Version 1.1: the file dqpt-rh-complexity-note_v1.zip was replaced to redact a local file path (a Windows user name) from its contents. No other change. REDACTION_NOTE.md lists the file hashes before and after. Wei et al. (Nat. Commun., 2026; preprint arXiv:2511.11199v1) report a quantum algorithm for ζ(½+it) and compare its cost against direct Riemann–Siegel summation. This note examines that comparison. The speedup claim is correct against the baselines the paper names, and correct for a single isolated evaluation. It is, however, applied to locating and verifying zeros, which is a bulk task, and the classical state of the art for bulk evaluation is amortized. Odlyzko–Schönhage (1988) amortizes to T^ε per point, for every ε > 0, over blocks of length T^(1/2) after one precomputation, where the quantum method offers no amortization mechanism at all and pays its per-point cost every time; Hiary's amortized method (Math. Comp., 2011) reaches the same T^(1/4) exponent classically fifteen years earlier. Neither is cited in the preprint's 47-item reference list — a finding scoped to arXiv:2511.11199v1 and to that version only, for reasons given below. The Hiary paper the preprint does cite is the later Annals method, not the amortized one. Odlyzko–Schönhage is present in the source bibliography database but uncited, one of 16 uncited entries of 65, a set that also contains Riemann (1859); no intent is inferred. One argument is developed and one is withdrawn. The withdrawn one is a precision penalty. A measurement made for this note over 6200 Gram points spanning six decades of height finds |Z| height-flat (fitted slope −0.00005941, R² = 0.000333), so the sampling resolution δ is a large constant factor, not an asymptotic penalty. What survives is practical rather than asymptotic and rests on a single datum, the most extreme value in the sources obtained here; no distribution for that tail is stated in any of them, so the datum is not an upper bound on the precision required. The argument that carries the weight is categorical rather than complexity-theoretic: amplitude estimation returns confidence intervals, whereas rigorous verification of the Riemann Hypothesis to a given height requires certified enclosures. Platt–Trudgian's accounting of their 3×10¹² computation supports this, attributing a 725× cost of rigour to interval arithmetic and sampling rate and explicitly not to precision. Separately, the paper's correspondence theorem is shown to hold at every zero in the strip, not only on the critical line; this is a statement about the construction, which is therefore coextensive with the Riemann Hypothesis rather than evidence bearing on it. Nothing here bears on the hypothesis itself, and nothing here disputes the physics. Every claim is traced to a hashed artifact and classified by the strength of its warrant. The bibliography findings are scoped to arXiv:2511.11199v1: the published record is an Accelerated Article Preview carrying no reference list, and the published version is known to differ from v1, so a recheck is registered against the edited version when it appears. Three blockers are left open and stated rather than closed — the unavailable published reference list, the absent tail distribution, and what actually sets the operative resolution for sign-change-based verification, given that precision was demonstrably not the binding classical constraint. An appendix registers one pre-specified, falsifiable prediction about quantum block-amortized cost.
Version 1.1: the file dqpt-rh-complexity-note_v1.zip was replaced to redact a local file path (a Windows user name) from its contents. No other change. REDACTION_NOTE.md lists the file hashes before and after. Wei et al. (Nat. Commun., 2026; preprint arXiv:2511.11199v1) report a quantum algorithm for ζ(½+it) and compare its cost against direct Riemann–Siegel summation. This note examines that comparison. The speedup claim is correct against the baselines the paper names, and correct for a single isolated evaluation. It is, however, applied to locating and verifying zeros, which is a bulk task, and the classical state of the art for bulk evaluation is amortized. Odlyzko–Schönhage (1988) amortizes to T^ε per point, for every ε > 0, over blocks of length T^(1/2) after one precomputation, where the quantum method offers no amortization mechanism at all and pays its per-point cost every time; Hiary's amortized method (Math. Comp., 2011) reaches the same T^(1/4) exponent classically fifteen years earlier. Neither is cited in the preprint's 47-item reference list — a finding scoped to arXiv:2511.11199v1 and to that version only, for reasons given below. The Hiary paper the preprint does cite is the later Annals method, not the amortized one. Odlyzko–Schönhage is present in the source bibliography database but uncited, one of 16 uncited entries of 65, a set that also contains Riemann (1859); no intent is inferred. One argument is developed and one is withdrawn. The withdrawn one is a precision penalty. A measurement made for this note over 6200 Gram points spanning six decades of height finds |Z| height-flat (fitted slope −0.00005941, R² = 0.000333), so the sampling resolution δ is a large constant factor, not an asymptotic penalty. What survives is practical rather than asymptotic and rests on a single datum, the most extreme value in the sources obtained here; no distribution for that tail is stated in any of them, so the datum is not an upper bound on the precision required. The argument that carries the weight is categorical rather than complexity-theoretic: amplitude estimation returns confidence intervals, whereas rigorous verification of the Riemann Hypothesis to a given height requires certified enclosures. Platt–Trudgian's accounting of their 3×10¹² computation supports this, attributing a 725× cost of rigour to interval arithmetic and sampling rate and explicitly not to precision. Separately, the paper's correspondence theorem is shown to hold at every zero in the strip, not only on the critical line; this is a statement about the construction, which is therefore coextensive with the Riemann Hypothesis rather than evidence bearing on it. Nothing here bears on the hypothesis itself, and nothing here disputes the physics. Every claim is traced to a hashed artifact and classified by the strength of its warrant. The bibliography findings are scoped to arXiv:2511.11199v1: the published record is an Accelerated Article Preview carrying no reference list, and the published version is known to differ from v1, so a recheck is registered against the edited version when it appears. Three blockers are left open and stated rather than closed — the unavailable published reference list, the absent tail distribution, and what actually sets the operative resolution for sign-change-based verification, given that precision was demonstrably not the binding classical constraint. An appendix registers one pre-specified, falsifiable prediction about quantum block-amortized cost.
