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Prospects of Quantum Error Mitigation for Quantum Signal Processing

T0 review · 3 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Zero-noise extrapolation recovers noiseless two-site correlations from QSP Hamiltonian simulation under local depolarizing noise at per-gate error rates of $10^{-3}$ and below, while at $10^{-2}$ it works only for short evolution times.

desk verdict Useful first map of ZNE+QSP under depolarizing noise, but the success claim is the best of six estimators, not a fixed protocol. read the letter →

arxiv 2505.05614 v1 pith:3P3KFVQA submitted 2025-05-08 quant-ph

Ugnė Liaubaitė , S. E. Skelton This is my paper · ORCID
classification quant-ph
keywords quantumerrormitigationzero-noiseextrapolationsignalprocessingHamiltoniansimulationdepolarizingnoisetransverse-fieldIsingmodelnear-termcomputingpost-selection
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether zero-noise extrapolation (ZNE) can make quantum signal processing (QSP) circuits useful on near-term hardware, where QSP is normally considered too deep and fragile to run. Using a low-precision QSP Hamiltonian simulation of a transverse-field Ising chain under local depolarizing noise, the authors show that at per-gate error rates $p=10^{-4}$ and $10^{-3}$, ZNE recovers the noiseless two-site correlation to within the algorithm's own $10^{-2}$ precision for evolution times $\tau\in[0.1,20]$, and that the higher-precision $10^{-4}$ setting also succeeds at $p=10^{-4}$. At $p=10^{-2}$, mitigation succeeds only for short times. The recovery regions come with a caveat: the QSP oracle is computed classically and applied as a single noiseless gate, so these are best-case QSP circuits. The paper maps which noise-depth-time regions make QSP plus ZNE a workable low-precision strategy.

What carries the argument

The central object is the QSP circuit with its explicit depth--precision relation: a target function such as $e^{i\tau H}$ is written as a truncated Jacobi--Anger polynomial $P(z)$ in the oracle eigenvalues $z=e^{i\theta}$, giving circuit depth $d_{\mathrm{QSP}}=2n d_o+1$ for polynomial degree $n$ and oracle depth $d_o$. Because ZNE scales the per-layer depolarizing noise, this depth formula turns the noisy expectation value into a controlled function of the noise parameter $p$, which is why an exponential extrapolation -- the fit that dominates in the successful cases -- can track layerwise error accumulation. The mechanism that sets the method's boundary is the approach to the fully mixed steady state: once $\langle O\rangle$ and its variance stop changing with $p$, there is no signal left to extrapolate, and ZNE fails independently of sampling cost.

What would settle it

Decompose the same QSP oracle into elementary gates (for example, via an LCU block-encoding or controlled rotations) and repeat the ZNE runs at $p=10^{-3}$ for $\tau\in[0.1,20]$: if the post-mitigation bias of the two-site correlation exceeds $10^{-2}$ anywhere in that interval, the successful-mitigation claim for realistic QSP circuits is refuted.

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Extended reading notes

Core claim

On the paper's own terms, the central finding is a regime map: with local depolarizing noise at rates $p=10^{-4}$ and $10^{-3}$, the ZNE-QSP protocol successfully recovers expectation values of interest over $\tau\in[0.1,20.0]$, with mean-squared error suppressed to the same order as the algorithmic error $O(10^{-2})$, and at $p=10^{-4}$ it also succeeds at target precision $O(10^{-4})$. At $p=10^{-2}$, the estimator stays within threshold only for short time intervals. Recovery persists for long evolution times up to $\tau=390$ (circuit depth 811) at the two lower noise levels, and ZNE handles QSP better than comparable-depth Trotter circuits. The authors attribute the relative success to QSP's built-in post-selection, which itself removes some noise, and they identify a hard limit: when depolarizing noise pushes the expectation value and its variance to the fully mixed steady state, ZNE cannot help even with an unlimited sample budget.

Load-bearing premise

The QSP oracle is computed classically and inserted as a single noiseless depth-one gate, so the simulated QSP circuits are far shorter and cleaner than any real hardware decomposition; if a physical oracle's internal noise and depth are large, the reported recovery regions shrink or disappear.

