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研究生： 李金彥
Chin-Yen Lee
論文名稱： Embedding the n-Qubit Projective Clifford Group into a Symmetric Group
指導教授： 俞韋亘
Wei-Hsuan Yu
口試委員:
學位類別： 博士
Doctor
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2025
畢業學年度： 113
語文別： 英文
論文頁數： 89
中文關鍵詞： Clifford群 、群表示 、標準形 、慣性子群 、排列表現
外文關鍵詞： Clifford group, presentation, normal form, inertia subgroup, permutation presentation
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    • n-量子比特Clifford群 Cliffn 和 n-量子比特射影Clifford群 Cn 在量子計算中扮演著關鍵角色，尤其在量子錯誤糾正和量子算法中。然而，即使對於較小的n值，文獻中也很少能找到直接構造這些群的方法。
    在本論文中，我們利用Clifford閘的標準形來確定Cn 生成元的中央化子。基於這些結果，我們提出將Cn 嵌入到Sym2(4n −1) 中的方法。此外，我們還證明了Cn 中z-閘的中央化子與Cn 的慣性子群同構。

    The n-qubit Clifford group, Cliffn, and the n-qubit projective Clifford group, Cn, play pivotal roles in quantum computing, especially in quantum error correction and quantum algorithms. However, methods for directly constructing these groups, even for small values of n, are
    rarely found in the literature.
    In this thesis, we use the normal form of Clifford gates to determine the centralizer of the generators of Cn. Based on these results, we propose an embedding of Cn into Sym2(4n −1). Furthermore, we show that the centralizer of the z-gate within Cn is isomorphic to the inertia subgroup of Cn.

    中文摘要 ............................................................................................. i 英文摘要 ............................................................................................. iii 1. Introduction ................................................................................. 3 2. Known Results on Clifford Groups............................................... 9 2.1 Normal form for Clifford operators . . . . . . . . . . . . . . 10 2.1.1 A presentation of Cn . . . . . . . . . . . . . . . . . . . . 17 2.1.2 Rewriting rules . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Normal subgroups of Cn . . . . . . . . . . . . . . . . . . . . 23 2.3 Irreducible representations of Cn . . . . . . . . . . . . . . . 25 3. Centralizers of Generators of Cn ................................................... 33 3.1 The centralizer of z1 gate in Cn . . . . . . . . . . . . . . . . 34 3.2 The centralizer of s1 gate in Cn . . . . . . . . . . . . . . . . 38 3.2.1 Application: A permutation representation of Cn . . . . . 41 3.2.2 A presentation of INn . . . . . . . . . . . . . . . . . . . 45 3.3 The centralizer of h1 gate in Cn . . . . . . . . . . . . . . . . 48 3.4 The centralizer of Λ1 gate in Cn . . . . . . . . . . . . . . . 50 參考文獻 ............................................................................................. 57 A. Basics of Representation Theory ................................................. 59 B. Supplement Equations for Proof 3.17 .......................................... 61 B.1 Equations for ℓ = 1 . . . . . . . . . . . . . . . . . . . . . . 61 B.2 Equations for ℓ = 2 . . . . . . . . . . . . . . . . . . . . . . 62 B.3 Equations for ℓ ≥ 3 . . . . . . . . . . . . . . . . . . . . . . 67 C. My Other Works During PhD ..................................................... 71

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    [Van] E. Van Den Berg, A simple method for sampling random Clifford operators, Proc. IEEE Int. Conf. Quantum Comput. Eng. (QCE), pp. 54–59 (2021).

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