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CONVOLUTIONAL DECODERS BASED ON QUANTUM ANNEALING

Ilyas Rysbekov

Published May 1, 2021• 0 citations
OpenAlex

Abstract

This research proposes a new method of decoding convolutional codes using quantum computers. The proposed method obtains maximum likelihood (ML) estimate of the transmitted codeword using quantum annealing (QA). The performance of the proposed method is assessed by its error performance and compared with the conventional Viterbi decoder on classical computers. The results verify the feasibility of QA for decoding convolutional codes. Furthermore, the execution time of both classical and quantum computers for decoding are compared and discussed.

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This research proposes a new method of decoding convolutional codes using quantum computers. The proposed method obtains maximum likelihood (ML) estimate of the transmitted codeword using quantum annealing (QA). The performance of the proposed method is assessed by its error performance and compared with the conventional Viterbi decoder on classical computers. The results verify the feasibility of QA for decoding convolutional codes. Furthermore, the execution time of both classical and quantum computers for decoding are compared and discussed.
A novel decoding algorithm is developed for general quantum convolutional codes. Exploiting useful ideas from classical coding theory, the new decoder introduces two innovations that drastically reduce the decoding complexity compared to the existing quantum Viterbi decoder. First, the new decoder uses an efficient linear-circuits-based mechanism to map a syndrome to a candidate vector, whereas the existing algorithm relies on a non-trivial lookup table. Second, the new algorithm is cleverly engineered such that only one run of the Viterbi algorithm suffices to locate the most-likely error pattern, whereas the existing algorithm must run the Viterbi algorithm many times. The efficiency of the proposed algorithm allows us to simulate and present the first performance curve of a general quantum convolutional code.
In the forthcoming 6G era, driven by the escalating demands for enhanced performance and reliability in communication systems, Low-Density Parity-Check (LDPC) codes emerge as a robust error correction coding scheme poised to effectively meet these requirements. In recent years, with the advancement of quantum computing, a proposal has emerged to integrate classical and quantum computing for the decoding of LDPC. The LDPC decoding can be treated as a Quadratic Unconstrained Binary Optimization (QUBO) problem, thereby allowing the utilization of the D-Wave 2000Q quantum annealer for solving it. Expanding on this groundwork, the paper proposes an approach to enhance quantum annealing performance using classical algorithms. Specifically, it involves performing quantum annealing on the problem first, followed by post-processed using Maximum Likelihood Estimation (MLE) for the multiple outputs, known as QAML. Furthermore, to fully exploit the advantages of quantum annealing, this paper incorporates the energy and frequency of occurrence of candidate codewords as prior information into the Maximum Likelihood Estimation. These are called QAMLE and QAMLF respectively, further enhancing decoding performance. The proposed method is evaluated and compared with direct Quantum Annealing (QA) and Quantum Annealing with post-processing using Minimum Distance (QAMD). Research findings indicate that QAML exhibits a lower bit error rate compared to QA and QAMD.
Quantum convolutional codes are predicted by many to offer higher error correction performance than quantum block codes of equivalent encoding complexity. However, to decode a quantum convolutional code is challenging, because the decoder does not have a measurement of the received codeword (due to the quantum mechanic rule “measurement destroys quantum states”), but only a measurement of the syndrome. Although quantum Viterbi decoding (QVD) has been proposed, its high complexity makes it rather difficult to implement or simulate. Exploiting useful ideas from classical coding theory, this paper develops a practical quantum syndrome decoder (QSD) by introducing two innovations that drastically reduce the decoding complexity compared to the existing QVD. The new decoder uses an efficient linear-circuits-based mechanism to map a syndrome to a candidate vector, obviating the need of a cumbersome lookup table. It is also cleverly engineered such that only one run of the Viterbi algorithm suffices to locate the most-likely error pattern, rather than have to run many rounds as in the previous QVD algorithm. The efficiency of the new decoder allows us to simulate and present the first performance curve of a general quantum convolutional code.
We formulate maximum likelihood (ML) channel decoding as a quadratic unconstraint binary optimization (QUBO) and simulate the decoding by the current commercial quantum annealing machine, D-Wave 2000Q. We prepared two implementations with Ising model formulations, generated from the generator matrix and the parity-check matrix respectively. We evaluated these implementations of ML decoding for low-density parity-check (LDPC) codes, analyzing the number of spins and connections and comparing the decoding performance with belief propagation (BP) decoding and brute-force ML decoding with classical computers. The results show that these implementations are superior to BP decoding in relatively short length codes, and while the performance in the long length codes deteriorates, the implementation from the parity-check matrix formulation still works up to 1k length with fewer spins and connections than that of the generator matrix formulation due to the sparseness of parity-check matrices of LDPC.
Quantum Annealing (QA) uses quantum fluctuations to search for a global minimum of an optimization-type problem faster than classical computer. To meet the demand for future internet traffic and mitigate the spectrum scarcity, this work presents the QA-aided maximum likelihood (ML) decoder for multi-user non-orthogonal multiple access (NOMA) networks as an alternative to the successive interference cancellation (SIC) method. The practical system parameters such as channel randomness and possible transmit power levels are taken into account for all individual signals of all involved users. The brute force (BF) and SIC signal detection methods are taken as benchmarks in the analysis. The QA-assisted ML decoder results in the same BER performance as the BF method outperforming the SIC technique, but the execution of QA takes more time than BF and SIC. The parallelization technique can be a potential aid to fasten the execution process. This will pave the way to fully realize the potential of QA decoders in NOMA systems.
Quantum communication system is influenced by environment noise in the whole process. Quantum error-correcting codes can solve this problem efficiently. The polynomial representation of a quantum state is defined. Based on the structure of the classical convolutional codes, the basis state of the quantum convolutional codes is transformed into the multiplication of an information polynomial by the generator polynomial. A new method is proposed to encode and decode these quantum codes and the networks can be realized with the polynomial multiplication circuits. Finally, we provide a maximum likelihood error estimation algorithm with complexity growing linearly.