arxiv: 2605.08251 · v1 · submitted 2026-05-07 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

The finite-shot help-harm boundary of zero-noise extrapolation

Vicenzo Scavino Alfaro

Authors on Pith no claims yet

Pith reviewed 2026-05-12 02:28 UTC · model grok-4.3

classification 🪐 quant-ph

keywords zero-noise extrapolationRichardson extrapolationquantum error mitigationfinite-shot regimemean-squared errorbias-variance tradeoff

0 comments

The pith

Zero-noise extrapolation switches from harmful to helpful only above a finite-shot mean-squared-error boundary set by initial bias reduction and variance penalty.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a finite-shot help-harm boundary for zero-noise extrapolation using fixed Richardson extrapolation. This is the lower crossing in local mean-squared error where ZNE stops increasing total error and starts decreasing it under limited measurement shots. A local expansion around the unmitigated estimator shows the boundary is controlled by the first squared-bias improvement and the first excess-variance penalty from the Richardson coefficients. The resulting scaling can be a shrinking power law, a fixed budget threshold, or no useful lower boundary, depending on whether the underlying measurements are deterministic stabilizer or variational energy estimates. Qiskit Aer simulations and IBM Quantum checks confirm the predicted separation between these measurement classes while highlighting protocol limits.

Core claim

We define a finite-shot help-harm boundary: the lower local mean-squared-error crossing where fixed Richardson ZNE changes from harmful to helpful. A local expansion shows that this boundary is governed by the first squared-bias improvement and first excess-variance penalty, producing either a shrinking power law, a budget threshold, or no shrinking lower boundary. Qiskit Aer simulations and variance-exponent fits support the predicted separation between deterministic stabilizer measurements and variational energy measurements.

What carries the argument

The finite-shot help-harm boundary, defined as the local mean-squared-error crossing point between the Richardson-extrapolated estimator and the unmitigated one, obtained from a first-order expansion in bias reduction and variance penalty.

If this is right

For deterministic stabilizer measurements the boundary shrinks as a power law with increasing shot budget.
For variational energy measurements the boundary can manifest as a fixed shot threshold below which ZNE increases total error.
Above the boundary ZNE delivers net error reduction despite the variance cost of coefficient splitting.
Readout-regime diagnostics can flag when hardware effects push the boundary outside the model's validity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Shot-allocation routines in variational algorithms could use an online estimate of this boundary to decide whether to apply ZNE.
The same bias-variance expansion approach could be applied to other extrapolation orders or to probabilistic error cancellation.
In circuits with gate-dependent or time-correlated noise the boundary location itself may become a function of circuit depth.

Load-bearing premise

The local expansion accurately models the mean-squared-error transition point and the simulations reliably separate deterministic stabilizer measurements from variational energy measurements without unaccounted hardware or protocol effects.

What would settle it

An experiment that measures mean-squared error for a known observable at multiple finite shot budgets under controlled noise, then checks whether the observed help-harm crossing scales exactly as predicted by the first bias-improvement and variance-penalty terms.

Figures

Figures reproduced from arXiv: 2605.08251 by Vicenzo Scavino Alfaro.

**Figure 2.** Figure 2: Distribution of QAOA random-graph slopes for 20 deterministic [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** Figure 3: QAOA depth study. The original fixed schedule does not recover the target regime at [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

**Figure 4.** Figure 4: Readout/SPAM as an effective measurement-regime change. With known- [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗

read the original abstract

Zero-noise extrapolation (ZNE) reduces noise-induced bias but can increase sampling variance through Richardson coefficients and shot splitting. We define a finite-shot help-harm boundary: the lower local mean-squared-error crossing where fixed Richardson ZNE changes from harmful to helpful. A local expansion shows that this boundary is governed by the first squared-bias improvement and first excess-variance penalty, producing either a shrinking power law, a budget threshold, or no shrinking lower boundary. Qiskit Aer simulations and variance-exponent fits support the predicted separation between deterministic stabilizer measurements and variational energy measurements, while readout-regime diagnostics and IBM Quantum checks delineate measurement-protocol and hardware-traceability limits.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines a finite-shot help-harm boundary for ZNE via local expansion and shows it separates stabilizer from variational cases in simulations, but the truncation may shift the actual crossing.

