Quantum Wave Simulation with Sources and Loss Functions

Andreas Fichtner; Cyrill B\"osch; Giacomo Aloisi; Malte Schade; Scott D. Keating

REVIEW 4 major objections 5 minor 3 cited by

Quantum Wave Simulation with Sources and Loss Functions

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper claims one Hamiltonian-simulation framework that simulates acoustic, Maxwell, and elastic waves with sources and $\ell^2$ losses, preserving a quartic speed-up over classical 3D solvers under global measurements and compact…

desk verdict A genuinely useful framework for simulating lossless wave equations on a quantum computer, with a plausible quartic speed-up in 3D, but the end-to-end loss pipeline and the optimality claim rest on gaps that need real fixes before the abstract's promises hold. read the letter →

arxiv 2411.17630 v3 pith:EO3VMDUG submitted 2024-11-26 quant-ph

Cyrill Bösch , Malte Schade , Giacomo Aloisi , Scott D. Keating , Andreas Fichtner This is my paper · ORCID

classification quant-ph MSC 81P6865M0635L05 PACS 03.67.Ac

keywords quantumwavesimulationHamiltoniananti-HermitianoperatoracousticequationMaxwell'sequationselasticl2lossfunctionspeed-up

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

This paper aims to show that the full pipeline of classical wave simulation—writing down a wave equation in heterogeneous, anisotropic media, adding sources, and comparing wavefields through an $\ell^2$ loss—can be moved onto a quantum computer without losing the theoretical runtime advantage. The target equations are the acoustic, Maxwell, and elastic wave equations, all rewritten as Schr\"odinger equations with anti-Hermitian Hamiltonians $H = iB^{-1/2}AB^{-1/2}$. The central quantitative claim is a quartic speed-up over classical solvers in three dimensions, under the conditions that measurements are sufficiently global and that sources and initial conditions are compactly supported (or rotationally symmetric). A sympathetic reader would care because wave-based inverse problems in seismology, imaging, and non-destructive testing are currently limited by the cost of numerical simulation.

What carries the argument

The load-bearing object is the anti-Hermitian structure of the wave operator: every equation is written as $\hat C = \hat B^{-1}\hat A$ with $B$ Hermitian positive definite and $A$ anti-Hermitian, and then conjugated by $B^{1/2}$ to obtain the Hamiltonian $H = iB^{-1/2}AB^{-1/2}$. Hamiltonian simulation (Theorem 1, adapted from Low\,–\,Chuang) supplies the runtime bound in terms of sparsity $d$, time $t$, and norm $\|H\|_{\max}$; the paper's Theorem 2 (subspace $\ell^2$-norm estimation) carries the measurement pipeline by expressing the loss on a subspace as the expectation of an observable built from four Pauli terms, achieving $O(M\log(1/\delta)/\epsilon)$ oracle calls. Source implementation rests on two mechanisms: classical pre-simulation of a source in a small fixed volume for compact pulses, and Lemma 3's initialization of rotationally covariant fields from a single ray using controlled rotations, with gate count $\tilde O(CN^{1/D})$.

What would settle it

Take a 3D inverse-problem scenario in the paper's setting, and compare the total runtime of the proposed pipeline (including the oracle that prepares the observed wavefield from measured data with no exploitable structure) against a classical finite-difference solver for the same grid and source. If loading the target field requires $\Omega(N)$ gates or queries, the end-to-end cost matches the classical $O(Nt)$ bound and the quartic speed-up disappears; counting those gates is a concrete check. A second check is to measure the sampling overhead of Theorem 2 for a subspace of small amplitude fraction, where the sample count should grow as the inverse squared amplitude rather than the claimed $O(1/\epsilon)$.

Watch

Extended reading notes

Core claim

The paper's central claim is that any linear, lossless wave equation of the form $dw/dt = B^{-1}Aw + s$, with $B$ Hermitian positive definite and $A$ anti-Hermitian, admits a natural quantum encoding: the similarity transform $w_Q = B^{1/2}w$ turns the evolution into a Schr\"odinger equation with Hamiltonian $H = iB^{-1/2}AB^{-1/2}$, so optimal sparse Hamiltonian simulation applies whenever the discretized operators are sparse and local. On this basis the paper argues that hybrid classical-quantum wave simulation in 3D runs in $\tilde{O}(t)$ time compared with $O(Nt)$ for classical solvers, giving a quartic speed-up that is optimal for time-domain solutions with local couplings. It further claims that subspace energies and $\ell^2$ losses between wavefields can be estimated with optimal sample complexity, that point sources can be initialized in $O(1)$ cost by classical pre-simulation in a small volume, and that boundary conditions expressible as linear constraints can be incorporated while preserving anti-Hermiticity.

Load-bearing premise

The protocol assumes an oracle that can prepare a quantum state encoding the target or observed wavefield (alongside the simulated field) at negligible cost; the paper does not analyze the cost of loading an unstructured target field into such an oracle, and if that loading cost scales with system size, the claimed speed-up in the inverse-problem setting would be cancelled.

Editorial extensions

If this is right

In 3D, full waveform simulations for acoustic, electromagnetic, and elastic media would run in time scaling polynomially with the grid's linear size rather than its number of grid points, a quartic improvement over classical solvers.
Subspace energies and $\ell^2$ wavefield misfits, the core quantities in waveform inversion, can be read out with optimal precision scaling, making misfit evaluation on a quantum computer as sample-efficient as theoretically possible.
Compactly supported pulse sources, including asynchronous multiple sources, can be implemented with $O(1)$ initialization cost, preserving the speed-up in realistic source scenarios.
Boundary conditions and linear constraints that preserve anti-Hermiticity, including mixed Dirichlet\,–\,Neumann settings, can be incorporated without breaking the Hamiltonian structure.

