{"id":"a33d6008-0f95-48a0-9fad-b9d6b6a8419a","arxiv_id":"2608.12712","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":5,"one_line_summary":"Depth-1 RY+CNOT+Z feature maps in denoisers can only represent even functions, so the Bayes-optimal denoiser's odd sine component sets an irreducible excess-risk floor.","lead":"This paper proves a parity floor for a family of quantum feature maps used in denoising: all reachable features are even, while the optimal denoiser also needs odd parts, so excess risk has an irreducible component. It introduces a closed-form torus benchmark and shows measured deficits are parity-limited rather than a sign of quantumness.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Parity-floor theorem is sound, but the empirical 'parity-dominated' attribution rests on an unquantified proxy; the direct odd-sector mass of the Bayes denoiser should be computed.","rationale":"Both Lemma 1 and Theorem 1 are correct as stated: the depth-1 RY+CNOT+Z features are even, the symmetric prior and kernel make L2(mu_sigma) parity-orthogonal, and the Bayes denoiser's sine component is odd, so the excess-risk decomposition follows by the Pythagorean identity. The reader's conditional verdict is appropriate. My stress-test pass found no flaw in the theorem or in the numerical controls that test it. The single load-bearing uncertainty is the attribution step: the paper's headline empirical claim is that the observed excess is parity-dominated, which is established by comparing the excess with P_hat_sigma, a proxy that equals the true floor only up to the difference between the even-reference approximation error and the full-reference approximation error. That difference is never quantified, and the negative residual at sigma=1.4 shows it is not negligible at all noise levels. The proposed check, directly estimating the odd-sector mass from the known prior, would settle the attribution without changing the theorem. Therefore the verdict remains conditional.","tokens_in":12750,"tokens_out":5440,"duration_ms":79955,"concrete_test":"Estimate the theoretical parity floor P_sigma = sum_i E_{theta_sigma}[(E[sin theta_{0,i} | theta_sigma])^2] directly from the known Gibbs prior and wrapped-Gaussian kernel, using a high-accuracy nested Monte Carlo or independent MCMC with multiple chains and N >= 100000. Compare these estimates with the parity proxy column and the measured depth-1 quantum excess at each sigma in Table 1. If |P_sigma - P_hat_sigma| is small and the measured excess lies at or above P_sigma with small residual, the parity-dominated attribution holds; if P_sigma differs materially from the proxy, the empirical conclusion must be revised even though the theorem stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1 and Lemma 1 are internally sound; the load-bearing weakness is the empirical instrument in Section 4. The parity proxy P_hat_sigma = R_even-ref - R_ref is used to claim that measured excess is parity-dominated, with P_hat_sigma approximately ||Pi_odd m_sigma||^2. But substituting R = R* + ||m - .||^2 gives P_hat_sigma = ||Pi_odd m_sigma||^2 + (||Pi_even m_sigma - e_ref||^2 - ||m_sigma - r_ref||^2). The paper never bounds the two reference-approximation errors; the selection of degree-3 for protocol consistency does not guarantee they cancel. The negative residual at sigma=1.4 (Tables 1-2) is a visible symptom: there the finite even reference is a worse anchor than the full reference, so the proxy can lie above both the floor and the measured excess. Since the headline 'parity-dominated' claim depends on this proxy rather than on Theorem 1 itself, the empirical conclusion is not yet supported. The prior and noise kernel are fully known, so the odd-sector mass can be computed directly and the proxy bypassed.