Clifford quantum computer and the Mathieu groups

TL;DR

This paper explores the group theoretical structure of the two-qubit Clifford group, revealing connections to Mathieu groups and Steiner systems, and discusses implications for quantum entanglement.

Contribution

It identifies specific normal subgroups within the two-qubit Clifford group and links them to Mathieu groups and Steiner systems, highlighting novel algebraic structures in quantum computing.

Findings

The inner group of the two-qubit Clifford group contains two notable normal subgroups.

The subgroup A'(6) stabilizes a hexad in the Steiner system S(3,6,22).

Outer automorphism groups relate to quantum entanglement features.

Abstract

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.