An exciton is a quasiparticle created by the association of an electron and a hole. Exciton-based systems are among the most promising candidates for use in solid-state quantum computation. Using the spherical geometry, I have obtained the exact wave function of the exciton lowest bound state. To obtain a full picture of this prototype system, I propose to study, in particular, the formation of the biexciton particle, which is a fundamental step towards the formation of a quantum gate, and to compute its binding energy using various approaches, such as perturbation theory or explicitly correlated methods. This will give new insights into the formation of such species, and for the realisation of two-qubit quantum gates.