Improved constant factors for qubitized Hamiltonian simulation

Danial Motlagh; Matthew Pocrnic

Improved constant factors for qubitized Hamiltonian simulation

T0 review · 0 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper proves that a degree-d Jacobi-Anger truncation with leading coefficient d = \alpha t + O((\alpha t)^{1/3} \log^{2/3}(1/\epsilon)) suffices for qubitized Hamiltonian simulation, bringing the query count's constant factor to…

desk verdict First rigorous leading-constant-1 bound for qubitized Hamiltonian simulation; the core proof is sound, with a few editorial fixes needed. read the letter →

arxiv 2608.02734 v1 pith:JFFSKKKQ submitted 2026-08-03 quant-ph

Matthew Pocrnic , Danial Motlagh This is my paper · ORCID

classification quant-ph

keywords HamiltoniansimulationqubitizationquantumsignalprocessingJacobi-AngerexpansionBesselfunctionsKapteyninequalityWatsonquerycomplexity

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

Quantum signal processing simulates $e^{-iHt}$ by polynomial approximation: the Jacobi-Anger expansion writes the evolution as a series of walk-operator terms with Bessel coefficients $J_k(\alpha t)$, and the practical question is where to truncate. This paper proves that a truncation of degree $d = \alpha t + \frac{(\alpha t)^{1/3}}{2}(3\ln(2/(\epsilon(1-e^{-\theta_0}))))^{2/3} + \tilde O((\alpha t)^{-1/3})$ is rigorously sufficient, so the leading constant in the number of walk-operator queries is $1$, not the $e/2$ or $2$ of earlier bounds. That reduces the estimated query overhead for any Hamiltonian simulation task built on qubitized quantum signal processing by a factor of about $e/2$, which matters because practical applications typically have $\alpha t$ between $10^4$ and $10^6$. The improvement comes from bounding the Bessel tail more carefully using Kapteyn's and Watson's inequalities.

What carries the argument

The load-bearing object is the truncation error $T_\epsilon(\tau,d) = \max_{x\in[-1,1]}|e^{i\tau x} - J_0(\tau) - 2\sum_{k=1}^{d-1} i^k J_k(\tau) T_k(x)|$. Lemma 3 bounds this by $2\sum_{k=d}^\infty |J_k(\tau)|$, invokes Kapteyn's inequality $|J_k(\tau)| \le ((\tau/k)e^{\sqrt{1-(\tau/k)^2}}/(1+\sqrt{1-(\tau/k)^2}))^k$, and transforms each term into $\exp(-\int_\tau^k \operatorname{arccosh}(s/\tau)\,ds)$; the tail is then dominated by a geometric series with ratio $e^{-\operatorname{arccosh}(d/\tau)}$, giving $T_\epsilon \le 2e^{-d\operatorname{arccosh}(d/\tau)+\sqrt{d^2-\tau^2}}/(1-e^{-\operatorname{arccosh}(d/\tau)})$. Setting $\theta=\operatorname{arccosh}(d/\tau)$ and using $\theta\cosh\theta - \sinh\theta \ge \theta^3/3$ reduces the condition $T_\epsilon\le\epsilon$ to a fixed-point inequality whose solution is Theorem 1; replacing Kapteyn's inequality by Watson's inequality inserts the prefactor $(2\pi\sqrt{d^2-\tau^2})^{-1/2}$ and yields the tighter Theorem 2. The integer ceiling and the $d+2$ query overhead from generalized quantum signal processing turn the degree bound into a query count.

What would settle it

Compute the true Jacobi-Anger tail $S(\tau,d)=2\sum_{k=d}^\infty |J_k(\tau)|$ over a dense grid of $\tau$ and $d$, such as $\tau\in[1,10^6]$ with $d$ just above $\tau$, and compare it against the Lemma 3 bound $2e^{-d\operatorname{arccosh}(d/\tau)+\sqrt{d^2-\tau^2}}/(1-e^{-\operatorname{arccosh}(d/\tau)})$; any single point where the bound falls below the true tail falsifies Lemma 3 and therefore both theorems. A more targeted check verifies whether $|J_k(\tau)| \le \exp(-\int_\tau^k \operatorname{arccosh}(s/\tau)\,ds)$ holds for all integer $k\ge\tau$ and all $\tau>0$.