Reading between the lines

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

The claimed speed-up is polynomial, not exponential, in the spatial dimension, and it requires sufficiently global measurements; for local measurements the sampling cost in Theorem 2 may dominate, so the practical regime is full-waveform or large-subvolume comparisons.
The oracle assumption for target wavefields is a genuine gap for inverse problems: if measured data are unstructured, state preparation cancels the speed-up; a structure-aware loading scheme would be needed to close it.
The windowing construction for rotationally symmetric sources suggests a generic technique: decompose a non-compact source into windows and synchronize via time-dilated Hamiltonians; this could extend to other linear PDEs with known homogeneous Green's functions.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

You should read this one if you care about quantum algorithms for PDEs. The paper does something real: it gives a clean B^{1/2} encoding that maps the acoustic, Maxwell, and elastic wave equations to Hamiltonian simulation, and it works through the practical pipeline — compact-pulse sources, asynchronous source superposition, subspace l2-loss estimation, and boundary conditions via linear constraints. There is a GitHub repository implementing the acoustic and Maxwell cases, which is more than most papers in this area ship. The central quartic speed-up for 3D, compact sources, and global measurements is consistent with the standard N^{1/D} scaling argument and with Babbush et al. on oscillator models; that part is plausible and not the problem.

The soft spots are concentrated where the paper reaches beyond what it proves. First, the runtime analysis in Section III writes O~(t) without the ||H||max factor from the Low–Chuang theorem; for finer grids ||H||max grows, and that changes the advertised speed-up. The claim that the quartic speed-up is optimal is asserted without a lower-bound proof — it may be true, but it's not established here. Second, Theorem 2's l2-loss protocol assumes a state preparation oracle U that loads the target or observed wavefield, and the cost of that oracle is never analyzed. The stress-test note lands: for receiver data on a 3D boundary, generic amplitude encoding costs ~N^{2/3}, which already exceeds the N^{1/3} quantum budget. The small-subspace escape (d = polylog N) is exactly the regime Section VI.A flags as sample-inefficient. So the end-to-end speed-up claim needs either a structured-data assumption on the target or a different measurement protocol. Third, the proof of Lemma 3 handwaves the rotation decomposition: it writes R_theta as a product over C-1 plane rotations but only supplies D-1 angular indices; for the elastic case (C=9, D=3) that needs real justification. Minor but real: Theorem 2 estimates times ||[a;b]||^2, but the normalized oracle U never reveals ||[a;b]||, so the normalization factor needs a separate estimation step or a different setup.

None of this is fatal to the core idea. The paper is honest about many limitations, the encoding is standard and well-explicated, and the code helps. It deserves a serious referee, but the referee should send it back for a revision that fixes the oracle-cost gap, the normalization issue, the ||H||max dependence, and the rotation lemma. If those are addressed, this becomes a solid reference for quantum wave simulation. I would not cite it in its current form, but I'd bring it to a reading group to argue about.

Recommendation: accept for peer review, with an expectation of major revision.

Referee Report

4 major / 5 minor

Summary. The manuscript proposes a quantum Hamiltonian-simulation framework for linear, anti-Hermitian (lossless) wave equations, covering acoustic waves, Maxwell's equations, and elastic waves in heterogeneous, anisotropic media. The central construction transforms the wave operator C = B^{-1}A into an anti-Hermitian Hamiltonian B^{-1/2}AB^{-1/2}, so that wave propagation becomes Schrödinger evolution. The paper then develops algorithms for measuring subspace l2-norms and energies, for implementing compactly supported and rotationally symmetric sources, and for incorporating boundary conditions by constrained elimination. The headline claim is that this pipeline achieves a quartic speed-up over classical solvers in 3D, under sufficiently global measurements and compactly supported sources, and that this speed-up is optimal for time-domain solutions because the discretized Hamiltonian has local couplings.

Significance. The wave-equation encoding itself is elegant and practically motivated: the B^{-1/2}AB^{-1/2} transformation is correctly derived for the acoustic, Maxwell, and elastic cases, and the paper gives concrete staggered-grid finite-difference constructions that preserve the required anti-Hermitian structure. The source-initialization strategy for compact pulses and the multi-state l2-loss measurement via Pauli observables are useful contributions, and the accompanying educational repository strengthens reproducibility. However, the central runtime claim rests on a cost accounting that is not fully justified as written, and the loss-measurement stage assumes a target-state preparation oracle whose cost is never analyzed. These issues are load-bearing for the abstract's quartic speed-up claim and for the end-to-end loss-estimation pipeline, so the manuscript needs substantial revision before the main claims are established.

major comments (4)