Editorial extensions

If this is right

  • At $p=10^{-4}$ and $10^{-3}$, QSP plus ZNE is a viable low-precision Hamiltonian simulation route for $\tau\in[0.1,20]$, so the technique is not limited to trivially shallow circuits.
  • At target precision $10^{-4}$ and $p=10^{-4}$, the same fixed shot budget ($5\times10^6$) suffices, so raising algorithmic precision does not necessarily require a larger sample budget in this regime.
  • Long-time QSP evolution (up to $\tau=390$) is mitigated at the two low noise levels, suggesting that noise strength relative to depth, rather than depth alone, is the barrier.
  • For $p=10^{-2}$, ZNE works only at short times, so users must select time windows carefully before trusting mitigated results.
  • ZNE fails completely once the system reaches the fully mixed steady state; this failure is not a sampling-cost problem and would survive an infinite shot budget.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If a real block-encoding oracle of depth $O(10)$ can be built with per-gate error near $10^{-4}$, the paper's regime map suggests QSP plus ZNE could outperform first-order Trotter at low precision and longer times; this transfers only if the oracle's own noise stays within the tested total-noise budget.
  • The variance plateau at large $p$ offers a cheap, circuit-agnostic diagnostic: when increasing the noise factor no longer changes the noisy expectation value, ZNE has no extrapolation signal and should be halted.
  • The dominance of the exponential fit follows from layerwise depolarizing noise; on hardware with biased or non-depolarizing noise, the fit ranking could change, so the reported recovery regions should be re-measured rather than assumed.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The manuscript numerically studies zero-noise extrapolation (ZNE) applied to a quantum signal processing (QSP) Hamiltonian simulation of a modified transverse-field Ising model under local depolarizing noise. The QSP oracle is implemented classically as a single noiseless gate of depth one, and the authors compare QSP with first-order Trotter simulation for the same low-precision targets and noise levels p = 1e-4, 1e-3, 1e-2. For each case they select, among six ZNE variants (three extrapolation fits and two noise-scaling sets), the one with the smallest mean-squared error relative to the known noiseless expectation value. They report successful mitigation for p = 1e-4 and p = 1e-3 over simulation times up to 20, partial success for p = 1e-2 at short times, and they identify a no-go regime in which the noisy state reaches the fully mixed steady state so that ZNE cannot help even with unlimited sampling. The paper also discusses sample-cost bounds and includes a numerical study for 4-, 6-, and 8-qubit chains.

Significance. If the result holds as stated, it would provide a useful numerical illustration that low-precision QSP Hamiltonian simulation can be error-mitigated by ZNE in regimes of moderate depth and low local depolarizing noise, and it would demarcate regimes where ZNE is fundamentally inapplicable. The manuscript is honest about its main simplification, makes its code available, and includes a concrete comparison with Trotter simulation. However, the central quantitative claim is currently established only for a best-case estimator selected using the unknown noiseless value and under an idealized noiseless depth-one oracle; both assumptions materially weaken the transferability of the reported recovery regions to any implementable ZNE-QSP protocol.

