Collective Bit Flipping-Based Decoding of Quantum LDPC Codes

TL;DR

The paper tackles decoding for variable-degree-3 quantum LDPC codes, which exhibit strong distance properties but suffer from high latency and deterioration under traditional iterative decoders due to trapping sets. It introduces a collective, low-latency decoding framework based on two-bit bit flipping (TBF), leveraging the structure of generalized hypergraph product codes to address both classical and quantum trapping sets via parallel decoders. By systematically designing decoders (e.g., D1, D4, D9, D24) and employing a recursive procedure to cover weight-5 error patterns within trapping sets, the method substantially improves error-floor performance over min-sum and layered decoding, often with only a few iterations. The approach shows strong empirical gains across several $d_v=3$ QLDPC codes, including better waterfall and no observed error floor for BB codes, underscoring its potential for low-latency, high-distance quantum decoding in practical devices.

Abstract

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 imposes stringent latency requirements and 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 (dv-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 dv-3 QLDPC codes.

Figures (12)

• Figure 1: Children trapping sets of the $(3,3)$ trapping set in \ref{['fig:33']} are the $(5,5)$ trapping set in \ref{['fig:55']}, the $(6,4)$ trapping set in \ref{['fig:64']}, and the $(8,6)$ trapping set in \ref{['fig:86']}. Each child of the $(3,3)$ trapping set is a juxtaposition of two, three, and four six-cycles, respectively. Odd-degree check nodes are represented by $\blacksquare$.
• Figure 2: The $(63,63)$ and $(49,49)$ trapping sets obtained by the the $(3, 3)$ trapping set in the $B^*$ and $A^*$ circulant matrices respectively. $B^*$ contains seven $(63,63)$ trapping sets and $A^*$ contains nine $(49,49)$ trapping sets. $(63,63)$ trapping sets are only interconnected with $(49,49)$ trapping sets and vice versa. Note that only degree-$3$ check nodes appear in the trapping sets. We use two styles of dotted lines to represent connections between remote nodes. The five $6$-cycles in a row, seen in \ref{['fig:stopping0']}, can be inferred from the polynomial $1+x+x^6$.
• Figure 3: The evolution of a $(4,4)$ trapping set \ref{['fig:44']} leading into the formation of a symmetric stabilizer \ref{['fig:60TS']}. The $(5,3)$ trapping set \ref{['fig:53']} is obtained after adding a single variable node and the $(6,0)$ trapping set is obtained after adding two variable nodes. The three line types used for the $(6,0)$ trapping set indicate that the lines are not interconnected. Notice that only variable nodes belonging to different circulant matrices are directly connected.
• Figure 4: Overview of the structure of $B1$ code. The symmetric stabilizer interconnects the two types of trapping sets. The $(63,63)$ and $(49,49)$ trapping sets are represented as yellow and green circles respectively, which indicate their symmetric structure.
• Figure 5: Uncorrectable weight-$3$ error configurations by BF decoder. These errors appear in smaller trapping sets within the $(63,63)$ and $(49,49)$ trapping sets. Variable nodes in error are marked with $\bullet$. Note that in this case, $\blacksquare$ corresponds to an unsatisfied check.

Theorems & Definitions (2)

• Definition 1
• Definition 2