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IEEE Conference on Industrial Electronics and Applications (ICIEA)

Entanglement-assisted nonbinary quantum LDPC codes with finite field method

Junhu Shao, Lin Zhou, Ying Sun

Published May 1, 2018• 3 citations
Publisher

Abstract

Based on quantum stabilizer formalism over multi-level quantum system, non-binary entanglement-assisted quantum coding theory was described in this paper. Then two classes of 2r-ary quantum quasi-cyclic LDPC codes were constructed by using algebraic finite field method over Fqwith q = 2r. By iteratively decoded with q-ary sum-product algorithm (QSPA), the constructed 2r-ary quantum LDPC codes have shown significantly better performances than the present binary quantum LDPC codes.

DOI
10.1109/iciea.2018.8397697

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Based on quantum stabilizer formalism over multi-level quantum system, non-binary entanglement-assisted quantum coding theory was described in this paper. Then two classes of 2r-ary quantum quasi-cyclic LDPC codes were constructed by using algebraic finite field method over Fqwith q = 2r. By iteratively decoded with q-ary sum-product algorithm (QSPA), the constructed 2r-ary quantum LDPC codes have shown significantly better performances than the present binary quantum LDPC codes.
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.
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
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, we present three algebraic methods for constructing structured nonbinary LDPC codes over finite fields, among which the first method is used to construct quasi-cyclic codes with girth at least 6 based on the automorphisms of finite fields, the second method gives a class of (4,ρ) quasi-cyclic codes with girth at least 8, the third method gives a class of codes with cycles limited. Simulation results show that the constructed codes perform very well over AWGN channel, and they have better performances or far lower computational complexities than the corresponding random Mackay codes or codes algebraically constructed by Lin.
We construct two families of non-binary entanglement assisted (EA) quasi-cyclic (QC) quantum low-density parity-check (QLDPC) codes over arbitrary finite fields, each possessing a precisely determined code rate. The first family is derived from a pair of non-binary classical QC-LDPC codes, designed such that the unassisted portion of the overall Tanner graph of the resulting EA-QC-QLDPC code is free of 4-cycles. The second family, on the other hand, is constructed from a single non-binary classical QC-LDPC code whose Tanner graph itself is 4-cycle-free. In developing the codes belonging to the first family, we employ a \emph{single Bell pair} to establish entanglement between the transmitter and the receiver, thereby minimizing the required entanglement resources. Furthermore, these constructions demonstrate that careful graph-based design can effectively balance error-correction performance with entanglement consumption, providing a practical approach for realizing efficient non-binary EA-QC-QLDPC codes.