major comments (3)
  1. [Sec. III and Appendix B 6] The success claim is not for a fixed ZNE protocol. Appendix B 6 states that six different extrapolation methods (three functional fits and two sets of scaling parameters) are considered for each case and that 'the one with the smallest MSE is chosen.' Because MSE is evaluated against the classically known noiseless expectation value, the bias curves in Fig. 2 and the statements in Sec. III that the p = 1e-4 and p = 1e-3 instances are 'successfully mitigated' report the minimum over six estimators, not the performance of a single implementable ZNE rule. A real user does not have the ideal value when choosing the extrapolation. Please report at least one fixed, pre-registered ZNE rule (for example, exponential fit with scaling factors [1,2,3] for all cases) and show the spread of all six variants; otherwise the central claim that 'the ZNE-QSP protocol is successful' is not established.
  2. [Sec. III and footnote 36] The benchmark observable appears to be unusually permissive. The chosen observable in Eq. (9) is a two-site correlation that remains close to unity for the simulated interval, and footnote 36 states that the error-tolerance parameter in the sample-budget calculation was 'set to 1, due to the trivial estimator being able to reach the following bias for our chosen observable.' If a trivial constant estimator already comes close to the target, then 'recovering' this observable to 1e-2 is not a strong demonstration of error mitigation. Please clarify whether footnote 36 refers to the sample-bound parameter or to the success threshold, state explicitly what the trivial estimator's bias is, and preferably include an observable with nontrivial time dependence so that the recovery claim tests the protocol rather than the near-constancy of the signal.
  3. [Sec. II C, Sec. IV, and Eq. (B1)] The entire numerical regime map assumes a noiseless depth-one QSP oracle. The authors correctly acknowledge this as 'a major simplification' and state that the results are 'a best-case limit on QSP circuits,' but the abstract and the concluding paragraph still present the p = 1e-3 recovery as a property of ZNE-QSP. Since the sampling lower bound in Eq. (B1) is exponential in circuit depth, the recoverable regions are exponentially sensitive to the oracle depth d_o. Please add an explicit sensitivity estimate for the reported success regions under a realistic d_o > 1, or, at minimum, restrict the abstract and the conclusions to the best-case-oracle scenario so that the claims do not overstate hardware relevance.
minor comments (6)
  1. [Eq. (6)] The notation E_O is used in Eq. (6) before it is defined; the text alternates between E_O, EO, and ⟨E_O⟩. Please define the estimator symbol once at the start of Sec. II B and use it consistently.
  2. [Appendix A 3, Eqs. (A23) and (A25)] The two depth formulas appear inconsistent: Eq. (A23) contains a term “1 + 4⌈r̃⌉d_o” while Eq. (A25) reads d_qsp(d_o) = 2 n d_o + 1. Please reconcile these expressions and state explicitly which formula generated the depth values reported in Fig. 9 and used in the mitigation simulations.
  3. [Sec. II D] Please specify the Trotter time step dt used to reach the stated O(10^-2) precision, since the Trotter depth and the QSP comparison in Figs. 1–2 depend on this choice.
  4. [Footnote 36] As written, the statement that the error-tolerance parameter “was set to 1” appears to contradict Sec. III, where ε_QEM = 1e-2 is introduced. Please clarify whether the footnote concerns the sample-bound parameter b_max rather than the success threshold, and if so make that distinction in the main text.
  5. [Figs. 1, 2, 13, 14] The axis labels and legend entries are difficult to read at the current font size, and Fig. 2 does not include a horizontal line at the stated ε_QEM = 1e-2 threshold. Adding the threshold line and enlarging the fonts would make the claimed success regions directly verifiable.
  6. [Throughout] There are several language and formatting issues, including “irregardless” in Appendix A 2, “T rotter” in the Sec. II D heading, inconsistent use of “O(...)” and “Ο(...)”, and a few unfinished equation fragments (e.g., Eq. (A17)). A careful proofreading pass is needed.

Circularity Check

1 steps flagged · score 4.0 of 10

The reported ZNE success is for the best of six extrapolation variants selected using the known noiseless value, so the protocol-level claim is partially circular.

  1. self definitional [Appendix B 5, Eq. (6); Figs. 1 and 2 captions ('Best performing QEM estimates')]
    "MSE[EO] = Var[EO] + b^2_EO. Bias b_EO is defined as the difference between the predicted result and the ideal result, EO−⟨O⟩ideal. ... We consider six different methods of extrapolation (three functional fits and two sets of scaling parameters) for each of our cases, and the one with the smallest MSE is chosen, see appendix B 6"

    The success metric (Eq. 6) is MSE = Var + (E_O − ⟨O⟩_ideal)^2, so selecting the extrapolation variant with the smallest MSE requires the noiseless expectation value that ZNE is supposed to predict. The bias curves in Fig. 2 and the statements that the p = 1e-4 and p = 1e-3 instances are 'successfully mitigated' are for the 'best performing QEM estimates' shown in Fig. 1, i.e., the variant chosen using the target. Section II B concedes: 'there is no definitive way to determine the most suitable fit without testing various fits and choosing one giving the least error.' A user cannot pre-specify the protocol from noisy data alone.

