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The Clifford theory of the n-qubit Clifford group

Kieran Mastel

TL;DR

This work develops a Clifford-theoretic bridge between the n-qubit Pauli group and the n-qubit Clifford group by relating representations of the normal abelian Pauli subgroup to the full Clifford group. It identifies a central extension structure where the inertia quotient is the affine symplectic group, enabling a surprising correspondence between irreducible characters of $\mathcal{C}_n$ and $\mathcal{C}_{n+1}$. The authors construct irreps of $\mathcal{C}_{n+1}$ from those of $\mathcal{C}_n$ via induction from the inertia group, leveraging the identical character tables of $Sp(2n,2)\ltimes\mathbb{Z}_2^{2n}$ and $\mathcal{C}_n$. This yields a practical, principled method to grow Clifford-group representations to higher qubit numbers, with potential impact on quantum information tasks relying on Clifford symmetries.

Abstract

The n-qubit Pauli group and its normalizer the n-qubit Clifford group have applications in quantum error correction and device characterization. Recent applications have made use of the representation theory of the Clifford group. We apply the tools of (the coincidentally named) Clifford theory to examine the representation theory of the Clifford group using the much simpler representation theory of the Pauli group. We find an unexpected correspondence between irreducible characters of the n-qubit Clifford group and those of the (n+1)-qubit Clifford group.

The Clifford theory of the n-qubit Clifford group

TL;DR

This work develops a Clifford-theoretic bridge between the n-qubit Pauli group and the n-qubit Clifford group by relating representations of the normal abelian Pauli subgroup to the full Clifford group. It identifies a central extension structure where the inertia quotient is the affine symplectic group, enabling a surprising correspondence between irreducible characters of and . The authors construct irreps of from those of via induction from the inertia group, leveraging the identical character tables of and . This yields a practical, principled method to grow Clifford-group representations to higher qubit numbers, with potential impact on quantum information tasks relying on Clifford symmetries.

Abstract

The n-qubit Pauli group and its normalizer the n-qubit Clifford group have applications in quantum error correction and device characterization. Recent applications have made use of the representation theory of the Clifford group. We apply the tools of (the coincidentally named) Clifford theory to examine the representation theory of the Clifford group using the much simpler representation theory of the Pauli group. We find an unexpected correspondence between irreducible characters of the n-qubit Clifford group and those of the (n+1)-qubit Clifford group.
Paper Structure (11 sections, 18 theorems, 77 equations, 7 tables)

This paper contains 11 sections, 18 theorems, 77 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representation theory
  4. Clifford theory
  5. The Pauli and Clifford groups
  6. Definitions
  7. The symplectic structure of the Clifford group
  8. The character table of the projective Pauli group
  9. The inertia subgroup
  10. The representation theory of the inertia quotient group
  11. Lifting irreducible characters to higher dimensional Clifford groups

Key Result

Theorem 2.3

Let $(W,\psi)$ be a representation of $H$, and $H$ be a subgroup of $G$. There exists a linear representation of $G$ induced by $\psi$ which we denote $\mathrm{Ind}_H^G\psi$ or $\mathrm{Ind}_H^GW$. This induced representation is unique up to isomorphism.

Theorems & Definitions (36)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Lemma 2.5
  • Theorem 2.6: Ceccherini-Silberstein2009 Theorem 2.1
  • Theorem 2.7: Clifford Correspondence
  • Corollary 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 26 more