Abstract
The second lesson was devoted to the properties of Pauli matrices and the description of quantum information primitives. While classical information processing is based on the very elementary Boolean group, quantum information is based on the non-commutative quaternion group, for which Pauli matrices provide a basis of representation. The quaternion group is in fact the smallest non-commutative group whose subgroups are all normal, i.e. invariant by conjugation with the other elements of the group. This fundamental mathematical property translates physically into the parallel character of observables and evolution operators in quantum mechanics. The geometry of the Bloch sphere, on which we can see very concretely quantum states and operators for a single qubit, has been covered in detail. In this lesson, we also introduced the notion of quantum register and that of generalized Pauli operators for several qubits. The universality of certain quantum gates, from which any algorithm can be realized, was also introduced. Finally, we discussed the reversible nature of quantum computation and the " musical staff notation ", in which each qubit is a line and single-qubit operations are presented like musical notes.
