One learned from Gottesman-Knill theorem that the Clifford model of quantum computing \cite{ Clark07} may be generated from a few quantum gates, the Hadamard, Phase and Controlled-Z gates, and efficiently simulated on a classical computer. We employ the group theoretical package GAP\cite{ GAP} for simulating the two qubit Clifford group $\mathcal{ C}_2$. 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 \cite{ Pauligraphs}. Now we find that the { \it inner} group ${ Inn}(\mathcal{ C}_2)=\mathcal{ C}_2/{ Center}(\mathcal{ C}_2)$ exactly contains two normal subgroups, one isomorphic to $\mathcal{ Z}_2^{ \times 4}$ (of order 16), and the second isomorphic to the parent $A'(6)$ (of order 5760) of the alternating group A(6). The group $A'(6)$ stabilizes an { \it hexad} in the Steiner system $S(3,6,22)$ attached to the Mathieu group M(22). Both groups A(6) and $A'(6)$ have an { \it outer} automorphism group $\mathcal{ Z}_2\times \mathcal{ Z}_2$, a feature we associate to two-qubit quantum entanglement.
Categories: quant-ph math-ph math.MP