Embedding the $n$-Qubit Projective Clifford Group into a Symmetric Group

TL;DR

This paper constructs a symmetric group containing the $n$-qubit projective Clifford group as a subgroup, using group-theoretic analysis of Clifford operators and their centralizers.

Contribution

It introduces a novel embedding of the $n$-qubit projective Clifford group into a symmetric group, expanding understanding of its algebraic structure.

Findings

Constructed a symmetric group ${ m Sym}_{2(4^n-1)}$ containing the Clifford group

Analyzed centralizers of key gates within the Clifford group

Provided a presentation of the inertia subgroup of $ $-qubit Clifford group

Abstract

In this paper, we construct a symmetric group ${\rm Sym}_{2(4^n-1)}$, which contains a subgroup isomorphic to the $n$-qubit projective Clifford group $\mathcal{C}_n$. To establish this result, we investigate the centralizers of the $z$ gate and the phase gate within the $n$-qubit projective Clifford group, utilizing the normal form of the Clifford operators. As a byproduct, we also provide a presentation of the inertia subgroup of $\mathcal{C}_n$.