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces CoupledPhaseTexture, a synthetic torus-diffusion denoising benchmark with analytic heat-kernel noising, and studies fixed (zero-trainable-parameter) quantum feature maps of the depth-1 RY+CNOT+Pauli-Z family. The central theoretical result is Theorem 1: for any even-sector feature class Q, the excess risk of the Bayes-optimal denoiser over the Bayes risk splits exactly into the odd-sector L2 mass of the diffused Bayes denoiser plus a within-sector approximation error, so the odd-sector mass is an irreducible, noise-scale-resolved lower bound for every even feature class. The proof is self-contained, uses only parity symmetry of the prior and noise, and requires no containment, linearity, or closedness assumption on Q. The paper then proposes a reference-relative diagnostic instrument that separates measured excess into a 'parity proxy' and a signed empirical residual, and reports that the depth-1 quantum map is parity-dominated on two priors, while classical cosine-only banks exhibit the same floor and sine-carrying banks match the numerical reference. The empirical attribution is the paper's main load-bearing claim, but it rests on an unquantified proxy, as the reader's report and the stress-test note both emphasize.","tokens_in":13002,"tokens_out":2426,"duration_ms":34243,"significance":"If Theorem 1 stands—and the proof in Section 3 and Appendix A is clean and correct—the paper delivers a genuinely parameter-free, analytically derived lower bound that is stronger than typical static representability statements: it applies to every even feature class, with no structural assumptions on Q, and it is re-derived at each noise scale. The benchmark itself is a useful, reproducible instrument for separating parity, within-sector approximation, and sample-complexity effects in fixed-map quantum denoisers, and the inclusion of matched classical controls, capacity counts, and explicit seed/config specifications is a strength. However, the headline empirical claim that the measured excess is 'parity-dominated' depends on identifying the parity proxy with the theoretical floor, and the manuscript does not quantify the difference between these two quantities. Because the prior and noise kernel are fully known, the odd-sector mass of the Bayes denoiser can be computed directly, which would make the attribution rigorous rather than proxy-based.","major_comments":[{"comment":"The parity proxy P̂σ = R_even-ref − R_ref is used to claim that measured excess is parity-dominated, but substituting the risk decomposition gives P̂σ = ||Π_odd m_σ||² + (||Π_even m_σ − e_ref||² − ||m_σ − r_ref||²), where e_ref is the selected even reference and r_ref the full numerical reference. The manuscript never bounds the two reference-approximation errors or shows they cancel. Since the degree-3 reference is selected for protocol consistency and is explicitly not the pointwise minimizer at every σ, the second difference is uncontrolled and can be negative, which is visible at σ=1.4 in Tables 1 and 2. This is load-bearing: the headline 'parity-dominated' claim depends on P̂σ accurately estimating ||Π_odd m_σ||², and without a bound or direct computation that identification is not established.","section":"Section 4, Eq. (parity proxy definition)"},{"comment":"The signed empirical residual δ̂σ,N is negative at σ=1.4 (Table 2: −0.0059, 95% CI [−0.007,−0.005]). The paper attributes this to the finite even reference not containing the quantum even feature span, but this is precisely the situation in which P̂σ can lie above both the true floor and the measured excess, making the proxy an unreliable anchor. Since the prior p(θ) and the wrapped-Gaussian kernel are fully specified in Section 2 and Appendix B, the direct quantity ||Π_odd m_σ||² = Σ_i E_{θ_σ}[(E[sin θ_i^0 | θ_σ])²] can be estimated to high accuracy; the authors should compute it and report it alongside P̂σ. If the direct computation does not match the proxy profile, the paper's central empirical conclusion would need revision.","section":"Section 5, Tables 1 and 2"},{"comment":"The two-parameter exponential fit c·exp(−aσ²) with R²=0.99 and 0.997 is presented as 'consistent with a leading odd harmonic dominating.' This is a descriptive fit, as the paper itself notes, and it does not add evidence for parity dominance because the same exponential decay could arise from the reference-approximation error terms in the proxy. The fit should either be compared with the directly computed odd-sector mass or explicitly presented as a qualitative consistency check only, not as corroboration of the attribution.","section":"Section 5, 'The observed parity-proxy profile is consistent with heat-kernel attenuation'"}],"minor_comments":[{"comment":"The definition of the signed empirical residual δ̂σ,N mixes a population reference-relative residual with a finite-sample effect, and the paper's explanation