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arXiv:2605.20346

Forced Gap Post-Selection for Quantum LDPC Codes and their Operations

Adam Wills, Theodore J. Yoder, Isaac Chuang

quant-phSubmitted May 19, 2026• Announced May 21, 2026• 0 citations• Revised Jun 23, 2026 (v2)
v1 · May 19, 2026v2 · Jun 23, 2026
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Abstract

We develop a simple and general post-selection strategy for high-rate quantum codes that is transferrable across decoders. After an initial baseline run, the decoder is re-run once per logical observable, and forced in these latter runs to provide a solution where the given observable has the complementary outcome. Shots are rejected that find logically complementary solutions with similar likelihoods compared to the baseline. Using the Relay-BP decoder, we benchmark the strategy on the $72$-qubit and $144$-qubit bivariate bicycle codes, as well as surgery gadgets for the latter. In comparison to previous post-selection strategies, our results offer an improved logical error rate by over a factor of $4$ on the same circuit and physical error rate, and at the same rate of post-selection. Our strategies are also lightweight, relying only on FPGA-friendly belief propagation, whereas the previous best used repeated rounds of a high-latency BP-OSD decoder.

Comments
v2: Updated funding acknowledgement

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We develop a simple and general post-selection strategy for high-rate quantum codes that is transferrable across decoders. After an initial baseline run, the decoder is re-run once per logical observable, and forced in these latter runs to provide a solution where the given observable has the complementary outcome. Shots are rejected that find logically complementary solutions with similar likelihoods compared to the baseline. Using the Relay-BP decoder, we benchmark the strategy on the 727272-qubit and 144144144-qubit bivariate bicycle codes, as well as surgery gadgets for the latter. In comparison to previous post-selection strategies, our results offer an improved logical error rate by over a factor of 444 on the same circuit and physical error rate, and at the same rate of post-selection. Our strategies are also lightweight, relying only on FPGA-friendly belief propagation, whereas the previous best used repeated rounds of a high-latency BP-OSD decoder.
Post-selection improves the logical reliability of quantum computation by rejecting shots which are likely to result in logical failure. We introduce graph neural networks that predict decoder failure without additional decoder executions, using only syndromes, and for high-rate quantum low-density parity-check (qLDPC) codes, existing belief-propagation posteriors. We evaluate rotated surface and bivariate bicycle (BB) code memories under uniform depolarising circuit-level noise. Retaining 90% of shots gives approximately 3700x and 740x reductions in logical error rates for [[72,12,6]] and [[144,12,12]] BB codes at realistic physical error rates. For the distance 5 surface code at physical noise p=0.003, our neural post-selector achieves logical error suppression comparable to the leading complementary gap method without running the decoder.
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 1.44×10−51.44\times10^{-5}1.44×10−5 to reduce the logical error rate by almost two orders of magnitude for the [[144,12,12]][[144,12,12]][[144,12,12]] bivariate bicycle code. These results establish argument reweighting as a practical and resource-efficient approach for enhancing quantum fault tolerance.
We propose a decoder for quantum low density parity check (LDPC) codes based on a beam search heuristic guided by belief propagation (BP). Our beam search decoder applies to all quantum LDPC codes and achieves different speed-accuracy tradeoffs by tuning its parameters such as the beam width. We perform numerical simulations under circuit level noise for the [[144,12,12]][[144, 12, 12]][[144,12,12]] bivariate bicycle (BB) code at noise rate p=10−3p=10^{-3}p=10−3 to estimate the logical error rate and the 99.9 percentile runtime and we compare with the BP-OSD decoder which has been the default quantum LDPC decoder for the past six years. A variant of our beam search decoder with a beam width of 64 achieves a 17×17\times17× reduction in logical error rate. With a beam width of 8, we reach the same logical error rate as BP-OSD with a 26.2×26.2\times26.2× reduction in the 99.9 percentile runtime. We identify the beam search decoder with beam width of 32 as a promising candidate for trapped ion architectures because it achieves a 5.6×5.6\times5.6× reduction in logical error rate with a 99.9 percentile runtime per syndrome extraction round below 1ms at p=5×10−4p=5 \times10^{-4}p=5×10−4. Remarkably, this is achieved in software on a single core, without any parallelization or specialized hardware (FPGA, ASIC), suggesting one might only need three 32-core CPUs to decode a trapped ion quantum computer with 1000 logical qubits.