full rationale

The numerical core is self-contained: noisy expectation values are simulated under local depolarizing noise, extrapolated to zero noise, and compared with classically computed ideal values; those ideal values are used for evaluation, not to set the extrapolation constants. The self-citations to Refs. [20] and [33] are prior technical QSP pre-processing inputs and are not invoked to justify ZNE recovery; QSP accuracy is numerically verified in Fig. 8. The oracle simplification, where the QSP oracle is classically computed and applied as a single gate, is explicitly acknowledged in Section II C and the Discussion and is a stated limitation rather than a circular step. The only circular-adjacent element is the post-hoc selection of the extrapolation method: Appendix B 5 chooses the variant with the smallest MSE against the ideal, and Figs. 1 and 2 report these 'best performing' estimates as the success of 'the ZNE-QSP protocol.' This makes the protocol-level success claim an in-sample best case rather than a fixed pre-registered rule. Because many individual variants, notably exponential fits with scaling [1,2,3], match the selected choices, and the extrapolations are computed from noisy data, the central low-noise recovery result retains independent content; however, the selection issue partially circularizes the 'protocol' wording. Footnote 36, stating that the error tolerance parameter was 'set to 1, due to the trivial estimator being able to reach the following bias for our chosen observable,' is an acknowledged looseness in the sample-bound analysis, not a circular derivation.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The numerical results rest on four free choices (normalization alpha, per-case extrapolation selection, fixed shot count, numerically chosen QSP degree) and on imported QSP theorems plus the local depolarizing noise model. No new physical entities are introduced. The most fragile input is the noiseless depth-one oracle, which is explicitly acknowledged by the authors.

free parameters (4)
  • Hamiltonian normalization alpha = 7/50
    Chosen to normalize H so that QSP can be applied. This changes the spectrum and dynamics, so all numerical results are specific to this value.
  • Extrapolation model and noise scaling factors = Exponential/Richardson/linear with scaling [1,2,3] or [1,1.25,1.5], varying by case (Table I)
    Selected per data set as the fit with smallest MSE, i.e., fitted to the data and to the known ideal value. This is not a fixed a priori protocol and biases the reported success regions.
  • Number of shots M = 5e6
    Fixed resource chosen ad hoc. It lies below the provable lower bound Me from [3], so success is not guaranteed and results depend on this specific sample count.
  • QSP polynomial degree n = Numerically selected per tau and target epsilon_QSP
    Chosen numerically rather than from the analytic bound, reducing depth by an average factor of 4.10-4.52. This directly controls noise accumulation and therefore affects ZNE performance.
assumptions (5)
  • standard math QSP decomposition theorem: any suitable Laurent polynomial with A^2+B^2 <= 1 has a QSP circuit (Lemma 1 from [18]).
    Background theorem used to justify that the Jacobi-Anger polynomial P_HS can be realized as a QSP circuit. Not proven in this paper.
  • standard math Jacobi-Anger truncation error bounds and the degree bound r(tau, epsilon) from [6,25].
    Used to set circuit depths and precision scaling in equations (A6)-(A12). These results are imported without reproof.
  • domain assumption ZNE validity: the noisy expectation value is infinitely differentiable with respect to the noise parameter for convex combinations of Pauli channels.
    Stated in Appendix B1 as the justification for extrapolating to zero noise. It is assumed rather than proved, and the paper notes non-analytic cases exist.
  • domain assumption Local depolarizing noise acts on every qubit in every circuit layer with the same parameter p.
    Standard but specific noise model. Quantitative results may not transfer to amplitude damping, correlated noise, or non-local error channels.
  • ad hoc to paper The QSP oracle can be applied as a single noiseless gate of depth one.
    The central simplification in Section II C. It avoids block-encoding and ancilla costs, making the QSP circuits artificially short and noise-free inside the oracle.

