Fault-tolerant quantum computers need fast and accurate error decoding. We present CoDecoder, a small neural frontend that composes with an existing decoder to improve accuracy and reduce mean decoding time. It predicts each shot's logical outcome, retains confident predictions and routes the remaining shots to the unchanged fallback. A logical-operator-aware output head scores shifted logical operators, combining equivalent logical lines on surface codes and deciding logical qubits jointly on bivariate-bicycle (BB) codes. Fused GPU inference and efficient routing reduce the overhead paid on every shot. In registered tests under circuit-level noise, composition reduces failures by 6.4-31.4% relative to correlated matching on surface codes of distance 7-17 and by 22-50% relative to full-DEM Relay-BP on two BB codes, with 2.3-124-fold speed-ups across these comparisons. With our re-implementations of the neural decoder Cascade, composition reduces decoding time in all four settings. A joint decision of both decoders on the routed shots keeps this speed and makes the composition more accurate than Cascade in all four, with 14-18% fewer failures on the [[72,12,6]] code. On the [[72,12,6]] BB code, a conventional head with independent logical outputs fails 3.8-5.0 times as often as our joint class score under matched training conditions. Per-shot analysis shows how composition gains accuracy. On retained shots, the frontend corrects more fallback errors than it introduces, even when it is less accurate overall.
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35 pages, 9 figures, 14 tables. Code, trained weights and the records of the registered tests: https://github.com/a85tract/CoDecoder
Fault-tolerant quantum computers need fast and accurate error decoding. We present CoDecoder, a small neural frontend that composes with an existing decoder to improve accuracy and reduce mean decoding time. It predicts each shot's logical outcome, retains confident predictions and routes the remaining shots to the unchanged fallback. A logical-operator-aware output head scores shifted logical operators, combining equivalent logical lines on surface codes and deciding logical qubits jointly on bivariate-bicycle (BB) codes. Fused GPU inference and efficient routing reduce the overhead paid on every shot. In registered tests under circuit-level noise, composition reduces failures by 6.4-31.4% relative to correlated matching on surface codes of distance 7-17 and by 22-50% relative to full-DEM Relay-BP on two BB codes, with 2.3-124-fold speed-ups across these comparisons. With our re-implementations of the neural decoder Cascade, composition reduces decoding time in all four settings. A joint decision of both decoders on the routed shots keeps this speed and makes the composition more accurate than Cascade in all four, with 14-18% fewer failures on the [[72,12,6]] code. On the [[72,12,6]] BB code, a conventional head with independent logical outputs fails 3.8-5.0 times as often as our joint class score under matched training conditions. Per-shot analysis shows how composition gains accuracy. On retained shots, the frontend corrects more fallback errors than it introduces, even when it is less accurate overall.
Foundation decoders, a class of high-capacity neural decoders, are leading candidates for fault-tolerant quantum computing, with accurate and efficient decoding at large code distances. However, their construction often faces a steep scaling barrier, as larger code distances rapidly amplify the cost of syndrome generation and neural optimization. To address this bottleneck, here we devise neural transfer unification (NTU), a unified framework for efficient foundation decoders. A central feature of NTU is its ability to align decoding tasks across code distances via algebraic structures shared by scalable code families, which enables knowledge learned on smaller codes to accelerate large-scale decoder training. We instantiate NTU as NTU-Transformer, a transformer-based neural decoder tailored for planar surface codes and bivariate bicycle codes. For planar surface codes under circuit-level noise, NTU-Transformer outperforms correlation-aware matching on the [[361,1,19]] code and further scales to the [[625,1,25]] code, where it exceeds standard matching through transfer adaptation. For the bivariate bicycle code with [[72,12,6]], it surpasses Relay-BP in the low-physical-error regime. These results establish our proposal as a scalable route to amortized cross-distance training of foundation decoders for fault-tolerant quantum processors.
Fault-tolerant quantum computers will depend crucially on the performance of the classical decoding algorithm which takes in the results of measurements and outputs corrections to the errors inferred to have occurred. Machine learning models have shown great promise as decoders for the surface code; however, this promise has not yet been substantiated for the more challenging task of decoding quantum low-density parity-check (QLDPC) codes. In this paper, we present a recurrent, transformer-based neural network designed to decode circuit-level noise on Bivariate Bicycle (BB) codes. For the [ [ 72 , 12 , 6 ] ] BB code, at a physical error rate of p = 0.1 % , our model achieves logical error rates almost 5 times lower than belief propagation with ordered statistics decoding (BP-OSD), and roughly 5 times larger than a most-likely error decoder. Moreover, while BP-OSD has a wide distribution of runtimes with significant outliers, our model has a consistent runtime and is an order-of-magnitude faster than the worst-case times from a benchmark BP-OSD implementation. On the [ [ 144 , 12 , 12 ] ] BB code, our model obtains worse logical error rates but maintains the speed advantage. These results provide initial evidence that machine learning decoders can out-perform conventional decoders on small QLDPC codes, but suggest more complex architectures and/or training procedures are necessary to scale to larger code sizes.