Wei et al. (Nat. Commun., 2026; preprint arXiv:2511.11199v1) report a quantum algorithm for ζ(½+it) and compare its cost against direct Riemann–Siegel summation. This note examines that comparison. The speedup claim is correct against the baselines the paper names, and correct for a single isolated evaluation. It is, however, applied to locating and verifying zeros, which is a bulk task, and the classical state of the art for bulk evaluation is amortized. Odlyzko–Schönhage (1988) amortizes to T^ε per point, for every ε > 0, over blocks of length T^(1/2) after one precomputation, where the quantum method offers no amortization mechanism at all and pays its per-point cost every time; Hiary's amortized method (Math. Comp., 2011) reaches the same T^(1/4) exponent classically fifteen years earlier. Neither is cited in the preprint's 47-item reference list — a finding scoped to arXiv:2511.11199v1 and to that version only, for reasons given below. The Hiary paper the preprint does cite is the later Annals method, not the amortized one. Odlyzko–Schönhage is present in the source bibliography database but uncited, one of 16 uncited entries of 65, a set that also contains Riemann (1859); no intent is inferred. One argument is developed and one is withdrawn. The withdrawn one is a precision penalty. A measurement made for this note over 6200 Gram points spanning six decades of height finds |Z| height-flat (fitted slope −0.00005941, R² = 0.000333), so the sampling resolution δ is a large constant factor, not an asymptotic penalty. What survives is practical rather than asymptotic and rests on a single datum, the most extreme value in the sources obtained here; no distribution for that tail is stated in any of them, so the datum is not an upper bound on the precision required. The argument that carries the weight is categorical rather than complexity-theoretic: amplitude estimation returns confidence intervals, whereas rigorous verification of the Riemann Hypothesis to a given height requires certified enclosures. Platt–Trudgian's accounting of their 3×10¹² computation supports this, attributing a 725× cost of rigour to interval arithmetic and sampling rate and explicitly not to precision. Separately, the paper's correspondence theorem is shown to hold at every zero in the strip, not only on the critical line; this is a statement about the construction, which is therefore coextensive with the Riemann Hypothesis rather than evidence bearing on it. Nothing here bears on the hypothesis itself, and nothing here disputes the physics. Every claim is traced to a hashed artifact and classified by the strength of its warrant. The bibliography findings are scoped to arXiv:2511.11199v1: the published record is an Accelerated Article Preview carrying no reference list, and the published version is known to differ from v1, so a recheck is registered against the edited version when it appears. Three blockers are left open and stated rather than closed — the unavailable published reference list, the absent tail distribution, and what actually sets the operative resolution for sign-change-based verification, given that precision was demonstrably not the binding classical constraint. An appendix registers one pre-specified, falsifiable prediction about quantum block-amortized cost.
We give quantum algorithms for estimating weighted exponential sums S(f,w,N)=∑k=0N−1wke2πif(k), with wk≥0, ∑kwk=1, and N=2n exponentially large. Under two explicit oracle assumptions -- efficiently computable prefix sums for the weights, enabling Grover--Rudolph state preparation, and a fixed-point circuit for f -- amplitude estimation yields S to additive error ε with O(ε−1log(1/γ)) oracle uses and polylog(N) gates per use; all bounds are full gate complexities, and the saving over classical sampling is quadratic in 1/ε. Applying this to the Riemann--Siegel formula, we prove that ζ(σ+it) in the critical strip can be estimated to accuracy δ with O(t(1−σ)/2δ−1) gates, hence O(t1/4δ−1) on the critical line, where O suppresses factors polylogarithmic in t/δ. At fixed accuracy this improves on the t1/2 Riemann--Siegel cost and on the best rigorous classical algorithm's t4/13+o(1); the quantum algorithm is advantageous precisely when δ≫t−3/52, which covers the accuracy needed to locate and count zeros. We show that a polylog(t) algorithm does not follow from these techniques -- undoing the normalization costs the ℓ1 mass ∑k≤Nk−σ=Θ(t(1−σ)/2) of the main sum -- and that Hiary-type block decompositions cannot improve the quantum query complexity. We also give an estimator for the magnitude of the amplitude sum ∣∑kak∣ of any efficiently preparable state, and review the required amplitude- and phase-estimation subroutines.