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Cite this review

Pith. "Pith review of Prospects of Quantum Error Mitigation for Quantum Signal Processing." pith.science (2026). https://pith.science/paper/3P3KFVQA

@misc{pith2026250505614,
  author       = {Pith},
  title        = {Pith review of: Prospects of Quantum Error Mitigation for Quantum Signal Processing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3P3KFVQA}},
  note         = {Machine review of arXiv:2505.05614}
}
read the original abstract

Quantum error mitigation (QEM) protocols have provably exponential bounds on the cost scaling; however, exploring which regimes QEM can recover usable results is still of sizable interest. The expected absence of complete error correction for near-term and intermediate-term quantum devices means that QEM protocols will remain relevant for devices with low enough error rates to attempt small examples of fault-tolerant algorithms. Herein, we are interested in the performance of QEM with a template for quantum algorithms, quantum signal processing (QSP). QSP-based algorithms are designed with an especially simple relation between the circuit depths and the algorithm's parameters, including the required precision. Hence, they may be a useful playground for exploring QEM's practical performance and costs for a range of controlled parameters. As a preliminary step in this direction, this work explores the performance of zero-noise-extrapolation (ZNE) on a Hamiltonian simulation algorithm designed within QSP under local depolarizing noise. We design a QSP-based Hamiltonian simulation of a modified Ising model under depolarizing noise for low precision and varying simulation times. We quantify for which noise and depth regimes our ZNE protocol can recover an approximation of the noiseless expectation value. We discuss existing bounds on the sample budget, eventually using a fixed number of shots. While this does not guarantee the success of QEM, it gives us usable results in relevant cases. Finally, we briefly discuss and present a numerical study on the region where ZNE is unusable, even given an unlimited sample budget.

Figures

Figures reproduced from arXiv: 2505.05614 by the authors.

Figure 2
Figure 2. FIG. 2: Biases of the best performing QEM for 4-qubit [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Higher-precision QSP. Biases of the best [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4: Long time evolution QSP. Best performing [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figures from the paper (9 more)
Figure 5
Figure 5. Figure 5: FIG. 5: Limiting scenarios for our 4 qubit QEM-QSP [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Decay of error with simulation time at a fixed [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Growth of error in the QSP decomposition step, [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Degrees and depths for [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Sufficient number of shots [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Bias vs simulation time, for both a fixed and [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: Limiting scenarios for our 4-qubit QEM-QSP [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13: Scaling plots for QEM-QSP. For 4-qubit (above left), 6-qubit (middle), and 8-qubit (above right) Ising [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14: Best performing mitigation for all qubits (4 [PITH_FULL_IMAGE:figures/full_fig_p016_14.png]

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    Appendix A: QSP Hamiltonian Simulation Herein, we summarize some well-known results on QSP and Hamiltonian simulation with QSP

    In our case, it was set to 1, due to the trivial estima- tor being able to reach the following bias for our chosen observable. Appendix A: QSP Hamiltonian Simulation Herein, we summarize some well-known results on QSP and Hamiltonian simulation with QSP

  29. [37]

    (A5) Following [6], cos(τx)−J0(τ)− RX k=0 (−1)kJ2k(τ)T2k(x) D0 ≤ϵ (A6) sin(τx)− RX k=0 (−1)kJ2k+1(τ)T2k+1(x) D0 ≤ϵ (A7) whenever 4 3√π e|τ| 4(R + 1) 2(R+1) ≤ϵ

    Jacobi-Anger expansion We begin with the Jacobi-Anger expansion for eiτx, eiτx = cos(τx) +i sin(τx) (A1) cos(τx) =J0(t) + ∞X k=1 (−1)kJ2k(τ)T2k(x) (A2) sin(τx) = ∞X k=1 (−1)kJ2k+1(τ)T2k+1(x) (A3) One can truncate the expansion at R and bound the re- sult on domainD0 = [−1, 1],...

