Trapping Sets of Iterative Decoders for Quantum and Classical Low-Density Parity-Check Codes

Abstract

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.