that its sign 'combines within-parity mismatch and finite-sample estimation effects' is clear but would benefit from a one-sentence formal statement of what δ̂σ,N converges to as N→∞.","section":"Section 4"},{"comment":"The MCMC mixing diagnostics report acceptance ≈0.53 and stability under 200→400→800 sweeps, but the table lists 'MCMC sweeps 200' as a fixed value; adding the chain-length stability numbers to the main text or a footnote would improve reproducibility confidence.","section":"Section 2, Table 8"},{"comment":"The statement that 'degree-3 is retained across the grid for protocol consistency, not as a pointwise minimizer' is important and should be repeated near Table 1, since readers may otherwise interpret the reference as the best achievable at each σ.","section":"Appendix D, Table 9"},{"comment":"In the abstract and Section 1, the phrase 'the floor is re-derived at each noise scale' is slightly misleading: Theorem 1 is derived once for a fixed σ and then evaluated at each σ; consider rephrasing to 're-evaluated' to avoid the impression that the proof changes with σ.","section":"Minor typographical issue"}],"recommendation":"major_revision","confidential_remarks":"The theorem is solid and the benchmark is well designed, but the paper's headline empirical claim is not yet supported because the parity proxy is unquantified. The fix is straightforward with the resources already in the paper: compute ||Π_odd m_σ||² directly from the known prior and heat kernel and compare with Table 1. I would also note that the paper's positioning relative to prior dequantization work is appropriate, but the significance section slightly overstates the empirical 'parity-dominated' conclusion relative to what the proxy can establish. The manuscript fits the journal's scope well and, after the direct computation is added, could be a valuable contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The serious result here is Theorem 1. It is a clean, containment-free decomposition of excess risk into the odd-sector L2 mass of the Bayes denoiser plus a within-sector approximation term, and it holds for any even feature class. Lemma 1 is equally clean: a depth-1 RY+CNOT+Z map is even-sector confined, so the theorem applies. The proof is self-contained, the scope is honestly delimited, and the paper correctly notes this is the RY/diffusion specialization of Schuld et al. (2021), with the floor and its sigma-profile as the new content. I would cite this theorem and the benchmark design without hesitation.\n\nThe benchmark itself is also a genuine contribution. CoupledPhaseTexture uses analytic heat-kernel noising on the torus, separates parity from within-sector approximation and sample complexity, and pairs the quantum map with matched classical controls. The feature counts are explicit, the ridge readout is standardized, and the paper repeatedly flags what is descriptive (the exponential fit) versus what is derived (the floor). That is good practice.\n\nThe soft spot is the empirical attribution in Section 4. The paper claims the measured excess is parity-dominated based on the proxy P_hat = R_even-ref - R_ref, which is taken to estimate ||Pi_odd m_sigma||^2. But as the stress-test note observes, substituting R = R* + ||m - .||^2 shows that P_hat actually equals the odd-sector mass plus a difference of two within-sector approximation errors. Those errors are never bounded, and the negative residual at sigma=1.4 in Tables 1-2 is a visible symptom that the even reference is not a reliable anchor there. Since the prior and noise kernel are fully known, the odd-sector mass can be computed directly (numerically, at least) and the proxy bypassed. Until that is done, the headline \"parity-dominated\" claim is suggestive but not fully supported. This is an addressable weakness, not a fatal one.\n\nOther issues are minor by comparison: the code is promised but not released, some tables lack uncertainty estimates, and the image external-validity transfer is too lightweight to bear much weight. The second-prior check is a nice touch and the re-uploading analysis is plausible, though the non-monotone depth trend would benefit from more than the three-graph conditioning table.