read the letter

The main thing to know is that this work defines a finite-shot help-harm boundary as the lower shot count where fixed Richardson ZNE stops increasing mean-squared error relative to the bare circuit. A local expansion in the leading bias reduction and variance penalty then predicts three regimes: a shrinking power-law boundary, a fixed budget threshold, or no boundary at all. Simulations back a clean split between deterministic stabilizer measurements and variational energy estimates, which aligns with the different noise sensitivities in those settings. That framing is new and gives a concrete rule for when to skip extrapolation under tight shot budgets on NISQ hardware. The derivation itself is straightforward and the Qiskit Aer runs plus IBM checks add some grounding. The soft spot is exactly the one the stress-test flags: the boundary is located where extrapolated MSE equals bare MSE, so any O(λ²) or higher terms that become comparable near that point will move or erase the predicted crossing. The abstract mentions variance-exponent fits but gives no residual checks of the truncated expansion against full simulated MSE at the observed crossings, and no error bars or exclusion criteria are described. If those checks are missing or weak in the full text, the claimed separation rests on thinner evidence than the local math suggests. This is for readers who run variational algorithms or stabilizer circuits on real devices and need a quick way to decide on mitigation. It deserves peer review because the practical question is well-posed and the math is transparent, even if the empirical validation needs tightening before publication.

Referee Report

3 major / 2 minor

Summary. The paper defines a finite-shot help-harm boundary for fixed Richardson zero-noise extrapolation (ZNE) as the lower local mean-squared-error (MSE) crossing where extrapolation transitions from harmful to helpful. A local expansion in the leading squared-bias improvement and first excess-variance penalty is used to derive the boundary's scaling, which can take the form of a shrinking power law, a total-shot budget threshold, or no lower boundary at all. Qiskit Aer simulations together with variance-exponent fits are presented as supporting a separation between deterministic stabilizer measurements and variational energy measurements, with additional readout-regime diagnostics and IBM Quantum hardware checks used to bound measurement-protocol and hardware effects.

Significance. If the local expansion remains accurate and the simulation evidence is robust, the work supplies a concrete, operationally useful criterion for deciding when ZNE is advisable under realistic finite-shot budgets—an issue of direct relevance to near-term quantum algorithms. The analytic separation between stabilizer and variational regimes is a clear strength, as is the emphasis on reproducible variance-exponent fits. These elements could help practitioners avoid counterproductive use of ZNE and guide future error-mitigation design.

major comments (3)

[Local expansion and boundary derivation] The first-order local expansion (governing the MSE crossing via squared-bias reduction and excess-variance penalty) may shift or eliminate the predicted boundary when O(λ²) or higher noise terms become comparable to the retained linear terms. The manuscript should report an explicit residual comparison of the truncated expansion against the full simulated MSE evaluated exactly at the numerically observed crossing points.
[Simulation results and variance-exponent fits] The claim that Qiskit Aer simulations and variance-exponent fits support the predicted separation between stabilizer and variational cases lacks supporting methodological detail: no information is given on data-exclusion rules, error-bar computation, or the precise fitting procedure. This omission prevents independent verification of the central empirical result.
[Hardware validation] The IBM Quantum hardware checks are invoked to delineate protocol and traceability limits, yet no quantitative metrics (e.g., observed versus predicted boundary locations, hardware-specific variance inflation factors) are supplied. Without these, it is unclear how strongly the hardware data corroborates the simulation-based claims.

minor comments (2)

[Abstract] The abstract would be strengthened by a single sentence stating the concrete noise models or circuit families used in the simulations.
[Throughout] Notation for bias, variance, and total MSE should be checked for consistency between the analytic sections and the figure captions.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive report. We address each major comment below and commit to revisions that directly incorporate the suggested improvements.

read point-by-point responses

Referee: [Local expansion and boundary derivation] The first-order local expansion (governing the MSE crossing via squared-bias reduction and excess-variance penalty) may shift or eliminate the predicted boundary when O(λ²) or higher noise terms become comparable to the retained linear terms. The manuscript should report an explicit residual comparison of the truncated expansion against the full simulated MSE evaluated exactly at the numerically observed crossing points.

