Quantum Constructions

Mirko Navara

Jian Liu

Quantum error-correcting codes protect quantum information against undesirable noise in quantum computation and quantum communication.It is one of important problems that constructing very good quantum codes.There are two methods that use binary codes to construct quantum codes,one is CSS construction,another is Steane construction.These two methods are all based on the construction of binary self-orthogonal codes.Combinational construction of binary self-orthogonal codes is investigated in this paper.From known codes chains of binary self-orthogonal codes and their dual codes chains with given dual distance,new binary self-orthogonal codes of dual distance three,four,five and six are constructed.Based on these results,for each satisfying,S-chains with parameters and are constructed.According to Steane's construction,many very good quantum codes of distance five and six are constructed by the obtained S-chains,some of these quantum codes are new and some of these quantum codes have improved parameters than previously known codes.

Ryutaroh Matsumoto, Tomohiko Uyematsu

Quantum error-correcting codes have attracted much attention. Among many research articles, the most general and systematic construction is the so called stabilizer code construction [6] or additive code construction [2], which constructs a quantum error-correcting code as an eigenspace of an Abelian subgroup S of the error group. Thereafter Calderbank et al. [3] proposed a construction of S from an additive code over the finite field F4 with 4 elements. These constructions work for tensor products of 2-state quantum systems. However Knill [8], [9] and Rains [13] observed that the construction [2], [6] can be generalized to n-state systems by an appropriate choice of the error basis. Rains [13] also generalized the construction [3] using additive codes over F4 to p-state quantum systems, but his generalization does not relate the problem of quantum code construction to classical error-correcting codes. We propose a construction of quantum error-correcting codes for p-state systems from classical error-correcting codes which is a generalization of [3]. Throughout this paper, p denotes a prime number andm a positive integer. This paper is organized as follows. In Sect. 2, we review the construction of quantum codes for nonbinary systems. In Sect. 3, we propose a construction of quantum codes for p-state systems from classical codes over Fp2 . In Sect. 4, we propose a construction of quantum codes for p-state systems from classical linear codes over Fp2m . In Sect. 5, we discuss a

L. A. Lugiato, A. Gatti, K.I. Petsas

Lingfei Jin, Chaoping Xing

There have been various constructions of classical codes from polynomial valuations in literature [2], [7], [8], [10], [11]. In this paper, we present a construction of classical codes based on polynomial construction again. One of the features of this construction is that not only the classical codes arisen from the construction have good parameters, but also quantum codes with reasonably good parameters can be produced from these classical codes. In particular, some new quantum codes are constructed (see Examples V.5 and V.6).

L.A. Lugiato, A. Gatti, K.I. Petsas