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Communications on Applied Nonlinear Analysis

New Quantum Codes using Monomial Matrix over Hermitian Dual-Containing Matrix Product Codes

Shivender Goswami

Published Feb 21, 2025• 0 citations
Publisher

Abstract

Quantum error correction (QEC) is beneficial for ensuring reliable quantum computation and communication by addressing the susceptibility of quantum states to errors from decoherence and noise. This study explores the use of Quantum Matrix Product Codes (MPCs) with an emphasis on monomial matrices and Hermitian duals to achieve efficient and robust error correction. We provide a detailed mathematical formulation of MPCs utilizing monomial matrices, emphasizing the importance of Hermitian duals in maintaining code integrity and ensuring effective error correction. The paper proposes a new criterion for utilizing the rank of the generator matrix associated with linear codes established from Hermitian dual-containing (HDC) of the MPCs. Then, using this criterion, a new set of quantum maximum-distance-separable (MDS) codes has been constructed with better code parameters and error-correction potential. For comparison purpose, different codes have been observed alongside with the obtained codes to ascertain the uniqueness and error correction capabilities.

DOI
10.52783/cana.v32.3963

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Quantum error correction (QEC) is beneficial for ensuring reliable quantum computation and communication by addressing the susceptibility of quantum states to errors from decoherence and noise. This study explores the use of Quantum Matrix Product Codes (MPCs) with an emphasis on monomial matrices and Hermitian duals to achieve efficient and robust error correction. We provide a detailed mathematical formulation of MPCs utilizing monomial matrices, emphasizing the importance of Hermitian duals in maintaining code integrity and ensuring effective error correction. The paper proposes a new criterion for utilizing the rank of the generator matrix associated with linear codes established from Hermitian dual-containing (HDC) of the MPCs. Then, using this criterion, a new set of quantum maximum-distance-separable (MDS) codes has been constructed with better code parameters and error-correction potential. For comparison purpose, different codes have been observed alongside with the obtained codes to ascertain the uniqueness and error correction capabilities.
The advancement of Quantum Error-Correcting (QEC) Maximum Distance Separable (MDS) codes holds substantial importance in practical applications, substantially augmenting the reliability and efficiency of quantum communication and computing. This paper introduces two new classes of QEC MDS codes, which are devised through the utilization of generalized Reed-Solomon (GRS) codes and the Hermitian construction approach. The novelty of our QEC MDS codes lies in their parameters being distinct from all previously reported codes. Moreover, most of our codes possess a considerably greater minimum distance in comparison to existing codes of the same length.
Matrix-product codes over finite fields are an important class of long linear codes by combining several commensurate shorter linear codes with a defining matrix over finite fields. The construction of matrix-product codes with certain self-orthogonality over finite fields is an effective way to obtain good qqq-ary quantum codes of large length. Specifically, it follows from CSS construction (resp. Hermitian construction) that a matrix-product code over Fq\mathbb{F}_{q}Fq​ (resp. Fq2\mathbb{F}_{q^{2}}Fq2​) which is Euclidean dual-containing (resp. Hermitian dual-containing) can produce a qqq-ary quantum code. In order to obtain such matrix-product codes, a common way is to construct quasi-orthogonal matrices (resp. quasi-unitary matrices) as the defining matrices of matrix-product codes over Fq\mathbb{F}_{q}Fq​ (resp. Fq2\mathbb{F}_{q^{2}}Fq2​). The usage of NSC quasi-orthogonal matrices or NSC quasi-unitary matrices in this process enables the minimum distance lower bound of the corresponding quantum codes to reach its optimum. This article has two purposes: the first is to summarize some results of this topic obtained by the author of this article and his cooperators in \cite{Cao2020Constructioncaowang,Cao2020New,Cao2020Constructionof}; the second is to add some new results on quasi-orthogonal matrices (resp. quasi-unitary matrices), Euclidean dual-containing (resp. Hermitian dual-containing) matrix-product codes and qqq-ary quantum codes derived from these newly constructed matrix-product codes.
Matrix-product codes over finite fields are an important class of long linear codes by combining several commensurate shorter linear codes with a defining matrix over finite fields. The construction of matrix-product codes with certain self-orthogonality over finite fields is an effective way to obtain good qqq-ary quantum codes of large length. This article has two purposes: the first is to summarize some results of this topic obtained by the author of this article and his cooperators in [10-12]; the second is to add some new results on quasi-orthogonal matrices (resp. quasi-unitary matrices), Euclidean dual-containing (resp. Hermitian dual-containing) matrix-product codes and qqq-ary quantum codes derived from these matrix-product codes.
The matrix-product (MP) code CA,k:=[C1,C2,…,Ck]⋅A\mathcal{C}_{A,k}:=[\mathcal{C}_{1},\mathcal{C}_{2},\ldots,\mathcal{C}_{k}]\cdot ACA,k​:=[C1​,C2​,…,Ck​]⋅A with a non-singular by column (NSC) matrix AAA plays an important role in constructing good quantum error-correcting codes. In this paper, we study the MP code when the defining matrix AAA satisfies the condition that AA†AA^†AA† is (D,τ)(D,τ)(D,τ)-monomial. We give an explicit formula for calculating the dimension of the Hermitian hull of a MP code. We provide the necessary and sufficient conditions that a MP code is Hermitian dual-containing (HDC), almost Hermitian dual-containing (AHDC), Hermitian self-orthogonal (HSO), almost Hermitian self-orthogonal (AHSO), and Hermitian LCD, respectively. We theoretically determine the number of all possible ways involving the relationships among the constituent codes to yield a MP code with these properties, respectively. We give alternative necessary and sufficient conditions for a MP code to be AHDC and AHSO, respectively, and show several cases where a MP code is not AHDC or AHSO. We provide the construction methods of HDC and AHDC MP codes, including those with optimal minimum distance lower bounds.
In this paper, from special code chainsC1⊇C2 ⊇C3\mathcal {C}_{1} \supseteq \mathcal {C}_{2}~\supseteq \mathcal {C}_{3}C1​⊇C2​ ⊇C3​such thatC1⊥h⊆C3\mathcal {C}^{\bot _{h}}_{1}\subseteq \mathcal {C}_{3}C1⊥h​​⊆C3​andC2⊥h⊆C2\mathcal {C}^{\bot _{h}}_{2}\subseteq \mathcal {C}_{2}C2⊥h​​⊆C2​, some Hermitian dual-containing (HDC) matrix-product (MP) codes are presented, whereC3\mathcal {C}_{3}C3​is not HDC. By studying some constacyclic codes of lengthsn=q2±1n=q^{2}\pm 1n=q2±1andn=q2−12n=\frac {q^{2}-1}{2}n=2q2−1​, we construct many HDC MP codes of length3n3n3n. Consequently, manyqqq-ary quantum codes with larger designed distanced≥q+1d\geq q+1d≥q+1are obtained from these MP codes, where4≤q≤94\leq q\leq 94≤q≤9.