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Elementary Constructions of Best Known Quantum Codes

Tran, Long, Nguyen, Trang, Aydin, Nuh

Published Jul 1, 2024• 0 citations
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Abstract

Building large-scale quantum computers is a very active area of research. A critical part of this effort is the ability to control quantum errors. Recently, many good quantum error correcting codes (QECC) over various finite fields Fq have been constructed from codes over extension rings or mixed alphabet rings via some version of a Gray map. We show that most of these codes can be obtained more directly from classical codes, including cyclic, constacyclic, quasicyclic (QC), quasi-twisted (QT), and polycyclic codes using the CSS construction, Hermitian construction, and Lisonek-Singh methods. We believe direct and more elementary methods should be preferred.

Related papers

Elementary Constructions of Best Known Quantum Codes

Nuh Aydın, Nguyen, Trang T. T., Tran, Long B.

Recently, many good quantum codes over various finite fields $F_q$ have been constructed from codes over extension rings or mixed alphabet rings via some version of a Gray map. We show that most of these codes can be obtained more directly from cyclic codes or their generalizations over $F_q$. Unless explicit benefits are demonstrated for the indirect approach, we believe that direct and more elementary methods should be preferred.

arXiv:2410.12167•Oct 16, 2024•0 citations

On Optimal and Quantum Code Construction from Cyclic Codes over FqPQ with Applications

Shakir Ali, Amal S. Alali, Pushpendra Sharma +3 more

The key objective of this paper is to study the cyclic codes over mixed alphabets on the structure of FqPQ, where P=Fq[v]⟨v3−α22v⟩ and Q=Fq[u,v]⟨u2−α12,v3−α22v⟩ are nonchain finite rings and αi is in Fq/{0} for i∈{1,2}, where q=pm with m≥1 is a positive integer and p is an odd prime. Moreover, with the applications, we obtain better and new quantum error-correcting (QEC) codes. For another application over the ring P, we obtain several optimal codes with the help of the Gray image of cyclic codes.

Entropy•Aug 2, 2023•4 citations

ON QUANTUM CODES CONSTRUCTION FROM CONSTACYCLIC CODES OVER THE RING I_q[u,v] / <u^2-a^2, v^2-a^2, uv-vu>

Shakir Ali, Pushpendra Sharma

This paper focuses on studying the properties of constacyclic codes, quantum error-correcting codes. The code is studied over a specific mathematical structure called the ring $\mathfrak{S}$, which is defined as $\mathfrak{S}=\mathfrak{I}_q[\mathfrak{u},\mathfrak{v}]/\langle \mathfrak{u}^2-\alpha^2,~ \mathfrak{v}^2-\alpha^2,~\mathfrak{u}\mathfrak{v}-\mathfrak{v}\mathfrak{u} \rangle$, where $\mathfrak{I}_q$ is a finite field of $q$ elements, $\alpha$ be the nonzero elements of the field $\mathfrak{I}_q$ and $q$ is a power of an odd prime $p$ such that $q=p^m, ~\textup{for}~ m \ge 1$. The paper also introduces a Gray map and use it to decompose constacyclic codes over the ring $\mathfrak{S}$ into a direct sum of constacyclic codes over $\mathfrak{I}_q$. We construct new and better quantum error-correcting codes over the ring $\mathfrak{S}$ (cf.; Table 1 and Table 2). Moreover, we also obtain best known linear codes as well as best dimension linear codes (cf.; Table 4).

Journal of the Indonesian Mathematical Society•Aug 21, 2024•0 citations

Constacyclic Codes over Group Ring (Zq[v])/G

Alireza Soleimani

Recently, codes over some special finite rings especially chain rings have been studied. More recently, codes over finite non-chain rings have been also considered. Study on codes over such rings or rings in general is motivated by the existence of some special maps called Gray maps whose images give codes over fields. Quantum error-correcting (QEC) codes play a crucial role in protecting quantum information. The construction of quantum codes via classical codes over 2 F was first introduced by Calderbank and Shor [4] and Steane [13] in 1996. This method, known as CSS construction, has received a lot of attention and it has allowed to find many good quantum stabilizer codes. Later, construction of quantum codes over larger alphabets from classical linear codes over q F has shown by Ketkar et al. in [10]. One direction of the main research in quantum error correction codes is constructing quantum codes that have large minimum distances [9] for a given size and length. In [14], based on classical quaternary constacyclic linear codes, some parameters for quantum codes are obtained. In [8, 9], respectively based on classical negacyclic and constacyclic linear codes some parameters for quantum MDS codes are presented. In this work, we determine self-dual and self-orthogonal codes arising from constacyclic codes over the group ring(Zq[v]/G

Theory of Approximation and Applications•Mar 1, 2019•0 citations

Unified Approach to Construct Quantum Error-correcting Code

Jianfa Qian, Wenping Ma

Quantum error-correcting codes play an important role in not only quantum communication but also quantum computation.All kinds of cyclic codes,for example,Hamming codes,BCH codes and Reed-Solomon codes et al.,constacyclic codes and quasi-cyclic codes have been used to construct quantum error-correcting codes.An unified approach to construct quantum error-correcting codes was presented by using quasi-twisted codes.A sufficient and necessary condition for quasi-twisted contained its dual codes,and a new method for constructing quasi-twisted codes was given.Moreover,new quantum quasi-twisted codes were obtained by using quasi-twisted codes.

