On LLR Mismatch in Belief Propagation Decoding of Overcomplete QLDPC Codes

TL;DR

The results demonstrate that initial LLR mismatch has a strong influence on the frame error rate (FER), particularly in the low noise regime, and shows that the optimal performance is not sharply localized: the FER remains largely insensitive over an extended region of mismatched LLRs.

Abstract

Belief propagation (BP) decoding of quantum low density parity check (QLDPC) codes is often implemented using overcomplete stabilizer (OS) representations, where redundant parity checks are introduced to improve finite length performance. Decoder behavior for such representations is governed primarily by finite iteration dynamics rather than asymptotic code properties. These dynamics are known to critically depend on the initialization of the decoder. In this paper, we investigate the impact of mismatched log likelihood ratios (LLRs) used for BP initialization on the performance of QLDPC codes with OS representations. Our results demonstrate that initial LLR mismatch has a strong influence on the frame error rate (FER), particularly in the low noise regime. We also show that the optimal performance is not sharply localized: the FER remains largely insensitive over an extended region of mismatched LLRs. This behavior motivates an interpretation of LLR mismatch as a regularization control parameter rather than a quantity that must be precisely matched to the quantum channel.

Figures (6)

• Figure 1: QEC decoding block diagram. The same OS GB$(126,28,126)$ construction ($k=28$, $n=m=126$) is used for both BP4 and its BP2 equivalent. The true channel depolarizing probability is $\varepsilon$. The LLR computation block uses $\varepsilon_{0}$.
• Figure 2: FER for BP2 on the OS GB$(126,28,126)$ code. LLR mismatch (with $\varepsilon_{0}=0.10$) is shown to improve finite iteration FER performance.
• Figure 3: FER for BP4 on the OS GB$(126,28,126)$ code. LLR mismatch (with $\varepsilon_{0}=0.10$) is shown to improve finite iteration FER performance.
• Figure 4: FER vs $\varepsilon$ for BP4 with $\ell^{\max}=4$, obtaining similar performance for different $\varepsilon_{0}$. Similar results (not shown) were observed for BP2.
• Figure 5: BP4 AO vs mismatched LLR. The blue shaded region denotes the AO standard deviation. (a) The optimum corresponds to a region of near equivalent priors. The inset shows that $\varepsilon_{0}=0.10$ is a pragmatic choice for BP4 over the simulated noise regime; (b) Sensitivity is dominated by the low noise subset.