In this letter, we propose a novel iterative decoding algorithm that exploits the degenerate nature of three different families of quantum low-density parity-check codes, i.e., surface, toric, and row-degree-4 bicycle codes. Such families of codes share harmful trapping sets that constitute symmetric stabilizers, making it impossible for any parallel-scheduled iterative message-passing decoder to converge even for error patterns of weight as low as two. By biasing subsets of nodes in the symmetric stabilizers, the decoder is able to converge to a valid error pattern. Furthermore, the proposed decoder has low decoding complexity - linear in the code’s blocklength - and a fully parallel schedule, making it suitable for low-latency efficient implementation.
In this letter, we propose a novel iterative decoding algorithm that exploits the degenerate nature of three different families of quantum low-density parity-check codes, i.e., surface, toric, and row-degree-4 bicycle codes. Such families of codes share harmful trapping sets that constitute symmetric stabilizers, making it impossible for any parallel-scheduled iterative message-passing decoder to converge even for error patterns of weight as low as two. By biasing subsets of nodes in the symmetric stabilizers, the decoder is able to converge to a valid error pattern. Furthermore, the proposed decoder has low decoding complexity - linear in the code’s blocklength - and a fully parallel schedule, making it suitable for low-latency efficient implementation.
In this paper, we propose a novel message-passing decoding approach that leverages the degeneracy of quantum low-density parity-check codes to enhance decoding performance, eliminating the need for serial scheduling or post-processing. Our focus is on two-block Calderbank-Shor-Steane (CSS) codes, which are composed of symmetric stabilizers that hinder the performance of conventional iterative decoders with uniform update rules. Specifically, our analysis shows that, under the isolation assumption, the min-sum decoder fails to converge when constant-weight errors are applied to symmetric stabilizers, as variable-to-check messages oscillate in every iteration. To address this, we introduce a decoding technique that exploits this oscillatory property by applying distinct update rules: variable nodes in one block utilize messages from previous iterations, while those in the other block are updated conventionally. Logical error-rate results demonstrate that the proposed de-coder significantly outperforms the normalized min-sum decoder and achieves competitive performance with belief propagation enhanced by order-zero ordered statistics decoding, all while maintaining linear complexity in the code's block length.
Practical quantum computations can require logical error rates of 10−12 or lower. Quantum low-density parity-check (QLDPC) codes are promising candidates for reaching such rates with low qubit overhead, provided that their syndromes can be decoded efficiently. Repeated noisy syndrome measurements introduce temporal degeneracy into the space-time decoding graph. We characterize the temporal kernel, whose vectors have zero total data parity, and show that it is generated by elementary temporal codewords of weight dv+2, where dv is the variable-node degree. For (dv,dc)-regular parity-check matrices with fixed degrees dv,dc>2 and girth greater than four, we derive counting bounds and construct degenerate error patterns of weight (dv+2)/2 for even dv and dv+1 for odd dv, with multiplicity Θ(nr) in the blocklength n and number of measurement rounds r. Under the isolation assumption, these patterns cause local failures of deterministic isotropic message-passing decoders under a flooding schedule. These failures depend on the node degrees rather than the minimum distance; for the bivariate bicycle (BB) codes considered, their weight is equal to four. Monte carlo simulations and fixed-weight importance sampling quantify harmful low-weight degenerate error patterns for the min-sum (MS) decoder beyond the isolation assumption. Motivated by the identified symmetries, we propose space-time anisotropic min-sum decoding with syndrome-guided realizations. Its complexity is linear in the number of Tanner graph edges for fixed maximum number of iterations and number of decoders. Numerical results on BB codes show threshold-like behavior acrossthe tested code sizes and improved failure probabilities relative to MS decoding with ordered statistics decoding post-processing.
In this paper, we propose a novel message-passing decoding approach that leverages the degeneracy of quantum low-density parity-check codes to enhance decoding performance, eliminating the need for serial scheduling or post-processing. Our focus is on two-block Calderbank-Shor-Steane (CSS) codes, which are composed of symmetric stabilizers that hinder the performance of conventional iterative decoders with uniform update rules. Specifically, our analysis shows that, under the isolation assumption, the min-sum decoder fails to converge when constant-weight errors are applied to symmetric stabilizers, as variable-to-check messages oscillate in every iteration. To address this, we introduce a decoding technique that exploits this oscillatory property by applying distinct update rules: variable nodes in one block utilize messages from previous iterations, while those in the other block are updated conventionally. Logical error-rate results demonstrate that the proposed decoder significantly outperforms the normalized min-sum decoder and achieves competitive performance with belief propagation enhanced by order-zero ordered statistics decoding, all while maintaining linear complexity in the code's block length.
