A great deal of theoretical activity has resulted from blending the fields of computer science and quantum mechanics. Out of this work has come the concept of a quantum computer, which promises to solve problems currently intractable for classical computers. This promise has, in turn, generated a large amount of effort directed toward investigating quantum computing experimentally. ¶ Quantum computing is difficult because fragile quantum superposition states of the computer’s register must be protected from the environment. This is made more difficult by the need to manipulate and measure these states. ¶ This thesis describes work that was carried out both to investigate and to demonstrate the utility of rare earth ion dopants for quantum computation. Dopants in solids are seen by many as a potential means of achieving scalable quantum computing. Rare earth ion dopants are an obvious choice for investigating such quantum computation. Long coherence times for both optical and nuclear spin transitions have been observed as well as optical manipulation of the spin states. The advantage that the scheme developed here has over nearly all of its competitors is that no complex nanofabrication is required. The advantages of avoiding nano-fabrication are two fold. Firstly, coherence times are likely to be adversely effected by the “damage” to the crystal structure that this manufacture represents. Secondly, the nano-fabrication presents a very serious difficulty in itself. ¶ Because of these advantages it was possible to perform two-qubit operations between independent qubits. This is the first time that such operations have been performed and presents a milestone in quantum computation using dopants in solids. It is only the second time two-qubit operations have been demonstrated in a solid. ¶ The experiments performed in this thesis were in two main areas: The first was the characterisation of hyperfine interactions in rare earth ion dopants; the second, simple demonstrations directly related to quantum computation. ¶ The first experiments that were carried out were to characterise the hyperfine interactions in Pr[superscript 3]+:Y[subscript 2]SiO[subscript 5]. The characterisation was the first carried out for the dopants in a site of such low symmetry. The resulting information about oscillator strengths and transition frequencies should prove indispensable when using such a system for quantum computation. It has already enabled an increase in the coherence times of nuclear spin transitions by two orders of magnitudes. ¶ The experiments directly related to the demonstration of quantum computation were all carried out using ensembles. The presence of a significant distribution of resonant frequencies, or inhomogeneous broadening, meant that many different sub-ensembles could be addressed, based on their resonant frequencies. Furthermore, the properties of the sub-ensembles could be engineered by optically pumping unwanted members to different hyperfine states away from resonance with the laser. ¶ A previously demonstrated technique for realising ensembles that could be used as single qubits was investigated and improved. Also, experiments were carried out to demonstrate the resulting ensembles’ utility as qubits. Further to this, ions from one of the ensembles were selected out, based on their interaction with the ions of another. Elementary two qubit operations were then demonstrated using these ensembles.