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

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 $\mathcal{C}_n$.