Quantum computers based on rare-earth-ion-doped crystals show promising properties in terms of scalability and connectivity if single ions can be used as qubits. Through simulations, we investigate gate operations on such qubits. We discuss how gate and system parameters affect gate errors, the required frequency bandwidth per qubit, and the risk for instantaneous spectral diffusion (ISD). Furthermore, we examine how uncertainties in the system parameters affect the gate errors, and how precisely the system needs to be known. We find gate errors for arbitrary single-qubit gates of $2.1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ when ISD is not considered and $3.4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ when we take heed to minimize it. Additionally, we construct two-qubit gates with errors ranging from $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\ensuremath{\rightarrow}3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ over a broad range of dipole-dipole interaction strengths.
We present a composite pulse controlled phase gate which, together with a bus architecture, improves the feasibility of a recent quantum computing proposal based on rare-earth-ion-doped crystals. The proposed gate operation is tolerant to variations between ions of coupling strengths, pulse lengths, and frequency shifts. In the absence of decoherence effects, it achieves worst case fidelities above
0.999 with relative variations in coupling strength as high as
10 and frequency shifts up to several percent of the resonant Rabi frequency of the laser used to implement the gate. We outline an experiment to demonstrate the creation and detection of maximally entangled states in the system.