Zero-Noise Extrapolation (ZNE)

Updated 5 December 2025

Zero-Noise Extrapolation (ZNE) is a quantum error mitigation technique that estimates ideal observables by deliberately amplifying noise in quantum circuits and extrapolating to the zero-noise limit.
It employs noise scaling methods such as unitary folding and pulse stretching to optimize measurement fidelity and suppress errors without requiring extra physical qubits.
By leveraging classical post-processing and controlled noise amplification, ZNE offers a hardware-efficient approach applicable to both NISQ devices and logical qubit systems.

Zero-Noise Extrapolation (ZNE) is a digital quantum error mitigation technique that estimates the ideal, noise-free expectation value of observables on a noisy quantum processor by deliberately amplifying the noise and extrapolating to the zero-noise limit. Unlike quantum error correction, ZNE does not require extra physical qubits, code overhead, or ancilla-driven protocols; instead, it uses classical post-processing and a sequence of quantum circuit executions under controlled, artificially increased noise. ZNE is widely studied in the context of NISQ devices, quantum simulation, quantum metrology, and extends to logical qubits, quantum annealing, and bosonic encodings.

1. Mathematical Foundations and Protocol Structure

ZNE operates under the assumption that the noisy outcome of a quantum circuit is a smooth function of a controllable noise-scaling parameter. For an observable $O$ and noise parameter $\lambda \geq 1$ , the measured expectation value $E(\lambda) = \mathrm{Tr}[O\rho(\lambda)]$ can be expanded as a low-order polynomial or analytic function in $\lambda$ : $E(\lambda) = E(0) + a_1\lambda + a_2\lambda^2 + \ldots + O(\lambda^{n+1})$ Given $n+1$ distinct noise scales $\{\lambda_k\}$ (with $\lambda_0=1$ and $\lambda_k>1$ ), the ideal value $E(0)$ is reconstructed via Richardson (polynomial) extrapolation: $E(0) = \sum_{k=0}^n c_k E(\lambda_k)$ where the coefficients $c_k$ are solutions of a Vandermonde linear system enforcing cancellation of the first $n$ order terms in $\lambda$ : $\sum_{k=0}^n c_k \lambda_k^m = \delta_{m, 0}, \quad m = 0,...,n$ This methodology generalizes to exponential ansätze ( $E(\lambda) = \alpha + \beta e^{-\gamma \lambda}$ ) where physically motivated, or to least-squares polynomial fits for variance mitigation (Mari et al., 2021, Mohammadipour et al., 28 Feb 2025).

2. Noise Scaling Techniques

Zero-noise extrapolation requires a procedure for scaling the effective noise. Several regimes and hardware architectures have motivated different approaches:

A. Unitary Folding (Gate Folding):

Each gate or block is replaced by an odd-length sequence $U(U^\dagger U)^n$ , increasing circuit depth and amplifying all sources of gate noise proportionally. Scale factor is $\lambda=2n+1$ (Giurgica-Tiron et al., 2020, Pelofske et al., 2023).
Global folding applies this to an entire circuit; local folding applies it at gate-level granularity.

B. Parameterized Pulse Stretching:

Relevant for Markovian noise, lengthening gate durations scales decoherence rates proportionally ( $\lambda$ ) (Schultz et al., 2022, Maupin et al., 2023).

C. Noise-Aware Folding:

The number and position of folds are optimized by leveraging hardware calibration data to balance accumulated noise across circuit links, improving accuracy when error rates are non-uniform (Hour et al., 23 Jan 2024, Sayapin et al., 4 Nov 2025).

D. Circuit Unoptimization:

Randomly increases depth by inserting gate-equivalent identities interleaved with random gates, providing exponentially many non-equivalent variants for noise amplification and bias averaging (Pelofske et al., 8 Mar 2025).

E. Analog Scaling in Quantum Annealing:

Anneal time and/or temperature are physically stretched for analog devices, scaling relaxation rates, with polynomial extrapolation in $\lambda$ (Amin et al., 2023).

F. Energy-Scaled ZNE (ES-ZNE):

For GKP bosonic codes, the mean photon number $\bar{n}$ (energy) is varied as the noise knob, and extrapolation is performed towards $\bar{n}\to\infty$ (Luo et al., 3 Dec 2025).

G. Layout-Based ZNE:

For systems with inhomogeneous error rates, spatially permuting circuit layout (cyclic layout permutations) provides an effective single-parameter scaling manifold (Sayapin et al., 4 Nov 2025).

3. Practical Procedures, Extrapolation Models, and Trade-Offs

A typical ZNE workflow consists of:

Choice of Noise Scales: Select a set $\{\lambda_k\}$ balancing increased error amplification and need for measurement precision; commonly, $n\lesssim 3$ is used due to rapid variance blowup (Mari et al., 2021, Mohammadipour et al., 28 Feb 2025).
Circuit Execution: For each $\lambda_k$ , build and execute the appropriately folded/noise-scaled circuit, accumulating $N_\mathrm{shots}$ measurements.
Statistical Processing: Fit measured data $\{(\lambda_k, E(\lambda_k))\}$ to a low-order polynomial, exponential, or hybrid model. Linear or quadratic fits are typical for weak noise, with Richardson-based analytic weights preferred for error cancellation.
Variance Amplification: The variance of the extrapolated estimate scales as $\sum_k c_k^2 \mathrm{Var}[E(\lambda_k)]$ and grows exponentially in $n$ ; thus, there is a bias-variance trade-off. Chebyshev-node selection and least-squares regression mitigate overfitting (Mohammadipour et al., 28 Feb 2025).
Error Budget and Shot Scaling: Achieving a total precision $\varepsilon$ requires $O(\varepsilon^{-\alpha})$ total shots, with $\alpha$ dictated by the node distribution and noise-scaling range.