  30. [38]

    to obtain a more concise lower bound for r(τ,ϵ ). To coax this expansion into the correct form for lemma 1, we use relations cos( jθ) = eiθ+e−iθ 2 , sin(jθ) = eiθ−e−iθ 2 to define Laurent polynomialsA(z),B(z) which are identical to the Jacobi-Anger truncation forz∈U(1), and ar...

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    We then construct our oracle, U =QDiagj {ei arccos(λj)} Q† (A16) This object is unitary and has eigenvalues ei arccos(λj ), so it is a suitable QSP oracle

    QSP Hamiltonian Simulation We begin by defining with Hamiltonian H with eigen- decomposition H = QDiag ({λj})Q†, where column j of Q defines eigenvector |λj⟩ and{λj} is the set of all eigenvalues. We then construct our oracle, U =QDiagj {ei arccos(λj)} Q† (A16) This object is ...

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    (A11), and then the relationship between our Hamiltonian simulation circuit depth and ϵapprox is dqsp≤ 4⌈˜r (|τ|,ϵapprox)⌉ + 1 + 4⌈˜r (|τ|,ϵapprox)⌉do

    Error and circuit depth Recall that n = n(t,ϵapprox) is defined from lemma 3 and eq. (A11), and then the relationship between our Hamiltonian simulation circuit depth and ϵapprox is dqsp≤ 4⌈˜r (|τ|,ϵapprox)⌉ + 1 + 4⌈˜r (|τ|,ϵapprox)⌉do. (A23) 0 5 10 15 20 20 40 60 80 100 Simul...

  33. [41]

    Success Probability of Hamiltonian Simulation The success probability of the QSP protocol is the com- ponent ofUQSPρinitU† QSP approximatingei arccos(H). We have to post-select the correct QSP-ancillary measure- ment, so the success probability is pQSP≈ 1 2e−iτHρeiτH = 1 2, (A...

  34. [42]

    If the expectation is infinitely differentiable with respect to λ, then one can obtain the expectation value expansion with a truncated Taylor expansion around λ = 0

    V alidity of ZNE for QSP The validity of the ZNE method for a given problem re- lies on the expansion of the noisy expectation value⟨O⟩λ as a function of noise parameter λ. If the expectation is infinitely differentiable with respect to λ, then one can obtain the expectation v...

  35. [43]

    A general lower bound for the sampling cost of any mitigation technique under the noise model we consider can be found in [3], and it is exponential in circuit depth

    Lower Bounds on Mitigation Cost There have been several works discussing the sampling cost of performing error mitigation techniques, including [1–3]. A general lower bound for the sampling cost of any mitigation technique under the noise model we consider can be found in [3],...

  36. [44]

    While depth for the same time τ depends on the method used, this does not alter the underlying derivations on sampling bounds

    Costs of T rotter vs QSP The sampling cost required for ZNE for the Trotter- based Hamiltonian simulation can be derived similarly to the QSP case, as all bounds are determined by noise level and depth. While depth for the same time τ depends on the method used, this does not ...

  37. [45]

    In this section, we will discuss the wider infeasibil- ity of ZNE by using physical intuition about the system in question

    F easible QEM regimes In appendix B 2, we discussed the sampling cost in- volved in performing error mitigation and the provable infeasibility of QEM for large circuits due to exponential costs. In this section, we will discuss the wider infeasibil- ity of ZNE by using physica...

  38. [46]

    Richardson extrapolation is a popular choice in ex- isting QEM literature due to its convergence behavior [21]

    Extrapolation T echniques Within the second step of ZNE, the extrapolation of noisy data, we will consider three functional fits. Richardson extrapolation is a popular choice in ex- isting QEM literature due to its convergence behavior [21]. Richardson’s estimation for the noi...

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    For each extrap- olation technique, we use three noise scaling steps, either [1, 1.25, 1.5] or [1, 2, 3]

    Numeric Comparison of Extrapolation T echniques In this section, we compare the performance of ZNE with three types of extrapolation functions: linear, Richardson, and exponential scaling. For each extrap- olation technique, we use three noise scaling steps, either [1, 1.25, 1...

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