Authors: We agree that the validity of the first-order truncation must be verified at the crossing points. In the revised manuscript we will add a direct residual comparison: at each numerically located crossing we evaluate both the truncated local expansion and the full simulated MSE, reporting the relative residual to quantify when higher-order terms remain negligible. revision: yes
Referee: [Simulation results and variance-exponent fits] The claim that Qiskit Aer simulations and variance-exponent fits support the predicted separation between stabilizer and variational cases lacks supporting methodological detail: no information is given on data-exclusion rules, error-bar computation, or the precise fitting procedure. This omission prevents independent verification of the central empirical result.

Authors: We apologize for the missing details. The revised manuscript will contain an explicit methods subsection stating: (i) data-exclusion rules (runs discarded only if variance diverged or convergence failed within 10^6 shots), (ii) error bars obtained by bootstrap resampling over 200 shot realizations, and (iii) the fitting procedure (weighted linear regression on log-variance versus log-noise strength, weights equal to the number of shots per point). These additions will permit full independent verification. revision: yes
Referee: [Hardware validation] The IBM Quantum hardware checks are invoked to delineate protocol and traceability limits, yet no quantitative metrics (e.g., observed versus predicted boundary locations, hardware-specific variance inflation factors) are supplied. Without these, it is unclear how strongly the hardware data corroborates the simulation-based claims.

Authors: We accept that quantitative metrics are required. The revision will include a table reporting, for each hardware run: the observed help-harm boundary location, the predicted location from the local expansion, and the measured variance inflation factor relative to the simulator. These numbers will be accompanied by a short discussion of the observed deviations. revision: yes

Circularity Check

0 steps flagged

Local expansion derives boundary scaling independently from its definition

full rationale

The paper explicitly defines the finite-shot help-harm boundary as the lower local MSE crossing where fixed Richardson ZNE changes from harmful to helpful. It then applies a local expansion in the leading squared-bias reduction and first excess-variance penalty to characterize the boundary's scaling (power law, threshold, or none). This is a direct mathematical derivation from the MSE expressions rather than a tautological redefinition or a fitted parameter presented as a prediction. Simulations and variance-exponent fits are offered only as supporting evidence, not as the source of the boundary or its scaling. No self-citations are load-bearing, and the derivation chain does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The local expansion and simulation support are presented without detailing underlying assumptions or fitted quantities.

pith-pipeline@v0.9.0 · 5396 in / 1113 out tokens · 32061 ms · 2026-05-12T02:28:16.925918+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
local expansion shows that this boundary is governed by the first squared-bias improvement and first excess-variance penalty, producing either a shrinking power law, a budget threshold, or no shrinking lower boundary
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
ΔMSE(ϵ, B) = D_p ϵ^{2p} − K_q ϵ^q / B + R_p(ϵ, B)

Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages · 3 internal anchors

[1]

Richardson, L.F.: IX. The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam.Philosophical Transactions of the Royal Society of London. Series A210, 307–357 (1911). https://doi.org/10.1098/rsta.1911.0009

work page doi:10.1098/rsta.1911.0009 1911
[2]

https://doi.org/10.1007/BF02165234 19

Bulirsch, R., Stoer, J.: Numerical treatment of ordinary differential equa- tions by extrapolation methods.Numerische Mathematik8, 1–13 (1966). https://doi.org/10.1007/BF02165234 19

work page doi:10.1007/bf02165234 1966
[3]

Temme, S

Temme, K., Bravyi, S., Gambetta, J.M.: Error mitigation for short- depth quantum circuits.Physical Review Letters119, 180509 (2017). https://doi.org/10.1103/PhysRevLett.119.180509

work page doi:10.1103/physrevlett.119.180509 2017
[4]

https://doi.org/10.1103/PhysRevX.7.021050

Li, Y., Benjamin, S.C.: Efficient variational quantum simulator incor- porating active error minimization.Physical Review X7, 021050 (2017). https://doi.org/10.1103/PhysRevX.7.021050

work page doi:10.1103/physrevx.7.021050 2017
[5]