Resource scaling is summarized below:

Extrapolation Order $n$	Circuits Needed	Shot Overhead per Setting	Total Variance Growth
1 (Linear)	2	$N_\mathrm{shots}$	Modest
2 (Quadratic)	3	$N_\mathrm{shots}$	$O(2^n)$
$n$ (Richardson/poly)	$n+1$	$N_\mathrm{shots}$	$O(2^n)$

(Mohammadipour et al., 28 Feb 2025, Mari et al., 2021, Giurgica-Tiron et al., 2020)

4. Extensions, Variants, and Hybrid Protocols

Several advanced forms of ZNE address hardware limitations, noise inhomogeneity, or shot overhead:

A. Purity-Assisted ZNE (pZNE):

Uses measured output-state purity $\operatorname{Tr} \rho^2$ to calibrate extrapolation, improving bias tolerance and extending the error-rate threshold above that of routine ZNE. Achieves $O(\lambda^2)$ bias suppression and remains well-conditioned up to noise probabilities $q_\lambda \lesssim 40\%$ (Jin et al., 2023).

B. Distance-Scaled ZNE (DS-ZNE) in Logical Qubits:

Leverages the exponential scaling of logical error with code distance; measurements are performed at several code distances, then extrapolated to infinitely long code distance (Wahl et al., 2023).

C. Block-Based ZNE for Structured Algorithms:

For quantum algorithms with repeated blocks (Grover's, QAOA), block-fidelity is directly characterized using shallow identity circuits; overall algorithmic success probability is reconstructed via an exponential-decay fit of the block event and mitigated using the (inferred) block fidelity, outperforming conventional full-circuit ZNE in deep/noisy regimes (Kim et al., 31 Jul 2025).

D. Modified/Folding-Free ZNE:

Mitigation can be performed analytically via knowledge of the infinite-noise (maximally mixed state) limit (reliability-based extrapolation), or decoherence decay laws (latency-aware exponential extrapolation), sometimes avoiding the run-time overhead of circuit folding (Patil et al., 2023).

E. Per-Outcome/Per-Group Extrapolation Function Selection:

In scenarios such as tomography or high-weight observables, per-measurement ansatz selection, possibly with estimator-guided or filter-based heuristic, yields higher aggregate fidelity than a global extrapolation model (Shi et al., 10 Mar 2024).

5. Limitations, Performance on Real Devices, and Best Practices

ZNE is limited by several fundamental and practical considerations:

Noise Model Fidelity: Markovian, time-uncorrelated noise allows strict polynomial scaling; time-correlated ( $1/f^\alpha$ ) or nonstationary noise challenges local folding and requires global folding or empirical validation of scaling (Schultz et al., 2022).
Applicability in High-Noise Regimes: ZNE’s effectiveness degrades as circuits become deep or noise is strong, with folded outputs converging to random baselines, rendering extrapolation ill-conditioned. Circuit cutting and partitioned ZNE or block-based approaches can mitigate this, if the original circuit is partitionable (Kim et al., 31 Jul 2025, Patil et al., 2023).
Shot Overhead and Variance: Increasing extrapolation order to cancel higher-order noise leads to exponential increase in required shots for given confidence due to the variance of the extrapolated estimator, necessitating careful order selection (Mari et al., 2021, Mohammadipour et al., 28 Feb 2025).
Hardware Inhomogeneity: Spatially varying noise is addressed by noise-aware folding, layout permutation averaging, and circuit unoptimization, improving ZNE performance on realistic hardware (Hour et al., 23 Jan 2024, Sayapin et al., 4 Nov 2025, Pelofske et al., 8 Mar 2025).
Integration with Other Mitigation Methods: ZNE is unified with probabilistic error cancellation and can be combined with readout mitigation, virtual distillation, or dynamical decoupling. Some approaches (hybrid NEPEC) interpolate between ZNE and PEC, balancing sampling cost and bias (Mari et al., 2021, Jin et al., 2023).
Quantum Metrology and Sensing: In quantum metrology and DC magnetometry, ZNE is effective in bias mitigation and sensitivity recovery away from optimal working points, but does not improve the fundamental shot-noise limited sensitivity (Zhao et al., 2021, Dyke et al., 26 Feb 2024).

Empirical studies confirm that ZNE suppresses expectation-value error by 1–2 orders of magnitude for VQE, QAOA, and quantum volume circuits, and increases the measured effective quantum volume beyond vendor benchmarks (Pelofske et al., 2023, Maupin et al., 2023, Pelofske et al., 8 Mar 2025).

6. Applications Beyond NISQ Devices

A. Fault-Tolerant Regimes:

ZNE extends to logical qubits via code distance scaling or energy scaling (bosonic codes), trading measurement overhead for reduced hardware overhead and increasing effective code distance or logical accuracy (Wahl et al., 2023, Luo et al., 3 Dec 2025).

B. Analog Quantum Simulation and Annealing:

ZNE is adapted to quantum annealers via zero-temperature and energy–time rescaling. These protocols have demonstrated significant extension of the coherence window in D-Wave superconducting processors (Amin et al., 2023).

C. Quantum Sensing:

ZNE has been analyzed in the context of quantum sensing (e.g., DC magnetometry), confirming limited sensitivity improvements but robust bias correction for large signals (Dyke et al., 26 Feb 2024).

References:

Zero-Noise Extrapolation remains a central, hardware-efficient quantum error mitigation approach for noisy quantum circuits, with ongoing methodological innovation and broad applicability from NISQ platforms to logical encodings and analog architectures.