Investigations on Automorphism Groups of Quantum Stabilizer Codes

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.