Available: https://doi.org/10.1103/PhysRevX.8.031027

Endo, S., Benjamin, S.C., Li, Y.: Practical quantum error mitigation for near-future appli- cations.Physical Review X8, 031027 (2018). https://doi.org/10.1103/PhysRevX.8.031027

work page doi:10.1103/physrevx.8.031027 2018
[6]

Quantum error mitigation,

Cai, Z., Babbush, R., Benjamin, S.C., Endo, S., Huggins, W.J., Li, Y., McClean, J.R., O’Brien, T.E.: Quantum error mitigation.Reviews of Modern Physics95, 045005 (2023). https://doi.org/10.1103/RevModPhys.95.045005

work page doi:10.1103/revmodphys.95.045005 2023
[7]

https://doi.org/10.7566/JPSJ.90.032001

Endo, S., Cai, Z., Benjamin, S.C., Yuan, X.: Hybrid quantum-classical algorithms and quantum error mitigation.Journal of the Physical Society of Japan90, 032001 (2021). https://doi.org/10.7566/JPSJ.90.032001

work page doi:10.7566/jpsj.90.032001 2021
[8]

In:2020 IEEE International Conference on Quantum Computing and Engineering (QCE), pp

Giurgica-Tiron, T., Hindy, Y., LaRose, R., Mari, A., Zeng, W.J.: Digital zero noise extrapolation for quantum error mitigation. In:2020 IEEE International Conference on Quantum Computing and Engineering (QCE), pp. 306–316 (2020). https://doi.org/10.1109/QCE49297.2020.00045

work page doi:10.1109/qce49297.2020.00045 2020
[9]

https://doi.org/10.1103/PhysRevA.102.012426

He, A., Nachman, B., de Jong, W.A., Bauer, C.W.: Zero-noise extrapolation for quantum- gate error mitigation with identity insertions.Physical Review A102, 012426 (2020). https://doi.org/10.1103/PhysRevA.102.012426

work page doi:10.1103/physreva.102.012426 2020
[10]

https://doi.org/10.1103/PhysRevA.105.042406

Pascuzzi, V.R., He, A., Bauer, C.W., de Jong, W.A.: Computationally efficient zero-noise extrapolation for quantum-gate-error mitigation.Physical Review A105, 042406 (2022). https://doi.org/10.1103/PhysRevA.105.042406

work page doi:10.1103/physreva.105.042406 2022
[11]

Physical Review A110, 062420 (2024)

Russo, A.E., Mari, A.: Quantum error mitigation by layerwise Richardson extrapolation. Physical Review A110, 062420 (2024). https://doi.org/10.1103/PhysRevA.110.062420

work page doi:10.1103/physreva.110.062420 2024
[12]

https://doi.org/10.1103/PhysRevA.106.062436

Krebsbach, M., Trauzettel, B., Calzona, A.: Optimization of Richardson ex- trapolation for quantum error mitigation.Physical Review A106, 062436 (2022). https://doi.org/10.1103/PhysRevA.106.062436

work page doi:10.1103/physreva.106.062436 2022
[13]

https://doi.org/10.22331/q-2025-11-14-1909

Mohammadipour, P., Li, X.: Direct analysis of zero-noise extrapolation: polynomial meth- ods, error bounds, and simultaneous physical-algorithmic error mitigation.Quantum9, 1909 (2025). https://doi.org/10.22331/q-2025-11-14-1909

work page doi:10.22331/q-2025-11-14-1909 1909
[14]

https://doi.org/10.22331/q-2023-06-06-1034

Bultrini, D., Gordon, M.H., Czarnik, P., Arrasmith, A., Cerezo, M., Coles, P.J., Cincio, L.: Unifying and benchmarking state-of-the-art quantum error mitigation techniques.Quantum 7, 1034 (2023). https://doi.org/10.22331/q-2023-06-06-1034

work page doi:10.22331/q-2023-06-06-1034 2023
[15]

https://doi.org/10.1038/s41534-022-00618-z

Takagi, R., Endo, S., Minagawa, S., Gu, M.: Fundamental limits of quantum error mitiga- tion.npj Quantum Information8, 114 (2022). https://doi.org/10.1038/s41534-022-00618-z

work page doi:10.1038/s41534-022-00618-z 2022
[16]