[Section III, paragraph following Theorem 1] The runtime derivation drops the ||H||max factor. The quoted Low-Chuang bound is O(t d ||H||max), but the text concludes that the Hamiltonian simulation runtime is O~(t) = O~(N^{1/D}). For the staggered-grid discretizations in the paper, ||H||max scales as 1/Δx = O(N^{1/D}) when the grid is refined over a fixed physical domain. If t denotes physical evolution time, then the product t||H||max must be carried through, and the stated O~(N^{1/D}) does not follow. If, instead, t is intended to denote the number of CFL time steps, so that t = T ||H||max, the paper should define t this way and reconcile it with the quoted theorem; as written, the two uses of t are conflated. Because the quartic speed-up is the abstract's central claim, this accounting must be corrected and stated unambiguously.
[Section IV, Theorem 2 and Eqs. (22)-(23)] Theorem 2 claims to estimate the unnormalized quantity l2_S = ||PS w1 + ... + PS wM||^2, but the proof estimates l2_S = <phi|O|phi> ||[w1;...;wM]||^2 and never provides a procedure for obtaining the normalization ||[w1;...;wM]||. For the two-field comparison in Eq. (23), this norm contains contributions from the target field outside the subspace S; the paper explicitly says the target field may be 'only defined or known within subvolume S' (Section IV, paragraph after Eq. (23)). In that case the full norm is unavailable, and the algorithm as stated computes only a normalized loss. The theorem statement and the loss-estimation claim need to be revised to either include a norm-estimation procedure or clearly restrict the claim to normalized quantities.
[Section IV, Theorem 2 and Section VI.A] The state preparation oracle U used in Theorem 2 is assumed but never instantiated or costed for the target/observed wavefield. Section III's oracle constraints concern the Hamiltonian H, not U. If w_target is obtained from a prior classical simulation or from receiver data on d points in 3D, generic amplitude encoding costs Omega(d polylog N) gates or queries. For d = Theta(N^{2/3}), this already exceeds the O~(N^{1/3}) simulation budget that the quartic speed-up claim requires; for d = polylog(N), the measurement is non-global and sample-inefficient by the paper's own limitation in Section VI.A(ii). Thus the loss-comparison stage can become the bottleneck, and the end-to-end speed-up claim is not established without an explicit model for how U is constructed and what it costs.
[Abstract and Section III] The statement that the quartic speed-up 'is optimal for time-domain solutions, as the Hamiltonian of the discretized wave equations has local couplings' is unsupported. No lower bound is proved for either the classical or the quantum side, and local coupling by itself does not imply a query lower bound for Hamiltonian simulation of wave equations. Either remove the optimality claim or provide a concrete lower-bound argument.

minor comments (5)

[Section II.C] There is a typo in the first sentence: 'In the interest a of clear notation' should read 'In the interest of clear notation'.
[Section III] The phrase 'all classical wave equation solvers' is stronger than the evidence provided; the comparison is with explicit grid-based solvers that cost O(Nt). Spectral or fast-summation methods can be faster in special cases, so the claim should be qualified to the generic heterogeneous local-time-stepping setting.
[Section III and Abstract] The terms 'quadratic', 'cubic', and 'quartic' speed-up are used without a definition of the asymptotic ratio being compared; because t is used both as evolution time and implicitly as a number of time steps, these labels should be defined precisely and connected to the corrected runtime expression.
[Appendix B.1, proof of Lemma 3] The proof assumes that the rotation R(S) can be decomposed into C-1 rotations in adjacent coordinate planes parameterized by the angular coordinates. For the elastic case (C=9) and for tensor-valued representations of SO(D), this decomposition is not immediate and should be justified or replaced by a more careful statement for the specific field components used.
[Figure 1] The caption of Figure 1 is very long and includes multiple procedural details already described in the text; consider shortening it and referring to the relevant algorithms in Sections V.B and V.C.

Circularity Check

0 steps flagged · score 0.0 of 10

No material circularity: the speed-up follows from applying independent Hamiltonian-simulation and amplitude-estimation results to an explicit anti-Hermitian mapping, with no fitted parameter or self-referential reduction.

full rationale

The derivation chain is self-contained against external results. The wave equation is mapped to a Schrödinger equation by the explicit similarity transform w_Q = B^{1/2}w and C_Q = B^{-1/2} A B^{-1/2}, so the Hamiltonian is constructed from the stated physics rather than assumed. The runtime bound uses Low and Chuang's sparse Hamiltonian simulation theorem (Theorem 1) on a Hamiltonian with O(1) sparsity, which is an independent external result. The l2-loss estimation in Theorem 2 reduces the subspace norm to expectation values of four Pauli observables under an explicitly constructed state |ψ> = P~_S (I ⊗ U)|0>, with sampling complexity inherited from Knill–Ortiz–Somma amplitude estimation; the oracle U is a stated input assumption, not a fitted quantity, and the theorem's content is the reduction, not the oracle's implementation cost. The source-initialization claims are similarly transparent: fixed-duration or short-duration pulse sources are pre-computed classically only over a constant-size volume, and rotationally covariant fields are prepared from a single reference ray using controlled rotations (Lemma 3), costing O~(N^{1/D}) gates. Self-citations such as [35] are used for standard integration-by-parts identities, which are independently verifiable and not load-bearing. No parameter is fitted and then renamed a prediction, and no cited 'uniqueness theorem' from the present authors is used to force a choice. Therefore the central claim has independent content and no significant circularity.

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

The central runtime claims rest on external Hamiltonian simulation theory, on block-structure assumptions for B, on globality of measurements, on the unproven rotational decomposition in Lemma 3, and on oracle assumptions for target state preparation. No free parameters are fitted to data, and no new physical entities are introduced.