This work presents a hardware-efficient and fully parallelizable decoder for quantum LDPC codes that leverages belief propagation (BP) with a speculative post-processing strategy inspired by classical Chase decoding algorithm. By monitoring bit-level oscillation patterns during BP, our method identifies unreliable bits and generates multiple candidate vectors to selectively flip syndromes. Each modified syndrome is then decoded independently using short-depth BP, a process we refer to as BP-SF (syndrome flip). This design eliminates the need for costly Gaussian elimination used in the current BP-OSD approaches. Our implementation achieves logical error rates comparable to or better than BP-OSD while offering significantly lower latency due to its high degree of parallelism for a variety of bivariate bicycle codes. Evaluation on the⟦144,12,12⟧\llbracket 144,12,12 \rrbracket[[144,12,12]]bivariate bicycle code shows that the proposed decoder reduces average latency to approximately 70% of BP-OSD. When post-processing is parallelized the average latency is reduced by 55% compared to the single process implementation, with the maximum latency reaching as low as 18%. These advantages make it particularly well-suited for real-time and resource-constrained quantum error correction systems.
Quantum error correction can be enhanced by post-selecting out runs that are likely to produce a logical failure, but the most accurate measures for that require costly decoder-level information. We introduce a practical, decoder-agnostic post-selection method that learns directly from syndrome data. The method trains a supervised classifier to distinguish between syndromes from low- and high-noise regimes, and then uses the classifier's output as an abort score for new runs, without requiring logical-error labels, correction operators, or code-specific likelihood calculations. We validate the approach in three complementary settings: circuit-level simulations of the Gross bivariate-bicycle code, code-capacity simulations of the surface code, and experimental logical magic-state distillation data from the QuEra neutral-atom processor. In the Gross and surface codes, learned syndrome post-selection reduces the conditional logical error rate at a fixed acceptance rate, with performance comparable to syndrome-weight filtering. For the surface code, the learned classifier reveals a post-selection transition distinct from the conventional decoding threshold. In the experimental data, the machine-learning score outperforms syndrome-weight post-selection and, when combined with logical-gap filtering, improves the output fidelity beyond using the logical gap alone. These results show that syndrome-only learning provides a scalable and hardware-compatible route to improving the reliability of quantum error correction.
Constant-rate quantum low-density parity-check (LDPC) codes promise fault-tolerant quantum computation with constant spatial overhead in the asymptotic limit. Nonetheless, discovering finite-length code instances with good practical performance remains challenging. We introduce a new family of quantum LDPC codes with design rate 1/51/51/5 and check weight 999 that approaches the teraquop memory regime per qubit-round with several hundred physical qubits, under idling-free circuit-level noise of strength 0.1%0.1\%0.1% and GPU-accelerated Relay-belief-propagation (Relay-BP) decoding with average latencies around 1-2 ms, a regime relevant to trapped-ion and neutral-atom processors. The construction involves the balanced product of classical LDPC codes with design rate 1/21/21/2 that share non-abelian Zℓ⋊Zm\mathbb{Z}_\ell \rtimes \mathbb{Z}_mZℓ​⋊Zm​ group symmetries, which may be of independent interest for classical error correction. We build syndrome extraction circuits tailored to reconfigurable atom arrays using a simple greedy scheduler, with single-round rearrangement times around 30-60 ms using present hardware specifications, and substantial room for future improvements. We also construct logical Pauli bases that are equivariant with respect to the group symmetry, which can significantly compress the design space for code surgery. Together, these results further advance the practicality of constant-rate quantum LDPC codes for near-term, fault-tolerant quantum computers.