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List Decoding and New Bicycle Code Constructions for Quantum LDPC Codes

Sheida Rabeti, Hessam Mahdavifar

TL;DR

The paper addresses the challenge of efficiently decoding quantum LDPC codes with low logical error rates. It introduces the MBBP-LD decoder, which runs multiple belief-propagation decoders on redundant parity-check representations and fuses their outputs with a Frequency-Weighted Decision rule, preserving $O(n)$ latency. It also proposes Univariate Bicycle (UB) codes to reduce the polynomial search space in code construction, and demonstrates substantial performance gains over BP-OSD on short QLDPC codes along with favorable complexity properties. Overall, the work provides a scalable decoding framework and new code constructions that enhance practical viability of quantum LDPC codes for fault-tolerant quantum computing.

Abstract

In this paper, we propose a new decoder, called the Multiple-Bases Belief-Propagation List Decoder (MBBP-LD), for Quantum Low-Density Parity-Check (QLDPC) codes. It extends the Multiple-Bases Belief-Propagation (MBBP) framework, originally developed for classical cyclic LDPC codes. The proposed method preserves the linear-time complexity of standard BP decoder while improving the logical error rate. To further reduce the logical error rate, a new decision rule is introduced for the post-processing list decoder, outperforming the conventional least-metric selector (LMS) criterion. For the recently developed and implemented bivariate bicycle (BB) code with parameters \([[144,12,12]]\), our proposed MBBP-LD decoder achieves up to 40\% lower logical error rate compared to the state-of-the-art decoder for short QLDPC codes, i.e., BP with ordered-statistics decoding (BP-OSD), while retaining the linear-time complexity of the plain BP decoder. In addition, we explore a new subclass of BB codes, that we refer to as the univariate bicycle (UB) codes, specifically with lower-weight parity checks (\(w=6,8\)). This reduces the polynomial search space for the code compared to general BB codes, i.e., by reducing the search space over two polynomial components in BB codes to just a single polynomial component in UB codes. Simulations demonstrate the promising performance of these codes under various types of BP decoders.

List Decoding and New Bicycle Code Constructions for Quantum LDPC Codes

TL;DR

The paper addresses the challenge of efficiently decoding quantum LDPC codes with low logical error rates. It introduces the MBBP-LD decoder, which runs multiple belief-propagation decoders on redundant parity-check representations and fuses their outputs with a Frequency-Weighted Decision rule, preserving latency. It also proposes Univariate Bicycle (UB) codes to reduce the polynomial search space in code construction, and demonstrates substantial performance gains over BP-OSD on short QLDPC codes along with favorable complexity properties. Overall, the work provides a scalable decoding framework and new code constructions that enhance practical viability of quantum LDPC codes for fault-tolerant quantum computing.

Abstract

In this paper, we propose a new decoder, called the Multiple-Bases Belief-Propagation List Decoder (MBBP-LD), for Quantum Low-Density Parity-Check (QLDPC) codes. It extends the Multiple-Bases Belief-Propagation (MBBP) framework, originally developed for classical cyclic LDPC codes. The proposed method preserves the linear-time complexity of standard BP decoder while improving the logical error rate. To further reduce the logical error rate, a new decision rule is introduced for the post-processing list decoder, outperforming the conventional least-metric selector (LMS) criterion. For the recently developed and implemented bivariate bicycle (BB) code with parameters , our proposed MBBP-LD decoder achieves up to 40\% lower logical error rate compared to the state-of-the-art decoder for short QLDPC codes, i.e., BP with ordered-statistics decoding (BP-OSD), while retaining the linear-time complexity of the plain BP decoder. In addition, we explore a new subclass of BB codes, that we refer to as the univariate bicycle (UB) codes, specifically with lower-weight parity checks (). This reduces the polynomial search space for the code compared to general BB codes, i.e., by reducing the search space over two polynomial components in BB codes to just a single polynomial component in UB codes. Simulations demonstrate the promising performance of these codes under various types of BP decoders.

Paper Structure

This paper contains 14 sections, 3 theorems, 11 equations, 3 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantum Stabilizer and CSS Codes
  4. Belief Propagation (BP) Decoding
  5. Generalized Bicycle (GB) Codes
  6. Multiple-Bases Belief-Propagation List-decoding (MBBP-LD)
  7. MBBP Decoding via Tree-Based Construction
  8. Decision Making (DM) Rule
  9. Univariate Bicycle (UB) Code
  10. Numerical Results
  11. Proposed Decoder Performance Comparison
  12. Complexity Analysis of MBBP-LD
  13. Performance of the Proposed UB Codes
  14. Conclusion

Key Result

Proposition 1

The dimension $k$ of the generalized bicycle code $[[2n, k]]$ defined by $a(x), b(x) \in \mathbb{F}_2[x]$ is given by: where $h(x) \triangleq \gcd(a(x), b(x), x^n - 1)$.

Figures (3)

  • Figure 1: MBBP-LD decoding with redundant-row construction.
  • Figure 2: Performance of the proposed MBBP-LD decoder under two different regimes (Fig. \ref{['fig:bp_144']} and Fig. \ref{['fig:bp_288']} have parallel and serial scheduling, respectively)
  • Figure 3: Performance of the proposed UB codes with $(I_{max}=1000, \beta=0, \tau=0.4)$.

Theorems & Definitions (6)

  • Proposition 1
  • Lemma 2: Subtree Size Bound
  • proof
  • Remark 1
  • Remark 2
  • Proposition 3: Frobenius Identity