assumptions (5)

standard math Hamiltonian simulation oracle: a sparse oracle for H with entries specified to m bits of precision, with query cost as given by Low-Chuang Theorem 1, applies directly to the discretized wave operator.
Section III uses Theorem 1 to bound quantum runtime; this is an external theorem, not derived in the paper.
domain assumption Material matrix B is diagonal or block-diagonal with block sizes independent of total system size, so B^{1/2} does not become dense and initial states remain efficiently preparable.
Section II.D states this requirement explicitly; it excludes general full-matrix anisotropic couplings beyond small constant-size blocks.
domain assumption Measurements are taken on subspaces that are either very small or very large relative to the full dimension, and the measured quantity carries enough probability mass to avoid exponential sampling cost.
Theorem 2 requires d=O(polylog L) or L-d=O(polylog L), and Section VI.A(ii) acknowledges the globality limitation.
ad hoc to paper For rotationally covariant fields, the representation R(S) of SO(D) is generated by C-1 plane rotations in adjacent coordinate planes parameterized by the D-1 angular coordinates.
Appendix B.1.c, Rotation Decomposition, uses this construction to build the initialization circuit. For Maxwell's equations with C=2D and R=diag(S,S), the stated adjacent-plane chain is not shown to generate the required rotations.
domain assumption Target or observed wavefields can be prepared by a state preparation oracle U at no more than polylog(N) or O~(N^{1/D}) cost.
Theorem 2 assumes U for [w1;...;wM], and Section III limits input information to O~(N^{1/D}), but no protocol or cost analysis is given for loading arbitrary observed or target fields.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Quantum Wave Simulation with Sources and Loss Functions." pith.science (2026). https://pith.science/paper/EO3VMDUG

@misc{pith2026241117630,
  author       = {Pith},
  title        = {Pith review of: Quantum Wave Simulation with Sources and Loss Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EO3VMDUG}},
  note         = {Machine review of arXiv:2411.17630}
}

abstract

We present a quantum algorithmic framework for simulating linear, anti-Hermitian (lossless) wave equations in heterogeneous, anisotropic, and time-independent media. This framework encompasses a broad class of wave equations, including the acoustic wave equation, Maxwell$'$s equations and the elastic wave equation. Our formulation is compatible with standard numerical discretization schemes and allows for the efficient implementation of multiple practically relevant time- and space-dependent sources. Furthermore, we demonstrate that subspace energies can be extracted and wave fields compared through an $l_2$ loss function, achieving optimal precision scaling with the number of samples taken. Additionally, we introduce techniques for incorporating boundary conditions and linear constraints that preserve the anti-Hermitian nature of the equations. Leveraging the Hamiltonian simulation algorithm, our framework achieves a quartic speed-up over classical solvers in 3D simulations, under conditions of sufficiently global measurements and compactly supported sources and initial conditions. This quartic speed-up is optimal for time-domain solutions, as the Hamiltonian of the discretized wave equations has local couplings. In summary, our framework provides a versatile approach for simulating wave equations on quantum computers, offering substantial speed-ups over state-of-the-art classical methods.

Figures

Figures reproduced from arXiv: 2411.17630 by the authors.

**Figure 2.** FIG. 2: Quantum circuits for efficient vector field initialization in 2D and 3D. The initial ray is prepared in the [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** FIG. 3: Staggered grid visualization for 2D [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4: Examples of different boundary condition implementations. [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

On the quantum computational complexity of classical linear dynamics with geometrically local interactions: Dequantization and universality
quant-ph 2025-05 conditional novelty 7.0 of 10

Short-time dynamics of geometrically local classical systems are dequantized and shown BPP-complete, while long-time dynamics are shown as powerful as exponential-time quantum computation.
Multi-Controlled Quantum Gates in Linear Nearest Neighbor
quant-ph 2025-05 conditional novelty 6.0 of 10

Multi-controlled X and SU(2) gates on linear-nearest-neighbor qubit arrays require at most 4k+8n-16 and 4k+8n-14 CNOT gates, respectively, improving earlier bounds.
Quantum simulation of elastic wave equations via Schr\"odingerisation
quant-ph 2025-05 conditional novelty 5.0 of 10

Elastic wave equations are rewritten as Schrödinger-type systems and simulated on quantum hardware in principle, with gate-complexity bounds showing exponential advantage in dimension under periodic boundaries.

Reference graph

Works this paper leans on

104 extracted references · 62 canonical work pages · cited by 3 Pith papers

[1]

This is achieved by:

Fixed Source Duration If the simulation domain grows while the source re- mains active over a fixed duration ∆T = T end−T start, the implementation complexity naturally stays O(1). This is achieved by:
[2]

Classically solving the forced wave equation over ∆T within a fixed volume around the source, suf- ficient to contain all emitted waves during this in- terval
[3]

Since the required initialization volume does not scale with N , only a constant number of grid points needs to be initialized, ensuring O(1) runtime complexity

Initializing the resulting sparse wave field on the QC. Since the required initialization volume does not scale with N , only a constant number of grid points needs to be initialized, ensuring O(1) runtime complexity
[4]

For example, increasing the grid res- olution by a factor of 2 (i.e., halving the spatial grid spacing ∆ x → ∆x 2 ) allows us to resolve higher spatial frequencies

Increasing Grid Resolution When refining the grid to simulate higher-frequency wave pulses, typically the source time function f (t) in- cludes higher frequencies, resulting in a duration in- versely proportional to the increased frequency band- width ∆ω: ∆T ∝ 1 ∆ω = 1 ωmax − ωmin , (27) where [ ωmin, ωmax] is the support of the Fourier trans- form ˜f (ω)...
[5]

Each source can be classically simulated within its respective volume, and the resulting wave fields are initialized on the QC as a single initial condition

Synchronous Activation If all sources are active over the same time interval, i.e., T start s = T start and T end s = T end for all s, the previous approach applies directly. Each source can be classically simulated within its respective volume, and the resulting wave fields are initialized on the QC as a single initial condition. The runtime scaling rema...
[6]

Let ws for s = 1 ,

Asynchronous Activation When activation times vary among different point sources, an additional step is required to synchronize them. Let ws for s = 1 , . . . , Sdenote the initial con- ditions from classically simulating each source over their 6 respective durations ∆Ts = T end s − T start s . The combined and energy-transformed initial state is given by...
[7]