\n\nWho is this for? Anyone working on quantum feature maps in generative models or on dequantization of angle-encoded circuits. It provides a reusable testbed and a clean theorem that sharpens the null result. It deserves a serious referee. My recommendation: accept for peer review with major revision on the empirical attribution, and encourage the authors to compute the odd-sector mass directly rather than relying on the proxy.","headline":"Sound parity-floor theorem and a genuinely reusable benchmark; the empirical attribution needs a direct computation of the odd-sector mass before the headline claim is fully supported.","tokens_in":13515,"tokens_out":1556,"would_cite":true,"duration_ms":23277,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.67.Lx"],"model":"deepseek-v4-flash","headline":"A parity floor is the binding constraint on depth-1 quantum denoisers.","keywords":["parity floor","quantum feature maps","diffusion denoisers","angle encoding","dequantization","heat kernel","Bayes risk","torus diffusion"],"falsifier":"Compute $\\|\\Pi_{\\mathrm{odd}}m_\\sigma\\|^2$ directly from an accurate estimate of the conditional expectations $E[\\sin\\theta_{0,i}\\mid\\theta_\\sigma]$ on the CoupledPhaseTexture prior: if any depth-1 even-feature map had measured excess below that value, Theorem 1 would be contradicted. For the empirical attribution, refit the even reference at degree 4 or 5 with a larger sample and check whether the parity proxy $\\hat{P}_\\sigma$ moves; a material drop would mean the reported floor is an artifact of the chosen reference.","tokens_in":12541,"feed_emoji":"⚛️","tokens_out":8549,"duration_ms":91503,"temperature":0.7,"pith_summary":"This paper identifies a structural reason a simple quantum feature map can fail at denoising: the map can only produce even functions of the encoded angles, while the part of the Bayes-optimal denoiser that uses sine components is odd. On the paper's torus-diffusion benchmark, the excess risk above the Bayes risk splits exactly into an unreachable odd-sector term and a within-sector approximation term. That first term, the parity floor, is an irreducible lower bound for every even feature class and varies with noise scale as the heat kernel washes out odd structure. The paper argues from measurements that this floor, rather than quantumness or capacity, dominates the gap for the depth-1 RY+CNOT+Pauli-Z family.","feed_headline":"Parity floor is the binding constraint on depth-1 quantum denoisers","feed_subtitle":"A theorem ties excess risk of angle-encoded quantum maps to an unreachable sine-sector floor.","key_machinery":"The load-bearing object is the parity decomposition of the Bayes denoiser under the inversion $\\theta\\mapsto-\\theta$ on the torus, with the forward noising given by the heat kernel. Lemma 1 shows that for a depth-1 RY+CNOT+Pauli-Z map, conjugating a $Z$-string by the encoder gives another $Z$-string, so every readout is $\\prod_{i\\in S'} \\cos a_i$ and hence even; evenness survives linear readout. Theorem 1 turns this into the exact excess-risk identity above, using orthogonality of even and odd subspaces under the parity-symmetric marginal law $\\mu_\\sigma$; the parity floor is the squared $L^2(\\mu_\\sigma)$ norm of the odd projection of the Bayes denoiser.","core_discovery":"On a symmetric prior and symmetric noise, the Bayes denoiser $m_\\sigma$ splits by parity: its $\\cos\\theta_0$-components are even and its $\\sin\\theta_0$-components are odd. A depth-1 RY+CNOT+Pauli-Z angle encoder is confined to the even sector, because every $Z$-string expectation is a product of cosines. Theorem 1 then states that for any even feature class $Q$, $R_\\sigma(Q)-R^\\star_\\sigma=\\|\\Pi_{\\mathrm{odd}}m_\\sigma\\|^2 + \\inf_{q\\in Q}\\|\\Pi_{\\mathrm{even}}m_\\sigma-q\\|^2$, where the first term is an exact, noise-scale-resolved lower bound that requires no containment, linearity, or closedness assumption on $Q$. What is new is the floor and its $\\sigma$-profile: the obstruction lives in the diffused denoising target, not in the static expressivity of the encoder.","pith_inferences":["The same parity argument should transfer to any parity-invariant fixed encoder on a symmetric noise kernel: the floor is a property of the target, so encoder comparisons should report even-sector reach separately from any odd-sector access.","A quantitative screening rule follows: any fixed map that cannot emit an odd function of the encoded angles is provably unable to beat the floor, so