Watch

Extended reading notes

Core claim

The central discovery is a rigorous, near-optimal bound on the Jacobi-Anger truncation degree needed to approximate $e^{-i\tau x}$ on $[-1,1]$. With $\theta_0 = (3/(\alpha t)\ln(2/\epsilon))^{1/3}$, Theorem 1 gives $d = \lceil \alpha t \cosh((\frac{3}{\alpha t}\ln\frac{2}{\epsilon(1-e^{-\theta_0})})^{1/3})\rceil$, which expands as $d = \alpha t + \frac{(\alpha t)^{1/3}}{2}(3\ln\frac{2}{\epsilon(1-e^{-\theta_0})})^{2/3} + \tilde O((\alpha t)^{-1/3})$. Theorem 2 refines this by using Watson's inequality instead of Kapteyn's, producing a bound that remains tight for smaller values of $\alpha t$. Since generalized quantum signal processing converts a degree-$d$ polynomial into exactly $d+2$ walk-operator queries, the final query count inherits the same leading constant $1$. The paper's numerical comparison shows the analytical bound tracking the exact Bessel tail closely in the practical range $\alpha t \approx 10^4$\textendash$10^6$.

Load-bearing premise

The load-bearing premise is the step in Section III, Eqs. (14)\textendash(16), where the Kapteyn bound on $|J_k(\tau)|$ is written as an equality with $\exp(-\int_\tau^k \operatorname{arccosh}(s/\tau)\,ds)$; that step is only valid as an inequality in the needed direction, and if the direction ever reversed, the geometric-series argument behind Eq. (23) and hence both main theorems would collapse.

Editorial extensions

If this is right

Every Hamiltonian simulation algorithm that calls qubitized quantum signal processing as a subroutine, including phase estimation, matrix inversion, Lindbladian time evolution, and first-quantized chemistry simulation, inherits a rigorous query-count reduction of about $e/2$ compared with the previous best bound.
For realistic values $\alpha t \approx 10^4$\textendash$10^6$, the new bound gives $d/\tau$ numerically so close to $1$ that further constant-factor work on the Jacobi-Anger truncation is unlikely to produce practically meaningful improvements.
The degree bound combines directly with generalized quantum signal processing, so the total number of block-encoding queries is $d+2$, with the same leading constant $1$.
Theorem 2's Watson-based bound serves as a rigorous replacement for the non-rigorous Airy-function asymptotic formula, remaining accurate across a wider range of $\alpha t$ including short time steps.

Reading between the lines

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

The same Bessel-tail majorization could be exported to other settings where Jacobi-Anger truncation errors are the bottleneck, such as fast multipole scattering calculations, yielding rigorous constant-factor improvements beyond Hamiltonian simulation; the paper does not discuss these applications.
The fixed-point recursion the authors mention in passing, substituting $\theta_1, \theta_2, \dots$, could be developed into an explicit iterative scheme with a proof of geometric convergence, which would tighten the remaining logarithmic gap for moderate $\alpha t$.
A direct pointwise check of the Kapteyn-to-exponential majorization across all $(\tau,k)$ would turn the equality in Eq. (16) into a certified inequality; if the true Bessel tail ever exceeded the Lemma 3 bound, the main theorems would need an amended prefactor, but the numerical evidence in Fig. 1 suggests the inequality direction is safe.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Here's my read of arXiv:2608.02734. The headline: the paper proves a rigorous Jacobi-Anger truncation degree with leading constant 1 for qubitized Hamiltonian simulation, improving the previous rigorous bound of e/2 from Jennings et al. The proof is real and the main theorems hold up. The authors acknowledge the Airy asymptotic in Babbush et al. already gave the leading 1 non-rigorously, so the novelty is the rigorous explicit bound, not the asymptotic discovery itself. That's fair.

The work is self-contained: Kapteyn and Watson inequalities, a geometric-series majorant, and a fixed-point argument pin down the degree. Lemmas 4 and 5 are sound; f and g are strictly decreasing and the fixed-point lower-bound trick works. I also checked the flagged issue in Lemma 3: the printed equation (14) looks like a false equality, but substituting r = tau/k gives exactly the exponential of -integral arccosh(s/tau) ds, so it's a typesetting mistake, not a gap in the argument. The subsequent bounds and the final theorem don't depend on the misprinted equality.

Credit where due: the numerical comparison to the exact Bessel tail is useful and shows the bound is tight in the practical tau range. The use of previous GQSP conversion is appropriately black-box. No code or data shipped, but the methods are described precisely enough to re-implement.

Soft spots, mostly editorial. The abstract says 'reducing the overhead estimates for all Hamiltonian simulation tasks' and 'future improvements will not be of practical significance.' Those claims outrun the body text, which focuses on qubitized/QSP simulation with a block-encoded Hamiltonian. The phrase 'loose by a logarithmic factor' in Section II is confusing. The figure caption cites Ref. [9] where it means Ref. [18]. None of these affect the math.

If I were refereeing, I'd recommend minor revision: fix Eq. (14), temper the abstract, correct the caption. The result is a legitimate improvement in the constant factor and worth publishing. A serious editor should send it to review. I'd cite it in resource estimates for qubitized simulation.

