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.
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 and Classical computers continue to work together in tight cooperation to solve difficult problems. The combination is thus suggested in recent times for decoding the Low Density Parity Check (LDPC) codes, for the next generation Wireless Communication systems. In this paper we have worked out the Quadratic Unconstrained Binary Optimization (QUBO) formulation for Rayleigh Fading channels for two different scenarios- channel state fully known and not known. The resultant QUBO are solved using D-Wave 2000Q Quantum Annealer and the outputs from the Annealer are classically postprocessed, invoking the notion of diversity. Simple minimum distance decoding of the available copies of the outputs led to improved performance, compared to picking the minimum-energy solution in terms of Bit Error Rate (BER). Apart from providing these results and the comparisons to fully classical Simulated Annealing (SA) and the traditional Belief Propagation (BP) based strategies, some remarks about diversity due to quantum processing are also spelt out.
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.
Low-Density Parity-Check (LDPC) codes, as a class of high-efficiency error-correcting codes with sparse parity-check matrices, have demonstrated enormous potential in the field of communications since their rediscovery, and have become a core coding scheme especially in the fifth-generation (5G) wireless communication systems. To address the decoding problem of LDPC codes in complex channel environments, this paper proposes an innovative quantum-classical hybrid decoding architecture. By transforming the decoding problem into a Quadratic Unconstrained Binary Optimization (QUBO) problem, the architecture leverages the parallel processing capability of coherent optical quantum computers to accelerate the iterative decoding process. The research focuses on two typical scenarios specified in the 5G standard: the high-throughput scenario (dominated by BG1 with long code length and high code rate) and the low-complexity scenario (dominated by BG2 with short code length and low code rate). Through establishing a rigorous mathematical model, a hybrid decoder based on quantum annealing and the belief propagation algorithm is designed in this paper, and the Bayesian optimization method is adopted to adaptively adjust key parameters such as the maximum number of iterations and damping factor. This research provides an important theoretical foundation and practical scheme for the practical application of quantum computing in the field of communications.
Decoding Low-Density Parity-Check (LDPC) codes is a fundamental problem in coding theory, and Belief Propagation (BP) is one of the most popular methods for LDPC code decoding. However, BP may encounter convergence issues and suboptimal performance, especially for short-length codes and in high-noise channels. The Quantum Approximate Optimization Algorithm (QAOA) is a type of Variational Quantum Algorithm (VQA) designed to solve combinatorial optimization problems by minimizing a problem-specific cost function. In this paper, we present a QAOA-based decoding framework for LDPC codes by formulating a decoding cost function that incorporates both parity-check constraints and soft channel reliability information. The resulting optimization problem is solved using QAOA to search for low-energy configurations corresponding to valid codewords. We test the proposed method through extensive numerical experiments and compare its performance with BP decoding. The experimental results demonstrate that the QAOA-based decoder achieves a higher probability of correctly recovering the transmitted codeword than BP across multiple experimental settings.
Maximum-likelihood (ML) decoding is an important and powerful method in communications to obtain the optimal performance of a channel code. However, ML decoding is a challenging problem whose complexity grows exponentially with the blocklength of the code, making it unfeasible to decode optimally for large codes. In this paper, we present a new approach to accelerate ML decoding by minimizing the number of ones in the code's underlying matrix representation. We formulate this minimization problem as an integer program and use a heuristic algorithm to solve it. Using these minimized matrices, we significantly reduce the runtime of different ML decoders for various codes, resulting in speedups of up to 81% compared to the runtime on the original matrices.
Quantum error correction (QEC) is indispensable for realizing fault-tolerant quantum computation, yet its effectiveness hinges critically on the classical decoding algorithm that interprets noisy syndrome measurements. Among all possible decoding strategies, maximum likelihood decoding (MLD) is provably optimal, since it identifies the logical group with largest likelihood by summing over all possible errors within logical class consistent with the observed syndrome. Despite its optimality, MLD is computationally intractable in general (#P-hard), motivating a rich landscape of exact and approximate algorithms. In this topical review, we provide a unified perspective on MLD by surveying recent advances through three complementary lenses: statistical mechanics, tensor networks, and artificial intelligence. From the statistical mechanics viewpoint, the MLD problem maps onto evaluating partition functions of disordered spin models, enabling exact solutions for certain codes and noise models as well as threshold estimation via phase-transition analysis. From the tensor network perspective, approximate contraction of tensor networks on the code's factor graph yields decoders that closely approach MLD accuracy with polynomial computational cost. From the artificial intelligence perspective, neural-network-based decoders, including autoregressive generative models and recurrent transformers, learn to approximate the MLD distribution from data, achieving high accuracy with the parallelism afforded by modern hardware accelerators. We discuss the connections among these three approaches, review their application to both simulated and experimental quantum hardware, and outline open challenges including real-time decoding, scalability to large code distances, and generalization to high-rate quantum low-density parity-check codes.