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International Journal of Quantum Information

SYMMETRIC AND ASYMMETRIC QUANTUM CODES

Kenza Guenda, Thomas Aaron Gulliver

Published Aug 1, 2013• 11 citations
Publisher

Abstract

The asymmetric CSS construction is extended to the Hermitian dual case. New infinite families of quantum symmetric and asymmetric codes are constructed. In particular, new quantum codes are obtained from binary BCH codes and MDS codes. These codes have known minimum distances and thus the relationship between the rate gain and minimum distance is given explicitly. The codes obtained are shown to have parameters better than those of previous codes. A number of known codes are special cases of the codes given here.

DOI
10.1142/s0219749913500470

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The asymmetric CSS construction is extended to the Hermitian dual case. New infinite families of quantum symmetric and asymmetric codes are constructed. In particular, new quantum codes are obtained from binary BCH codes and MDS codes. These codes have known minimum distances and thus the relationship between the rate gain and minimum distance is given explicitly. The codes obtained are shown to have parameters better than those of previous codes. A number of known codes are special cases of the codes given here.
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In this paper, we discuss two problems on asymmetric quantum error-correcting codes (AQECCs). The first one is on the construction of a [[12, 1, 5/3]]2asymmetric quantum code, we show an impure [[12, 1, 5/3 ]]2exists. The second one is on the construction of AQECCs from binary cyclic codes, we construct many families of new asymmetric quantum codes with dz> δmax+1 from binary primitive cyclic codes of length n = 2m-1, where δmax= 2⌈m/2⌉-1 is the maximal designed distance of dual containing narrow sense BCH code of length n = 2m-1. A number of known codes are special cases of the codes given here. Some of these AQECCs have parameters better than the ones available in the literature.
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.
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.