Solving wave propagation problems via geometric quantum state preparation on dispersion manifolds

Amit Ben-kish; Jonathan Nemirovsky; Lee Peleg; Netanel Barel; Yakov Solomons; Yotam Shapira

REVIEW

Solving wave propagation problems via geometric quantum state preparation on dispersion manifolds

Yakov Solomons, Lee Peleg, Netanel Barel, Jonathan Nemirovsky, Amit Ben-Kish, Yotam Shapira

Quantum Physics · Submitted 9 Sep 2026 · 38 references

Pith's verdict: A genuinely new geometric state-preparation trick for the singular Helmholtz equation; the exponential speedup is conditional on a constant ring-width scaling that is numerically plausible but not rigorously proven.

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CitationAmit Ben-kish, Jonathan Nemirovsky, Lee Peleg, Netanel Barel, Yakov Solomons, Yotam Shapira. "Solving wave propagation problems via geometric quantum state preparation on dispersion manifolds." arXiv:2609.10282 (2026). https://pith.science/paper/W6G5SVPU

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Editor's note

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The core idea is the real contribution: instead of preparing the full Fourier state and post-selecting onto the resonant manifold — which costs an exponential number of repetitions — the algorithm directly prepares a uniform superposition over a narrow ring around k = ω/c, using a quantum analogue of the Bresenham circle algorithm, then imprints source phases and Fourier-transforms back. Casting the singular Helmholtz problem as geometric state preparation is new as far as I can tell, and the circuit construction is explicit enough to take seriously.

What the paper does well: the success probability is computed step by step in Appendix A; the gate count O(n^2) is plausible given the cited arithmetic circuits; and the 64×64 classical emulation shows overlap error 0.01 against the exact solution for two emitters, with the correct interference pattern. The authors also honestly compare against the screened-Poisson approach of Ref. [16] and explain why it fails for the singular case. No circularity or fitted parameters. Citation pattern is clean.

The main soft spot is exactly the one flagged in the stress test. The domain-size independence of the success probability, Eq. (30), follows only if the ring width n_r and regularization width n_eps remain constant as L grows. Fig. 5 shows n_r flattening for target errors 0.05 and 0.01 over a few orders of magnitude, but the largest domain is only a few hundred wavelengths and there is no analytic error bound for truncating the principal-value tail, which decays as 1/|k−ω/c|. So the speedup is real if the constant-width scaling holds; the evidence is numerical, not proof. I would not call this a fatal flaw, but the authors should either prove the scaling or explicitly state it as a conjecture backed by numerics. A referee should push on this.

Minor issues: the overall success probability for their example is about 2×10^-3, times 1/N_s for sources — constant, but not negligible; amplitude-estimation overhead for extracting physical observables is not discussed; and the numerics are emulations with no code or data released, so reproducibility is limited.

Who this is for: anyone working on quantum PDE solvers, particularly frequency-domain wave problems. It deserves a serious referee; I would send it to review and ask for a sharper treatment of the ring-width scaling.

Referee report

Major objections (3)

[Section III.B, Fig. 5, and Appendix A] The algorithm's exponential speedup rests on the claim that the resonant ring width n_r (in Fourier grid spacings) can be held constant as the computational domain size L increases. Figure 5 shows this behavior for L up to a few hundred wavelengths, but no rigorous analytical bound is provided for the truncation error introduced by cutting off the inverse operator's tail. The expression for the truncation error in the continuum limit (Eq. A20 and surrounding text) is derived under the assumption that the ring is narrow, but it does not quantify the error that is actually committed by discarding principal-value contributions from outside the ring. For classically intractable domain sizes, it is conceivable that the required n_r would grow, reducing or eliminating the exponential advantage. An analytical error estimate, or at least a scaling argument with explicit constants, is needed to substantiate the central claim.
[Eq. (27) and Eq. (A20)] The success probability expression (Eq. 27) is derived in the continuous limit using the approximation k ≈ ω/c + x. While this is valid near resonance, the finite-resolution effects of the discretized grid are not accounted for. The discrete sum in Eq. (A19) is replaced by an integral with a radial Jacobian k dk, but the cancellation of the Jacobian relies on the symmetry of the interval and the evenness of |q(k)|^2. For finite n_r (especially small n_r), the discrete sum may not exactly match the integral approximation, and the scaling of the success probability with domain size may have a residual dependence that is not captured. The authors should either provide a discrete version of the success probability calculation or bound the error of the continuous approximation.
[Section III.C, complexity analysis] The gate-count analysis states a total of O(n^2) two-qubit gates, citing the square-root and reciprocal circuits from Ref. [12]. However, the complexity of the reciprocal circuit is given as O(n^2) in that reference, but the constant factors and the number of ancillary qubits are non-negligible for practical implementation. Additionally, the analysis assumes that the ring width n_r and the arithmetic precision n_q are constants independent of n. While this is plausible for a fixed target accuracy, the argument should explicitly state how n_q scales with n to maintain a fixed relative error in the solution. If n_q grows with n (e.g., to resolve the phase factor e^{i(kx)} accurately), the O(n^2) scaling may be affected.

