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Journal of Xidian University

Construction and decoding of nonbinary quantum LDPC codes

Lin Zhou

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

Nonbinary quantum codes are more suitable for error-correction in multi-level quantum systems.Based on stabilizer formalism over the finite field,a class of nonbinary quantum LDPC codes is presented in this paper.And the BP iterative decoding algorithm for these codes is described under the quantum depolarizing channel with the Monte Carlo simulation method.For example,a class of CSS structure 4-ary quantum LDPC codes is given with code rates 1/2 and 1/4.Compared with the present binary quantum LDPC codes of equivalent codelength,the channel threshold with 10-5FER is improved from 0.016 to 0.025.

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Nonbinary quantum codes are more suitable for error-correction in multi-level quantum systems.Based on stabilizer formalism over the finite field,a class of nonbinary quantum LDPC codes is presented in this paper.And the BP iterative decoding algorithm for these codes is described under the quantum depolarizing channel with the Monte Carlo simulation method.For example,a class of CSS structure 4-ary quantum LDPC codes is given with code rates 1/2 and 1/4.Compared with the present binary quantum LDPC codes of equivalent codelength,the channel threshold with 10-5FER is improved from 0.016 to 0.025.
Quantum error correction codes are needed to protect quantum systems from interference. An important family of quantum error correction codes are stabilizer codes. Many stabilizer codes in the literature belong to a subclass of stabilizer codes, referred to as CSS codes since they are based on the so-called CSS formalism. CSS codes have received significant attention because the dual-containing condition for the CSS formalism is relatively straightforward to satisfy and also because it is easy to produce CSS codes from classical error correction codes. However, some stabilizer codes in the literature show superior error correction performance than existing CSS codes, and are based on binary low-density parity-check (LDPC) codes. Since it has been shown that CSS codes based on nonbinary LDPC codes have good performance over quantum qubit channels, in this paper we investigate stabilizer codes based on nonbinary LDPC codes. In particular, we propose constructions of stabilizer codes based on nonbinary quasi-cyclic low-density parity-check (QC-LDPC) codes for qubit channels. The simulation results show that our QC-LDPC stabilizer codes decoded by a nonbinary sum-product algorithm have better performance than their binary counterparts.
A generic method for constructing quantum LDPC codes is presented. We first explain how to overcome the difficulty of finding a set of low weight generators for the stabilizer group of the code. Our approach is based on a graph representation of the generators of the stabilizer group and on a simple local rule to ensure commutativity. We provide several specific examples of quantum LDPC codes obtained by our method, together with numerical simulations over the depolarizing channel and the erasure channel.
We propose a method of constructing quantum LDPC codes from multiplicative groups of order p - 1, where p = 4n + 1 is a prime for some positive integer n. The proposed quantum LDPC codes of non-CSS structure are constructed from a pair of classical regular quasi-cyclic (QC) LDPC codes. We show that these classical regular QC-LDPC codes without cycles of length 4 are orthogonal with respect to the symplectic inner product. Moreover, the proposed construction method yields a large number of new quantum LDPC codes with various code lengths and rates. The performance of the proposed quantum LDPC codes over quantum depolarizing channels with an iterative belief-propagation decoding algorithm is evaluated by simulations.
In the late 1950's and early 1960's, finite fields were successfully used to construct linear block codes, especially cyclic codes, with large minimum distances for correcting random errors with algebraic decoding, such as Bose-Chaudhuri-Hocqenghem (BCH) and Reed-Solomon (RS) codes. Recently it has been shown that finite fields can also be used successfully to construct binary quasi-cyclic (QC)-LDPC codes that perform very well not only over the AWGN channel but also over the binary erasure channel with iterative decoding, besides being efficiently encodable. This paper is concerned with constructions of nonbinary QC-LDPC codes based on finite fields
This paper presents an algebraic method for constructing structural nonbinary LDPC codes based on the prime difference over finite field. The approach gives a class of quasi-cyclic (QC)-LDPC codes with girth at least 6. Especially, the approach can construct a class of short length code and the decoding complexity is low. Simulation results show that the constructed codes decoded with iterative decoding perform well over the AWGN channel using BPSK transmission in terms of bit-error probability and rate of decoding convergence, collectively.