On character table of Clifford groups

Abstract

Based on a presentation of $\mathcal{C}_n$ and the help of [GAP], we construct the character table of the Clifford group $\mathcal{C}_n$ for $n=1,2,3$. As an application, we can efficiently decompose the (higher power of) tensor product of the matrix representation in those cases. Our results recover some known results in [HWW, WF] and reveal some new phenomena. We prove that when $n \geq 3$, (1) the trivial character is the only linear character for $\mathcal{C}_n$ and hence $\mathcal{C}_n$ equals to its commutator subgroup, (2) the $n$-qubit Pauli group $\mathcal{P}_n$ is the only proper non-trivial normal subgroup of $\mathcal{C}_n$, (3) the matrix representation $\mathcal{M}_{2^n}$ is a faithful representation for $\mathcal{C}_n$. As a byproduct, we give a presentation of the finite symplectic group $Sp(2n,2)$ in terms of generators and relations.