Clifford quantum computer and the Mathieu groups

One learned from Gottesman-Knill theorem that the Clifford model of quantum computing [11] may be generated from a few quantum gates, the Hadamard, π/4-Phase and Controlled-Z gates, and efficiently simulated on a classical computer. We employ the group theoretical package GAP[10] for simulating the two qubit Clifford group C2. We already found that the symmetric group S(6), aka the automorphism group of the generalized quadrangle W (2), controls the geometry of the two-qubit Pauli graph [14]. Sixfold symmetry is also revealed in the inner Clifford group Inn(C2) = C2/Center(C2). It contains two normal subgroups, one isomorphic to Z 2 , and the second isomorphic to the semi-direct product U6 = Z 2 ⋊A(6) (of order 5760). The group U(6) stabilizes an hexad in the Steiner system S(3, 6, 22) attached to the Mathieu group M(22). Both groups A(6) and U6 have an outer automorphism group Z2 ×Z2, a feature we associate to two-qubit quantum entanglement.