Quantum error mitigation on noisy devices is limited not only by residual bias but also by the shot noise and calibration errors introduced by the mitigation procedure itself. We derive finite-shot mean-square-error boundaries for probabilistic error cancellation (PEC), Clifford data regression (CDR), and no mitigation for noisy Pauli-observable estimates. Exact PEC removes the target bias under an exact noise inverse at the price of a quasi-probability variance overhead, whereas population linear CDR can have smaller target-shot variance but retains a calibration floor when the training and target noise responses do not match. This competition yields a finite CDR-dominant operating window whose upper endpoint scales as $B_{\mathrm{PEC}=\mathrm{CDR}}(p)\propto 1/(δ_1^2p)$, where $δ_1$ is the first-order CDR calibration mismatch. We further prove a target-response projection theorem showing that response-blind affine CDR removes the first-order bias only when the target noise response is affine in the ideal target value; otherwise a nonzero projection error gives an irreducible local calibration floor. The same mean-square-error formulation extends to second-order calibration, commuting Pauli Hamiltonians, finite CDR training shots, and residual PEC model bias. A closed-form two-qubit calculation and QAOA simulations support the predicted no-mitigation, CDR-dominant, and PEC-dominant regimes.
Quantum error mitigation on noisy devices is limited not only by residual bias but also by the shot noise and calibration errors introduced by the mitigation procedure itself. We derive finite-shot mean-square-error boundaries for probabilistic error cancellation (PEC), Clifford data regression (CDR), and no mitigation for noisy Pauli-observable estimates. Exact PEC removes the target bias under an exact noise inverse at the price of a quasi-probability variance overhead, whereas population linear CDR can have smaller target-shot variance but retains a calibration floor when the training and target noise responses do not match. This competition yields a finite CDR-dominant operating window whose upper endpoint scales as BPEC=CDR(p)∝1/(δ12p), where δ1 is the first-order CDR calibration mismatch. We further prove a target-response projection theorem showing that response-blind affine CDR removes the first-order bias only when the target noise response is affine in the ideal target value; otherwise a nonzero projection error gives an irreducible local calibration floor. The same mean-square-error formulation extends to second-order calibration, commuting Pauli Hamiltonians, finite CDR training shots, and residual PEC model bias. A closed-form two-qubit calculation and QAOA simulations support the predicted no-mitigation, CDR-dominant, and PEC-dominant regimes.
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.
The Noisy Intermediate-Scale Quantum (NISQ) era is characterized by high error rates that fundamentally limit the utility of quantum algorithms. Existing error mitigation techniques—Zero-Noise Extrapolation (ZNE), Probabilistic Error Cancellation (PEC), and Clifford Data Regression (CDR)—achieve varying degrees of success but lack a unified signal-processing perspective. Motivated by the recently established isomorphism between Kraus operator representations and discrete impulse decompositions in digital signal processing (DSP), we present a novel quantum error mitigation framework that treats noisy quantum channels as linear time-invariant systems amenable to deconvolution. We adapt classical Wiener filtering theory to the quantum domain via the Liouville superoperator representation, formulating error mitigation as a linear minimum mean-square error (LMMSE) estimation problem in the space of density operators. Experimental validation across five representative NISQ scenarios demonstrates that the Quantum Wiener Filter (QWF) significantly outperforms ZNE, PEC, and CDR in four of five test cases, achieving 66–93% reduction in observable error for GHZ state preparation under amplitude damping, VQE energy estimation under thermal relaxation, random Clifford circuits under depolarizing noise, and QAOA optimization under crosstalk. The method exhibits statistically significant superiority (Wilcoxon signed-rank test, p < 10⁻⁴) across 30 independent trials per scenario. The fifth scenario—an adversarial state-dependent noise model—validates the Markovian assumption underlying the deconvolution framework by demonstrating controlled failure when channel linearity is violated. Our results establish that the DSP-QM isomorphism enables practical error mitigation with superior fidelity recovery (mean fidelity 0.96–0.99 vs. 0.78–0.98 for baselines) at modest computational overhead (1.2× shot budget). This work bridges quantum information theory and classical signal processing, opening a new avenue for NISQ algorithm optimization.Keywords: Quantum Error Mitigation, NISQ, Wiener Filter, Signal Processing, Qiskit.
Quantum error mitigation (QEM) is critical for harnessing the potential of near-term quantum devices. Particularly, QEM protocols can be designed based on machine learning, where the mapping between noisy computational outputs and ideal ones can be learned on a training set consisting of Clifford circuits or near-Clifford circuits that contain only a limited number of non-Clifford gates. This learned mapping is then applied to noisy target circuits to estimate the ideal computational output. In this work, we propose a learning-based error mitigation framework called Clifford Perturbation Data Regression (CPDR), which constructs training sets by Clifford circuits with small perturbations. Specifically, these circuits are parameterized quantum circuits, where the rotation angles of the gates are restricted to a narrow range, ensuring that the gates remain close to Clifford gates. This design enables the efficient simulation of the training circuits using the Sparse Pauli Dynamics method. As a result, CPDR is able to utilize training sets with a better diversity to train the model, compared with previous learning-based QEM models that construct training sets with only Clifford or near-Clifford circuits. Numerical simulations on small-scale Ising model circuits demonstrate that the performance of CPDR dramatically outperforms that of existing methods such as Zero-Noise Extrapolation and learning-based Probabilistic Error Cancellation. Furthermore, using the experimental data from IBM's 127-qubit Eagle processor, our findings suggest that CPDR demonstrates improved accuracy compared to the original mitigation results reported in [Nature 618, 500 (2023)].
Error mitigation is essential for the practical implementation of quantum algorithms on noisy intermediate-scale quantum (NISQ) devices. This work explores and extends Clifford Data Regression (CDR) to mitigate noise in quantum chemistry simulations using the Variational Quantum Eigensolver (VQE). Using the H4 molecule with the tiled Unitary Product State (tUPS) ansatz, we perform noisy simulations with the ibm torino noise model to investigate in detail the effect of various hyperparameters in CDR on the error mitigation quality. Building on these insights, two improvements to the CDR framework are proposed. The first, Energy Sampling (ES), improves performance by selecting only the lowest-energy training circuits for regression, thereby further biasing the sample energies toward the target state. The second, Non-Clifford Extrapolation (NCE), enhances the regression model by including the number of non-Clifford parameters as an additional input, enabling the model to learn how the noisy-ideal mapping evolves as the circuit approaches the optimal one. Our numerical results demonstrate that both strategies outperform the original CDR.
Achieving near-term quantum advantage will require effective methods for mitigating hardware noise. Data-driven approaches to error mitigation are promising, with popular examples including zero-noise extrapolation (ZNE) and Clifford data regression (CDR). Here, we propose a scalable error mitigation method that conceptually unifies ZNE and CDR. Our approach, called variable-noise Clifford data regression (vnCDR), significantly outperforms these individual methods in numerical benchmarks. vnCDR generates training data first via near-Clifford circuits (which are classically simulable) and second by varying the noise levels in these circuits. We employ a noise model obtained from IBM's Ourense quantum computer to benchmark our method. For the problem of estimating the energy of an 8-qubit Ising model system, vnCDR improves the absolute energy error by a factor of 33 over the unmitigated results and by factors of 20 and 1.8 over ZNE and CDR, respectively. For the problem of correcting observables from random quantum circuits with 64 qubits, vnCDR improves the error by factors of 2.7 and 1.5 over ZNE and CDR, respectively.
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.