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Quantum error correction through constacyclic codes over mixed alphabet

Mohd Azmi, Mohammad Fareed Ahmad, Nadeem ur Rehman +1 more

Let q=pr, where p is an odd prime and r be a positive integer, 𝒮=𝔽q[u]〈u2−1〉 and 𝒯=𝔽q[u,v]〈u2−1,v2−1,uv=vu〉 and Ω=(ω1,ω2,ω3)∈𝔽q𝒮𝒯⁠. The algebraic structure of constacyclic codes over 𝔽q𝒮𝒯−Ω–constacyclic code of block length (l, m, n) is examined in this paper. The construction of quantum error-correcting codes (QECCs) from constacyclic codes over 𝔽q𝒮𝒯 is also covered in this paper and we derive some new QECCs.

AIP conference proceedings•Jan 1, 2026•0 citations
Building large-scale quantum computers is a very active area of research. A critical part of this effort is the ability to control quantum errors. Recently, many good quantum error correcting codes (QECC) over various finite fields Fq have been constructed from codes over extension rings or mixed alphabet rings via some version of a Gray map. We show that most of these codes can be obtained more directly from classical codes, including cyclic, constacyclic, quasicyclic (QC), quasi-twisted (QT), and polycyclic codes using the CSS construction, Hermitian construction, and Lisonek-Singh methods. We believe direct and more elementary methods should be preferred.
Recently, many good quantum codes over various finite fields FqF_qFq​ have been constructed from codes over extension rings or mixed alphabet rings via some version of a Gray map. We show that most of these codes can be obtained more directly from cyclic codes or their generalizations over FqF_qFq​. Unless explicit benefits are demonstrated for the indirect approach, we believe that direct and more elementary methods should be preferred.
The key objective of this paper is to study the cyclic codes over mixed alphabets on the structure of FqPQ, where P=Fq[v]⟨v3−α22v⟩ and Q=Fq[u,v]⟨u2−α12,v3−α22v⟩ are nonchain finite rings and αi is in Fq/{0} for i∈{1,2}, where q=pm with m≥1 is a positive integer and p is an odd prime. Moreover, with the applications, we obtain better and new quantum error-correcting (QEC) codes. For another application over the ring P, we obtain several optimal codes with the help of the Gray image of cyclic codes.
This paper focuses on studying the properties of constacyclic codes, quantum error-correcting codes. The code is studied over a specific mathematical structure called the ring S\mathfrak{S}S, which is defined as S=Iq[u,v]/⟨u2−α2, v2−α2, uv−vu⟩\mathfrak{S}=\mathfrak{I}_q[\mathfrak{u},\mathfrak{v}]/\langle \mathfrak{u}^2-\alpha^2,~ \mathfrak{v}^2-\alpha^2,~\mathfrak{u}\mathfrak{v}-\mathfrak{v}\mathfrak{u} \rangleS=Iq​[u,v]/⟨u2−α2, v2−α2, uv−vu⟩, where Iq\mathfrak{I}_qIq​ is a finite field of qqq elements, α\alphaα be the nonzero elements of the field Iq\mathfrak{I}_qIq​ and qqq is a power of an odd prime ppp such that q=pm, for m≥1q=p^m, ~\textup{for}~ m \ge 1q=pm, for m≥1. The paper also introduces a Gray map and use it to decompose constacyclic codes over the ring S\mathfrak{S}S into a direct sum of constacyclic codes over Iq\mathfrak{I}_qIq​. We construct new and better quantum error-correcting codes over the ring S\mathfrak{S}S (cf.; Table 1 and Table 2). Moreover, we also obtain best known linear codes as well as best dimension linear codes (cf.; Table 4).
Recently, codes over some special finite rings especially chain rings have been studied. More recently, codes over finite non-chain rings have been also considered. Study on codes over such rings or rings in general is motivated by the existence of some special maps called Gray maps whose images give codes over fields. Quantum error-correcting (QEC) codes play a crucial role in protecting quantum information. The construction of quantum codes via classical codes over 2 F was first introduced by Calderbank and Shor [4] and Steane [13] in 1996. This method, known as CSS construction, has received a lot of attention and it has allowed to find many good quantum stabilizer codes. Later, construction of quantum codes over larger alphabets from classical linear codes over q F has shown by Ketkar et al. in [10]. One direction of the main research in quantum error correction codes is constructing quantum codes that have large minimum distances [9] for a given size and length. In [14], based on classical quaternary constacyclic linear codes, some parameters for quantum codes are obtained. In [8, 9], respectively based on classical negacyclic and constacyclic linear codes some parameters for quantum MDS codes are presented. In this work, we determine self-dual and self-orthogonal codes arising from constacyclic codes over the group ring(Zq[v]/G
Quantum error-correcting codes play an important role in not only quantum communication but also quantum computation.All kinds of cyclic codes,for example,Hamming codes,BCH codes and Reed-Solomon codes et al.,constacyclic codes and quasi-cyclic codes have been used to construct quantum error-correcting codes.An unified approach to construct quantum error-correcting codes was presented by using quasi-twisted codes.A sufficient and necessary condition for quasi-twisted contained its dual codes,and a new method for constructing quasi-twisted codes was given.Moreover,new quantum quasi-twisted codes were obtained by using quasi-twisted codes.