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

Chin-Yen Lee

Abstract

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 $\mathcal{C}_n$.

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

Abstract

In this paper, we construct a symmetric group , which contains a subgroup isomorphic to the -qubit projective Clifford group . To establish this result, we investigate the centralizers of the gate and the phase gate within the -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 .

Paper Structure

This paper contains 8 sections, 18 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Normal form for Clifford operators
  3. Rewriting rules
  4. The centralizer of $z_1$ gate in $\mathcal{C}_n$
  5. The centralizer of $s_1$ gate in $\mathcal{C}_n$
  6. Permutation representations of projective Clifford groups
  7. Presentations of $\mathcal{I}\mathcal{N}_n$ and $\mathcal{I}\mathcal{N}_n/\mathcal{P}_n$
  8. A presentation of projective Clifford groups

Key Result

Theorem 1

Let $V$ be the conjugacy class of $s_1$ in $\mathcal{C}_n$ and let $\sigma_g$ denote the permutation induced by the conjugate action of $g \in \mathcal{C}_n$ on $V$. Then the map $\mathcal{C}_n \to {\rm Sym}_{V}: g \mapsto \sigma_g$ is a faithful homomorphism. Moreover, the size of $V$ is equal to $

Theorems & Definitions (34)

  • Definition 1.1
  • Definition 1.2
  • Theorem
  • Remark 1.3
  • Theorem
  • Theorem 2.1
  • Theorem 4.1
  • Lemma 4.2
  • Theorem 4.3
  • proof
  • ...and 24 more