Quantum structures in qudit spaces

The Toffoli gate and the Hadamard gate give rise to an approximately universal set of quantum computational gates. We study the algebraic properties of extensions of this system by introducing the notion of the extended Shi-Aharonov structure. The quotient of this structure is isomorphic to a structure based on a particular set of real numbers (the Bloch hypersphere).The aim of this paper is to bring together researchers working in different areas (quantum circuits, algebra, geometry) opening up new perspectives. The algebraic approach may be useful in circuit analysis.