Zero-noise extrapolation (ZNE) is a widely used quantum error mitigation technique that artificially amplifies circuit noise and then extrapolates the results to the noise-free circuit. A common ZNE approach is Richardson extrapolation, which relies on polynomial interpolation. Despite its simplicity, efficient implementations of Richardson extrapolation face several challenges, including approximation errors from the non-polynomial behavior of noise channels, overfitting due to polynomial interpolation, and exponentially amplified measurement noise. This paper provides a comprehensive analysis of these challenges, presenting bias and variance bounds that quantify approximation errors. Additionally, for any precision ε , our results offer an estimate of the necessary sample complexity. We further extend the analysis to polynomial least squares-based extrapolation, which mitigates measurement noise and avoids overfitting. Finally, we propose a strategy for simultaneously mitigating circuit and algorithmic errors in the Trotter-Suzuki algorithm by jointly scaling the time step size and the noise level. This strategy provides a practical tool to enhance the reliability of near-term quantum computations. We support our theoretical findings with numerical experiments.
Zero-noise extrapolation (ZNE) is a widely used quantum error mitigation technique that artificially amplifies circuit noise and then extrapolates the results to the noise-free circuit. A common ZNE approach is Richardson extrapolation, which relies on polynomial interpolation. Despite its simplicity, efficient implementations of Richardson extrapolation face several challenges, including approximation errors from the non-polynomial behavior of noise channels, overfitting due to polynomial interpolation, and exponentially amplified measurement noise. This paper provides a comprehensive analysis of these challenges, presenting bias and variance bounds that quantify approximation errors. Additionally, for any precision ε , our results offer an estimate of the necessary sample complexity. We further extend the analysis to polynomial least squares-based extrapolation, which mitigates measurement noise and avoids overfitting. Finally, we propose a strategy for simultaneously mitigating circuit and algorithmic errors in the Trotter-Suzuki algorithm by jointly scaling the time step size and the noise level. This strategy provides a practical tool to enhance the reliability of near-term quantum computations. We support our theoretical findings with numerical experiments.
Zero-noise extrapolation (ZNE) mitigates errors in near-term quantum devices by extrapolating measurements obtained at amplified noise levels to estimate noise-free expectation values. In practice, commonly used extrapolation models are fitted without enforcing physical constraints, which can yield predictions outside the valid range of quantum observables.
In this work, we introduce physically bounded variants of polynomial, exponential, and polynomial--exponential extrapolation models by explicitly parameterizing the zero-noise estimate and constraining it during optimization. We evaluate the approach using a large synthetic benchmark comprising 180,000 circuits and approximately 3.6 million ZNE experiments generated under realistic device noise models derived from IBM quantum backends. We also perform preliminary validation on real quantum hardware using GHZ and W-state circuits.
Across the synthetic benchmark, bounded extrapolation substantially reduces unphysical predictions and improves the stability of exponential- and polynomial--exponential-family models, whereas polynomial models show little difference between bounded and unbounded variants. Hardware experiments show similar qualitative behaviour: bounded models generally avoid pathological extrapolations and often provide a more reliable balance between accuracy and usable coverage. At the same time, the results highlight practical limitations of current devices, including stronger-than-expected noise effects and variability not fully captured by simulation models. These results suggest that enforcing physical constraints during extrapolation improves the reliability of ZNE and that this approach can be incorporated into existing workflows with minimal modification.
Quantum-gate errors are a significant challenge for achieving precision measurements on noisy intermediate-scale quantum (NISQ) computers. This paper focuses on zero-noise extrapolation (ZNE), a technique that can be implemented on existing hardware, studying it in detail and proposing modifications to existing approaches. In particular, we consider identity insertion methods for amplifying noise because they are hardware agnostic. We build a mathematical formalism for studying existing ZNE techniques and show how higher order polynomial extrapolations can be used to systematically reduce depolarizing errors. Furthermore, we introduce a method for amplifying noise that uses far fewer gates than traditional methods. This approach is compared with existing methods for simulated quantum circuits. Comparable or smaller errors are possible with fewer gates, which illustrates the potential for empowering an entirely new class of moderate-depth circuits on near term hardware.
