On character table of Clifford groups
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866910815507972096 |
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| author | Lee, Chin-Yen Yu, Wei-Hsuan Peng, Yung-Ning Lai, Ching-Jui |
| author_facet | Lee, Chin-Yen Yu, Wei-Hsuan Peng, Yung-Ning Lai, Ching-Jui |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_14850 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On character table of Clifford groups Lee, Chin-Yen Yu, Wei-Hsuan Peng, Yung-Ning Lai, Ching-Jui Representation Theory Mathematical Physics Quantum Physics 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. |
| title | On character table of Clifford groups |
| topic | Representation Theory Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/2309.14850 |