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China Communications

Constructing non-binary asymmetric quantum codes via graphs

Zhi Ma, Leng Riguang, Zhengchao Wei, Zhong Shuqin

Published Feb 1, 2013• 1 citation
Publisher

Abstract

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.

DOI
10.1109/cc.2013.6472857

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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.
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.
Abstract. It is recently conjectured in quantum information processing that phase-shift errors occur with high probability than qubit-flip errors, hence the former is more disturbing to quantum information than the later one. This leads us to construct asymmetric quantum error controlling codes to protect quantum information over asymmetric channels, Pr Z ≥ Pr X. In this paper we present two generic methods to derive asymmetric quantum cyclic codes using the generator polynomials and defining sets of classical cyclic codes. Consequently, the methods allow us to construct several families of asymmetric quantum BCH, RS, and RM codes. Finally, the methods are used to construct families of asymmetric subsystem codes. 1
Asymmetric quantum codes are useful for quantum channels in which phase and bit errors occur with different probabilities, since the two distances, dz and dx, can be controlled separately. We develop a permutation-paired matrix-product construction for such codes over Fq. The main task is to build classical code pairs C,D⊆Fq2kn satisfying the Hermitian inclusion D⊥H⊆C, while keeping explicit dimension and distance bounds. Let A∈Fq2k×k be a non-singular-by-columns (NSC) matrix with AA†=DPτ, where D is an invertible diagonal and Pτ corresponds to an involution τ. For C=[C1,…,Ck]A and D=[D1,…,Dk]A, we prove D⊥H=[Dτ(1)⊥H,…,Dτ(k)⊥H]A. Thus, the global inclusion D⊥H⊆C is equivalent to the shorter paired inclusions Dτ(i)⊥H⊆Ci. This yields asymmetric quantum codes with parameters [[kn,∑i=1k(ri+si)−kn,dz/dx]]q, where the bounds for dz and dx follow from NSC matrix-product distance estimates. For nested maximum distance separable (MDS) constituents, the paired conditions reduce to ri+sτ(i)≥n, giving explicit infinite families. Concrete τ-OD matrices and numerical examples show that nontrivial permutations can increase the quantum dimension while preserving prescribed lower bounds for dz and dx.
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.