(A2) By introducing |ϕ⟩ = [ a; b]/||[a; b]||, we can write Eq

Proof of Lemma 4 The squared l2-norm on this subspace is defined as l2 2,S = LX i (PS a)i − (PS b)i 2 . (A2) By introducing |ϕ⟩ = [ a; b]/||[a; b]||, we can write Eq. (A2) as an expected value estimation l2 2,S = ⟨ϕ|O|ϕ⟩ ||[a; b]||2, (A3) where the Hermitian observable O is given by O2L×2L = PS −PS −PS PS . (A4) The observable O can be recognized as a dif...
[8]

Consider a quantum state |ϕ⟩ = [w1; w2

Proof of Theorem 2 Building on Lemma 4, we can estimate general l2- norms of arbitrary many sub-states. Consider a quantum state |ϕ⟩ = [w1; w2; . . .; wM ]/||[w1; w2; . . .; wM ]||, (A16) with wi ∈ CL, where M is the number of sub-states. We aim to estimate a squared l2-norm, l2 Σ, on a subspace S defined by the projector PS , such that l2 Σ,S = ||PS w1 +...

Show all 104 references

[9]

W eighted l2-norms To compare the states in the standard basis, we utilize a weighted inner product which reverses the transforma- tion with B1/2: Corollary 5(Weighted l2-norm). Given Theorem 2, we can estimate a weighted l2-norm with O(V log(1/δ)/ϵ) calls to the oracle U (alo...
[10]

Rotational Covariance Since wsym is rotationally covariant around x0 = 0, we have R(S)wsym(x) = wsym(Sx), ∀x, S, (B1) where S ∈ SO(D) uniquely decomposes R(S) ∈ SO(C)

Proof of Lemma 3 a. Rotational Covariance Since wsym is rotationally covariant around x0 = 0, we have R(S)wsym(x) = wsym(Sx), ∀x, S, (B1) where S ∈ SO(D) uniquely decomposes R(S) ∈ SO(C). b. Grid Construction We discretize the space around x0 into a grid as follows: • Radial C...
[11]

For Θ = 2 k, the total number of angular qubits is (D − 1)k

Binary Representation of Angular Indices: Each angular coordinate θi is discretized into Θ di- visions and represented using k = log 2(Θ) qubits. For Θ = 2 k, the total number of angular qubits is (D − 1)k
[12]

Specifically, we perform rotations in the (i, i+ 1)-planes for i = 1,

Rotation Decomposition: To rotate the initial vector wsym(xa,0) to all possible directions ˜uθ, we decompose the rotation Rθ ∈ SO(C) into a se- quence of C − 1 rotations. Specifically, we perform rotations in the (i, i+ 1)-planes for i = 1, . . . , C− 1: Rθ = C−1Y i=1 Ri,i+1(θ...
[13]

The rotation angles θ(m) i are defined as: θ(m) i = π 2m , (B7) so that the cumulative rotation angle θi =Pk m=1 bi,mθ(m) i , where bi,m ∈ {0, 1} is the mth bit of θi

Sequence of Controlled Rotations: For each plane (i, i+ 1), we apply controlled rotation gates CRi,i+1(θ(m) i ) in the real plane, where m indexes the bits in the binary representation of the angular coordinate θi. The rotation angles θ(m) i are defined as: θ(m) i = π 2m , (B7...
[14]

12 (a) 2D Initialization circuit

Control and T arget Qubits: The control qubits are the angular index qubits |θ⟩, and the target qubits are the field component qubits |c⟩. 12 (a) 2D Initialization circuit. The full vector field is constructed from a single ray in [1 , 0] direction. (b) 3D Initialization circu...
[15]

This function realizes overlapping windows that satisfy Eq

Windowing While different choices of windows are possible, in our implementation we use a symmetric double sigmoid func- tion Wj(t) = 1 1 + e−z·t − 1 1 + e−z·(t−(τj+1−τj )) , (B9) where z defines the steepness of the sigmoid functions. This function realizes overlapping window...
[16]

In 2D, we have three fields: u, vx, and vy, discretized on different grid points within a rectangular domain Ω = [ x0, x1] × [y0, y1]

2D Staggered Grid We now demonstrate how the staggered grid is con- structed in practice for the D = 2 case, though the same procedure can be extended to the general D-dimensional case. In 2D, we have three fields: u, vx, and vy, discretized on different grid points within a r...
[17]

Generalization to FD schemes with higher orders of accuracy is straight- forward since the symmetry of the FD operators is re- tained

Discrete Gradient and Divergence Operators, and their Anti-T ranspose Relationship We construct the discretized gradient and divergence operators in 2D based on the first-order accurate cen- tral FD scheme introduced earlier. Generalization to FD schemes with higher orders of ...
[18]

A visual representation is shown in Figure 4a, where the actual boundaries of the domain are depicted with a dashed line

Implementation of Boundary Conditions The staggered grid described in the previous section naturally implements Neumann BCs, as the velocities in the direction perpendicular to the domain’s boundaries are assumed to be vanishing, i.e., v(x) · n(x) = 0 for all x ∈ ∂Ω, where n(x...
[19]

Fichtner, Full Seismic Waveform Modelling and Inver- sion, Advances in Geophysical and Environmental Me- chanics and Mathematics (Springer, 2011)

A. Fichtner, Full Seismic Waveform Modelling and Inver- sion, Advances in Geophysical and Environmental Me- chanics and Mathematics (Springer, 2011)

2011
[20]

Cesca and F

S. Cesca and F. Grigoli, in Advances in Geophysics, Vol. 56, edited by R. Dmowska (Elsevier, 2015) pp. 169– 228

2015
[21]

