M. Planat: Cliff. quantum computer and Mathieu groups

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.




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