On character table of Clifford groups

TL;DR

This paper constructs the character tables of Clifford groups for n=1,2,3, enabling efficient decomposition of tensor products and revealing structural properties such as normal subgroups and representation faithfulness.

Contribution

It provides explicit character tables for small Clifford groups, offers new insights into their structure, and presents a presentation of the symplectic group $Sp(2n,2)$.

Findings

The trivial character is the only linear character for n≥3.

The n-qubit Pauli group is the only proper non-trivial normal subgroup.

The matrix representation is faithful for n≥3.

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.