Referee Report

0 major / 5 minor

Summary. The paper proves new rigorous bounds on the Jacobi-Anger truncation degree needed for quantum signal processing (QSP) / qubitized Hamiltonian simulation. For τ = αt and error ε, Theorem 1 gives d = ⌈τ cosh((3/τ ln(2/(ε(1−e^{−θ0}))))^{1/3})⌉ with θ0 = (3/τ ln(2/ε))^{1/3}, which expands as d = τ + (τ^{1/3}/2)(3 ln(2/(ε(1−e^{−θ0}))))^{2/3} + Õ(τ^{−1/3}); Theorem 2 sharpens this using Watson's inequality. The proofs use Kapteyn's and Watson's Bessel-function inequalities, a geometric-series majorant of the tail, and a fixed-point argument to linearize the transcendental bound. The leading query-count constant is 1, improving on prior bounds that gave about e/2 times the leading constant, and the paper reports numerical comparisons against the exact Bessel tail computed with SciPy.

Significance. If the theorems are correct, this is a useful constant-factor improvement for resource estimates in QSP/qubitized Hamiltonian simulation. The central derivation is self-contained: it uses standard Bessel inequalities and an explicit numerical benchmark, and the asymptotic form (d ≈ τ + O(τ^{1/3} log^{2/3}(1/ε))) is the expected near-optimal behavior. The paper is incremental (constant factors rather than asymptotic class), but constant-factor reductions are practically meaningful for fault-tolerant resource estimation. The apparent concern about Eq. (14) does not land: substituting r = τ/k = sech u shows that the Kapteyn factor equals exp(−k arccosh(k/τ) + √(k²−τ²)) exactly, so the displayed equality is correct. The main remaining issues are editorial clarity and scope overstatement in the abstract.

minor comments (5)

[Section III, proof of Lemma 4] The fixed-point paragraph is logically correct but too compressed: from θ0 = inf f(θ) it follows that f(θ0) ≥ θ* and therefore every θ ≥ f(θ0) satisfies the sufficient condition θ ≥ f(θ), because f is decreasing and f(θ0) ≥ θ0. The current wording ('using the property θ ≥ f(θ0)') asks the reader to reconstruct this argument; please state it explicitly. The analogous step for θ_lo in the proof of Lemma 5 would also benefit from an explicit monotonicity sentence.
[Figure 1 caption] The legend cites 'Ref [9] Cor. 62' and 'Ref [2] Lemma. 5', but the corresponding bibliography entries are Ref. [18] (Gilyén et al.) and Ref. [7] (Jennings et al.); please correct the citation numbers.
[Abstract and Section I] The phrase 'reducing the overhead estimates for all Hamiltonian simulation tasks on quantum computers' overstates the scope: the result applies to QSP/GQSP-based (qubitized) Hamiltonian simulation, not to every Hamiltonian simulation method (e.g., Trotter or product-formula approaches). Please reword to 'qubitized Hamiltonian simulation' or 'QSP-based Hamiltonian simulation'.
[Section III, first paragraph] There is a typo: 'Kapetyn' should be 'Kapteyn'. Also, in Section II, 'more closely tracts the error' should be 'more closely tracks the error'.
[Section III, Eq. (14)] For the record, the equality in Eq. (14) is correct; no correction is needed. The stress-test concern about a possible inequality direction does not arise because the hyperbolic substitution is exact.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the degree-bound derivation is self-contained; the only author-overlapping citation is an external GQSP black-box conversion.

full rationale

The paper's central claim is the improved Jacobi-Anger truncation degree bound. The derivation chain is self-contained: Lemma 3 starts from the standard Kapteyn inequality (external Ref. [19]), transforms the bound into exp(-∫ arccosh(s/τ) ds), and sums a geometric majorant series; Lemma 4 solves the resulting transcendental inequality via a fixed-point argument; Lemma 5 tightens the bound using Watson's inequality (external Ref. [20]). No fitted parameter is renamed as a prediction, and no ansatz is smuggled in via a self-citation. The only place where the authors cite their own prior work is the GQSP query-count conversion, 'applying the art of [17]', used to translate the polynomial degree d into d+2 walk-operator queries. That conversion is a published, external theorem used as a black box; it is not the source of the constant-factor improvement, and the central degree-bound proof does not depend on it. The SciPy comparison in Fig. 1 is an external numerical benchmark, not an input to the proof. The possible typographical issue in Eq. (14) concerns the printed identity, not the direction of the inequality, and does not constitute circularity.

