A note on the stabilizer formalism via noncommutative graphs

TL;DR

This work unifies stabilizer quantum error correction with the noncommutative-graph framework by constructing operator-system subspaces $\mathcal{V}_{M_0}$ from unitary group representations. For Abelian $G$, it establishes an anticlique criterion: if $\mathcal{V}_{M_0}$ is an operator system and $P=\prod_{g\in G}P_{i_g}(g)$ has rank at least $2$, then $P$ is an anticlique for $\mathcal{V}_{M_0}$, i.e., $P\mathcal{V}_{M_0}P=\mathbb{C}P$. Furthermore, when $G$ is Abelian and $-I_2^{\otimes n}\notin G$, the span of all such $\mathcal{V}_{M_0}$ coincides with the set of correctable Pauli errors outside the normalizer $N(G)$ plus the identity: $\operatorname{span}\{\mathcal{V}_{M_0} : \mathcal{V}_{M_0}\ \text{operator-system}\} = \operatorname{span}\{(\mathcal{P}_n \setminus N(G)) \cup I_2^{\otimes n}\}$. The authors further show how a stabilizer formalism emerges in this language, and how a basis change to a logical Pauli frame yields the conventional stabilizer error-detection picture. This provides a conceptual bridge between stabilizer codes and noncommutative-graph theory, with potential implications for code design within this operator-system framework.

Abstract

In this short note we formulate a stabilizer formalism in the language of noncommutative graphs. The classes of noncommutative graphs we consider are obtained via unitary representations of compact groups, and suitably chosen operators on finite-dimensional Hilbert spaces. Furthermore, in this framework, we generalize previous results in this area for determining when such noncommutative graphs have anticliques.

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• Definition 1
• Definition 2
• Remark 1
• Theorem 3
• proof
• Definition 3
• Theorem 4
• Theorem 5