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Frontiers in Physics

Construction of quaternary quantum error-correcting codes via orthogonal arrays

Shanqi Pang, Fuyuan Yang, Rong Yan, Jiao Du, Tian‐Yin Wang

Published Feb 24, 2023• 4 citations
Publisher

Abstract

This paper presents a method based on orthogonal arrays of constructing pure quaternary quantum error-correcting codes. As an application of the method, some infinite classes of quantum error-correcting codes with distances 2, 3, and 4 can be obtained. Moreover, the infinite class of quantum codes with distance 2 is optimal. The advantage of our method also lies in the fact that the quantum codes we obtain have less items for a basis quantum state than existing ones.

DOI
10.3389/fphy.2023.1148398

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This paper presents a method based on orthogonal arrays of constructing pure quaternary quantum error-correcting codes. As an application of the method, some infinite classes of quantum error-correcting codes with distances 2, 3, and 4 can be obtained. Moreover, the infinite class of quantum codes with distance 2 is optimal. The advantage of our method also lies in the fact that the quantum codes we obtain have less items for a basis quantum state than existing ones.
In this paper, by using the Hamming distance, we establish a relation between quantum error-correcting codes ((N,K,d+1))s and orthogonal arrays with orthogonal partitions. Therefore, this is a generalization of the relation between quantum error-correcting codes ((N,1,d+1))s and irredundant orthogonal arrays. This relation is used for the construction of pure quantum error-correcting codes. As applications of this method, numerous infinite families of optimal quantum codes can be constructed explicitly such as ((3,s,2))s for all si≥3, ((4,s2,2))s for all si≥5, ((5,s,3))s for all si≥4, ((6,s2,3))s for all si≥5, ((7,s3,3))s for all si≥7, ((8,s2,4))s for all si≥9, ((9,s3,4))s for all si≥11, ((9,s,5))s for all si≥9, ((10,s2,5))s for all si≥11, ((11,s,6))s for all si≥11, and ((12,s2,6))s for all si≥13, where s=s1⋯sn and s1,…,sn are all prime powers. The advantages of our approach over existing methods lie in the facts that these results are not just existence results, but constructive results, the codes constructed are pure, and each basis state of these codes has far less terms. Moreover, the above method developed can be extended to construction of quantum error-correcting codes over mixed alphabets.
By using difference schemes, orthogonal partitions and a replacement method, some new methods to construct pure quantum error-correcting codes are provided from orthogonal arrays. As an application of these methods, we construct several infinite series of quantum error-correcting codes including some optimal ones. Compared with the existing binary quantum codes, more new codes can be constructed, which have a lower number of terms (i.e., the number of computational basis states) for each of their basis states.
The method of constructing quantum error-correcting code by the quaternary self-orthogonal sub-code chains and L-chains are investigated in this paper.Random searching method is used to find quaternary self-orthogonal codes and these self-orthogonal codes could form some self-orthogonal code chains.By using combinatorial method,quantum error-correcting codes are constructed in light of L-chains which are constructed by dual of these self-orthogonal codes.The code chains of these self-orthogonal sub-codes and the L-chains which are obtained from the dual of these self-orthogonal sub-codes of length n between 20 and 36 and n=40,45,50,55,60 and dual distance five and six are determined.Some quantum codes of distance five and six are constructed by the obtained L-chains,some quantum error-correcting codes are new.
A heuristic algorithm is designed to searching for good higher dimensional self-orthogonal codes over GF(A) from low dimensional self-orthogonal codes. Many self-orthogonal codes of length 20 les n les 36 and dual distance 5 or 6 are obtained, and several have improved dual distance. Consequently, using these self-orthogonal codes and their dual codes, some linear quantum codes of minimum distance five or six for such length n are obtained, and three of these codes [[24, 8, 5]], [[25, 9, 5]], [[27, 7, 6]] are record-breaking. Quaternary code, self-orthogonal code, linear quantum error-correcting code.
By using construction arithmetic,the subcodes of quaternary self-dual codes with code length 10≤n≤20 are studied and seve-ral self-orthogonal codes' matrixes with dual distance 3,4,5 and 6 are constructed.The series of the self-dual and self-orthogonal chains are constructed by these matrixes and the subcodes' L-chains with maximal dual distance are deduced.In the end,some quantum error correct codes are constructed by using these quaternary subcodes' L-chains and the construction theory of additive quantum error correct code.Some quantum error correct codes with good parameters are better than previously known.
The construction of quantum error-correcting codes (QECCs) with good parameters is a hot topic in the area of quantum information and quantum computing. Quantum maximum distance separable (QMDS) codes are optimal because the minimum distance cannot be improved for a given length and code size. The QMDS codes over mixed alphabets are rarely known even if the existence and construction of QECCs over mixed alphabets with minimum distance more than or equal to three are still an open question. In this paper, we define an mmm-QMDS code over mixed alphabets, which is a generalization of QMDS codes. We establish a relation between mmm-QMDS codes over mixed alphabets and asymmetrical orthogonal arrays (OAs) with orthogonal partitions. Using this relation, we propose a general method to construct mmm-QMDS codes. As applications of this method, numerous infinite families of mmm-QMDS codes over mixed alphabets can be constructed explicitly. Compared with existing codes, the constructed codes have more flexibility in the choice of parameters, such as the alphabet sizes, length and dimension of the encoding state.