{"id":"3cbb9423-a5c8-4fad-8640-884603869030","arxiv_id":"2505.05614","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"ZNE can recover noiseless expectation values for short, low-noise QSP Hamiltonian simulations, but a steady-state regime makes it fail at higher noise, independent of sample budget.","lead":"This paper simulates zero-noise extrapolation on quantum-signal-processing circuits for a small Ising Hamiltonian and finds that the mitigation recovers the noiseless expectation value for low depolarizing noise and short simulation times. It also identifies a noise-driven steady-state regime where ZNE stops working even with unlimited samples.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Post-hoc selection of the best of six ZNE variants, using the known noiseless value, underlies the reported success; a fixed pre-registered extrapolation rule may not meet the 1e-2 bias threshold.","rationale":"Concern and why it is load-bearing. The central quantitative claim ('p=1e-4 and p=1e-3 instances are successfully mitigated') is presented as a statement about ZNE. But the evaluation protocol picks the best-performing extrapolation method and noise-scaling schedule for each data set after comparing against the ideal value (App. B 6). Six estimators are tried; the minimum-MSE one is reported. Consequently, the reported bias is an envelope over six post-hoc fitted procedures. For a real deployment, no ground truth is available, so the paper does not specify how a user would choose among linear, Richardson, and exponential fits or between [1,2,3] and [1,1.25,1.5] scalings. This is a validity concern even within the idealized noiseless-oracle setting, and it is exactly the setting of the strongest claim. The oracle simplification is explicitly acknowledged as a best-case limit and thus does not threaten the internal numerical observation; the post-hoc model selection does, because it makes the central claim unfalsifiable in a practical sense: some variant may always be found that fits the data. The trivial-observable footnote reinforces that the benchmark is permissive: a constant estimator already achieves the bias used to set the error tolerance, so the discriminating power of the chosen O is questionable. The proposed test—fixing one extrapolation rule and recomputing the bias—settles whether the success survives without ground-truth access. If it does not, the paper's contribution can still be valuable as a best-case feasibility benchmark, which is why the reader's CONDITIONAL verdict remains appropriate; no verdict change is warranted on the basis of this stress test alone.","tokens_in":19846,"tokens_out":10090,"duration_ms":109251,"concrete_test":"Using the released code, re-run the 4-qubit Section III simulations for p = 1e-4, 1e-3, 1e-2 with one fixed extrapolation rule—e.g., exponential fit with scaling factors [1,2,3] for all τ—and without evaluating any alternative fit against ⟨O⟩_ideal. Recompute the bias curves from Fig. 2 and compare against the 1e-2 threshold. If bias exceeds 1e-2 anywhere in the currently reported success regions, the headline claim is an artifact of post-hoc selection and the paper should be reframed as a best-case benchmark rather than a ZNE protocol demonstration.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central success claim is not for a fixed ZNE protocol. Appendix B 6 states: '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.' This selection uses the known noiseless expectation value, which a real user of ZNE does not have. The bias curves in Fig. 2 and the 'successful mitigation' statements in Section III are therefore the best case over six estimators, not the performance of any single implementable ZNE procedure. The protocol-defining question—which fit and scaling to use—is left unresolved; the paper's own Section II B admits 'there is no definitive way to determine the most suitable fit without testing various fits and choosing one giving the least error.' Footnote 36 compounds the concern: the error tolerance parameter was 'set to 1, due to the trivial estimator being able to reach the following bias for our chosen observable,' indicating that the chosen observable is unusually easy to predict and the benchmark is permissive. The load-bearing issue is internal: even granting the idealized noiseless oracle, the numerical demonstration does not establish that a pre-specified ZNE rule recovers the correlation to 1e-2.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":20009,"tokens_out":8368,"duration_ms":90298,"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":[{"comment":"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.","section":"Sec. III and Appendix B 6"},{"comment":"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.","section":"Sec. III and footnote 36"},{"comment":"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.","section":"Sec. II C, Sec. IV, and Eq. (B1)"}],"minor_comments":[{"comment":"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.","section":"Eq. (6)"},{"comment":"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.","section":"Appendix A 3, Eqs. (A23) and (A25)"},{"comment":"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.","section":"Sec. II D"},{"comment":"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.","section":"Footnote 36"},{"comment":"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.","section":"Figs. 1, 2, 13, 14"},{"comment":"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.