Minor objections (6)

[Section I, paragraph 3] The phrase 'exponential quantum advantage' is used loosely. The authors should clarify that they are comparing with classical algorithms that scale polynomially in the number of grid points, not with all possible classical methods. Also, the claim of 'exponential compression of the system degrees of freedom' is standard for quantum encoding but should be stated precisely.
[Section II.C, Fig. 1] The circuit diagram for the screened Poisson equation shows a single controlled rotation step, but the text mentions that the reciprocal value q(k) is computed via quantum arithmetic and then encoded. The figure should indicate the presence of the arithmetic work registers and the uncomputation step for clarity.
[Section III.A, Eq. (23)] The factor q(k) in Eq. (23) is defined in Eq. (20) as a complex quantity. The normalization factor N_r in (23) should be defined as sum_{k in ring} |q(k)|^2, but this is not explicitly stated until after Eq. (25). Please move or add a definition earlier.
[Section III.B, Fig. 4] The overlap error 1 - |<u_C|u_Q>| is plotted. It would be helpful to also show the L2 norm of the difference between the full wave fields (not just the normalized states), as the scaling of the error with ring width for the unnormalized solution is relevant for applications.
[Appendix A, Eq. (A21)] The final success probability expression contains the factor ρh/ (32 n_r). The quantity ρ = ω/c has dimensions of inverse length, and h is the grid spacing. The product ρh is dimensionless and of order 1 (since h ~ λ/4). The authors should comment on the typical magnitude of this factor and how it affects the overall repetition count.
[General] There are several minor typos: 'Helmholtz' is misspelled as 'Helmholtz' in the abstract (though correct in the main text); Eq. (28) should read N_s (subscript s); the word 'approximating' is misspelled in the text after Eq. (11). These should be corrected.

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The paper

The paper claims

Direct preparation of resonant Fourier states eliminates exponential overhead in quantum wave-equation solvers.

The reading

This paper presents a quantum algorithm for solving the Helmholtz equation in the frequency domain. The central challenge is that the Fourier-space inverse operator is singular on the resonance surface, making standard Fourier-space methods require an exponentially large number of post-selection trials that grows with the computational domain size. The authors overcome this by preparing a quantum state directly on the resonant manifold—a narrow ring around the dispersion surface—and then encoding source positions through Fourier phases. The success probability of the resulting algorithm depends only on the number of sources, not on the domain size, so the exponential compression of spatial degrees of freedom is preserved. More broadly, the work recasts the solution of hyperbolic PDEs with singular Fourier support as a problem of geometric quantum state preparation.

What carries the argument

Geometric quantum state preparation on the dispersion manifold: a quantum circuit that first produces a uniform superposition over a narrow ring of Fourier modes centered on the resonant circle, using a quantum analogue of the Midpoint circle algorithm together with quantum square-root and comparator circuits, then imprints the phase factors $e^{-i(k_x x_j + k_y y_j)}$ for each source via controlled phase gates, and finally applies an inverse quantum Fourier transform to obtain the spatial wave field.

Extended reading notes

Core claim

The paper claims that by directly preparing a quantum state supported on the resonant manifold (the set of Fourier modes satisfying $k^2\approx\omega^2/c^2$) and encoding the source-dependent phases, one can solve the discretized Helmholtz equation with $O(n^2)$ gates on $2n$ spatial qubits and a post-selection success probability that is independent of the computational domain size. This eliminates the exponential overhead that would arise from projecting a full Fourier-space state onto the resonant surface. The method replaces the condition-number-dependent complexity of generic quantum linear-system solvers with the cost of preparing a geometrically defined ring state in Fourier space.

Load-bearing premise

The algorithm assumes that the required width of the resonant ring (which sets the approximation accuracy) remains constant as the computational domain size grows, but this is only supported by numerical evidence up to a few hundred wavelengths and lacks a rigorous analytical proof.

