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Journal of Lightwave Technology

LDPC-Coded $M$-ary PSK Optical Coherent State Quantum Communication

Ivan B. Djordjević

Published Mar 1, 2009• 18 citations
Publisher

Abstract

This paper addresses two important problems of interest for deep-space optical communications, quantum inter-chip/intra-chip optical communications and quantum-key distribution: 1) the problem of codedM-ary phase-shift keying optical coherent-state quantum communication in the absence of background radiation, and 2) the problem of coded binary coherent state communication in the presence of background radiation. In both cases large girth low-density parity-check (LDPC) codes are used as channel codes. With girth-10 LDPC-code of rate 0.8 and BPSK, the bit-error ratio of 10-8can be achieved for average number of signal photons of 0.73, resulting in net-effective coding gain of 7.63 dB.

DOI
10.1109/jlt.2008.2004566

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•Jan 1, 2011•2 citations
LDPC-Coded MMM-ary PSK Optical Coherent State Quantum Communication
This paper addresses two important problems of interest for deep-space optical communications, quantum inter-chip/intra-chip optical communications and quantum-key distribution: 1) the problem of codedM-ary phase-shift keying optical coherent-state quantum communication in the absence of background radiation, and 2) the problem of coded binary coherent state communication in the presence of background radiation. In both cases large girth low-density parity-check (LDPC) codes are used as channel codes. With girth-10 LDPC-code of rate 0.8 and BPSK, the bit-error ratio of 10-8can be achieved for average number of signal photons of 0.73, resulting in net-effective coding gain of 7.63 dB.
The problem of communication using optical coherent quantum states, in the presence of background radiation, is considered. Two modulation formats are studied, on-off keying (OOK) and M-ary pulse-position modulation (PPM). The bit-error-rate performance improvement due to low-density parity-check coding is reported. For OOK, it is assumed that the coherent state signal has a random phase. For an average number of noise photons N = 0.1, the required number of signal photons per information bit is six in the case of coded quantum OOK. For the same level of noise (N = 0.1) and assuming that signal phase is known, coded 16-ary PPM requires only 1.21 of signal photons per information bit.
Low-density parity-check code (LDPC) not only has good performance approaching the Shannon limit, but also has low decoding complexity and flexible structure. It is a research hot-spot in the field of channel coding in recent years and has a wide range of application prospects in optical communication systems. In this paper, the decoding aspects and performance of LDPC codes are analyzed and compared according to the bit error rate (BER) of LDPC codes. The computer simulation was carried out under additive white Gaussian noise (AWGN) channel and binary phase shift keying (BPSK) modulation. Through theoretical analysis and simulation results, this paper explores the way of multi-rate LDPC decoding.
The parity-check matrix of a nonbinary (NB) low-density parity-check (LDPC) code over Galois field GF(q) is constructed by assigning nonzero elements from GF(q) to the 1s in corresponding binary LDPC code. In this paper, we state and prove a theorem that establishes a necessary and sufficient condition that an NB matrix over GF(q), constructed by assigning nonzero elements from GF(q) to the 1s in the parity-check matrix of a binary quasi-cyclic (QC) LDPC code, must satisfy in order for its null-space to define a nonbinary QC-LDPC (NB-QC-LDPC) code. We also provide a general scheme for constructing NB-QC-LDPC codes along with some other code construction schemes targeting different goals, e.g., a scheme that can be used to construct codes for which the fast-Fourier-transform-based decoding algorithm does not contain any intermediary permutation blocks between bit node processing and check node processing steps. Via Monte Carlo simulations, we demonstrate that NB-QC-LDPC codes can achieve a net effective coding gain of 10.8 dB at an output bit error rate of 10-12. Due to their structural properties that can be exploited during encoding/decoding and impressive error rate performance, NB-QC-LDPC codes are strong candidates for application in optical communications.
We present an optical low-density parity-check (LDPC) decoder suitable for implementation above 100 Gbits/s, which provides large coding gains when based on large-girth LDPC codes. We show that a basic building block, the probabilities multiplier circuit, can be implemented using a Mach-Zehnder interferometer, and we propose corresponding probabilistic-domain sum-product algorithm (SPA). We perform simulations of a fully parallel implementation employing girth-10 LDPC codes and proposed SPA. The girth-10 LDPC(24015,19212) code of the rate of 0.8 outperforms the BCH(128,113)xBCH(256,239) turbo-product code of the rate of 0.82 by 0.91 dB (for binary phase-shift keying at 100 Gbits/s and a bit error rate of 10(-9)), and provides a net effective coding gain of 10.09 dB.
To meet the increasing requirement on the transmission characteristics of optical communication systems,based on the structure of parity-check matrix of Richardson-Urbanke algorithm,a lowercomplexity constructing method is proposed for the quasi-cyclic low-density parity-check(QC-LDPC) codes in the optical communication systems,and a QC-LDPC(4288,4020) code with 93.7% higher code-rate is also constructed.Simulation shows that the constructed QC-LDPC(4288,4020) code has net coding gain of about 1.7 dB over the classic RS(255,239) code which has been widely used for optical communication systems and 0.3 dB more than that of SCG-LDPC(3969,3720) code with the same 93.7% higher code-rate,and the NCG of the QC-LDPC(4288,4020) code is about 1.3 dB away from the Shannon limit at bit error rate BER of 10-7;furthermore,the error floor of QC-LDPC(4288,4020) code is lower than that of the randomly constructed LDPC(4288,4020) with the column weight of 3.
The optimal distributions on a channel input that yield the random coding exponent, the expurgation exponent, and the zero-rate reliability function for a discrete classical-quantum channel with M-ary Phase Shift Keying (PSK) coherent state signal are analytically derived. In each case, the optimal distributions are given by a uniform distribution over the channel input alphabet. As a result, the expressions of the error exponents for M-ary PSK coherent state signal are simplified. By using this result, typical behavior of the error exponents is sketched in the case of M = 16.