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Cryptography and Communications

Quantum stabiliser codes via concatenation of idempotents in commutative group algebras

Kai Lin Ong

Published Oct 25, 2025• 1 citation
Publisher

Abstract

Abstract Group algebras offer a rich algebraic perspective for the study of classical codes. Through this lens, idempotents have been shown to characterise properties of their associated classical codes. However, attempts to extend the use of idempotents to the construction of quantum stabiliser codes have remained limited, arguably due to insufficient variation in their coefficients. Therefore, this paper proposes a new method of concatenating idempotents from commutative quaternary group algebras and studies the constructed stabiliser codes. The resulting stabilisers are binary subspaces of the respective group algebras, each generated by a single element whose components are distinct idempotents that together exhibit sufficient coefficient variation. Notably, these generators may be units, which leads to an interesting deviation from the conventional setting of idempotents as zero-divisors. The potential of these generators in stabiliser formalism are studied via an underlying group involution. It is proven that the parameters of the quantum stabiliser codes can be characterised by the idempotents involved.

DOI
10.1007/s12095-025-00846-y

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Abstract Group algebras offer a rich algebraic perspective for the study of classical codes. Through this lens, idempotents have been shown to characterise properties of their associated classical codes. However, attempts to extend the use of idempotents to the construction of quantum stabiliser codes have remained limited, arguably due to insufficient variation in their coefficients. Therefore, this paper proposes a new method of concatenating idempotents from commutative quaternary group algebras and studies the constructed stabiliser codes. The resulting stabilisers are binary subspaces of the respective group algebras, each generated by a single element whose components are distinct idempotents that together exhibit sufficient coefficient variation. Notably, these generators may be units, which leads to an interesting deviation from the conventional setting of idempotents as zero-divisors. The potential of these generators in stabiliser formalism are studied via an underlying group involution. It is proven that the parameters of the quantum stabiliser codes can be characterised by the idempotents involved.
Abstract Certain classical codes can be viewed isomorphically as ideals of group algebras, while studying their algebraic structures help extracting the code properties. Research has shown that this was remarkably efficient in the case when the code generators are idempotents. In quantum error correction, the theory of stabilizer formalism requires classical self-orthogonal additive codes over the finite field GF(4), which, via the lens of group algebras, are essentially F2F_2F2​ F 2 -submodules over GF(4). Therefore, this paper provides a classification on idempotents in commutative group algebra GF(4)G, followed by a criterion that allows idempotents to generate stabilizer subgroups. Later, the construction of quantum stabilizer codes is done in the case when G is a cyclic group CnC_nCn​ C n , for n=2m−1n=2^m-1n=2m−1 n = 2 m - 1 and n=2m+1n=2^m+1n=2m+1 n = 2 m + 1 . Quantum bounds on their burst error minimum distance are subsequently determined.
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 minimum distances 5, 7 and 8 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.
The structural properties and construction methodologies of quantum codes (QCs) defined over the finite field Z_3, the finite field with three elements, are examined in this work. The primary focus is on utilizing constacyclic codes within a specific algebraic structure the finite commutative non-chain ring R=Z3+uZ3+vZ3+uvZ3, where the indeterminates u and v satisfy the relations u2=1, v2=1, and uv=vu. We provide a set of idempotent generators of the ring and define linear codes and calculated some self-inverse units. The relationship between R and Z34 is established using a gray map. Constacyclic (CC) codes are broken down into cyclic (C) codes and negacyclic (N) codes in order to determine the parameters of QCs over Z_3. Several QCs of arbitrary lengths are developed as an application. In this paper, we calculated those various parameters of QCs which are better than existing parameters of QCs. This paper contains only those units which have self-inverse.
In this short letter, our main objective is to construct good quantum stabilizer codes over small fields, which have a more potential possibility to be employed in actual quantum processors. We firstly consider a suitable class of two-generators quasi-cyclic (QC) codes and determine their parameters. Then by virtue of their symplectic dual algebraic structure, we derive a sufficient condition to be symplectic dual-containing of these codes. Subsequently, an explicit construction of stabilizer codes from quasi-cyclic codes is presented. As for computational results, we provide many stabilizer codes with good parameters over binary, ternary and quaternary fields that all can not be deduced by the quantum Gilbert-Varshamov (GV) bound. In the binary case, our several codes strictly improve the lower bounds on the minimum distance in current records. In the ternary and quaternary occasions, numerous codes fill some gaps or have better parameters than the current results.
The concept of generalized concatenated quantum codes (GCQC) provides a systematic way for constructing good quantum codes from short component codes. We introduce a stabilizer formalism for GCQCs, which is achieved by defining quantum coset codes. This formalism offers a new perspective for GCQCs and enables us to derive a lower bound on the code distance of stabilizer GCQCs from component codes parameters, for both non-degenerate and degenerate component codes. Our formalism also shows how to exploit the error-correcting capacity of component codes to design good GCQCs efficiently.
The concept of generalized concatenated quantum codes (GCQC) provides a systematic way for constructing good quantum codes from short component codes. We introduce a stabilizer formalism for GCQCs, which is achieved by defining quantum coset codes. This formalism offers a new perspective for GCQCs and enables us to derive a lower bound on the code distance of stabilizer GCQCs from component codes parameters, for both non-degenerate and degenerate component codes. Our formalism also shows how to exploit the error-correcting capacity of component codes to design good GCQCs efficiently.