Editorial extensions

If this is right

Quantum simulation of large-scale wave propagation (e.g., 3D fields spanning $10^{18}$–$10^{21}$ grid points) becomes feasible with roughly 100 logical qubits and a constant number of repetitions, provided the number of sources is small.
The same approach extends to any hyperbolic PDE whose Fourier-space solution is supported on a lower-dimensional dispersion manifold, such as anisotropic or elastic wave equations, turning the computational task into a geometric state-preparation problem.
Reflecting boundary conditions (Dirichlet or Neumann) can be incorporated via odd or even extensions with only one additional qubit per spatial dimension, so the method applies to realistic bounded domains.
The gate complexity $O(n^2)$ and the $T$-gate count $O(n^2)$ are polylogarithmic in the number of grid points ($2^{2n}$), offering an exponential quantum speedup over classical solvers that scale polynomially with the grid size.

Reading between the lines

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

The constant ring-width assumption is only numerically verified up to domain sizes of several hundred wavelengths; a rigorous analytical bound on the required width as a function of domain size and accuracy would solidify the asymptotic claim.
The algorithm as stated handles point sources with simple Fourier phases; extending it to arbitrary source distributions (e.g., volumetric or smooth sources) would require additional efficient Fourier representations, which may not always be available.
For problems with many sources ($N_s$ large), the success probability drops as $1/N_s$, but the authors anticipate that $N_s$ is small in practice; an adaptive or amplitude-amplification strategy could mitigate this dependence if needed.
The approach could be combined with Hamiltonian simulation methods for time-domain problems by treating the frequency-domain solution as a building block, potentially offering speedups in transient-wave scenarios.

Original abstract

We present a quantum algorithm for solving partial differential equations through a linear-system formulation, focusing on wave propagation problems described by the discretized Helmholtz equation in frequency domain. Although quantum linear-system solvers offer exponential compression of the system degrees of freedom, their runtime complexity is generally governed by the condition number of the discretized operator. Exploiting the analytic structure of the differential operator can provide an alternative to explicit matrix inversion, as illustrated for the screened Poisson equation through an explicit quantum circuit. For the Helmholtz equation, however, the inverse operator becomes singular on the dispersion surface $k^2=\omega^2/c^2$, rendering direct Fourier-space state-preparation methods exponentially inefficient. We address this challenge by directly preparing quantum states supported on the resonant manifold and encoding source locations through Fourier phases. The resulting algorithm eliminates the exponentially large overhead associated with post-selection on the resonant manifold, yielding a success probability that depends linearly on the number of sources and is independent of the computational domain size. More generally, our approach applies to hyperbolic differential equations whose Fourier-space solutions possess singular support on dispersion manifolds, recasting their solution as a problem of geometric quantum state preparation.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,687 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.49 · human participants 0.01 · controlled comparison 0.08 · randomized assignment 0.01 · states quantitative result 0.62 · compares with prior results 0.82 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We present a quantum algorithm for solving partial differential equations through a linear-system formulation, focusing on wave propagation problems described by the discretized Helmholtz equation in frequency domain. Although quantum linear-system solvers offer exponential compression of the system degrees of freedom, their runtime complexity is generally governed by the condition number of the discretized operator. Exploiting the analytic structure of the differential operator can provide an alternative to explicit matrix inversion, as illustrated for the screened Poisson equation through an explicit quantum circuit. For the Helmholtz equation, however, the inverse operator becomes singular on the dispersion surface $k^2=\omega^2/c^2$, rendering direct Fourier-space state-preparation methods exponentially inefficient. We address this challenge by directly preparing quantum states supported on the resonant manifold and encoding source locations through Fourier phases. The resulting algorithm eliminates the exponentially large overhead associated with post-selection on the resonant manifold, yielding a success probability that depends linearly on the number of sources and is independent of the computational domain size. More generally, our approach applies to hyperbolic differential equations whose Fourier-space solutions possess singular support on dispersion manifolds, recasting their solution as a problem of geometric quantum state preparation.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present a quantum algorithm for solving partial differential equations through a linear-system formulation, focusing on wave propagation problems described by the discretized Helmholtz equation in frequency domain.	contribution 0.99	0.92	0.09	asserted 1.00
s1	Although quantum linear-system solvers offer exponential compression of the system degrees of freedom, their runtime complexity is generally governed by the condition number of the discretized operator.	background 0.96	0.08	0.03	asserted 0.91
s2	Exploiting the analytic structure of the differential operator can provide an alternative to explicit matrix inversion, as illustrated for the screened Poisson equation through an explicit quantum circuit.	background 0.92	0.56	0.05	asserted 0.68
s3	For the Helmholtz equation, however, the inverse operator becomes singular on the dispersion surface $k^2=\omega^2/c^2$, rendering direct Fourier-space state-preparation methods exponentially inefficient.	background 0.51	0.31	0.05	asserted 1.00
s4	We address this challenge by directly preparing quantum states supported on the resonant manifold and encoding source locations through Fourier phases.	method 0.79	0.92	0.19	asserted 1.00
s5	The resulting algorithm eliminates the exponentially large overhead associated with post-selection on the resonant manifold, yielding a success probability that depends linearly on the number of sources and is independent of the computational domain size.	result 0.96	0.92	0.54	asserted 1.00
s6	More generally, our approach applies to hyperbolic differential equations whose Fourier-space solutions possess singular support on dispersion manifolds, recasting their solution as a problem of geometric quantum state preparation.	interpretation 0.50	0.82	0.39	asserted 0.98

