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Communications in Theoretical Physics

Quantum BCH Codes Based on Spectral Techniques

Guo Ying, Zeng Gui-Hua

Published Feb 1, 2006• 1 citation
Publisher

Abstract

When the time variable in quantum signal processing is discrete, the Fourier transform exists on the vector space of n -tuples over the Galois field F 2 , which plays an important role in the investigation of quantum signals. By using Fourier transforms, the idea of quantum coding theory can be described in a setting that is much different from that seen that far. Quantum BCH codes can be defined as codes whose quantum states have certain specified consecutive spectral components equal to zero and the error-correcting ability is also described by the number of the consecutive zeros. Moreover, the decoding of quantum codes can be described spectrally with more efficiency.

DOI
10.1088/0253-6102/45/2/019

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When the time variable in quantum signal processing is discrete, the Fourier transform exists on the vector space of n -tuples over the Galois field F 2 , which plays an important role in the investigation of quantum signals. By using Fourier transforms, the idea of quantum coding theory can be described in a setting that is much different from that seen that far. Quantum BCH codes can be defined as codes whose quantum states have certain specified consecutive spectral components equal to zero and the error-correcting ability is also described by the number of the consecutive zeros. Moreover, the decoding of quantum codes can be described spectrally with more efficiency.
当在量信号处理的时间变量是分离的时， Fouriertransform 在 Galois 域 F_2 上在 n 元组的向量空间存在，它在量信号的调查起一个重要作用。由使用 Fourier 变换，编码理论的量的想法能在与看见那不同的一个背景被描述那远。QuantumBCH 代码能被定义为状态有其量的代码某些指定连续光谱到零的部件平等者和改正错误的能力被数字也描述连续零。而且，译码量代码能与更多的效率幽灵似地被描述。
Quantum error correcting codes (QECCs) play an important role in preventing quantum information decoherence. Good quantum stabilizer codes were constructed by classical error correcting codes. In this paper, Bose-Chaudhuri-Hocquenghem (BCH) codes over finite fields are used to construct quantum codes. First, we try to find such classical BCH codes, which contain their dual codes, by studying the suitable cyclotomic cosets. Then, we construct nonbinary quantum BCH codes with given parameter sets. Finally, a new family of quantum BCH codes can be realized by Steane's enlargement of nonbinary Calderbank-Shor-Steane (CSS) construction and Hermitian construction. We have proven that the cyclotomic cosets are good tools to study quantum BCH codes. The defining sets contain the highest numbers of consecutive integers. Compared with the results in the references, the new quantum BCH codes have better code parameters without restrictions and better lower bounds on minimum distances. What is more, the new quantum codes can be constructed over any finite fields, which enlarges the range of quantum BCH codes.
Galois rings are regarded as "building blocks" of a finite commutative ring with identity. There have been many papers on classical error correction codes over Galois rings published. As an important warm-up before exploring quantum algorithms and quantum error correction codes over Galois rings, we study the quantum Fourier transform (QFT) over Galois rings and prove it can be efficiently preformed on a quantum computer. The properties of the QFT over Galois rings lead to the quantum algorithm for hidden linear structures over Galois rings.
. We introduce a new class of quantum error--correcting codes derived from (classical) Reed--Solomon codes over finite fields of characteristic two. Quantum circuits for encoding and decoding based on the discrete cyclic Fourier transform over finite fields are presented. 1 Introduction During the last years it has been shown that computers taking advantage of quantum mechanical phenomena outperform currently used computers. The striking examples are integer factoring in polynomial time (see [18]) and finding pre--images of an n--ary Boolean function ("searching") in time O( p 2 n ) (see [12]). Quantum computers are not only of theoretical nature---there are several suggestions how to physically realize them (see, e. g., [6, 7]). On the way towards building a quantum computer, one very important problem is to stabilize quantum mechanical systems since they are very vulnerable. A theory of quantum error--correcting codes has already been established (see [15]). Nevertheless, the pr...
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.
Galois fields are constantly gaining importance in quantum computing due to their wide usage in quantum error correction algorithms, and so it becomes relevant to define the QFT over Galois fields because of its main role in many of the most important quantum algorithms. The present article illustrates how to generalize the QFT so it can be applied over Galois fields and explains several examples of the application of the QFT over the simplest Galois fields. In particular the QFT will be defined for the Galois fields F2, F4 and GF(9), also the application of the QFT and the operations involved will be made using the Maple mathematical software.