Igel, Computational Seismology: A Practical Introduc- tion (Oxford University Press, 2017)

H. Igel, Computational Seismology: A Practical Introduc- tion (Oxford University Press, 2017)

2017
[22]

S. J. Arrowsmith, D. T. Trugman, J. MacCarthy, K. J. Bergen, D. Lumley, and M. B. Magnani, Reviews of Geo- physics 60, e2021RG000769 (2022)

2022
[23]

R. G. Pratt, L. Huang, N. Duric, and P. Littrup, in Medi- cal Imaging 2007: Physics of Medical Imaging, Vol. 6510 (SPIE, 2007) pp. 1523–1534

2007
[24]

Gemmeke and N

H. Gemmeke and N. V. Ruiter, Nuclear Instruments and Methods in Physics Research Section A: Acceler- ators, Spectrometers, Detectors and Associated Equip- ment 580, 1057 (2007)

2007
[25]

J. M. Sanches, A. F. Laine, and J. S. Suri, eds., Ul- trasound Imaging: Advances and Applications(Springer, 2012)

2012
[26]

Ould Naffa, M

S. Ould Naffa, M. Goueygou, B. Piwakowski, and F. Buyle-Bodin, Ultrasonics 40, 247 (2002)

2002
[27]

Kim and W

K. Kim and W. R. Wagner, Annals of Biomedical Engi- neering 44, 621 (2016)

2016
[28]

Lopez, R

A. Lopez, R. Bacelar, I. Pires, T. G. Santos, J. P. Sousa, and L. Quintino, Additive Manufacturing 21, 298 (2018)

2018
[29]

M. P. Bendsoe and O. Sigmund, Topology Optimization: Theory, Methods, and Applications(Springer Science & Business Media, 2013)

2013
[30]

R. E. Christiansen and O. Sigmund, JOSA B 38, 496 (2021)

2021
[31]

A. K. Sharma, M. Kosta, G. Shmuel, and O. Amir, Com- posite Structures 280, 114846 (2022)

2022
[32]

Bordiga, E

G. Bordiga, E. Medina, S. Jafarzadeh, C. B¨ osch, R. P. Adams, V. Tournat, and K. Bertoldi, Nature Materials , 1 (2024)

2024
[33]

Yilmaz, Seismic Data Analysis (Society of Explo- ration Geophysicists, 2001)

¨O. Yilmaz, Seismic Data Analysis (Society of Explo- ration Geophysicists, 2001)

2001
[34]

Fichtner and J

A. Fichtner and J. Trampert, Geophysical Journal Inter- national 187, 1604 (2011)

2011
[35]

Rawlinson, A

N. Rawlinson, A. Fichtner, M. Sambridge, and M. K. Young, in Advances in Geophysics, Vol. 55, edited by R. Dmowska (Elsevier, 2014) pp. 1–76. 18

2014
[36]

Betancourt (2018), arXiv:1701.02434

M. Betancourt (2018), arXiv:1701.02434

2018 arXiv
[37]

M. K. Sen and R. Biswas, GEOPHYSICS 82, R119 (2017)

2017
[38]

Fichtner, A

A. Fichtner, A. Zunino, and L. Gebraad, Geophysical Journal International 216, 1344 (2019)

2019
[39]

I. E. Ulrich, C. Boehm, A. Zunino, and A. Fichtner, in Medical Imaging 2022: Ultrasonic Imaging and Tomog- raphy, Vol. 12038 (SPIE, 2022) pp. 48–60

2022
[40]

M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information, 10th ed. (Cambridge univer- sity press, 2010)

2010
[41]

Montanaro, npj Quantum Information 2, 1 (2016)

A. Montanaro, npj Quantum Information 2, 1 (2016)

2016
[42]

Cerezo, A

M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Nature Reviews Physics3, 625 (2021)

2021
[43]

J. M. Martyn, Z. M. Rossi, A. K. Tan, and I. L. Chuang, PRX Quantum 2, 040203 (2021)

2021
[44]

A. M. Childs and N. Wiebe (2012), arXiv:1202.5822

2012 arXiv
[45]

G. H. Low and I. L. Chuang, Phys. Rev. Lett. 118, 010501 (2017)

2017
[46]

G. H. Low and I. L. Chuang, Quantum 3, 163 (2019)

2019
[47]

G. H. Low, Y. Su, Y. Tong, and M. C. Tran, PRX Quan- tum 4, 020323 (2023)

2023
[48]

P. C. S. Costa, S. Jordan, and A. Ostrander, Physical Review A 99, 012323 (2019)

2019
[49]

A. Suau, G. Staffelbach, and H. Calandra, ACM Trans- actions on Quantum Computing 2, 2:1 (2021)

2021
[50]

Koukoutsis, K

E. Koukoutsis, K. Hizanidis, A. K. Ram, and G. Vahala, Physical Review A 107, 042215 (2023)

2023
[51]

Y. Sato, R. Kondo, I. Hamamura, T. Onodera, and N. Yamamoto (2024), arXiv:2402.18398

2024 arXiv
[52]

Y. Sato, H. Tezuka, R. Kondo, and N. Yamamoto (2024), arXiv:2407.05019

2024 arXiv
[53]

Schade, C

M. Schade, C. B¨ osch, V. Hapla, and A. Fichtner, Geo- physical Journal International 238, 321 (2024)

2024
[54]

Babbush, D

R. Babbush, D. W. Berry, R. Kothari, R. D. Somma, and N. Wiebe, Physical Review X 13, 041041 (2023)

2023
[55]

