Asymptotically good quantum codes

552

560

One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. In past years, many good quantum error-correcting codes had been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes over Fq in terms of classical codes over F q 2 is provided that generalizes the well-known notion of additive codes over F4 of the binary case. This paper derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum BCH codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper.

Section: Conclusion and Open Problemsmentioning

confidence: 99%

Nonbinary Stabilizer Codes Over Finite Fields

Ketkar

Klappenecker

Kumar

et al. 2006

552

560

“…As already mentioned, this issue was first treated in [4]. One important ingredient of the code construction in [4] is a sequence of polynomially constructible algebraic geometry (AG) codes.…”

Section: A Polynomial Constructions Of Quantum Codesmentioning

confidence: 99%

“…This has allowed us to utilize many results from coding theory. For example, quantum codes constructible in polynomial time are presented in [4] based on developments of algebraic geometry codes. In the present paper, we propose a method for concatenating quantum codes, which will be obtained by developing Forney's idea of concatenated codes [5].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Concatenated Quantum Codes Constructible in Polynomial Time: Efficient Decoding and Error Correction

Hamada

2008

Abstract-A method for concatenating quantum errorcorrecting codes is presented. The method is applicable to a wide class of quantum error-correcting codes known as CalderbankShor-Steane (CSS) codes. As a result, codes that achieve a high rate in the Shannon theoretic sense and that are decodable in polynomial time are presented. The rate is the highest among those known to be achievable by CSS codes. Moreover, the best known lower bound on the greatest minimum distance of codes constructible in polynomial time is improved for a wide range.

“…Ashikhmin et al [13] and Chen et al [14] constructed asymptotically good quantum codes based on algebraic geometry codes. Later, Matsumoto [15] improved the bound of Ashikhmin et al [13].…”

Section: Shannxi 710071 Chinamentioning

confidence: 99%

A Family of Asymptotically Good Quantum Codes Based on Code Concatenation

Xing

Wang

2009

We explicitly construct an infinite family of asymptotically good concatenated quantum stabilizer codes where the outer code uses CSS-type quantum Reed-Solomon code and the inner code uses a set of special quantum codes. In the field of quantum error-correcting codes, this is the first time that a family of asymptotically good quantum codes is derived from bad codes. Its occurrence supplies a gap in quantum coding theory. As in classical coding theory, we want to construct quantum codes with large minimum distance. More generally, we want to construct asymptotically good 1