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A new construction of quantum error-correcting codes from a condition of equivalency

Sun Guang-ren

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

A new method to construct quantum error-correcting codes was given by introducing representation number from a condition of equivalency,and a type of quantum error-correcting codes was proved to exist with combining CRSS codes.

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By using difference schemes, orthogonal partitions and a replacement method, some new methods to construct pure quantum error-correcting codes are provided from orthogonal arrays. As an application of these methods, we construct several infinite series of quantum error-correcting codes including some optimal ones. Compared with the existing binary quantum codes, more new codes can be constructed, which have a lower number of terms (i.e., the number of computational basis states) for each of their basis states.
The theory of quantum error correction was established more than a decade ago as the primary tool for fighting decoherence in quantum information processing. Although great progress has already been made in this field, limited methods are available in constructing new quantum error correction codes from old codes. Here we exhibit a simple and general method to construct new quantum error correction codes by nesting certain quantum codes together. The problem of finding long quantum error correction codes is reduced to that of searching several short length quantum codes with certain properties. Our method works for all length and all distance codes, and is quite efficient to construct optimal or near optimal codes. Two main known methods in constructing new codes from old codes in quantum error-correction theory, the concatenating and pasting, can be understood in the framework of nested quantum error correction codes.
This paper provides new instances of quantum error-correcting codes. According to recent results by Nakayama and Hagiwara, by using two sets of binary sequences that satisfy three conditions, we can construct quantum error-correcting codes for single deletion errors. However, only two instances of two sets that satisfy the three conditions have been found to date. The present study constructs two infinite series of sets that satisfy the three conditions. In other words, this paper constructs new infinite series of quantum error-correcting codes for single deletion errors.
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.
The essential insight of quantum error correction was that quantum information can be protected by suitably encoding this quantum information across multiple independently erred quantum systems. Recently it was realized that, since the most general method for encoding quantum information is to encode it into a subsystem, there exists a novel form of quantum error correction beyond the traditional quantum error correcting subspace codes. These new quantum error correcting subsystem codes differ from subspace codes in that their quantum correcting routines can be considerably simpler than related subspace codes. Here we present a class of quantum error correcting subsystem codes constructed from two classical linear codes. These codes are the subsystem versions of the quantum error correcting subspace codes which are generalizations of Shor's original quantum error correcting subspace codes. For every Shor-type code, the codes we present give a considerable savings in the number of stabilizer measurements needed in their error recovery routines.