Version 4 (2026-08-25). Two additions from the correction campaign, carried as dated in-body correction records. The (2,5) entry, printed as a prediction in v3, is now census-exact -- and it was the wall law's first divergence test. At the constructed maximal-wall stratum point the stiff transverse count is exactly 10, so alpha_stratum(2,5) = 5; a fresh 60k ensemble's calibrated read is consistent with 5, with alpha = 10 (the superseded single-wall count's prediction) excluded at p = 0.000 and demonstrated power ~ 1; and a stratum-conditional paired ladder tracks the exact rank-10 Hessian form to <= 0.002 while a rank-20 comparator separates by >= 2.7 on identical draws. The law m_C(2,d) = 2(d-1) is census-exact at d = 3, 4, 5, now including the first point where it diverges (by 2x) from the superseded count. The (3,4) entry, printed open in v3, is now a PAIR of statements, and neither alone is alpha(3,4). The posted product-wall counting landed census-exact (alpha_stratum(3,4) = 5, so the asymptotic exponent satisfies alpha(3,4) <= 5), while the accessible window at n = 3e4 is steeper (calibrated 90 percent set [5.4, 6.9], with the previously read 4.0-4.1 excluded at power ~ 1): the codim-10 stratum's prefactor has not engaged at reachable depth, and window and asymptote have visibly decoupled. This is the scope lesson of the (d >= 3) x (d' > d) regime -- census plus conditional ladder measure the law, global windows measure the pre-asymptotic shoulder -- and it retro-explains the earlier fresh-seed ladders' instability as shoulder reads. Version 3 (2026-08-20). One SUPERSESSION: the qutrit-qutrit exponent closes. Versions 1 and 2 left alpha(3,3) open, v1 by a small-sample fit reading 3.25 and v2 by the bracket [2.3, 3.0] read off a descending window ladder. Both readings were artifacts of the estimator problem for which the pair-type table was already corrected in v2. The descent is attenuation tracking sample count rather than tail depth: an ensemble drawn from an EXACT alpha = 3.0 law descends by 0.62 across the ladder against the real data's 0.34, so a descending ladder is evidence FOR a constant exponent, not against one. Replacing the window fit with a calibrated Monte-Carlo test and inverting it over candidate exponents gives alpha(3,3) = 3.0 at p = 0.517 with a 90 percent confidence set [2.75, 3.10]; v1's 3.25 is excluded at p = 0.017. Power against alternatives is 0.990 / 0.590 / 0.120 / 0.575 / 0.990 / 1.000 at true alpha = 2.50 / 2.75 / 3.00 / 3.25 / 3.50 / 4.00. An independent re-audit at n = 10^4 reads flat near three on two seeds. Disclosed defect, carried in the paper: the test's own calibration row missed at the low edge (rejection rate 0.207 at a nominal 0.10 when the truth is alpha = 2.50, 4.4 standard errors high), so the lower end of the confidence set is softer than printed. This ensemble separates 3.0 from 3.25 with about 58 percent power and has none to separate 3.0 from 2.95. The (3,3) CARRIER family remains unattributed: the exponent is settled, the stratum that carries the accessible tail there is not. Version 2 (2026-08). Changes from v1: the pair-type exponent table is corrected (v1's alpha(2,4) = 2.08 was a 50-sample order-statistics fluctuation, placed at the 0.15th percentile of its own estimator's sampling distribution; three independent codebases now agree on the wall law alpha(2,d) = d, census-exact at d = 3 and 4); the transverse constant 1/12 is derived in closed form (diagonal cost 1/6, halved by the congruent partner's equal-cost correction); the (2,3) tail's carrier stratum is fully attributed (an isotropic-intertwiner family sharing the exponent 3); the qutrit-qutrit exponent is honestly bracketed open at [2.3, 3.0]; and the dispersion rule (tail-exponent fits print bootstrap dispersion or do not print) is adopted. The v2 ledger appendix carries the correction anatomy. Why is the multipartite certification constant heavy-tailed, where does the tail live, and what kills it? Localization: the credential pencil is exactly block-diagonal over party-subset supports; the tail is the worst pair; geometry statistics reconstruct from pair statistics (14/14 factor-band comparisons). The law: a frame-congruence theorem: whenever the two parties' operation frames subtend equal Bloch angles, the rotation intertwining them is an exact common kernel direction; the zero set is codimension one at generic angles, transverse growth is universally quadratic (mu = (Delta theta)^2/12), and the qubit-pair tail exponent is exactly 1/2 (the previously measured 0.39 is the finite-window slope of this law). The dichotomy: pairs containing a d >= 3 party measure exponents 2.08 to 3.69 because higher-dimensional operation pairs carry many local invariants that must all match: freedom helps the certifier. Engineering: heterogeneous dimension profiles tame the tail; qubit-pair presence splits all measured geometries into two clean classes; screening is a five-minute two-party calculation. Pre-registered; two entry gates that caught wrong conventions and all deaths ledgered.