Decoding Correlated Errors in Quantum LDPC Codes

TL;DR

This work addresses decoding correlated errors in quantum LDPC codes under circuit-level noise by introducing GARI, a graph augmentation and rewiring framework that eliminates 4-cycles involving $Y$-type errors to enable effective belief-propagation-like inference. By reformulating the decoding problem with an augmented matrix $ ilde D_{XYZ}$ and employing a normalized min-sum decoder with a hybrid serial-layered schedule, the authors achieve high accuracy with low latency, particularly when using a 24-decoder ensemble. On bivariate bicycle codes of distances 6, 10, and 12, GARI-NMS matches or surpasses prior best decoders (e.g., BPOSD, XYZ-Relay-BP) and, for distance-12, attains a logical error rate per round of $(6.70 ext{±} 1.93) imes 10^{-9}$ at a physical error rate of $10^{-3}$. FPGA implementations demonstrate real-time decoding with an average per-round latency of 273 ns and sub-microsecond latency in 99.99% of cases, signaling a viable path to practical fault-tolerant quantum computation.

Abstract

We introduce a decoding framework for correlated errors in quantum LDPC codes under circuit-level noise. The core of our approach is a graph augmentation and rewiring for interference (GARI) method, which modifies the correlated detector error model by eliminating 4-cycles involving Y-type errors, while preserving the equivalence of the decoding problem. We test our approach on the bivariate bicycle codes of distances 6, 10, and 12. A normalized min-sum decoder with a hybrid serial-layered schedule is applied on the transformed graph, achieving high accuracy with low latency. Performance is further enhanced through ensemble decoding, where 24 randomized normalized min-sum decoders run in parallel on the transformed graph, yielding the highest reported accuracy (on par with XYZ-Relay-BP) with unprecedented speed for the tested codes under uniform depolarizing circuit level noise. For the distance 12 (gross) code, our approach yields a logical error rate of $(6.70 \pm 1.93) \times 10^{-9}$ at a practical physical error rate of $10^{-3}$. Furthermore, preliminary FPGA implementation results show that such high accuracy can be achieved in real time, with a per-round average decoding latency of 273 ns and sub-microsecond latency in 99.99% of the decoding instances.

Figures (7)

• Figure 1: Logical error rate performance of our method (GARI-NMS-ensemble), compared with BPOSD and XYZ-Relay-BP-5 muller2025improvedbeliefpropagationsufficient. We consider BB codes with $[[n,k,d]]$ parameters equal to $[[72,12,6]]$, $[[90,8,10]]$ and $[[144,12,12]]$, under uniform depolarizing circuit level noise with a physical error rate of 0.001. We plot logical error rates against the BPT efficiency of the code, defined as $kd^2/n$, which accounts for the non-uniform variation of the $n,k,d$ parameters, and is inspired by the Bravyi-Poulin-Terhal bound bravyi2010tradeoffs. For BPOSD, we report the best achievable performance obtained by fine-tuning both the MP decoder and its underlying message schedule. The XYZ-Relay-BP-5 decoder uses 301 DMem-BP decoders in series with up to 18080 decoding iterations, and memory strength parameters taken from muller2025improvedbeliefpropagationsufficient. It searches for five different solutions and returns the one with the lowest weight. The GARI-NMS-ensemble decoder employs 24 NMS decoders in parallel for BB codes with distances 10 and 12, with each decoder performing up to 400 iterations. We adopt a minimum-latency approach, in which decoding stops as soon as one decoder in the ensemble converges, and the corresponding solution is then returned. For distance 6, only a single GARI-NMS decoder is used.
• Figure 2: (a) A 4-cycle in a decoding graph: error and check nodes are depicted as circles and squares, respectively; edges forming the 4-cycle are shown in bold, other incident edges are shown as dashed. (b) A biclique in the decoding graph; only edges participating in the biclique are shown. The MP inference is not exact, due to the presence of the 4-cycles. (c) The GARI transform eliminates all 4-cycles forming the biclique, by performing a variable change that introduces a new error node and a new check node in the graph (depicted in blue), as explained in the main text. If the eliminated 4-cycles were the only cycles in the bipartite graph, the new graph would be cycle-free, and the MP inference would be exact. The error and check node labels in (b) and (c) are included to illustrate the connection with the decoding-matrix perspective in Section \ref{['sec:matrix']} and should be disregarded unless specifically referenced in the main text.
• Figure 3: High-level architecture of the GARI MP decoder. MP processing moves upward, completing one iteration after first updating the bottom-part unit, followed by the $D_X$ and $D_Z$ units, which are updated in parallel. The input consists of prior probabilities for $(e_Z, e_X, e_Y)$, and the output of the estimated $\bar{e}_X$ error.
• Figure 4: Logical error rate per-round vs. physical error rate for various BB codes under uniform depolarising circuit-level noise, using Stim circuits from either gong2024lowlatencyiterativedecodingqldpc or github_relaybp. We prefix decoder names with the decoding matrix they operate on, using GARI for $\accentset{}{D}_{XYZ}$. The hyperparameters of the decoders are discussed in the main text. Note that we consider the best performing BPOSD decoder, operating on either $D_{XYZ}$ or $D_{Z}$, depending on the code, as explained in the main text. For GARI-NMS decoders, the normalization factor is set to 0.96875 in subfigures (a)-(c), and 0.9921875 in (d). For codes with distances 10 and 12, the GARI-NMS-ensemble decoder uses 24 GARI-NMS decoders in parallel.
• Figure 5: Logical error rate performance of the proposed method compared with the Tesseract decoder introduced in beni2025tesseractsearchbaseddecoderquantum, for the SI1000 noise model with a physical error rate of 0.001. We plot the logical error rate versus the BPT code efficiency $k d^2 / n$, for BB codes with parameters $[[72,12,6]]$, $[[90,8,10]]$, and $[[144,12,12]]$, corresponding to efficiencies of 6, 8.89, and 12, respectively (see also Fig. \ref{['fig:logical-error-rate-comp']}). The data for the BPOSD and Tesseract decoders are taken from beni2025tesseractsearchbaseddecoderquantum. GARI-NMS uses a normalization factor of 0.9921875, and ensemble decoders for distances 10 and 12 use 48 GARI-MP decoders in parallel.