https://doi.org/10.1103/PhysRevLett.131.210602

Takagi, R., Tajima, H., Gu, M.: Universal sampling lower bounds for quantum error mitigation.Physical Review Letters131, 210602 (2023). https://doi.org/10.1103/PhysRevLett.131.210602

work page doi:10.1103/physrevlett.131.210602 2023
[17]

https://doi.org/10.1103/PhysRevLett.131.210601 20

Tsubouchi, K., Sagawa, T., Yoshioka, N.: Universal cost bound of quantum error miti- gation based on quantum estimation theory.Physical Review Letters131, 210601 (2023). https://doi.org/10.1103/PhysRevLett.131.210601 20

work page doi:10.1103/physrevlett.131.210601 2023
[18]

https://doi.org/10.1038/s41534-023-00707-7

Qin, D., Chen, Y., Li, Y.: Error statistics and scalability of quantum error mitigation for- mulas.npj Quantum Information9, 35 (2023). https://doi.org/10.1038/s41534-023-00707-7

work page doi:10.1038/s41534-023-00707-7 2023
[19]

https://doi.org/10.1038/s41567-024-02536-7

Quek, Y., Stilck França, D., Khatri, S., Meyer, J.J.: Exponentially tighter bounds on limitations of quantum error mitigation.Nature Physics20, 1648–1658 (2024). https://doi.org/10.1038/s41567-024-02536-7

work page doi:10.1038/s41567-024-02536-7 2024
[20]

https://doi.org/10.1103/PhysRevLett.127.200505

Piveteau, C., Sutter, D., Bravyi, S., Gambetta, J.M., Temme, K.: Error mitigation for universal gates on encoded qubits.Physical Review Letters127, 200505 (2021). https://doi.org/10.1103/PhysRevLett.127.200505

work page doi:10.1103/physrevlett.127.200505 2021
[21]

Exploiting degener- acy in belief propagation decoding of quantum codes,

Piveteau, C., Sutter, D., Woerner, S.: Quasiprobability decompositions with reduced sam- pling overhead.npj Quantum Information8, 12 (2022). https://doi.org/10.1038/s41534- 022-00517-3

work page doi:10.1038/s41534- 2022
[22]

https://doi.org/10.1103/PRXQuantum.3.010345

Suzuki, Y., Endo, S., Fujii, K., Tokunaga, Y.: Quantum error mitigation as a universal error reduction technique: applications from the NISQ to the fault-tolerant quantum computing eras.PRX Quantum3, 010345 (2022). https://doi.org/10.1103/PRXQuantum.3.010345

work page doi:10.1103/prxquantum.3.010345 2022
[23]

Cambridge University Press, Cambridge (2010)

Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Informa- tion, 10th anniversary edn. Cambridge University Press, Cambridge (2010). https://doi.org/10.1017/CBO9780511976667

work page doi:10.1017/cbo9780511976667 2010
[24]

Improved simulation of stabilizer circuits.Physical Review A, 70(5), 2004

Aaronson, S., Gottesman, D.: Improved simulation of stabilizer circuits.Physical Review A 70, 052328 (2004). https://doi.org/10.1103/PhysRevA.70.052328

work page doi:10.1103/physreva.70.052328 2004
[25]

A Quantum Approximate Optimization Algorithm

Farhi, E., Goldstone, J., Gutmann, S.: A quantum approximate optimization algorithm. arXiv:1411.4028 (2014). https://doi.org/10.48550/arXiv.1411.4028

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1411.4028 2014
[26]

https://doi.org/10.22331/q-2022-07-07-759

Farhi, E., Goldstone, J., Gutmann, S., Zhou, L.: The quantum approximate optimization algorithm and the Sherrington-Kirkpatrick model at infinite size.Quantum6, 759 (2022). https://doi.org/10.22331/q-2022-07-07-759

work page doi:10.22331/q-2022-07-07-759 2022
[27]

Cerezoet al., Variational quantum al- gorithms, Nat

Cerezo, M., Arrasmith, A., Babbush, R., Benjamin, S.C., Endo, S., Fujii, K., McClean, J.R., Mitarai, K., Yuan, X., Cincio, L., Coles, P.J.: Variational quantum algorithms.Nature Reviews Physics3, 625–644 (2021). https://doi.org/10.1038/s42254-021-00348-9

work page doi:10.1038/s42254-021-00348-9 2021
[28]

