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A Joint Code and Belief Propagation Decoder Design for Quantum LDPC Codes

Sisi Miao, Jonathan Mandelbaum, Holger Jäkel, Laurent Schmalen

TL;DR

This work tackles the difficulty of decoding quantum LDPC codes with belief propagation in the presence of short Tanner graph cycles. It introduces CAMEL, a joint code and ensemble BP decoder that confines 4-cycles to a single variable node and mitigates their impact via decimation across four parallel decoders, enabling effective quaternary BP. Two concrete CAMEL code constructions are developed: quasi-cyclic (QC) codes and finite geometry (FG) codes; both yield codes with minimum distance around $d \approx \sqrt{n}$ and robust BP decoding on depolarizing channels. Numerical results show large decoding gains over conventional BP and competitive performance against more complex decoders at comparable rates. The approach promises practical, low-latency quantum error correction with scalable, cycle-aware code families.

Abstract

Quantum low-density parity-check (QLDPC) codes are among the most promising candidates for future quantum error correction schemes. However, a limited number of short to moderate-length QLDPC codes have been designed and their decoding performance is sub-optimal with a quaternary belief propagation (BP) decoder due to unavoidable short cycles in their Tanner graphs. In this paper, we propose a novel joint code and decoder design for QLDPC codes. The constructed codes have a minimum distance of about the square root of the block length. In addition, it is, to the best of our knowledge, the first QLDPC code family where BP decoding is not impaired by short cycles of length 4. This is achieved by using an ensemble BP decoder mitigating the influence of assembled short cycles. We outline two code construction methods based on classical quasi-cyclic codes and finite geometry codes. Numerical results demonstrate outstanding decoding performance over depolarizing channels.

A Joint Code and Belief Propagation Decoder Design for Quantum LDPC Codes

TL;DR

This work tackles the difficulty of decoding quantum LDPC codes with belief propagation in the presence of short Tanner graph cycles. It introduces CAMEL, a joint code and ensemble BP decoder that confines 4-cycles to a single variable node and mitigates their impact via decimation across four parallel decoders, enabling effective quaternary BP. Two concrete CAMEL code constructions are developed: quasi-cyclic (QC) codes and finite geometry (FG) codes; both yield codes with minimum distance around and robust BP decoding on depolarizing channels. Numerical results show large decoding gains over conventional BP and competitive performance against more complex decoders at comparable rates. The approach promises practical, low-latency quantum error correction with scalable, cycle-aware code families.

Abstract

Quantum low-density parity-check (QLDPC) codes are among the most promising candidates for future quantum error correction schemes. However, a limited number of short to moderate-length QLDPC codes have been designed and their decoding performance is sub-optimal with a quaternary belief propagation (BP) decoder due to unavoidable short cycles in their Tanner graphs. In this paper, we propose a novel joint code and decoder design for QLDPC codes. The constructed codes have a minimum distance of about the square root of the block length. In addition, it is, to the best of our knowledge, the first QLDPC code family where BP decoding is not impaired by short cycles of length 4. This is achieved by using an ensemble BP decoder mitigating the influence of assembled short cycles. We outline two code construction methods based on classical quasi-cyclic codes and finite geometry codes. Numerical results demonstrate outstanding decoding performance over depolarizing channels.
Paper Structure (7 sections, 3 theorems, 18 equations, 4 figures, 2 tables)

This paper contains 7 sections, 3 theorems, 18 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. CAMEL: Joint Code and Decoder Design
  4. Quasi-Cyclic QLDPC Codes
  5. QLDPC Codes from Finite Geometries
  6. Numerical Results
  7. Conclusion

Key Result

Theorem 1

Consider two classical binary linear codes $\mathcal{C}_1$ and $\mathcal{C}_2$ with parameters $[n,k_1,d_1]$ and $[n,k_2,d_2]$. Their PCM are $\bm{H}_X\in \mathbb{F}_2^{(n-k_1)\times n}$ and $\bm{H}_Z\in \mathbb{F}_2^{(n-k_2)\times n}$, respectively. If $\mathcal{C}_2^{\perp}\subseteq \mathcal{C}_1$ an $[[n,k_1+k_2-n,\min\{d_1,d_2\}]]$QSC can be constructed.

Figures (4)

  • Figure 1: Block diagram of the proposed ensemble decoder.
  • Figure 2: FER vs. depolarizing probability $\varepsilon$ curves when decoding an E4 $[[273,111,17]]$ QLDPC code from an EG using different decoding algorithms.
  • Figure 3: FER vs. depolarizing probability $\varepsilon$ curves for the quasi-cyclic QLDPC codes shown in Tab. \ref{['tab:QC_codes']} and the reference code R1.
  • Figure 4: FER vs. depolarizing probability $\varepsilon$ curves for QLDPC codes constructed using EG shown in Tab. \ref{['tab:FG_codes']} and the reference code R2.

Theorems & Definitions (6)

  • Theorem 1
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Example