Error detection can suppress noise at a lower sampling cost than probabilistic error cancellation (PEC), but leaves residual logical errors. We combine the two approaches to mitigate these errors while lowering the cost of PEC. We introduce a spacetime Pauli-Lindblad representation and use it to construct the post-selected logical noise perturbatively from physical noise and syndrome information. Individually undetected faults contribute at first order, while detected faults with canceling syndromes generate higher-order terms coupling distinct spacetime locations. To reconcile the nonlinearity of post-selection with the linearity of PEC, we formulate the combined protocol in terms of unnormalized post-selected maps, with normalization performed only after averaging. On the ibm_aachen superconducting quantum processor, Clifford experiments validate the post-selected noise model and its perturbative scaling. We then demonstrate error detection with first-order PEC for transverse-field Ising dynamics using 22 data and 27 check qubits. The combined protocol recovers the mean magnetization within statistical uncertainty through six Trotter steps and reduces the inferred sampling overhead by up to a factor of 63 relative to PEC without post-selection.
Error detection can suppress noise at a lower sampling cost than probabilistic error cancellation (PEC), but leaves residual logical errors. We combine the two approaches to mitigate these errors while lowering the cost of PEC. We introduce a spacetime Pauli-Lindblad representation and use it to construct the post-selected logical noise perturbatively from physical noise and syndrome information. Individually undetected faults contribute at first order, while detected faults with canceling syndromes generate higher-order terms coupling distinct spacetime locations. To reconcile the nonlinearity of post-selection with the linearity of PEC, we formulate the combined protocol in terms of unnormalized post-selected maps, with normalization performed only after averaging. On the ibm_aachen superconducting quantum processor, Clifford experiments validate the post-selected noise model and its perturbative scaling. We then demonstrate error detection with first-order PEC for transverse-field Ising dynamics using 22 data and 27 check qubits. The combined protocol recovers the mean magnetization within statistical uncertainty through six Trotter steps and reduces the inferred sampling overhead by up to a factor of 63 relative to PEC without post-selection.
Probabilistic error cancellation (PEC) is unbiased but suffers exponential sampling overhead set by noise-weighted circuit volume, whereas quantum error-detecting codes (QEDCs) remove many physical faults by stabilizer post-selection but leave an undetectable logical residue. We exploit this complementarity by using post-selection to map physical noise to a weaker accepted logical channel, and then applying PEC only to the residual channel. The resulting feedback-free QED+PEC scheme interleaves Clifford logical blocks, stabilizer measurements, post-selection, and probabilistic cancellation on accepted trajectories, without real-time decoding or active recovery. A key complication is that post-selection correlates accepted fault branches through stabilizer-commutation constraints, so the sparse Pauli-Lindblad factorization underlying bare PEC no longer applies directly. We therefore construct the inverse channel perturbatively: for fixed order K, only accepted fault branches up to order K are retained, reducing preprocessing from 2m branches to O(mK) per block. The order-K protocol cancels the normalized post-selected channel through degree K, leaving a per-block error O(WK+1) that accumulates at most linearly. For logical GHZ-state preparation with the [[n,n−2,2]] Iceberg code under circuit-level depolarizing noise and ideal stabilizer measurements, first-order QED+PEC reaches n=200 physical qubits and lowers sampling overhead by three to four orders of magnitude relative to standard PEC while maintaining F≃0.956. Syndrome-noise tests show that readout-only flips mainly increase post-selection cost, whereas noisy GHZ-assisted global stabilizer extraction can remove the advantage. This identifies a discrete-Zeno trade-off: cheap detection reshapes the effective channel PEC must invert, rather than simply adding overhead.
Near-term quantum workloads demand error management, yet the two lightest-weight techniques, Quantum Error Detection (QED) and Probabilistic Error Cancellation (PEC), have complementary cost profiles whose joint architectural design space remains unexplored. QED encodes logical qubits and discards error-flagged runs, filtering noise with low qubit overhead but leaving residual errors; PEC can correct these in software, but at exponential cost in noise strength. If QED efficiently reduces per-gate noise, PEC's cost savings can outweigh QED's discard overhead; realizing this, however, requires solving two system-level design challenges.
First, the \textit{QED interval} -- how often detection cycles are inserted -- is a tunable architectural parameter governing the cost-accuracy tradeoff. We derive an efficiency condition and show that the canonical one-cycle-per-gate frequency does not achieve break-even in any code we evaluate, while optimized intervals on high-rate Iceberg codes do. Second, we discover that naive PEC+QED integration \textit{degrades} accuracy below the QED-only baseline. The root cause is a transient error profile in the first detection cycle that corrupts PEC's noise model. We develop \textit{steady-state extraction}, a co-designed characterization protocol that isolates steady-state error behavior, reducing estimation bias by up to 10.2×. On a [[6,4,2]] Iceberg code running QAOA (p=4--8) with a fixed shot budget, PEC+QED achieves 2--11× lower absolute error and up to 31× lower MSE versus PEC on physical qubits, with per-interval savings compounding over interval depth.
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.
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.
Quantum error mitigation is regarded as a possible path to near-term quantum utility. The methods under the quantum error mitigation umbrella term, such as probabilistic error cancellation (PEC), zero-noise extrapolation (ZNE) or Clifford data regression (CDR) are able to significantly reduce the error for the estimation of expectation values, although at an exponentially scaling cost, i.e., in the sampling overhead. In this work, we present a method to reduce the sampling overhead of PEC through Pauli error propagation combined with classical preprocessing. Our findings indicate that this method significantly reduces sampling overheads for Clifford circuits, leveraging the well-defined interaction between the Clifford group and Pauli noise. Additionally, we show that the method is applicable to non-Clifford circuits, though with more limited effectiveness, primarily constrained by the number of non-Clifford gates present in the circuit. We further provide examples of Clifford sub-circuits commonly encountered in relevant calculations, such as resource state generation in measurement-based quantum computing.
Until fault-tolerance becomes implementable at scale, quantum computing will heavily rely on noise mitigation techniques. While methods such as zero noise extrapolation with probabilistic error amplification (ZNE-PEA) and probabilistic error cancellation (PEC) have been successfully tested on hardware recently, their scalability to larger circuits may be limited. Here, we introduce the tensor-network error mitigation (TEM) algorithm, which acts in post-processing to correct the noise-induced errors in estimations of physical observables. The method consists of the construction of a tensor network representing the inverse of the global noise channel affecting the state of the quantum processor, and the consequent application of the map to informationally complete measurement outcomes obtained from the noisy state. TEM does therefore not require additional quantum operations other than the implementation of informationally complete POVMs, which can be achieved through randomised local measurements. The key advantage of TEM is that the measurement overhead is quadratically smaller than in PEC. We test TEM extensively in numerical simulations in different regimes. We find that TEM can be applied to circuits of twice the depth compared to what is achievable with PEC under realistic conditions with sparse Pauli-Lindblad noise, such as those in [E. van den Berg et al., Nat. Phys. (2023)]. By using Clifford circuits, we explore the capabilities of the method in wider and deeper circuits with lower noise levels. We find that in the case of 100 qubits and depth 100, both PEC and ZNE fail to produce accurate results by using ∼105 shots, while TEM succeeds.