The pursuit of practical quantum utility on near-term quantum processors is critically challenged by their inherent noise. Quantum error mitigation (QEM) techniques are leading solutions to improve computation fidelity with relatively low qubit-overhead, while full-scale quantum error correction remains a distant goal. However, QEM techniques incur substantial measurement overheads, especially when applied to families of quantum circuits parameterized by classical inputs. Focusing on zero-noise extrapolation (ZNE), a widely adopted QEM technique, here we devise the surrogate-enabled ZNE (S-ZNE), which leverages classical learning surrogates to perform ZNE entirely on the classical side. Unlike conventional ZNE, whose measurement cost scales linearly with the number of circuits, S-ZNE requires only constant measurement overhead for an entire family of quantum circuits, offering superior scalability. Theoretical analysis indicates that S-ZNE achieves accuracy comparable to conventional ZNE in many practical scenarios, and numerical experiments on up to 100-qubit ground-state energy and quantum metrology tasks confirm its effectiveness. Our approach provides a template that can be effectively extended to other quantum error mitigation protocols, opening a promising path toward scalable error mitigation.
Abstract Quantum error mitigation aims to reduce errors in quantum systems and improve accuracy. Zero‐noise extrapolation (ZNE) is a commonly used method, where noise is amplified, and the target expectation is extrapolated to a noise‐free point. However, ZNE relies on assumptions about error rates based on the error model. In this study, a purity‐assisted zero‐noise extrapolation (pZNE) method is utilized to address limitations in error rate assumptions and enhance the extrapolation process. The pZNE is based on the Pauli diagonal error model implemented using the Pauli twirling technique. Although this method does not significantly reduce the bias of routine ZNE, it extends its effectiveness to a wider range of error rates where routine ZNE may face limitations. In addition, the practicality of the pZNE method is verified through numerical simulations and experiments on the online quantum computation platform, Quafu . Comparisons with routine ZNE and virtual distillation methods show that biases in extrapolation methods increase with error rates and may become divergent at high error rates. The bias of pZNE is slightly lower than routine ZNE, while its error rate threshold surpasses that of routine ZNE. Furthermore, for full density matrix information, the pZNE method is more efficient than the routine ZNE.
Variational Quantum Eigensolvers (VQE) promise molecular ground-state energies on today's noisy intermediate-scale quantum (NISQ) processors, but device noise biases the estimated energy severely — under a device-calibrated noise model, the hydrogen molecule's energy error grows from 0.000 mHa (noiseless) to 303.6 mHa (noisy), roughly 190× the "chemical accuracy" threshold of 1.6 mHa that computational chemistry requires. Error-mitigation techniques repair such biases in software, but published results usually report each technique in isolation, making it impossible to know which mitigation to buy with a fixed shot budget. This work answers that question with a controlled comparison. Under identical circuits, noise profiles and shot budgets, I measure the bias and variance of four estimator stacks — raw sampling, twirled-readout extinction (TREX-style calibrated readout inversion), zero-noise extrapolation (ZNE), and ZNE+TREX — across independent random seeds. Three findings emerge. (1) ZNE is a bias–variance tradeoff, not a free lunch: a 7-point Richardson extrapolant reduces the deterministic bias 3200-fold (303.6 → 0.094 mHa) but amplifies statistical noise by a measured factor of up to 25× per run, while TREX removes readout bias with no variance penalty. (2) ZNE has a finite operating window in circuit depth: error recovery falls 99.97% → 99.1% → 95.5% as CX count grows 56 → 112 → 168, and is exhausted entirely for a 1616-CX lithium-hydride circuit at near-term error rates. (3) A noise-bias-tailored repetition code corrects the dominant error axis with logical error rates matching theory (3p², 10p³, 35p⁴ for distances 3, 5, 7) at strictly linear gate overhead (gates = 6d − 6). The results reframe mitigation selection as budget allocation on an empirically measured bias–variance frontier and provide open-source tooling to reproduce every number.
Reliable benchmarking of Quantum Error Mitigation (QEM) requires distinguishing genuine improvements from artefacts of the post-processing arithmetic. In this paper, we expose a failure mode in Richardson Zero-Noise Extrapolation (ZNE), a widely used technique routinely (and often implicitly) relied upon in benchmarks and experiments. When noise amplification operates beyond usable signals - a regime that is quickly reached on current hardware for non-trivial circuits - we show that the extrapolation no longer reflects the underlying physics, but collapses into a fixed rescaling of a single noisy measurement, producing a bogus apparent improvement that is independent of noise amplification. This poses a rarely considered threat to the validity of many empirical evaluations in quantum computing.
Measurements on real hardware (IQM Euro-Q-Exa) confirm this collapse with ordinary folding alone: as circuit depth erodes the signal, the reported estimate decouples from the truth and overshoots the ideal by up to 21%. We further introduce a matched-cost "garbage-folding" negative control that carries no usable signal yet reports a larger apparent improvement than genuine folding - showing that the magnitude of an improvement is not evidence of its correctness - alongside a zero-cost check flagging the artefact from data a benchmark already holds. We distil both into a short reporting checklist for ZNE benchmarks.