**Clifford quantum computer and the Mathieu groups**

by Michel Planat, PEPS STIC CNRS Collaboration(s)

**Abstract.** One learned from Gottesman-Knill theorem that the Clifford model of quantum computing [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[GAP] for simulating the two qubit Clifford group *C*_2 (Footnote) . 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 [Pauligraphs]. Now we find that the *inner* group Inn(*C*_2)=*C*_2/Center(*C*_2) exactly contains two normal subgroups, one isomorphic to *Z*_2^{times 4} (of order 16), and the second isomorphic to the parent *A*‘(6) (of order 5760) of the alternating group (6). The group *A*‘(6) stabilizes an *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 *outer* automorphism group *Z*_2\times *Z*_2, a feature we associate to two-qubit quantum entanglement.

Footnote: The two qubit Clifford group *C*_2 is group within Clifford spaces.

Sources:

http://fr.arxiv.org/abs/0711.1733v1

http://hal.archives-ouvertes.fr/hal-00186745/fr/