Fault-tolerant quantum computers use decoders to monitor for errors and find a plausible correction. A decoder may provide a decoder confidence score (DCS) to gauge its success. We adopt a swim distance DCS, computed from the shortest path between syndrome clusters. By contracting tensor networks, we compare its performance under phenomenological noise to the well-known complementary gap and find that both reliably estimate the logical error probability (LEP) in a decoding window. We explore ways to use this to mitigate the LEP in entire logical circuits. For shallow circuits, we just abort if any decoding window produces an exceptionally low DCS: for a distance-13 surface code under circuit-level noise, rejecting a mere 0.1% of possible DCS values improves the entire circuit's LEP by more than 5 orders of magnitude. For larger algorithms comprising up to billions of windows, DCS-based rejection remains effective for enhancing observable estimation. Moreover, one can use the DCS to assign each circuit's output a unique LEP, and use it as a basis for maximum likelihood estimation. This can reduce the effects of noise by an order of magnitude at no quantum cost; methods can be combined for further improvements.
Fault-tolerant quantum computers use decoders to monitor for errors and find a plausible correction. A decoder may provide a decoder confidence score (DCS) to gauge its success. We adopt a swim distance DCS, computed from the shortest path between syndrome clusters. By contracting tensor networks, we compare its performance under phenomenological noise to the well-known complementary gap and find that both reliably estimate the logical error probability (LEP) in a decoding window. We explore ways to use this to mitigate the LEP in entire logical circuits. For shallow circuits, we just abort if any decoding window produces an exceptionally low DCS: for a distance-13 surface code under circuit-level noise, rejecting a mere 0.1% of possible DCS values improves the entire circuit's LEP by more than 5 orders of magnitude. For larger algorithms comprising up to billions of windows, DCS-based rejection remains effective for enhancing observable estimation. Moreover, one can use the DCS to assign each circuit's output a unique LEP, and use it as a basis for maximum likelihood estimation. This can reduce the effects of noise by an order of magnitude at no quantum cost; methods can be combined for further improvements.
Quantum error mitigation (QEM) is typically viewed as a suite of practical techniques for today's noisy intermediate-scale quantum devices, with limited relevance once fault-tolerant quantum computers become available. In this work, we challenge this conventional wisdom by showing that QEM can continue to provide substantial benefits in the era of quantum error correction (QEC), and in an even more efficient manner than it does on current devices. We introduce a framework for logical-level QEM that leverages soft information naturally produced by QEC decoders, requiring no additional data, hardware modifications, or runtime overhead beyond what QEC protocols already provide. Within this framework, we develop and analyze three logical-level QEM techniques: post-selection and runtime abort policies, probabilistic error cancellation, and zero-noise extrapolation. Our techniques reduce logical error rates by more than 100x while discarding fewer than 0.1% of shots; they also provide in situ characterization of logical channels for QEM protocols. As a proof of principle, we benchmark our approach using a surface-code architecture and two state-of-the-art decoders based on tensor-network contraction and minimum-weight perfect matching. We evaluate logical-level QEM on random Clifford circuits and molecular simulation algorithms and find that, compared to previous approaches relying on QEC only or QEC combined with QEM, we can achieve up to 87.4% spacetime overhead savings. Our results demonstrate that logical-level QEM with QEC decoder soft information can reliably improve logical performance, underscoring the efficiency and usefulness of QEM techniques for fault-tolerant quantum computers.
To utilize quantum error-correcting codes, a decoder must infer the logical sector from the measured syndrome. Beyond a hard logical decision, some decoders provide soft information estimating its reliability. For minimum-weight perfect matching (MWPM), a common confidence measure is the complementary, or logical, gap. Neural-network decoders trained with binary cross entropy naturally output a different score, the logit, whose relation to the matching gap remains unclear. Here, we test whether the logit of a graph neural network (GNN) decoder behaves as a learned proxy for the logical gap. Using a pretrained GNN for the rotated surface code under uniform circuit-level noise [PRR, 7(2):023181, 2025], we compare its logit with the MWPM gap shot by shot on identical syndromes. The native GNN decoder-logit pair yields a lower post-selected logical error rate than the native MWPM decoder-gap pair over the operationally relevant range of acceptance rates. Exchanging the scores between decoders isolates score discrimination from hard-decision accuracy and shows that both contribute to the advantage. At larger distances each score best ranks failures of its own decoder, whereas at small distance, where the bounded matching gap saturates, the learned confidence better discriminates MWPM failures than the gap itself. Finally, GNN confidence is substantially better described by the posterior logistic form implied by an ideal log-likelihood ratio, while the MWPM gap shows systematic departures. Thus, a neural-network decoder trained only on syndromes and logical labels learns a gap-like confidence proxy whose native scale more closely follows the expected posterior calibration form, supporting learned soft output when MWPM gap estimates are unavailable, costly, or insufficiently expressive.