candidate routes to advantage must either break parity symmetry or carry odd readouts aligned with the target's odd spectrum.","A direct testable extension is to run the same decomposition on a prior that breaks $\\theta\\mapsto-\\theta$ symmetry; Theorem 1 would not apply, and the instrument could then separate parity misalignment from odd-sector absence."],"forward_implications":["Every even-feature fixed map pays the same parity floor at each noise scale; choosing a different entangler or readout order only changes the within-sector term.","A classical cosine-only bank is floored to essentially the same level as the depth-1 quantum even map, while a classical bank that adds the sine sector matches the numerical reference: the measured deficit is parity, not quantumness.","Above roughly $N=300$ samples, the quantum even map and its classical even counterpart track each other and remain separated from the sine-carrying bank, so the gap is representational rather than a sample-complexity artifact.","Re-uploading the data breaks exact evenness but does not reliably close the floor; the excess rises non-monotonically with depth while conditioning improves, consistent with representation misalignment rather than capacity."],"supporting_citations":[{"why":"supplies the Fourier-representation viewpoint that the single-layer angle encoding is a truncated Fourier feature map, which the floor theorem specializes to diffusion denoising.","marker":"Schuld et al., 2021"},{"why":"prior fair-comparison study whose measured parity with classical controls motivated the mechanistic question this paper answers.","marker":"Kim & Yoo, 2026"},{"why":"fixes the conditional denoiser formulation of diffusion that the Bayes predictor $m_\\sigma$ instantiates.","marker":"Ho et al., 2020"},{"why":"supplies the score-denoiser connection and heat-kernel noising context used to read the floor's $\\sigma$-profile.","marker":"Song et al., 2021"},{"why":"defines the learning-from-quantum-data regime that the paper leaves open as a possible route around the floor.","marker":"Huang et al., 2022"},{"why":"taxonomy of classically tractable feature maps used to place the depth-1 encoder and to delimit the simulable regime.","marker":"Masot-Llima et al., 2025"}],"fun_headline_variants":["Even-feature quantum denoisers hit parity floor from missing sine sector","Depth-1 quantum denoiser excess split: odd sine target unreachable","Parity floor in quantum denoisers: exact lower bound from sine sector","Angle-encoded denoisers confined to even sector, sine floor exact"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The theorem itself assumes only that the feature class $Q$ lies in the even sector; the paper's headline attribution that the measured excess is parity-dominated additionally assumes the selected degree-3 cosine-only reference tracks the best even approximation of the Bayes target closely enough that the empirical residual is a faithful measure of the within-sector gap.","fun_headline_variants_meta":{"raw":{"variants":["Even-feature quantum denoisers hit parity floor from missing sine sector","Depth-1 quantum denoiser excess split: odd sine target unreachable","Parity floor in quantum denoisers: exact lower bound from sine sector","Angle-encoded denoisers confined to even sector, sine floor exact"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000626,"raw_usage":{"total_tokens":2951,"prompt_tokens":1054,"completion_tokens":1897,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":670,"completion_tokens_details":{"reasoning_tokens":1816}},"tokens_in":670,"tokens_out":1897,"duration_ms":15571,"temperature":1.0,"reasoning_tokens":1816,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T01:00:15.731895+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $\\|\\Pi_{\\mathrm{odd}}m_\\sigma\\|^2$ directly from an accurate estimate of the conditional expectations $E[\\sin\\theta_{0,i}\\mid\\theta_\\sigma]$ on the CoupledPhaseTexture prior: if any depth-1 even-feature map had measured excess below that value, Theorem 1 would be contradicted. For the empirical attribution, refit the even reference at degree 4 or 5 with a larger sample and check whether the parity proxy $\\hat{P}_\\sigma$ moves; a material drop would mean the reported floor is an artifact of the chosen reference.","supporting_citations":[],"review_version":1}