Quantum low-density parity-check (QLDPC) codes with asymptotically non-zero rates are prominent candidates for achieving fault-tolerant quantum computation, primarily due to the low operational depth of their syndrome-measurement circuits. Numerous studies advocate the necessity of fast decoders to fully harness the capabilities of QLDPC codes, thus driving the focus towards designing low-complexity iterative decoders. However, empirical investigations indicate that such iterative decoders are susceptible to having a high error floor when decoding QLDPC codes. The main objective of this paper is to analyze the decoding failures of thehypergraph-product(HGP) andlifted-product(LP) codes and to design decoders that mitigate these failures, thus achieving a reduced error floor. The suboptimal performance of these codes can predominantly be ascribed to two structural phenomena: (1) stabilizer-induced trapping sets (TS), which correspond to stabilizer-induced subgraphs in the Tanner graphs, and (2) classical trapping sets (TS), which originate from the classical codes used in the construction of HGP and LP codes. The dynamics of stabilizer-induced TSs are examined, and a straightforward modification of iterative decoders is proposed to circumvent these TSs. Moreover, this work proposes a systematic methodology for designing decoders that can circumvent the classical TSs in both HGP and LP codes by deriving them from decoders capable of avoiding the TSs in the parent classical LDPC codes. When decoders that can avoid stabilizer-induced TSs are run in parallel with those that can mitigate the effect of classical TSs, the logical error rate improves significantly in the error-floor region.

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.

Quantum low-density parity-check (QLDPC) codes have been proven to achieve higher minimum distances at higher code rates than surface codes. However, this family of codes must cope with the stringent latency constraints imposed by quantum technology and tends to exhibit poor performance under iterative decoding, especially when the variable degree is low. In this work, we improve both the error correction performance and decoding latency of variable degree-3 ($d_{v}$-3) QLDPC codes under iterative decoding. Firstly, we perform a detailed analysis of the structure of a well-known family of QLDPC codes, i.e., hypergraph product-based codes. Then, we propose a decoding approach that stems from the knowledge of harmful configurations apparent in these codes. Our decoding scheme is based on applying a modified version of bit flipping (BF) decoding, namely two-bit bit flipping (TBF) decoding, which adds more degrees of freedom to BF decoding. The granularity offered by TBF decoding helps us design sets of decoders that operate in parallel and can collectively decode error patterns appearing in harmful configurations of the code, thus addressing both the latency and performance requirements. Finally, simulation results demonstrate that the proposed decoding scheme surpasses other iterative decoding approaches for various$d_{v}$-3 QLDPC codes.

We introduce a low-complexity message-passing quantum error correction algorithm for decoding Quantum Low-Density Parity-Check (QLDPC) stabilizer codes. The proposed decoder operates on the quaternary stabilizer graph but only exchanges binary messages. This leads to a significantly reduced complexity compared to other quaternary belief propagation (BP) algorithms that pass floating-point messages. The efficacy of the proposed decoder is evaluated by providing decoding examples, performance metrics using Monte-Carlo simulations, and complexity analysis. Despite its reduced complexity, the performance loss of the proposed decoder is modest compared to floating-point parallel quaternary decoders for a Calderbank-Shor-Steane (CSS) code family. In particular, experiments obtained over the [[1054, 140, 20]] lifted product (LP) Tanner code demonstrated that for low error rates (< 0.01), the proposed quaternary-binary message-passing decoder approaches the performance of quaternary BP by converging in almost the same number of iterations while requiring less complex operations. Additionally, for non-CSS codes, our decoder performs similarly as quaternary floating-point decoders despite its lower complexity.

