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Computer Knowledge and Technology

A Construction Method of Quantum Low-density-parity-check Codes Based on Stabilizer Codes

Jing Liu

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

Based on the classical LDPC construction method and quantum error correction techniques,a construction method of quantum low-density-parity-check codes based on stabilizer codes is discussed method in this paper.We provide quantum code (12,3) obtained by our methed as example to verify this construction, together with a numerical simulation of the performance of quantum code (32,12) and quantum code (64,24)over the depolarizing channel.

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Based on the classical LDPC construction method and quantum error correction techniques,a construction method of quantum low-density-parity-check codes based on stabilizer codes is discussed method in this paper.We provide quantum code (12,3) obtained by our methed as example to verify this construction, together with a numerical simulation of the performance of quantum code (32,12) and quantum code (64,24)over the depolarizing channel.
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.
This article develops a general method for constructing entanglement-assisted quantum low-density parity-check (LDPC) codes, which is based on combinatorial design theory. Explicit constructions are given for entanglement-assisted quantum error-correcting codes with many desirable properties. These properties include the requirement of only one initial entanglement bit, high error-correction performance, high rates, and low decoding complexity. The proposed method produces several infinite families of codes with a wide variety of parameters and entanglement requirements. Our framework encompasses the previously known entanglement-assisted quantum LDPC codes having the best error-correction performance and many other codes with better block error rates in simulations over the depolarizing channel. We also determine important parameters of several well-known classes of quantum and classical LDPC codes for previously unsettled cases.
Quantum error correction is the basic technique for realization of quantum communication and quantum computation. So far, the theories of quantum error correction have become more and more perfect, and many counterpart of classical error correction coding technique have been found in quantum area. In this article, we proposed a new method of construction of parity check matrix of quantum low density parity check (LDPC) codes which based on classical quasi-cyclic LDPC codes, and selected the quantum code (3,8)(16,6) as the example to illustrate the construction of quantum code. We also gave its completing circuit by controlled-not gate and Hadamard gate.
Based on the classical LDPC construction method and quantum error correction techniques,a construction method of quantum low-density-parity-check codes is discussed and a novel coset searching algorithm which is important in the construction method is developed in this paper.We select quantum code(3,8)(16,6) as an example to verify this algorithm and compare it with the conventional one.The result shows that the algorithm can improve the encoding efficiency and reduce complexity with the same performance as the conventional one.At the same time,the algorithm can slove the problem of storage of quantum codes.
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.
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.