Clifford quantum computer and the Mathieu groups
Résumé
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 $\mbox{Inn}(\mathcal{C}_2)=\mathcal{C}_2/\mbox{Center}(\mathcal{C}_2)$ 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.
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