Early fault-tolerant quantum algorithms are a key step in scalable quantum computing, but are ultimately constrained by residual, uncorrectable logical errors. Logical quantum error mitigation can eliminate this bias, however its standard requirements include prior noise characterization, more complex experiments, and substantial sampling overhead. Although no mitigation technique can simultaneously overcome all three constraints, we demonstrate that the first two---noise learning and need for modified circuits---can be completely bypassed for logical circuits by leveraging only syndrome records. Specifically, we introduce virtual error cancellation: a method that, unlike previous bias-free approaches, operates entirely in classical post-processing on universal quantum circuits. We develop and numerically test end-to-end protocols that reduce the sampling overhead beyond fundamental limitations of existing syndrome-aware mitigation techniques, including syndrome-based postselection. In particular, we propose a scheme that pairs the low-latency decoder used in the experiment with a high-complexity decoder run in post-processing. In numerical simulations of a distance-7 rotated surface code, we show that our full-stack dual-decoder algorithm exhibits an error that diminishes as the number of samples increases, capable of reaching more than three orders of magnitude of suppression compared to the decoded logical error rate in an experimentally achievable setting. These results establish syndrome records as a resource for correcting universal computational estimates after quantum execution.
Quantum error-correcting decoders provide both corrections and confidence estimates for the logical-error risk of individual measurement records. Such confidence can guide postselection, but it depends on the accuracy of the decoder noise model. When the physical device drifts away from that model, confidence measures such as the logical gap can become miscalibrated. We show that this mismatch can itself provide useful information. On IBM quantum hardware, syndrome statistics drift across consecutive acquisition windows, and the resulting detector histories differ substantially from those predicted by the stationary detector error model used for decoding. We observe a similar hardware-model mismatch in independent Google surface-code memory data. We exploit this effect by assigning additional rejection weight to detector histories that are unlikely under the assumed model. For minimum-weight perfect matching (MWPM), the minimum weight W provides an inexpensive proxy for syndrome surprisal. On held-out IBM repetition-code memories, likelihood-aware postselection reduces the retained logical-error probability (LEP) by 10.6% at d=7 and 28.8% at d=9, both at 15% rejection. On Google ten-round d=5 XZZX surface-code memory data, it reduces retained LEP by 75.4% at 90% rejection. Applied to Google's magic-state cultivation experiment with the Tesseract decoder, it reduces tomography infidelity by 34.7% with 30% additional grafting-stage rejection, or increases accepted yield by 30.4% at a target fidelity of 97.0%. Burst-noise simulations show that the advantage persists as code distance and syndrome-extraction volume increase. The method reuses quantities already produced during logical-gap decoding and requires only an additional fitting step.
Accurate noise estimation is essential for fault-tolerant quantum computing, since decoding per formance depends critically on the fidelity of circuit-level noise parameters. We introduce a differ entiable Maximum Likelihood Estimation (dMLE) framework that learns these parameters directly from experimental syndrome data using likelihood gradients, with no decoder in the training loop. Mapping syndrome likelihoods to statistical-mechanical models enables exact and efficient evalua tion via planar solvers for repetition codes and tensor networks (TNs) for surface codes, extending the exact treatment to code distance d = 5 with a small number of rounds, and, via spatial subsam pling combined with time-translation symmetry, to larger codes and larger round counts. Across four experimental superconducting datasets, dMLE reduces logical error rates by up to 30.6% for repetition codes. On the Sycamore d = 3 surface code, a TN decoder using the dMLE prior attains a logical error rate comparable to that reported for state-of-the-art AlphaQubit, using standard syndromes alone without auxiliary leakage data. On the larger d = 5 surface code the dMLE prior lowers the logical error rate of the TN and Tesseract decoders by 7.18% and 5.23% relative to a prior optimized by reinforcement learning; at d = 7 it lowers that of Tesseract by 4.56% and 4.83% in the X and Z bases.
Broad applications of quantum computers will require error correction (EC). However, hardware roadmaps indicate that physical qubit numbers will remain limited in the foreseeable future, leading to residual logical errors that constrain the size and accuracy of achievable computations. Recent work suggested logical error mitigation (LEM), which applies known error mitigation (EM) methods to logical errors, eliminating their effect at the cost of a runtime overhead.
We introduce syndrome-aware logical error mitigation (SALEM), which mitigates logical errors conditioned on the error syndromes measured during error correction. The runtime overhead of SALEM is exponentially lower than that of LEM schemes which do not make use of syndrome data, enabling substantially larger circuit volumes that can be executed accurately. Compared to the routinely used combination of error correction and syndrome rejection (post-selection), SALEM increases the size of reliably executable computations by orders of magnitude. In the practical setting where space and time overheads are fixed and error reduction methods are compared by their resulting estimation errors, we observe a surprising phenomenon: SALEM, which tightly combines EC with EM, can outperform physical EM even above the standard fault-tolerance (pseudo) threshold. Thus, SALEM can make use of EC in regimes of physical error rates where EC is commonly deemed useless.