Quantum computing in the NISQ era and beyond.Quantum, 2:79, 2018

Preskill, J.: Quantum computing in the NISQ era and beyond.Quantum2, 79 (2018). https://doi.org/10.22331/q-2018-08-06-79

work page internal anchor Pith review doi:10.22331/q-2018-08-06-79 2018
[29]

Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Al´ an Aspuru-Guzik

Bharti, K., Cervera-Lierta, A., Kyaw, T.H., Haug, T., Alperin-Lea, S., Anand, A., Degroote, M., Heimonen, H., Kottmann, J.S., Menke, T., Mok, W.K., Sim, S., Kwek, L.C., Aspuru- Guzik, A.: Noisy intermediate-scale quantum algorithms.Reviews of Modern Physics94, 015004 (2022). https://doi.org/10.1103/RevModPhys.94.015004

work page doi:10.1103/revmodphys.94.015004 2022
[30]

Nature Communications5(1), 4213 (2014) https://doi.org/ 10.1038/ncomms5213

Peruzzo, A., McClean, J., Shadbolt, P., Yung, M.H., Zhou, X.Q., Love, P.J., Aspuru-Guzik, A., O’Brien, J.L.: A variational eigenvalue solver on a photonic quantum processor.Nature Communications5, 4213 (2014). https://doi.org/10.1038/ncomms5213

work page doi:10.1038/ncomms5213 2014
[31]

The theory of variational hybrid quantum-classical algorithms,

McClean, J.R., Romero, J., Babbush, R., Aspuru-Guzik, A.: The theory of varia- tional hybrid quantum-classical algorithms.New Journal of Physics18, 023023 (2016). https://doi.org/10.1088/1367-2630/18/2/023023

work page doi:10.1088/1367-2630/18/2/023023 2016
[32]

Nature Communications9(1) (2018) https:// doi.org/10.1038/s41467-018-07090-4

McClean, J.R., Boixo, S., Smelyanskiy, V.N., Babbush, R., Neven, H.: Barren plateaus in quantum neural network training landscapes.Nature Communications9, 4812 (2018). https://doi.org/10.1038/s41467-018-07090-4 21

work page doi:10.1038/s41467-018-07090-4 2018
[33]

Nature549(7671), 242–246 (2017) https://doi.org/10.1038/nature23879

Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets.Nature549, 242–246 (2017). https://doi.org/10.1038/nature23879

work page doi:10.1038/nature23879 2017
[34]

Greenberger, Michael A

Greenberger, D.M., Horne, M.A., Zeilinger, A.: Going beyond Bell’s theorem. In: Kafatos, M. (ed.)Bell’s Theorem, Quantum Theory and Conceptions of the Universe, pp. 69–72. Springer, Dordrecht (1989). https://doi.org/10.1007/978-94-017-0849-4_10

work page doi:10.1007/978-94-017-0849-4_10 1989
[35]

https://doi.org/10.1103/PhysRevLett.65.1838

Mermin, N.D.: Extreme quantum entanglement in a superposition of macro- scopically distinct states.Physical Review Letters65, 1838–1840 (1990). https://doi.org/10.1103/PhysRevLett.65.1838

work page doi:10.1103/physrevlett.65.1838 1990
[36]

Quantum computing with Qiskit

Javadi-Abhari, A., Treinish, M., Krsulich, K., Wood, C.J., Lishman, J., Gacon, J., Martiel, S., Nation, P.D., Bishop, L.S., Cross, A.W., Johnson, B.R., Gambetta, J.M.: Quantumcom- puting with Qiskit. arXiv:2405.08810 (2024). https://doi.org/10.48550/arXiv.2405.08810

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2405.08810 2024
[37]

https://doi.org/10.22331/q-2022-08-11-774

LaRose, R., Mari, A., Kaiser, S., Karalekas, P.J., Alves, A.A., Czarnik, P., El Mandouh, S., Gordon, M.H., Hindy, Y., Robertson, A., Thakre, P., Shammah, N., Zeng, W.J.: Mitiq: a software package for error mitigation on noisy quantum computers.Quantum6, 774 (2022). https://doi.org/10.22331/q-2022-08-11-774

work page doi:10.22331/q-2022-08-11-774 2022
[38]

