Decomposition of Pauli groups via weak central products

TL;DR

This paper presents a decomposition of Pauli groups on qudits using weak central products, revealing their structure and aiding in identifying abelian subgroups, with implications for quantum error correction.

Contribution

It introduces a novel decomposition method for Pauli groups via weak central products, extending to lifted groups relevant in quantum error correction.

Findings

Decomposition of Pauli groups into weak central products

Identification of abelian subgroups within Pauli groups

Factorization of lifted Pauli groups for quantum codes

Abstract

For any $m \ge 1$ and odd prime power $\mathtt{q}=\mathtt{p}^m$, for $\mathtt{q}=2$, and for any $n \ge 1$, we show a result of decomposition for Pauli groups $\mathcal{P}_{n,\mathtt{q}}$ in terms of weak central products. This can be used to describe the underlying structure of Pauli groups on $n$ qudits of dimension $\mathtt{q}$ and enables us to identify abelian subgroups of $\mathcal{P}_{n,\mathtt{q}}$. As a consequence of our main results, we show a similar factorisation for the so--called `lifted' Pauli groups, recently introduced by Gottesman and Kuperberg in the context of error-correcting codes in quantum information theory.