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Journal of Combinatorial Designs

Quantum twisted codes

J�rgen Bierbrauer, Yves Edel

Published Jan 1, 2000• 154 citations
Publisher

Abstract

A major contribution of [1] is a reduction of the problem of correcting errors in quantum computations to the construction of codes in binary symplectic spaces. This mechanism is known as the additive or stabilizer construction. We consider an obvious generalization of these quantum codes in the symplectic geometry setting and obtain general constructions using our theory of twisted BCH-codes (also known as Reed–Solomon subspace subcodes). This leads to families of quantum codes with good parameters. Moreover, the generator matrices of these codes can be described in a canonical way. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 174–188, 2000

DOI
10.1002/(sici)1520-6610(2000)8:3<174::aid-jcd3>3.0.co;2-t

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A major contribution of [1] is a reduction of the problem of correcting errors in quantum computations to the construction of codes in binary symplectic spaces. This mechanism is known as the additive or stabilizer construction. We consider an obvious generalization of these quantum codes in the symplectic geometry setting and obtain general constructions using our theory of twisted BCH-codes (also known as Reed–Solomon subspace subcodes). This leads to families of quantum codes with good parameters. Moreover, the generator matrices of these codes can be described in a canonical way. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 174–188, 2000
Stabilizer codes form an important class of quantum error correcting codes which have an elegant theory, efficient error detection, and many known examples. Constructing stabilizer codes of length nnn is equivalent to constructing subspaces of Fpn×Fpn\mathbb{F}_p^n \times \mathbb{F}_p^nFpn​×Fpn​ which are "isotropic" under the symplectic bilinear form defined by ⟨(a,b),(c,d)⟩=aTd−bTc\left\langle (\mathbf{a},\mathbf{b}),(\mathbf{c},\mathbf{d}) \right\rangle = \mathbf{a}^{\mathrm{T}} \mathbf{d} - \mathbf{b}^{\mathrm{T}} \mathbf{c}⟨(a,b),(c,d)⟩=aTd−bTc. As a result, many, but not all, ideas from the theory of classical error correction can be translated to quantum error correction. One of the main theoretical contribution of this article is to study stabilizer codes starting with a different symplectic form. In this paper, we concentrate on cyclic codes. Modifying the symplectic form allows us to generalize the previous known construction for linear cyclic stabilizer codes, and in the process, circumvent some of the Galois theoretic no-go results proved there. More importantly, this tweak in the symplectic form allows us to make use of well known error correcting algorithms for cyclic codes to give efficient quantum error correcting algorithms. Cyclicity of error correcting codes is a "basis dependent" property. Our codes are no more "cyclic" when they are derived using the standard symplectic forms (if we ignore the error correcting properties like distance, all such symplectic forms can be converted to each other via a basis transformation). Hence this change of perspective is crucial from the point of view of designing efficient decoding algorithm for these family of codes. In this context, recall that for general codes, efficient decoding algorithms do not exist if some widely believed complexity theoretic assumptions are true.
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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.
Abstract The development of quantum codes and entanglement-assisted quantum codes with improved parameters through classical codes has been a critical area in both quantum communications and quantum information theory. In this paper, we introduce a novel approach using a family of one-generator quasi-twisted codes of index 3 and provide an extended version of them that maintain self-orthogonality, enabling the creation of stabilizer quantum codes. Computational findings include the construction of binary stabilizer codes with good parameters. Additionally, we propose techniques to design maximal-entanglement entanglement-assisted quantum codes based on this class of quasi-twisted codes. As a practical outcome, we find some optimal quantum codes and good maximal-entanglement entanglement-assisted quantum codes derived from these class of quasi-twisted codes and their extension.
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