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from proof or derivation 0.94.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

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S sections
T body sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Quantum implementation of the inverse operator for the screened Poisson equation in Fourier space. The Quantum

**Figure 2.** FIG. 2. Illustration of the (a) real and (b) imaginary parts

**Figure 3.** FIG. 3. Quantum circuit for the geometric quantum state preparation algorithm. The momentum registers are first initialized

**Figure 4.** FIG. 4. Accuracy of the resonant-ring approximation. (a)

**Figure 5.** FIG. 5. Minimum resonant-ring width required to achieve

**Figure 6.** FIG. 6. Demonstration of the geometric quantum state

**Figure 7.** FIG. 7. Quantum circuit implementing the quantum analogue of the Midpoint (Bresenham) circle algorithm. The dashed block

**Figure 8.** FIG. 8. Construction of the finite-width resonant ring.

Figures reproduced from arXiv:2609.10282 by the authors.

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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [16]

A. Gu, S. Jin, and C. Ma, Quantum simulation of helmholtz equations via schr¨ odingerization, arXiv preprint arXiv:2507.23547 (2025)

View on Pith Pith review arXiv 2025

[2] [12]

Y. Cao, A. Papageorgiou, I. Petras, J. Traub, and S. Kais, Quantum algorithm and circuit design solving the pois- son equation, New Journal of Physics15, 013021 (2013)

View on Pith 2013

[3] [1]

P. C. Costa, S. Jordan, and A. Ostrander, Quantum algo- rithm for simulating the wave equation, Physical Review A99, 012323 (2019)

View on Pith 2019

[4] [2]

The prepared arc is then coherently extended up to the outer radius r+ =ω/c+r/2,producing a uniform superposition over the corresponding sector of the resonant ring. The re- maining sectors are subsequently generated by exploiting the symmetry of the circle, yielding a uniform superposi- tion over all momentum components contained within the resonant ring...

View on Pith

[5] [3]

A. Suau, G. Staffelbach, and H. Calandra, Practical quantum computing: Solving the wave equation using a quantum approach, ACM Transactions on Quantum Computing2, 1 (2021)

View on Pith 2021

[6] [4]

Y. Sato, R. Kondo, I. Hamamura, T. Onodera, and N. Yamamoto, Hamiltonian simulation for hyperbolic partial differential equations by scalable quantum cir- cuits, Physical Review Research6, 033246 (2024)

View on Pith 2024

[7] [5]

H. Tezuka and Y. Sato, Quantum algorithm for electro- magnetic field analysis, International Journal for Numer- ical Methods in Engineering127, e70344 (2026)

View on Pith 2026

[8] [6]

A. Shringi, H.-C. Wu, A. Shokry, X. Li, and M. T. Kan- demir, Structure-preserving quantum simulation of wave equations on a trapped-ion processor, arXiv preprint arXiv:2607.28499 (2026)

View on Pith Pith review arXiv 2026

[9] [7]

R. Babbush, D. W. Berry, R. Kothari, R. D. Somma, and N. Wiebe, Exponential quantum speedup in simulat- ing coupled classical oscillators, Physical Review X13, 041041 (2023)

View on Pith 2023

[10] [8]

N. Luangsirapornchai, P. Sanglaor, A. Sornsaeng, S. Bressan, T. Chotibut, K. Suksen, and P. Chongstit- vatana, Practical quantum circuit implementation for simulating coupled classical oscillators, IEEE Access (2025)

View on Pith 2025

[11] [9]

A. W. Harrow, A. Hassidim, and S. Lloyd, Quantum al- gorithm for linear systems of equations, Physical review letters103, 150502 (2009)

View on Pith 2009

[12] [10]

A. M. Childs, J.-P. Liu, and A. Ostrander, High-precision quantum algorithms for partial differential equations, Quantum5, 574 (2021)

View on Pith 2021

[13] [11]

D. W. Berry, High-order quantum algorithm for solving linear differential equations, Journal of Physics A: Math- ematical and Theoretical47, 105301 (2014)

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