Investigations on Automorphism Groups of Quantum Stabilizer Codes

TL;DR

This paper explores the automorphism groups of quantum stabilizer codes, introducing new concepts and examples, and investigates how these symmetries relate to the codes' error-correcting capabilities and transitivity properties.

Contribution

It introduces weak and Clifford-twisted automorphism groups for stabilizer codes, provides examples linking automorphism groups to Mathieu groups, and proves nonexistence results for highly transitive automorphism groups.

Findings

Clifford-twisted automorphism groups may connect to Mathieu groups.

Highly transitive automorphism groups are incompatible with certain stabilizer codes.

New automorphism group concepts deepen understanding of quantum code symmetries.

Abstract

The stabilizer formalism for quantum error-correcting codes has been, without doubt, the most successful at producing examples of quantum codes with strong error-correcting properties. In this paper, we discuss strong automorphism groups of stabilizer codes, beginning with the analogous notion from the theory of classical codes. Two weakenings of this concept, the weak automorphism group and Clifford-twisted automorphism group, are also discussed, along with many examples highlighting the possible relationships between the types of "automorphism groups". In particular, we construct an example of a $[[10,0,4]]$ stabilizer code showing how the Clifford-twisted automorphism groups might be connected to the Mathieu groups. Finally, nonexistence results are proved regarding stabilizer codes with highly transitive strong and weak automorphism groups, suggesting a potential inverse relationship between the error-correcting properties of a quantum code and the transitivity of those automorphism groups.