Recent work has shown that syndrome information obtained during error-corrected quantum computation can be utilized for logical error mitigation (LEM), yielding exponential reductions in runtime overhead compared to "external'' LEM methods, which do not make use of syndrome data. However, these syndrome-aware LEM methods (SALEM) generally require adaptive circuit operations based on real-time processing of syndromes. Here, we introduce syndrome-based offline logical error mitigation (SOLEM), a framework that utilizes syndrome data entirely in post-processing, without any modification to the given error-corrected circuit. In this sense, SOLEM is "built into'' error-corrected circuits, and is in fact simpler to implement than external LEM protocols. The basic idea of SOLEM is to use naturally occurring syndrome-conditioned logical error channels in place of deliberate circuit modifications. We develop unbiased and heuristic estimators within this framework, including offline logical versions of probabilistic error cancellation (PEC) and zero-noise extrapolation (ZNE), as well as a global-regression estimator. SOLEM's fully offline nature leads to three significant advantages. First, SOLEM can post-process syndromes with a more powerful decoder than the one used in real time. Second, multiple estimators can be constructed from the same measured data and optimized using empirical statistical errors and related performance metrics. Third, logical error characterization is needed only in post-processing, allowing characterization data to be collected during circuit execution and reducing sensitivity to hardware drift. Remarkably, SOLEM estimators can achieve these advantages while attaining a runtime overhead which is exponentially lower than that of external LEM and is comparable to that of corresponding SALEM estimators.
Recent work has shown that syndrome information obtained during error-corrected quantum computation can be utilized for logical error mitigation (LEM), yielding exponential reductions in runtime overhead compared to "external'' LEM methods, which do not make use of syndrome data. However, these syndrome-aware LEM methods (SALEM) generally require adaptive circuit operations based on real-time processing of syndromes. Here, we introduce syndrome-based offline logical error mitigation (SOLEM), a framework that utilizes syndrome data entirely in post-processing, without any modification to the given error-corrected circuit. In this sense, SOLEM is "built into'' error-corrected circuits, and is in fact simpler to implement than external LEM protocols. The basic idea of SOLEM is to use naturally occurring syndrome-conditioned logical error channels in place of deliberate circuit modifications. We develop unbiased and heuristic estimators within this framework, including offline logical versions of probabilistic error cancellation (PEC) and zero-noise extrapolation (ZNE), as well as a global-regression estimator. SOLEM's fully offline nature leads to three significant advantages. First, SOLEM can post-process syndromes with a more powerful decoder than the one used in real time. Second, multiple estimators can be constructed from the same measured data and optimized using empirical statistical errors and related performance metrics. Third, logical error characterization is needed only in post-processing, allowing characterization data to be collected during circuit execution and reducing sensitivity to hardware drift. Remarkably, SOLEM estimators can achieve these advantages while attaining a runtime overhead which is exponentially lower than that of external LEM and is comparable to that of corresponding SALEM estimators.
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.
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 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.
Recent progress in fault-tolerant quantum computing suggests that leveraging error-syndrome information at the logical layer can substantially improve performance, including the estimation of logical observables from noisy states. In this work, based on quantum estimation theory, we develop an information-theoretic framework to quantify the utility of error syndromes for noisy logical observable estimation. We distinguish two operational regimes of such syndrome-aware protocols: classical protocols, in which the logical measurement basis is fixed and syndrome information is used only in classical post-processing, and quantum protocols, in which the logical quantum control can be tailored to depend on the observed error syndrome. For classical syndrome-aware protocols, we prove a universal limitation: on average, syndrome information can improve the effective logical error rate by at most a factor of two, implying at most a quadratic reduction in sampling overhead. In contrast, once syndrome-conditioned quantum control is permitted, we demonstrate that the effective logical error rate decays exponentially with the number of code blocks. These findings provide fundamental guidance for designing future fault-tolerant architectures that actively exploit syndrome records rather than discarding them after decoding.
We present a general framework for applying linear quantum error mitigation (QEM) techniques directly to physical qubits within a logical qubit to suppress logical errors. By exploiting the linearity of quantum error correction (QEC), we demonstrate that any linear QEM method$\unicode{x2014}$including probabilistic error cancellation (PEC), zero-noise extrapolation (ZNE), and symmetry verification$\unicode{x2014}$can be integrated into the physical layer without requiring modifications to the subsequent QEC decoder. Applying this framework to memory experiments using PEC, we analytically prove and numerically verify that the leading-order contribution to the logical error can be removed, increasing the effective code distance by 2. Our simulations on repetition and rotated surface codes show that a distance-3 code with physical-level PEC achieves logical error rates lower than or similar to a distance-5 unmitigated code while using 40% and 64% fewer qubits, respectively. These results establish physical-level QEM as a widely compatible and resource-efficient strategy for enhancing logical performance in early fault-tolerant architectures.
Error mitigation is essential for unlocking the full potential of quantum algorithms and accelerating the timeline toward quantum advantage. As quantum hardware progresses to push the boundaries of classical simulation, efficient and robust error mitigation methods are becoming increasingly important for producing accurate and reliable outputs. However, existing error-mitigation approaches face a fundamental tradeoff between practical performance and reliability: heuristic methods such as zero-noise extrapolation (ZNE) enjoy faster runtime but lack accuracy guarantees, while rigorous techniques such as probabilistic error cancellation (PEC) provide unbiased estimates at prohibitive computational cost. We introduce a characterization-based, rigorously-grounded quantum error mitigation and error suppression framework (QESEM) that resolves this tradeoff by leveraging the accuracy guarantees of quasi-probabilistic mitigation with dramatically reduced overhead. We explain the innovative methods underlying QESEM and demonstrate its capabilities in the largest utility-scale error mitigation experiment based on an unbiased method. This experiment simulates the kicked transverse field Ising model with far-from-Clifford parameters on an IBM Heron device. We further validate QESEM's versatility across arbitrary quantum circuits and devices through high-accuracy error-mitigated molecular VQE circuits executed on IBM Heron and IonQ trapped-ion devices. Compared with multiple variants of the widely used zero-noise extrapolation method, QESEM consistently achieves higher accuracy while avoiding the prohibitive runtime overhead associated with PEC. These results mark a significant step forward in accuracy and reliability for running quantum circuits on current devices across diverse applications. Finally, we provide projections of QESEM's performance on near-term devices toward quantum advantage.