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Entropy

Construction of Binary Quantum Error-Correcting Codes from Orthogonal Array

Shanqi Pang, Hanxiao Xu, Mengqian Chen

Published Jul 19, 2022• 10 citations
Publisher

Abstract

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.

DOI
10.3390/e24071000

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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.
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.
Quantum error-correcting codes protect quantum information against undesirable noise in quantum computation and quantum communication.It is one of important problems that constructing very good quantum codes.There are two methods that use binary codes to construct quantum codes,one is CSS construction,another is Steane construction.These two methods are all based on the construction of binary self-orthogonal codes.Combinational construction of binary self-orthogonal codes is investigated in this paper.From known codes chains of binary self-orthogonal codes and their dual codes chains with given dual distance,new binary self-orthogonal codes of dual distance three,four,five and six are constructed.Based on these results,for each satisfying,S-chains with parameters and are constructed.According to Steane's construction,many very good quantum codes of distance five and six are constructed by the obtained S-chains,some of these quantum codes are new and some of these quantum codes have improved parameters than previously known codes.
This paper provides new instances of quantum error-correcting codes. According to recent results by Nakayama and Hagiwara, by using two sets of binary sequences that satisfy three conditions, we can construct quantum error-correcting codes for single deletion errors. However, only two instances of two sets that satisfy the three conditions have been found to date. The present study constructs two infinite series of sets that satisfy the three conditions. In other words, this paper constructs new infinite series of quantum error-correcting codes for single deletion errors.
In this paper, we present a characterization of (binary and non-binary) quantum error-correcting codes. Based on this characterization, we introduce a method to construct p p -ary quantum codes using Boolean functions satisfying a system of certain quadratic relations. As a consequence of the construction, we are able to construct quantum codes of minimum distance 2 2 . In particular, we produce a class of binary quantum ( ( n , 2 n − 2 − 1 2 ( n − 1 ( n − 1 ) / 2 ) , 2 ) ) ((n,2^{n-2}-\frac 12{n-1\choose (n-1)/2},2)) -codes for odd length n ≥ 5 n\ge 5 . For n ≥ 11 n\ge 11 , this improves the result by Rains in Quantum codes of minimal distance two , 1999, showing the existence of binary quantum ( ( n , 3 ⋅ 2 n − 4 , 2 ) ) ((n,3\cdot 2^{n-4},2)) -codes for odd n ≥ 5 n\ge 5 . Moreover, our binary quantum