Quantum low-density parity-check (QLDPC) codes provide non vanishing rates, distance scaling with the blocklength of the code, and facilitate fast iterative decoding because of their sparsity. However, in practice iterative decoding fails to exploit the distance of the code, because it cannot resolve the symmetries imposed by degeneracy. In this work, we provide a graph theoretic characterization of degeneracy for the family of generalized bicycle (GB) codes. This viewpoint shows that harmful degenerate error patterns persist whenever they remain related by automorphisms preserved by the decoder. Motivated by symmetry breaking via graph coloring, we compare three coloring approaches: no coloring, block-coloring, and edge-coloring. For GB codes, we show that edge-coloring can eliminate all automorphisms in low-weight stabilizer-induced subgraphs. We practically realize the coloring schemes as isotropic, block- anisotropic and edge-anisotropic min-sum (MS) decoding. Experimental results show that edge anisotropic min-sum decoding obtains improved performance over isotropic and block anisotropic decoding for several GB codes in a small number of iterations.
Quantum low-density parity-check codes are promising candidates for low-overhead fault-tolerant quantum computing, but degeneracy is known to impair the convergence of belief-propagation (BP) decoding of these codes. In this work, we show that appending linearly independent rows to a check matrix of a stabilizer code can reduce the search space for a valid degenerate solution. Motivated by this, we extend the recently proposed affine subcode ensemble decoding technique from the classical to the quantum setting. Moreover, we employ overcomplete matrices for each decoding path. Monte-Carlo simulations on toric and generalized bicycle codes demonstrate improved convergence and reduced logical error rate.
Protecting logical information in the form of a classical bit or a quantum bit (qubit) is an essential step in ensuring fault-tolerant classical or quantum computation. Error correction codes and their decoders perform this step by adding redundant information that aids the decoder to recover or protect the logical information even in the presence of noise. Low-density parity-check (LDPC) codes have been one of the most popular error correction candidates in modern communication and data storage systems. Similarly, their quantum analogues, quantum LDPC codes are being actively pursued as excellent prospects for error correction in future fault-tolerant quantum systems due to their asymptotically non-zero rates, sparse parity check matrices, and efficient iterative decoding algorithms. This dissertation deals with failure configurations, known as \emph{trapping sets} of classical and quantum LDPC codes when decoded with iterative message passing decoding algorithms, and the \emph{error floor phenomenon} - the degradation of logical error rate performance at low physical noise regime. The study of quantum trapping sets will enable the construction of better quantum LDPC codes and also help in modifying iterative quantum decoders to achieve higher fault-tolerant thresholds and lower error floors. Towards this goal, the dissertation also presents iterative decoders for classical and quantum LDPC codes using the \emph{deep neural network framework}, novel iterative decoding algorithms, and a decoder-aware \emph{expansion-contraction method} for error floor estimation. In this dissertation, we first establish a systematic methodology by which one can identify and classify \emph{quantum trapping sets} (QTSs) according to their topological structure and decoder used. For this purpose, we leverage the known harmful configurations in the Tanner graph, called \emph{trapping sets} (TSs), from the classical error correction world. The conventional definition of a trapping set of classical LDPC codes is generalized to address the syndrome decoding scenario for quantum LDPC codes. Furthermore, we show that the knowledge of QTSs can be used to design better quantum LDPC codes and decoders. In the context of the development of novel decoders, we extend the stochastic resonance based decoders to quantum LDPC codes, propose iteration-varying message passing decoders with their message update rules learned by neural networks tuned for low logical error rate, and present a syndrome based generalized belief propagation algorithm for tackling convergence failure of iterative decoders due to the presence of short cycles. Our analysis of TSs of a layered decoding architecture clearly reveals the dependence of the harmfulness of TSs (classical or quantum) on the iterative decoder, and thus on the error floor estimates. We present a computationally efficient method for estimating error floors of LDPC codes over the binary symmetric channel without any prior knowledge of its trapping sets. The sub-graph expansion-contraction method is a general procedure for TS characterization, which lists all harmful error patterns up to a given weight for the LDPC code and decoder. Based on this decoder-aware trapping set characterization for LDPC codes, we propose a model-driven deep neural network (DNN) framework that unfolds the decoding iterations, to design the \emph{decoder diversity of finite alphabet iterative decoders (FAIDs)}. Our decoder diversity DNN-FAID delivers excellent waterfall performance along with a low error floor.