Quantum stabilizer codes from Abelian and non-Abelian groups association schemes

TL;DR

This paper introduces a novel method for constructing binary quantum stabilizer codes using association schemes derived from Abelian and non-Abelian groups, enabling the creation of codes with various distances and parameters.

Contribution

The paper presents a new construction method for binary quantum stabilizer codes based on association schemes from both Abelian and non-Abelian groups, expanding the toolkit for quantum code design.

Findings

Constructed binary stabilizer codes up to 40 qubits.

Presented codes with distances 5 and 7 with good quantum parameters.

Demonstrated the effectiveness of Abelian group schemes for flexible code construction.

Abstract

A new method for the construction of the binary quantum stabilizer codes is provided, where the construction is based on Abelian and non-Abelian groups association schemes. The association schemes based on non-Abelian groups are constructed by bases for the regular representation from U6n, T4n, V8n and dihedral D2n groups. By using Abelian group association schemes followed by cyclic groups and non-Abelian group association schemes a list of binary stabilizer codes up to 40 qubits is given in tables 4, 5, and 10. Moreover, several binary stabilizer codes of distances 5 and 7 with good quantum parameters is presented. The preference of this method specially for Abelian group association schemes is that one can construct any binary quantum stabilizer code with any distance by using the commutative structure of association schemes