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Journal of Nanjing University of Posts and Telecommunications

Construction of Quantum CSS Codes Based on Euclidean Geometry

LI Miao-miao

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

Based on the lines and points in finite geometry,a method to construct the low density parity check codes is discussed in this paper.According to the characteristics of these codes,the dual codes of this kind of LDPC codes are obtained by splitting the rows or columns of the check matrix.A novel construction method of quantum low-density-parity-check code is developed according to the theory of quantum CSS codes.Quantum code(15,4) is selected as an example to illustrate the availability of this construction method,and its performance is analyzed under the bit-flipping channel.It is shown that this method is easier to construct the dual codes,and the performance of quantum codes constructed is superior to that of classical ones.

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arXiv:0712.4115•Globecom•Jan 1, 2008•47 citations
Based on the lines and points in finite geometry,a method to construct the low density parity check codes is discussed in this paper.According to the characteristics of these codes,the dual codes of this kind of LDPC codes are obtained by splitting the rows or columns of the check matrix.A novel construction method of quantum low-density-parity-check code is developed according to the theory of quantum CSS codes.Quantum code(15,4) is selected as an example to illustrate the availability of this construction method,and its performance is analyzed under the bit-flipping channel.It is shown that this method is easier to construct the dual codes,and the performance of quantum codes constructed is superior to that of classical ones.
A new class of quantum low-density parity-check (LDPC) codes whose parity-check matrices are dual-containing matrices constructed based on lines of Euclidean geometries (EGs) is presented. The parity-check matrices of our quantum codes contain one and only one 4-cycle in every two rows and have better distance properties. However, the classical parity-check matrix constructed from EGs does not satisfy the condition of dual-containing. In some parameter conditions, parts of the rows in the matrix maybe have not any nonzero element in common. Notably, we propose four families of fascinating structure according to changes in all the parameters, and the parity-check matrices are adopted to satisfy the requirement of dual-containing. Series of matrix properties are proved. Construction methods of the parity-check matrices with dual-containing property are given. The simulation results show that the quantum LDPC codes constructed by this method perform very well over the depolarizing channel when decoded with iterative decoding based on the sum-product algorithm. Also, the quantum codes constructed in this paper outperform other quantum codes based on EGs.
With the development of quantum error correction techniques, many counterparts of classical error correction coding techniques have been found in quantum area. As the quantum counterpart of a good error correction code in classical communications, quantum low density parity check code becomes a promising coding technique in quantum error correction area. In this paper, a construction method of quantum low density parity check code (quantum LDPC) based on Projective Geometry is proposed, and the quantum code QLDPC [21,6] is selected as an example to illustrate the whole construction procedures. By numerical simulation, the error correction performance of QLDPC [21,6] is discussed over the bit-flipping channel. It is shown that this method is available and has some advantages.
In this paper, a construction method of quantum low density parity check codes (quantum LDPC) which based on classical quasi-cyclic sparse sequence is proposed, and the quantum code QLDPC(3,8)(16,6) and QLDPC(3,8)(64,24) are selected as examples to illustrate this construction method. The performances of this kind quantum LDPC codes are discussed over the bit-flipping channel by numerical simulations. The results show that these quantum codes have good ability to correct error in quantum channel.
CSS code is a simple,efficient quantum error correction code construction,and has been found applications in construction of varieties of quantum codes,so far.In this paper,we propose a new construction method of quantum parity check matrix codes based on CSS by taking advantage of the method used in classical quasi-cyclic LDPC codes.Using quick encoding algorithm,we can obtain the quantum code based on this proposed method.At last,a quantum code(3,8)(16,6) is selected as the example to illustrate the process of construction,and its encoding circuit by Controlled-not gate and Hadard gate is also presented.
Low-density parity check (LDPC) codes are a significant class of classical codes with many applications. Several good LDPC codes have been constructed using random, algebraic, and finite geometries approaches, with containing cycles of length at least six in their Tanner graphs. However, it is impossible to design a self-orthogonal parity check matrix of an LDPC code without introducing cycles of length four. In this paper, a new class of quantum LDPC codes based on lines and points of finite geometries is constructed. The parity check matrices of these codes are adapted to be self- orthogonal with containing only one cycle of length four in each pair of two rows. Also, the column and row weights, and bounds on the minimum distance of these codes are given. As a consequence, these codes can be encoded using shift-register encoding algorithms and can be decoded using iterative decoding algorithms over various quantum depolarizing channels.