Efficient Approximate Degenerate Ordered Statistics Decoding for Quantum Codes via Reliable Subset Reduction

Efficient and scalable decoding of quantum codes is essential for high-performance quantum error correction. In this work, we introduce Reliable Subset Reduction (RSR), a reliability-driven preprocessing framework that leverages belief propagation (BP) statistics to identify and remove highly reliable qubits, substantially reducing the effective problem size. Additionally, we identify a degeneracy condition that allows high-order OSD to be simplified to order-0 OSD. By integrating these techniques, we present an ADOSD algorithm that significantly improves OSD efficiency. Our BP+RSR+ADOSD framework extends naturally to circuit-level noise and can handle large-scale codes with more than $10^4$ error variables. Through extensive simulations, we demonstrate improved performance over MWPM and Localized Statistics Decoding for a variety of CSS and non-CSS codes under the code-capacity noise model, and for rotated surface codes under realistic circuit-level noise. At low physical error rates, RSR reduces the effective problem size to as little as 1% (e.g., for $ε=0.001$ in surface-code DEM), enabling higher-order OSD with drastically reduced computational complexity. These results highlight the practical efficiency and broad applicability of the BP+ADOSD framework for both theoretical and realistic quantum error correction scenarios.

💡 Research Summary

The paper tackles the long‑standing challenge of scalable quantum error‑correction decoding by introducing a two‑stage framework that combines belief‑propagation (BP) statistics with an approximate degenerate ordered‑statistics decoder. The first stage, Reliable Subset Reduction (RSR), exploits both the stability of hard‑decision bits across BP iterations and the confidence of final soft beliefs to identify a “reliable” subset of qubits. By fixing these highly reliable variables, the original linear system (2 n binary variables with n‑k independent parity constraints) is dramatically reduced in dimension—from n + k down to a small residual r that can be as low as 1 % of the original size for physical error rates ε≈10⁻³. This reduction is achieved with only O(n) additional work, essentially re‑using the messages already computed by BP4 (the quaternary BP that respects X/Z/Y correlations).

The second stage applies an OSD variant (OSD4) to the remaining uncertain variables, ordering them by the reliabilities supplied by RSR. Classical OSD would generate a list of candidate error vectors by Gaussian elimination followed by flips of low‑reliability bits; OSD4 follows the same procedure but operates on the reduced problem, making high‑order OSD computationally feasible even for codes with >10⁴ error variables. Crucially, the authors recognize that quantum codes are highly degenerate: many error candidates belong to the same stabilizer coset and therefore have identical logical effect. They derive a “degeneracy‑aware pruning” condition that discards flips corresponding to stabilizer multiplication, because such flips do not change the logical syndrome. When all remaining flips lie in the trivial logical coset, high‑order OSD is unnecessary, and a zero‑order Approximate Degenerate OSD (ADOSD4) suffices. This insight reduces the list size and eliminates unnecessary high‑order computations.

The framework is extended to realistic circuit‑level noise by converting the detector error model (DEM) generated by the STIM simulator into a binary linear system. The same BP‑RSR‑ADOSD pipeline can then be applied without modification, handling both data‑ and syndrome‑errors in the phenomenological model. Extensive simulations were performed on a diverse set of quantum LDPC and topological codes: BB codes, rotated toric and surface codes, (6.6.6) and (4.8.8) color codes, twisted XZZX toric codes, generalized hypergraph‑product (GHP) codes, and lift‑connected surface (LCS) codes. Under the code‑capacity depolarizing channel, MBP4 + ADOSD4 consistently outperformed prior BP‑OSD hybrids, ambiguity clustering, and localized statistics decoding, achieving lower logical error rates down to ~10⁻⁶ and higher thresholds for topological families. In the DEM circuit‑level setting for rotated surface codes, the proposed method surpassed the minimum‑weight perfect‑matching (MWPM) decoder and the localized statistics decoder, attaining an improved threshold of approximately 0.76 %.

A key quantitative result is the dramatic dimensionality reduction achieved by RSR: for ε ≤ 0.005 the effective problem size drops below 4 % of the original, and for ε = 0.001 it falls below 1 %. This enables the use of high‑order OSD (order ≥ 2) with manageable computational cost, a regime previously inaccessible to OSD‑based post‑processing. The overall computational budget remains modest—at most ten BP iterations (typically 3–6) followed by a single Gaussian elimination on the reduced system—making the approach suitable for real‑time decoding in large‑scale quantum processors.

Complexity analysis shows RSR adds only linear overhead, while the OSD step scales as O(r³) for Gaussian elimination plus O(r·L) for list generation, where r is the reduced variable count and L the list size. Because r is often two orders of magnitude smaller than n, the total runtime is comparable to, or better than, state‑of‑the‑art decoders such as MWPM, especially for highly degenerate LDPC codes where MWPM is not directly applicable.

The paper acknowledges limitations: the choice of reliability thresholds in RSR must be tuned per code and noise level, and at higher physical error rates the fraction of reliably fixed qubits diminishes, reducing the benefit of dimensionality reduction. Nevertheless, the combination of BP‑derived reliability, aggressive yet principled variable fixing, and degeneracy‑aware OSD constitutes a powerful, general‑purpose decoding strategy that bridges the gap between theoretical optimality and practical scalability for both code‑capacity and realistic circuit‑level quantum error correction.