Iterative decoders for finite length quantum low-density parity-check (QLDPC) codes are impacted by short cycles, detrimental graphical configurations known as trapping sets (TSs) present in a code graph as well as symmetric degeneracy of errors. In this paper, we develop a systematic methodology by which quantum trapping sets (QTSs) can be defined and categorized according to their topological structure. Conventional definition of a TS from classical error correction is generalized to address the syndrome decoding scenario for QLDPC codes. We show that QTS information can be used to design better QLDPC code and decoder. For certain finite-length QLDPC codes, frame error rate improvements of two orders of magnitude in the error floor regime are demonstrated without needing any post-processing steps.

In this paper, we present a computationally efficient method for estimating error floors of low-density parity-check (LDPC) codes over the binary symmetric channel (BSC) without any prior knowledge of its trapping sets (TSs). Given the Tanner graph G of a code, and the decoding algorithm V, the method starts from a list of short cycles in G, and expands each cycle by including its sufficiently large neighborhood in G. Variable nodes of the expanded sub-graphs EXP are then corrupted exhaustively by all possible error patterns, and decoded by V operating on EXP. Union of support of the error patterns for which V fails on each EXP defines a subset of variable nodes that is a TS. The knowledge of the minimal error patterns and their strengths in each TSs is used to compute an estimation of the frame error rate. This estimation represents the contribution of error events localized on TSs, and therefore serves as an accurate estimation of the error floor performance of V at low BSC cross-over probabilities. We also discuss trade-offs between accuracy and computational complexity. Our analysis shows that in some cases the proposed method provides a million-fold improvement in computational complexity over standard Monte-Carlo simulation.

Quantum low-density parity check (QLDPC) codes are promising in realization of scalable, fault tolerant quantum memory for computation. Many of the QLDPC codes constructions suffer from unavoidable short cycles in their Tanner graph which degrade the decoding performance of the belief propagation (BP) algorithm. In this paper, we propose a syndrome based generalized belief propagation (GBP) algorithm for decoding of quantum LDPC codes and analyze how the proposed algorithm escapes from short cycle trapping sets effectively compared to the BP algorithm. Simulation results show improved decoding performance of the GBP algorithm over BP for the dual containing Calderbank, Shor and Steane (CSS) codes when cycles of length 4 are considered in the region based approach.

In this paper, we present how the decoding performance of layered decoder can be analyzed using trapping sets. Surprisingly, a simple horizontal layered Gallager-B decoder breaks all weight-3 error patterns on a (5,3) trapping set successfully, resulting in a steeper slope in frame error rate (FER) curves compared to a flooding schedule decoder. Theoretical validation of the results is also done using a semi- analytical method for computing the error floors of LDPC codes on a binary symmetric channel decoded using layered Gallager-B algorithm.

We introduce a stochastic resonance based decoding paradigm for quantum codes using an error correction circuit made of a combination of noisy and noiseless logic gates. The quantum error correction circuit is based on iterative syndrome decoding of quantum low-density parity check codes, and uses the positive effect of errors in gates to correct errors due to decoherence. We analyze how the proposed stochastic algorithm can escape from short cycle trapping sets present in the dual containing Calderbank, Shor and Steane (CSS) codes. Simulation results show improved performance of the stochastic algorithm over the deterministic decoder.

We investigate into the loopy belief propagation algorithm for binary low density parity check (LDPC) codes having cycles of small girth. Independence assumption among messages passed, assumed reasonable in all configurations of graphs, fails the most in graphical structures with short cycles. We investigate into this limitation and propose a modified algorithm, by considering dependency in the probability domain. This improves the performance of decoding over such graphs when compared to the original message passing algorithm at higher signal-to-noise ratio (SNR), thereby, yielding lower error floors.

We investigate into the limitations of the sum-product algorithm in the probability domain over graphs with isolated short cycles. By considering the statistical dependency of messages passed in a cycle of length 4, we modify the update equations for the beliefs at the variable and check nodes. We highlight an approximate log domain algebra for the modified variable node update to ensure numerical stability. At higher signal-to-noise ratios (SNR), the performance of decoding over graphs with isolated short cycles using the modified algorithm is improved compared to the original message passing algorithm (MPA).