Simple, Efficient, and Generic Post-Selection Decoding for qLDPC codes

TL;DR

The paper tackles the challenge of achieving low logical-error rates in quantum error correction with limited qubit resources by introducing argument reweighting, a simple post-selection strategy compatible with maximum-likelihood-type decoders across qLDPC codes. It uses a second decoding round under a reweighted error model to detect high-confidence syndrome outcomes, employing gap or ratio tests and accepting only when corrections agree (PEC), with extensions to two- and three-round logical-error criteria (2R-LEC, 3R-LEC). Across circuit-level simulations on rotated surface codes and BB codes with decoders like MWPM and BP variants, the method achieves two- to three-orders-of-magnitude reductions in logical error rates at modest rejection rates, notably reducing the $[[144,12,12]]$ BB code logical error rate from $9.08\times10^{-7}$ to $1.41\times10^{-8}$ at $r\approx 1.44\times10^{-5}$. The approach is scalable, adaptable to sliding-window decoding, and offers a practical pathway to fault-tolerant quantum computation by balancing accuracy and resource overhead, including applications to modular distillation, gate teleportation, and error mitigation via spacetime-noise inversion.

Abstract

Quantum error correction is indispensable for scalable quantum computation. Although encoding logical qubits substantially enhances noise resilience, achieving logical error rates low enough for practical algorithms remains challenging on existing hardware. Here we introduce argument reweighting, a simple and broadly applicable post-selection decoding strategy that boosts the performance of maximum-likelihood-type decoders, including minimum-weight perfect matching and belief-propagation families. The method suppresses logical errors by performing additional decoding rounds under reweighted error models, enabling acceptance of high-confidence syndrome outcomes. Circuit-level simulations across multiple decoders and qLDPC codes show that argument reweighting substantially suppresses logical errors, requiring a rejection rate of only $1.44\times10^{-5}$ to reduce the logical error rate by almost two orders of magnitude for the $[[144,12,12]]$ bivariate bicycle code. These results establish argument reweighting as a practical and resource-efficient approach for enhancing quantum fault tolerance.

Figures (11)

• Figure 1: (a) Efficiency of post-selection decoding. A post-selection scheme trades a rejection rate for a reduction in the logical error rate. The scheme is deemed more efficient if it achieves the same logical-error suppression at a smaller rejection rate. (b) Schematic illustration of argument reweighting (shown with the physical-error criterion). In a maximum-likelihood-type decoder, the correction corresponds to the most likely error configuration among all those compatible with the measured syndrome, which are represented by horizontal bars. The first decoding round uses the original error model and outputs the candidate correction (highlighted in magenta). In the second round, the error model is modified to suppress the likelihood of this candidate, and decoding is performed again. The circuit run is accepted if and only if the second-round output agrees with the first; otherwise, it is rejected.
• Figure 2: Total logical error rate after $d$ rounds of parity-check measurements, where $d$ denotes the code distance. We simulated rotated surface codes ($d=5, 9, 11$) surface_horsman_2012 and four BB code instances ($[[18,4,4]]$, $[[72,12,6]]$, $[[90,8,10]]$, and $[[144,12,12]]$) highthreshold_bravyi_2024coprime_wang_2025. Results are obtained using the ratio test variant defined in Eq.(\ref{['eq:reweight']}). Further details regarding the simulation are provided in Appendix \ref{['app:details']}.
• Figure 3: Efficiency of various post-selection strategies applied to the BP-LSD decoder. The performance of four strategies---3R-LEC, error-cluster statistics (ECS), correction weight (CW), and detector density (DD) efficient_lee_2025---is compared across three BB code instances. For each code, we estimate the rejection rates (per $d$ parity-check cycles) required to reduce the logical error rate by one and two orders of magnitude. We find that our method (3R-LEC) consistently requires the lowest rejection rate among all post-selection strategies considered. We note that the rejection rates for 3R-LEC are reported as conservative upper bounds, whereas those for other strategies are calculated precisely (subject to statistical error); see Appendix \ref{['app:details']} for further details.
• Figure 4: Performance with increasing reweighting-decoding rounds. Results are shown for the [[72,12,6]] BB code using the BP-OSD decoder.
• Figure 5: Logical error suppression factor. For each decoder and code, the suppression factor is defined as the logical error rate at a given rejection rate divided by the logical error rate without post-selection.