Y. Ito, Y. Tanaka, and K. Fujii (2023), arXiv:2311.15628

2023 arXiv
[56]

S. Danz, M. Berta, S. Schr¨ oder, P. Kienast, F. K. Wil- helm, and A. Ciani (2024), arXiv:2405.08694

2024
[57]

S. Jin, N. Liu, and C. Ma, ESAIM: Mathematical Mod- elling and Numerical Analysis 58, 1853 (2024)

2024
[58]

S. Jin, N. Liu, and Y. Yu, Physical Review A108, 032603 (2023)

2023
[59]

C. Ma, S. Jin, N. Liu, K. Wang, and L. Zhang (2024), arXiv:2411.10999

2024 arXiv
[60]

S. G. Johnson, Notes on the algebraic structure of wave equations (2007)

2007
[61]

Mostafazadeh, International Journal of Geometric Methods in Modern Physics 07, 1191 (2010)

A. Mostafazadeh, International Journal of Geometric Methods in Modern Physics 07, 1191 (2010)

2010
[62]

Kaltenbacher, Fundamental equations of acoustics, in Computational Acoustics(Springer International Pub- lishing, 2018) p

M. Kaltenbacher, Fundamental equations of acoustics, in Computational Acoustics(Springer International Pub- lishing, 2018) p. 12

2018
[63]

D. J. Griffiths, Introduction to Electrodynamics (Cam- bridge University Press, 2023)

2023
[64]

S. G. Johnson and J. D. Joannopoulos, Photonic Crys- tals: The Road from Theory to Practice(Springer Science & Business Media, 2001)

2001
[65]

Jackson, Classical Electrodynamics(Wiley, 2021)

J. Jackson, Classical Electrodynamics(Wiley, 2021)

2021
[66]

Voigt, Lehrbuch der Kristallphysik (Vieweg und Teubner Verlag, 1966)

W. Voigt, Lehrbuch der Kristallphysik (Vieweg und Teubner Verlag, 1966)

1966
[67]

Scheibner, A

C. Scheibner, A. Souslov, D. Banerjee, P. Sur´ owka, W. T. M. Irvine, and V. Vitelli, Nature Physics 16, 475 (2020)

2020
[68]

Aki and P

K. Aki and P. G. Richards, Quantitative Seismology, 2nd Ed. (University Science Books, 2002)

2002
[69]

Landau, L

L. Landau, L. Pitaevskii, A. Kosevich, and E. Lifshitz, Theory of Elasticity: Volume 7 , Bd. 7 (Butterworth- Heinemann, 2012)

2012
[70]

Hoepffner, Implementation of boundary conditions (2007)

J. Hoepffner, Implementation of boundary conditions (2007)

2007
[71]

Banjai and S

L. Banjai and S. Sauter, SIAM Journal on Numerical Analysis 47, 227 (2009)

2009
[72]

Qian and W

Z.-G. Qian and W. C. Chew, IEEE Transactions on An- tennas and Propagation 57, 3594 (2009)

2009
[73]

J.-G. Wei, Z. Peng, and J.-F. Lee, Radio Science 47, 1 (2012)

2012
[74]

Omar and D

S. Omar and D. Jiao, IEEE Transactions on Microwave Theory and Techniques 63, 897 (2015)

2015
[75]

Smith, S

F. Smith, S. Tsynkov, and E. Turkel, Journal of Scientific Computing 81, 1181 (2019)

2019
[76]

A. W. Harrow and A. Y. Wei, in Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA), Proceedings (Society for Industrial and Applied Mathematics, 2019) pp. 193–212

2020
[77]

Rall and B

P. Rall and B. Fuller, Quantum 7, 937 (2023)

2023
[78]

Krovi (2024), arXiv:2404.07303

H. Krovi (2024), arXiv:2404.07303

2024 arXiv
[79]

Koukoutsis, K

E. Koukoutsis, K. Hizanidis, A. K. Ram, and G. Vahala, Future Generation Computer Systems 159, 221 (2024)

2024
[80]

Z. Hu, R. Xia, and S. Kais, Scientific Reports 10, 3301 (2020)

2020
[81]

Nassar, H

H. Nassar, H. Chen, A. N. Norris, and G. L. Huang, Physical Review B 97, 014305 (2018)

2018
[82]

H. Chen, L. Yao, H. Nassar, and G. Huang, Physical Review Applied 11, 044029 (2019)

2019
[83]

Galiffi, R

E. Galiffi, R. Tirole, S. Yin, H. Li, S. Vezzoli, P. A. Huido- bro, M. G. Silveirinha, R. Sapienza, A. Al` u, and J. B. Pendry, Advanced Photonics 4, 014002 (2022)

2022
[84]

B¨ osch, A

C. B¨ osch, A. Fichtner, and M. Serra-Garcia, Physical Re- view B 110, 064307 (2024)

2024
[85]

Kikuchi, C

Y. Kikuchi, C. Mc Keever, L. Coopmans, M. Lubasch, and M. Benedetti, npj Quantum Information 9, 1 (2023)

2023
[86]

Wright, C

L. Wright, C. Mc Keever, J. T. First, R. Johnston, J. Tillay, S. Chaney, M. Rosenkranz, and M. Lubasch, Physical Review Research 6, 043169 (2024)

2024
[87]

Acharya (2024), arXiv:2408.13687

R. Acharya (2024), arXiv:2408.13687

2024 arXiv
[88]

R. Iten, R. Colbeck, I. Kukuljan, J. Home, and M. Chri- standl, Physical Review A 93, 032318 (2016)

2016
[89]

