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Computer Engineering and Applications Journal

Construction of quantum codes from line graph of cube.

Lina Zhang

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

Quantum error-correcting codes play an important role in not only quantum communication but also quantum computation.Many good quantum error-correcting codes have been constructed by using classical linear codes,for example,Hamming codes,BCH codes,RS codes and Reed-Muller codes.A new method to obtain quantum codes and asymmetric quantum codes is presented.Based on line graph of the n-cube,infinite families of new quantum codes and asymmetric quantum codes are presented.Moreover,if the length of asymmetric quantum codes is large,the asymmetric quantum codes are able to correct quantum errors with great asymmetry.

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Quantum error-correcting codes play an important role in not only quantum communication but also quantum computation.Many good quantum error-correcting codes have been constructed by using classical linear codes,for example,Hamming codes,BCH codes,RS codes and Reed-Muller codes.A new method to obtain quantum codes and asymmetric quantum codes is presented.Based on line graph of the n-cube,infinite families of new quantum codes and asymmetric quantum codes are presented.Moreover,if the length of asymmetric quantum codes is large,the asymmetric quantum codes are able to correct quantum errors with great asymmetry.
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.
In this paper we extend to asymmetric quantum error-correcting codes (AQECC) the construction methods, namely: puncturing, extending, expanding, direct sum and the (u|u + v) construction. By applying these methods, several families of asymmetric quantum codes can be constructed. Consequently, as an example of application of quantum code expansion developed here, new families of asymmetric quantum codes derived from generalized Reed-Muller (GRM) codes, quadratic residue (QR), Bose-Chaudhuri-Hocquenghem (BCH), character codes and affine-invariant codes are constructed.
Recently, the theory of quantum error control codes has been extended to include quantum codes over asymmetric quantum channels - qubit-flip and phase-shift errors may have equal or different probabilities. Previous work in constructing quantum error control codes has focused on code constructions for symmetric quantum channels. In this paper we establish a method to construct asymmetric quantum codes based on classical codes. We derive, once again, families of asymmetric quantum codes from classical BCH and RS codes over finite fields. Particularly, we present interesting asymmetric quantum codes based on nonprimitive narrow-sense BCH codes with parameters [[n, k, dz/dx]]qfor certain values of code lengths, dimensions, and various minimum distance. Finally, our constructions are well explained by an illustrative example.
Long quantum codes using projective Reed–Muller codes are constructed. Projective Reed–Muller codes are evaluation codes obtained by evaluating homogeneous polynomials at the projective space. We obtain asymmetric and symmetric quantum codes by using the CSS construction and the Hermitian construction, respectively. We provide entanglement-assisted quantum error-correcting codes from projective Reed–Muller codes with flexible amounts of entanglement by considering equivalent codes. Moreover, we also construct quantum codes from subfield subcodes of projective Reed–Muller codes.
Recently, the theory of quantum error control codes has been extended to subsystem codes over symmetric and asymmetric quantum channels -- qubit-flip and phase-shift errors may have equal or different probabilities. Previous work in constructing quantum error control codes has focused on code constructions for symmetric quantum channels. In this paper, we develop a theory and establish the connection between asymmetric quantum codes and subsystem codes. We present families of subsystem and asymmetric quantum codes derived, once again, from classical BCH and RS codes over finite fields. Particularly, we derive an interesting asymmetric and symmetric subsystem codes based on classical BCH codes with parameters [[n,k,r,d]]q[[n,k,r,d]]_q[[n,k,r,d]]q​, [[n,k,r,dz/dx]]q[[n,k,r,d_z/d_x]]_q[[n,k,r,dz​/dx​]]q​ and [[n,k′,0,dz/dx]]q[[n,k',0,d_z/d_x]]_q[[n,k′,0,dz​/dx​]]q​ for arbitrary values of code lengths and dimensions. We establish asymmetric Singleton and Hamming bounds on asymmetric quantum and subsystem code parameters; and derive optimal asymmetric MDS subsystem codes. Finally, our constructions are well explained by an illustrative example. This paper is written on the occasion of the 50th anniversary of the discovery of classical BCH codes and their quantum counterparts were derived nearly 10 years ago.
The theory of quantum error correcting codes is a primary tool for fighting decoherence and other quantum noise in quantum communication and quantum computation. Recently, the theory of quantum error correcting codes has developed rapidly and been extended to protect quantum information over asymmetric quantum channels, in which phase-shift and qubit-flip errors occur with different probabilities. In this paper, we generalize the construction of symmetric quantum codes via graphs (or matrices) to the asymmetric case, converting the construction of asymmetric quantum codes to finding matrices with some special properties. We also propose some asymmetric quantum Maximal Distance Separable (MDS) codes as examples constructed in this way.