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

The central claim rests on standard Bessel inequalities and known quantum-signal-processing theorems. No parameters are fitted to data and no new physical or mathematical entities are introduced.

assumptions (5)

standard math Kapteyn's inequality (Eq. 8) is a valid upper bound on |J_k(τ)| for 0 ≤ τ ≤ k.
Imported from Ref. [19]; used in Lemma 3 to bound individual Bessel terms.
standard math Watson's inequality (Eq. 37) is a valid upper bound on J_ν(νz) for all ν>0 and 0<z≤1.
Imported from Ref. [20]; used in Lemma 5 to improve the tail bound.
standard math The Chebyshev polynomials T_k(x) satisfy |T_k(x)| ≤ 1 on [-1,1].
Used in Eq. (11)-(12) to bound the truncation error by the sum of absolute Bessel coefficients.
standard math The Jacobi-Anger expansion converges uniformly and represents e^{iτx} on [-1,1].
Used in Eq. (2)-(4) as the starting point.
domain assumption The GQSP theorem from Ref. [17] converts a degree-d polynomial approximation into d+2 queries to the block encoding.
External quantum-algorithmic result, not re-derived; determines the query count in Theorems 1 and 2.

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Cite this review

Pith. "Pith review of Improved constant factors for qubitized Hamiltonian simulation." pith.science (2026). https://pith.science/paper/JFFSKKKQ

@misc{pith2026260802734,
  author       = {Pith},
  title        = {Pith review of: Improved constant factors for qubitized Hamiltonian simulation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JFFSKKKQ}},
  note         = {Machine review of arXiv:2608.02734}
}

abstract

Quantum signal processing (QSP) serves as the asymptotically optimal technique for Hamiltonian simulation on a quantum computer. By approximating the time evolution operator via the Jacobi-Anger expansion, the Hamiltonian simulation problem reduces to a problem in polynomial approximation theory: find a sufficient degree-$d$ polynomial series to approximate $e^{-i\tau x}$ on $[-1,1]$ within error $\epsilon$. While $d\in\tilde{\mathcal{O}}(\tau)$ is known to be asymptotically optimal, there exists a gap between state-of-the-art bounds and the optimal constant multiplicative factor, which is approximately equal to 1. Here, we close this gap almost entirely, to the point where possible future improvements will not be of practical significance. Our improvement resides in a careful treatment of the Bessel tail in the Jacobi-Anger series using Kapteyn's and Watson's inequalities, thereby reducing the overhead estimates for all Hamiltonian simulation tasks on quantum computers by a factor of $\approx e/2$.

Figures

Figures reproduced from arXiv: 2608.02734 by the authors.

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Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [9]

Jennings, M

D. Jennings, M. Lostaglio, S. Pallister, A. T. Sornborger, and Y. Suba¸ sı, Efficient quantum linear solver algorithm with detailed running costs, arXiv:2305.11352v1 (2023)

arXiv 2023

[2] [18]

Gily´ en, Y

A. Gily´ en, Y. Su, G. H. Low, and N. Wiebe, Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics, Proceedings of the 51st annual ACM SIGACT symposium on theory of computing , 193 (2019)

work page 2019

[3] [1]

G. H. Low and I. L. Chuang, Optimal Hamiltonian simulation by quantum signal processing, Physical Review Letters118, 010501 (2017)

work page 2017

[4] [2]

Mukhopadhyay, T

P. Mukhopadhyay, T. F. Stetina, and N. Wiebe, Quantum simulation of the first-quantized Pauli-Fierz Hamiltonian, PRX Quantum5, 010345 (2024)

work page 2024

[5] [3]

R. A. Lang, P. Jain, J. M. Arrazola, and D. Motlagh, Quantum algorithm for simulating non-adiabatic dynamics at metallic surfaces, arXiv:2601.16264 (2026)

arXiv 2026

[6] [4]

Motlagh, R

D. Motlagh, R. A. Lang, P. Jain, J. A. Campos-Gonzalez-Angulo, W. Maxwell, T. Zeng, A. Aspuru-Guzik, and J. M. Arrazola, Quantum algorithm for vibronic dynamics: case study on singlet fission solar cell design, Quantum Science and Technology10, 045048 (2025)

work page 2025

[7] [5]

Stetina and N

T. Stetina and N. Wiebe, First-quantized quantum simulation of non-relativistic QED with emergent topologically pro- tected Coulomb interactions, arXiv:2508.19343 (2025)

arXiv 2025

[8] [6]

M. E. Morales, L. Pira, P. Schleich, K. Koor, P. C. Costa, D. An, A. Aspuru-Guzik, L. Lin, P. Rebentrost, and D. W. Berry, Quantum linear system solvers: A survey of algorithms and applications, Reviews of Modern Physics98, 025005 (2026)

work page 2026