Localized statistics decoding for quantum low-density parity-check codes

TL;DR

Quantum LDPC codes offer lower overhead than the surface code, but practical decoding has been a barrier. This work introduces LSD, a parallel, reliability-guided inversion decoder that factorizes the decoding problem into local clusters and solves them concurrently using on-the-fly PLU elimination, achieving performance on par with BP+OSD while significantly improving runtime and hardware compatibility. Across surface codes, hypergraph product codes, and bivariate bicycle codes, LSD (with potential higher-order reprocessing) matches or surpasses existing soft-information-guided decoders and scales favorably in sub-threshold regimes. The approach enables real-time syndrome processing on specialized hardware and opens avenues for efficient, scalable quantum error correction in experimental settings.

Abstract

Quantum low-density parity-check codes are a promising candidate for fault-tolerant quantum computing with considerably reduced overhead compared to the surface code. However, the lack of a practical decoding algorithm remains a barrier to their implementation. In this work, we introduce localized statistics decoding, a reliability-guided inversion decoder that is highly parallelizable and applicable to arbitrary quantum low-density parity-check codes. Our approach employs a parallel matrix factorization strategy, which we call on-the-fly elimination, to identify, validate, and solve local decoding regions on the decoding graph. Through numerical simulations, we show that localized statistics decoding matches the performance of state-of-the-art decoders while reducing the runtime complexity for operation in the sub-threshold regime. Importantly, our decoder is more amenable to implementation on specialized hardware, positioning it as a promising candidate for decoding real-time syndromes from experiments.

Figures (12)

• Figure 1: Illustration of the factorization of the decoding problem on a $5 \times 10$ surface code patch. Below the threshold, errors are typically sparsely distributed on the decoding graph and form small clusters with disjoint support.
• Figure 2: Reliability-based weighted cluster growth example for the surface code. (a) The syndrome of an error is indicated as red square vertices. The fault nodes are colored to visualize their error probabilities obtained from belief propagation pre-processing. (b) Clusters after the first two growth steps. In the guided cluster growth strategy, fault nodes are added individually to the local clusters. The order of adding the first two fault nodes to each cluster is random since both have the same probability due to the presence of degenerate errors. (c) After an additional growth step, the two clusters are merged and the combined cluster is valid. (d) Legend for the used symbols.
• Figure 3: Cluster size statistics of the $\llbracket 144, 12, 12 \rrbracket$ bivariate bicycle code of Ref. bravyi_high-threshold_2024 under circuit-level noise with strength $p$. Markers show the mean of the distribution while shapes are violin plots of the distribution obtained from $10^5$ samples. Yellow distributions show statistics for the optimal factorization while the blue distributions show statistics for the factorization returned by BP+LSD. We show in panel (a) the distribution of the maximum cluster size $\kappa$ and in (b) the distribution of the cluster count, $\nu$, for each decoding sample. Markers and distributions are slightly offset from the actual error rate to increase readability.
• Figure 4: Comparison of various decoders guided by belief propagation for decoding rotated surface codes of distance $d$ subject to circuit-level depolarizing noise parameterized by a single parameter, called the physical error rate$p$, see \ref{['ssec:noise_model']} for details. We use Stim to perform a surface_code:rotated_memory_z experiment for $d$ syndrome extraction cycles with single and two-qubit error probabilities $p$. (a) The performance of BP+OSD-0 that matches the performance of the proposed decoder. (b) The performance of a BeliefFind decoder that shares a cluster growth strategy with the proposed decoder. (c) Performance of the proposed BP+LSD decoder. The shading indicates hypotheses whose likelihoods are within a factor of 1000 of the maximum likelihood estimate, similar to a confidence interval.
• Figure 5: Below threshold logical error rate $p_L$ of a family of $\llbracket 25s^2, s^2 \rrbracket$ constant-rate hypergraph product codes decoded with of the BP+LSD decoder. We simulate $N_c = 12$ rounds of syndrome extraction cycles under circuit-level noise with physical error rate $p$ and apply a $(3, 1)-$overlapping window technique to enable fast and accurate single-shot decoding, see \ref{['ssec:noise_model']} for details. The shading indicates hypotheses whose likelihoods are within a factor of 1000 of the maximum likelihood estimate, similar to a confidence interval. Dashed lines are an exponential fit with a linear exponent to the numerically observed error rates.

Theorems & Definitions (4)

• Definition 4.1: Clusters
• Definition 4.2: Cluster sub-matrix
• Definition 4.3: Cluster-boundary and candidate fault nodes
• Definition 4.4: Cluster collisions