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is an exploratory numerical study with honest caveats about the idealized oracle, and the self-citations to the QSP preprocessing work are legitimate prior technical inputs rather than a circularity concern. The main gap is that the reported recovery is the best over six ZNE estimators chosen using the known ideal value, which means the paper does not yet demonstrate a fixed, usable ZNE protocol. This is fixable within the manuscript's scope by changing the evaluation to a pre-registered or conservative rule and by adding an oracle-depth sensitivity analysis, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Three things to know. This is the first explicit test of standard ZNE on QSP Hamiltonian simulation; that combination is genuinely new in the cited literature. It is an honest preliminary feasibility study, with code on GitHub and a clear statement of what is idealized. The most solid takeaway is the steady-state no-go: for local depolarizing noise, once the circuit reaches the fully mixed state, ZNE cannot work regardless of sample count. That observation is independent of the extrapolation choices and is worth keeping.\n\nThe main success claim—that ZNE recovers the two-site correlation within 1e-2 bias for p=1e-4 and 1e-3—needs a caveat. As the paper says in Appendix B6, the reported curves are the best of six extrapolation variants (three fits, two scaling sets), chosen after seeing the noiseless value. That is not a pre-specified protocol. A user who has to pick a fit and scaling before seeing the answer may not meet the threshold. The stress-test note has this right. The paper admits the choice problem in Section IIB but leaves it unresolved. Additionally, the oracle is applied as a single noiseless depth-one gate (Section IIC), so the circuit depths are much shorter than any hardware decomposition, and footnote 36 concedes the observable is easy to predict: the trivial estimator already reaches the allowed bias. These three choices point in the same direction: the quantitative regime map is optimistic.\n\nI would not call this a fatal flaw. The paper labels itself a preliminary study and a best-case limit, and the authors say so in the Discussion. The numerics are reproducible and the reasoning is clear. The problem is that the headline claim, \"successful mitigation,\" is stated for a selection procedure, not for an implementable ZNE rule. That is a real gap between the claim and the demonstration.\n\nWho gets value: anyone benchmarking QEM on structured circuits or thinking about which noise/depth regimes are worth trying. The paper is a useful starting point, not a protocol recommendation.\n\nMy recommendation: send it to peer review, conditional on the authors either fixing a single extrapolation rule (and reporting the spread over fits) or reframing the contribution explicitly as a best-case benchmark rather than a success claim for ZNE. The steady-state section can stay as is.","headline":"Useful first map of ZNE+QSP under depolarizing noise, but the success claim is the best of six estimators, not a fixed protocol.","tokens_in":20620,"tokens_out":2127,"would_cite":false,"duration_ms":23560,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["quantum error mitigation","zero-noise extrapolation","quantum signal processing","Hamiltonian simulation","depolarizing noise","transverse-field Ising model","near-term quantum computing","post-selection"],"falsifier":"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.","tokens_in":19534,"feed_emoji":"⚛️","tokens_out":10665,"duration_ms":91358,"temperature":0.7,"pith_summary":"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.","feed_headline":"Zero-noise extrapolation recovers QSP results under 1e-3 noise","feed_subtitle":"At 1e-4 and 1e-3 per-gate noise, the corrected expectation value stays within 1e-2 bias for evolution times up to 20.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces the QSP Hamiltonian simulation template whose circuits are being error-mitigated.","marker":"[5]"},{"why":"Supplies the Jacobi-Anger degree bounds and QSVT/block-encoding framework used to set precision and circuit depth.","marker":"[6]"},{"why":"Gives the Laurent-QSP decomposition and error conditions used to build the QSP sequence.","marker":"[18]"},{"why":"Provides the Fejer-Wilson pre-processing used to compute the QSP rotation bases.","marker":"[20]"},{"why":"Introduces zero-noise extrapolation, the mitigation protocol whose success is being assessed.","marker":"[21]"},{"why":"Supplies the looser sampling bound adapted into the QSP-specific shot estimate.","marker":"[2]"},{"why":"Supplies the tighter exponential sample-cost bound that motivates using a fixed number of shots.","marker":"[3]"},{"why":"Sets the first-order Trotter error scaling used to build the low-precision Trotter baseline for comparison.","marker":"[28]"}],"fun_headline_variants":["Zero-noise extrapolation saves QSP at low noise, fails at high","QSP error mitigation works at 1e-3 noise but hits a wall","ZNE recovers QSP results at 1e-4 and 1e-3 noise, not at 1e-2","QSP noise floor: ZNE helps below 1e-2 but not beyond"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Zero-noise extrapolation saves QSP at low noise, fails at high","QSP error mitigation works at 1e-3 noise but hits a wall","ZNE recovers QSP results at 1e-4 and 1e-3 noise, not at 1e-2","QSP noise floor: ZNE helps below 1e-2 but not beyond"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000613,"raw_usage":{"total_tokens":2904,"prompt_tokens":1056,"completion_tokens":1848,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":672,"completion_tokens_details":{"reasoning_tokens":1750}},"tokens_in":672,"tokens_out":1848,"duration_ms":11874,"temperature":1.0,"reasoning_tokens":1750,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:01:45.216418+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Haah, Product Decomposition of Periodic Functions in Quantum Signal Processing, Quantum 3, 190 (2019)","cited_arxiv_id":null,"evidence_quote":"Gives the Laurent-QSP decomposition and error conditions used to build the QSP sequence."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the tighter exponential sample-cost bound that motivates using a fixed number of shots."}],"review_version":1}