are not yet at or below the thresholds
required for fault-tolerant quantum computation (FTQC). Using quantum optimal
control theory based on the Krotov method, we show here that fast and
high-fidelity single-qubit and two-qubit gates in the universal quantum gate
set for FTQC, taking into account the effects of the leakage state, nearby
noise qubits and distant bath spins, can be achieved with errors less than
those required by the threshold theorem of FTQC.