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.
Erasure qubits reduce overhead in fault-tolerant quantum error correction (QEC) by converting dominant faults into detectable errors known as erasures. They have demonstrated notable improvements in thresholds and scaling in surface and Floquet code memories. In this work, we use erasure qubits on Bivariate Bicycle (BB) codes from the quantum low-density parity-check (QLDPC) regime. Owing to their sparse structure and favorable rate-distance trade-offs, BB codes are practical candidates for QEC. We introduce BiBiEQ, a novel framework that compiles a given BB code into an erasure-aware memory circuit C_E. This erasure circuit C_E comprises erasure checks (ECs), resets, and erasures spread over a user-specified erasure check schedule (2EC, 4EC). BiBiEQ converts this erasure circuit C_E into the stabilizer circuit C for general-purpose decoding. BiBiEQ provides two engines for this conversion, BiBiEQ-Exact and BiBiEQ-Approx. BiBiEQ-Exact preserves the joint-erasure correlations and serves as our accuracy benchmark, while BiBiEQ-Approx uses an independence approximation to accelerate large sweeps and expose accuracy-throughput trade-offs. Using BiBiEQ, we decode the stabilizer circuits to get a per-round logical error rate (LER) for the BB codes and quantify the effect of the EC schedules on the correctable operating region below the pseudo-threshold. The 4EC schedule keeps the accuracy of both engines close to one another, making BiBiEQ-Approx a reliable proxy for BiBiEQ-Exact for faster sweeps. Below the pseudo-threshold, the code distance (d) hop from distance (d) 6 to 10 yields a drop in LER by 10-17x larger than distance (d) 10 to 12, showing that most gains are realized by d=10.
An efficient decoder is essential for quantum error correction, and data-driven neural decoders have emerged as promising, flexible solutions. Here, we introduce a diffusion model framework to infer logical errors from syndrome measurements in quantum low-density parity-check codes. Using the bivariate bicycle code with realistic circuit-level noise, we show that masked diffusion decoders are more accurate, often faster on average, and always faster in the worst case than other state-of-the-art decoders, including belief propagation with ordered statistics decoding (BP-OSD) and autoregressive neural decoders. We show that by using fewer diffusion steps during inference one can gain significant speed at minimal cost in accuracy. By examining the factored attention from our trained neural network we find that, despite being trained solely on paired samples of syndrome-logical errors, this diffusion decoder learns the structure of the quantum codes. We also compare both masked and continuous diffusion decoders on code-capacity noise models, finding that masked diffusion decoders scale better than continuous diffusion decoders.
Quantum low-density parity-check (qLDPC) codes reduce the qubit overhead of fault-tolerant quantum computation by an order of magnitude, but their decoding is harder than its classical counterpart: because many physical errors are equivalent up to stabilizers, the degenerate maximum-likelihood (ML) decoder must compare the probabilities of entire equivalence classes of errors, that is, partition functions, rather than single errors. The workhorse decoder BP+OSD sidesteps degeneracy heuristically and offers no guarantees. We treat degenerate decoding as probabilistic inference in an undirected graphical model: the probability of each logical class is the partition function of an unconstrained, strictly positive Markov random field over the code's check variables, a construction that generalizes the random-bond Ising mapping of the surface code to arbitrary CSS codes and to spacetime decoding with measurement errors and circuit-level noise. On this model we build two decoders. The first estimates all class partition functions by annealed importance sampling with common random numbers and attaches to every decision a certificate of optimality: a paired bootstrap test, or, composed with constant-factor estimators such as WISH, an exact optimality proof. The second is region-based: the Bethe free energy, whose bias cancels between classes, reproduces exact ML decoding on every tested surface-code instance at millisecond cost, and enlarging the regions to elimination clusters makes exact degenerate ML decoding of the [[72,12,6]] bivariate bicycle code feasible. Across surface codes and the bivariate bicycle codes [[72,12,6]] and [[144,12,12]], under code-capacity, phenomenological, and circuit-level noise, the sampling decoder matches or exceeds BP+OSD while certifying the bulk of its decisions, and the certificate flags exactly the syndromes on which any fast decoder should be distrusted.