The Annals of Statistics7(1), 1–26 (1979) https://doi.org/10.1214/aos/1176344552

Efron, B.: Bootstrap methods: another look at the jackknife.The Annals of Statistics7, 1–26 (1979). https://doi.org/10.1214/aos/1176344552

work page doi:10.1214/aos/1176344552 1979
[39]

doi:10.1017/CBO9780511802843 , isbn =

Davison, A.C., Hinkley, D.V.: Bootstrap Methods and Their Application. Cambridge Uni- versity Press, Cambridge (1997). https://doi.org/10.1017/CBO9780511802843

work page doi:10.1017/cbo9780511802843 1997
[40]

https://doi.org/10.22331/q-2021-09-21-548

Cai, Z.: Quantum error mitigation using symmetry expansion.Quantum5, 548 (2021). https://doi.org/10.22331/q-2021-09-21-548

work page doi:10.22331/q-2021-09-21-548 2021
[41]

https://doi.org/10.1103/PRXQuantum.2.040330

Strikis, A., Qin, D., Chen, Y., Benjamin, S.C., Li, Y.: Learning-based quantum error miti- gation.PRX Quantum2, 040330 (2021). https://doi.org/10.1103/PRXQuantum.2.040330

work page doi:10.1103/prxquantum.2.040330 2021
[42]

Czarnik, A

Czarnik, P., Arrasmith, A., Coles, P.J., Cincio, L.: Error mitigation with Clifford quantum- circuit data.Quantum5, 592 (2021). https://doi.org/10.22331/q-2021-11-26-592

work page doi:10.22331/q-2021-11-26-592 2021
[43]

https://doi.org/10.1109/TQE.2023.3305232

Russo, A.E., Mari, A., Shammah, N., LaRose, R.: Testing platform-independent quantum error mitigation on noisy quantum computers.IEEE Transactions on Quantum Engineering 4, 1–18 (2023). https://doi.org/10.1109/TQE.2023.3305232

work page doi:10.1109/tqe.2023.3305232 2023
[44]

and Gambetta, Jay M

Bravyi, S., Sheldon, S., Kandala, A., McKay, D.C., Gambetta, J.M.: Mitigating mea- surement errors in multiqubit experiments.Physical Review A103, 042605 (2021). https://doi.org/10.1103/PhysRevA.103.042605

work page doi:10.1103/physreva.103.042605 2021
[45]

https://doi.org/10.1103/PRXQuantum.2.040326

Nation, P.D., Kang, H., Sundaresan, N., Gambetta, J.M.: Scalable mitigation of measurement errors on quantum computers.PRX Quantum2, 040326 (2021). https://doi.org/10.1103/PRXQuantum.2.040326

work page doi:10.1103/prxquantum.2.040326 2021
[46]

Available: http://dx.doi.org/10.1038/s41586-019-1040-7

Kandala, A., Temme, K., Corcoles, A.D., Mezzacapo, A., Chow, J.M., Gambetta, J.M.: Error mitigation extends the computational reach of a noisy quantum processor.Nature 567, 491–495 (2019). https://doi.org/10.1038/s41586-019-1040-7

work page doi:10.1038/s41586-019-1040-7 2019
[47]

https://doi.org/10.1038/s41567-022-01914-3

Kim, Y., Wood, C.J., Yoder, T.J., Merkel, S.T., Gambetta, J.M., Temme, K., Kandala, A.: Scalable error mitigation for noisy quantum circuits produces competitive expectation values.Nature Physics19, 752–759 (2023). https://doi.org/10.1038/s41567-022-01914-3

work page doi:10.1038/s41567-022-01914-3 2023
[48]

Zenodo (2026)

Scavino Alfaro, V.: Finite-shot ZNE boundary validation supplement, version 1.0.1. Zenodo (2026). https://doi.org/10.5281/zenodo.20057161 22

work page doi:10.5281/zenodo.20057161 2026