He, M.-X

Y. He, M.-X. Luo, E. Zhang, H.-K. Wang, and X.-F. Wang, International Journal of Theoretical Physics 56, 2350 (2017)

2017
[90]

Zindorf and S

B. Zindorf and S. Bose (2024), arXiv:2404.02279

2024 arXiv
[91]

E. A. Martinez, T. Monz, D. Nigg, P. Schindler, and R. Blatt, New Journal of Physics 18, 063029 (2016)

2016
[92]

Goel and J

N. Goel and J. K. Freericks, Quantum Science and Tech- nology 6, 044010 (2021)

2021
[93]

Y. Kim, A. Morvan, L. B. Nguyen, R. K. Naik, C. J¨ unger, L. Chen, J. M. Kreikebaum, D. I. Santiago, and I. Siddiqi, Nature Physics 18, 783 (2022)

2022
[94]

Knill, G

E. Knill, G. Ortiz, and R. D. Somma, Physical Review A 75, 012328 (2007)

2007
[95]

Gleinig and T

N. Gleinig and T. Hoefler, in 2021 58th ACM/IEEE De- sign Automation Conference (DAC)(2021) pp. 433–438

2021
[96]

Malvetti, R

E. Malvetti, R. Iten, and R. Colbeck, Quantum 5, 412 (2021)

2021
[97]

T. M. L. de Veras, L. D. da Silva, and A. J. da Silva, Quantum Information Processing 21, 204 (2022). 19

2022
[98]

Zhang, T

X.-M. Zhang, T. Li, and X. Yuan, Physical Review Let- ters 129, 230504 (2022)

2022
[99]

R. Vale, T. M. D. Azevedo, I. C. S. Ara´ ujo, I. F. Araujo, and A. J. da Silva, IEEE Transactions on Computer- Aided Design of Integrated Circuits and Systems 43, 802 (2024)

2024
[100]

Virieux, Geophysics 49, 1933 (1984)

J. Virieux, Geophysics 49, 1933 (1984)

1984
[101]

R. L. Higdon, Journal of Computational Physics 101, 386 (1992)

1992
[102]

Berenger, Journal of Computational Physics 114, 185 (1994)

J.-P. Berenger, Journal of Computational Physics 114, 185 (1994)

1994
[103]

R. G. Clapp, Reverse time migration with random boundaries, in SEG Technical Program Expanded Ab- stracts 2009(Society of Exploration Geophysicists, 2009) pp. 2809–2813

2009
[104]

Shen and R

X. Shen and R. G. Clapp, Geophysics 80, R351 (2015)

2015

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

This is achieved by:

Fixed Source Duration If the simulation domain grows while the source re- mains active over a fixed duration ∆T = T end−T start, the implementation complexity naturally stays O(1). This is achieved by:

[2] [2]

Classically solving the forced wave equation over ∆T within a fixed volume around the source, suf- ficient to contain all emitted waves during this in- terval

[3] [3]

Since the required initialization volume does not scale with N , only a constant number of grid points needs to be initialized, ensuring O(1) runtime complexity

Initializing the resulting sparse wave field on the QC. Since the required initialization volume does not scale with N , only a constant number of grid points needs to be initialized, ensuring O(1) runtime complexity

[4] [4]

For example, increasing the grid res- olution by a factor of 2 (i.e., halving the spatial grid spacing ∆ x → ∆x 2 ) allows us to resolve higher spatial frequencies

Increasing Grid Resolution When refining the grid to simulate higher-frequency wave pulses, typically the source time function f (t) in- cludes higher frequencies, resulting in a duration in- versely proportional to the increased frequency band- width ∆ω: ∆T ∝ 1 ∆ω = 1 ωmax − ωmin , (27) where [ ωmin, ωmax] is the support of the Fourier trans- form ˜f (ω)...

[5] [5]

Each source can be classically simulated within its respective volume, and the resulting wave fields are initialized on the QC as a single initial condition

Synchronous Activation If all sources are active over the same time interval, i.e., T start s = T start and T end s = T end for all s, the previous approach applies directly. Each source can be classically simulated within its respective volume, and the resulting wave fields are initialized on the QC as a single initial condition. The runtime scaling rema...

[6] [6]

Let ws for s = 1 ,

Asynchronous Activation When activation times vary among different point sources, an additional step is required to synchronize them. Let ws for s = 1 , . . . , Sdenote the initial con- ditions from classically simulating each source over their 6 respective durations ∆Ts = T end s − T start s . The combined and energy-transformed initial state is given by...

[7] [7]

(A2) By introducing |ϕ⟩ = [ a; b]/||[a; b]||, we can write Eq

Proof of Lemma 4 The squared l2-norm on this subspace is defined as l2 2,S = LX i (PS a)i − (PS b)i 2 . (A2) By introducing |ϕ⟩ = [ a; b]/||[a; b]||, we can write Eq. (A2) as an expected value estimation l2 2,S = ⟨ϕ|O|ϕ⟩ ||[a; b]||2, (A3) where the Hermitian observable O is given by O2L×2L = PS −PS −PS PS . (A4) The observable O can be recognized as a dif...

[8] [8]

Consider a quantum state |ϕ⟩ = [w1; w2

Proof of Theorem 2 Building on Lemma 4, we can estimate general l2- norms of arbitrary many sub-states. Consider a quantum state |ϕ⟩ = [w1; w2; . . .; wM ]/||[w1; w2; . . .; wM ]||, (A16) with wi ∈ CL, where M is the number of sub-states. We aim to estimate a squared l2-norm, l2 Σ, on a subspace S defined by the projector PS , such that l2 Σ,S = ||PS w1 +...