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Computer Knowledge and Technology

Construction Quantum CSS Codes Based on Finite Geometries

Xun Chun-ling

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

Using the low density parity check matrices created by finite geometric approach with its dual codes,which are constructed by splitting the rows of the LDPC check matrices,a quantum LDPC codes construction method is proposed based on finite geometries in this paper. Finally,this method is proved by theory deduction adequately.

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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.
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.
Low density parity check (LDPC) codes with iterative decoding based on belief propagation (IDBP) achieve astonishing error performance close to the Shannon limit. Until now there has been no known method for constructing these Shannon limit approaching codes systematically. Good LDPC codes are largely generated by computer search. As a result, the encoding of long LDPC codes is in general very complex. This paper presents the first algebraic method for constructing LDPC codes systematically based on finite analytic geometries. Four classes of finite geometry LDPC codes with relatively good minimum distances are constructed. These codes are either cyclic or quasi-cyclic and therefore their encoding can be implemented with simple linear feedback shift registers. Long finite geometry LDPC codes have been constructed and they achieve an error performance only a few tenths of a dB away from the Shannon limit. Finite geometry LDPC codes are strong competitors to turbo codes for error control in communication and digital data storage systems.
Quantum error correcting codes are essential to achieve fault-tolerant quantum computation. This work introduces two constructions of high-rate, dual-containing (DC) Calderbank--Shor--Steane low-density parity-check (LDPC) codes based on quasi-dyadic matrices. We characterize the automorphism group of such codes, investigate their minimum distance behavior, and provide several theoretical results on their cycle properties. Monte Carlo simulations under depolarizing and phenomenological noise show better finite-length logical error rates than the considered DC benchmark codes and competitive performance against several state-of-the-art quantum LDPC code families. Finally, we employ an automorphism-ensemble belief propagation decoder to improve their decoding performance.
In this study, a new method for constructing low-density parity-check (LDPC) codes is presented. This construction is based on permutation matrices which come from a finite abstract group and hence the codes constructed in this manner are called group permutation low-density parity-check (GP-LDPC) codes. A necessary and sufficient condition under which a GP-LDPC code has a cycle is given and some properties of these codes are investigated. A class of flexible-rate GP-LDPC codes without cycles of length four is also introduced. Simulation results show that GP-LDPC codes perform very well with the iterative decoding and can outperform their random-like counterparts.