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Paper 1

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

A. Naghipour, M. A. Jafarizadeh, S. Shahmorad

Year

2014

Journal

arXiv preprint

DOI

arXiv:1407.6228

arXiv

1407.6228

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

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Paper 2

The injective norm of CSS quantum error-correcting codes

Stephane Dartois, Gilles Zémor

Year

2025

Journal

arXiv preprint

DOI

arXiv:2510.23736

arXiv

2510.23736

In this paper, we compute the injective norm - a.k.a. geometric entanglement - of standard basis states of CSS quantum error-correcting codes. The injective norm of a quantum state is a measure of genuine multipartite entanglement. Computing this measure is generically NP-hard. However, it has been computed exactly in condensed-matter theory - notably in the context of topological phases - for the Kitaev code and its extensions, in works by Orús and collaborators. We extend these results to all CSS codes and thereby obtain the injective norm for a nontrivial, infinite family of quantum states. In doing so, we uncover an interesting connection to matroid theory and Edmonds' intersection theorem.

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