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REVIEW 3 major objections 7 minor 61 references

Structure-preserving Fourier circuits on a trapped-ion processor keep subdomain kinetic-energy dynamics of 1D/2D acoustic waves and variable-mass Dirac evolution within a few percent of classical references at encoded sizes up to 4096.

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2026-07-31 05:25 UTC pith:CFYMO5PY

load-bearing objection Solid H2-2 hardware benchmark for structured wave PDEs: real size jump and an exact 2D bright–dark block reduction, scoped honestly to low-bandwidth inputs and coarse KE observables. the 3 major comments →

arxiv 2607.28499 v1 pith:CFYMO5PY submitted 2026-07-30 quant-ph

Structure-Preserving Quantum Simulation of Wave Equations on a Trapped-Ion Processor

Abhishek Shringi , Hsuan-Cheng Wu , Ahmed Shokry , Xiantao Li , Mahmut Taylan Kandemir This is my paper
classification quant-ph
keywords quantum simulationwave equationquantum Fourier transformtrapped-ion processorbright-dark reductionDirac equationstructure-preserving discretizationkinetic energy observable
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper asks whether near-term quantum hardware can already track physically meaningful wave dynamics, not just run tiny toy circuits. The authors build Fourier-based, structure-preserving circuits for the one- and two-dimensional acoustic wave equations and for a one-dimensional Dirac model with a stepwise mass, and run them on the Quantinuum H2-2 trapped-ion processor. Instead of trying to reconstruct every field value, they estimate subdomain kinetic energies directly from measurement samples on grids as large as 1024 points in 1D and 32 by 32 in 2D. Across those problems the hardware traces match classical kinetic-energy curves with mean absolute errors from about 0.006 to 0.024. The point for a sympathetic reader is concrete: for structured, low-bandwidth wave problems, accurate observable dynamics can still be resolved at thousands of encoded degrees of freedom on present trapped-ion machines.

Core claim

On Quantinuum H2-2, structure-preserving QFT-based circuits for 1D and 2D acoustic waves and Strang-split variable-mass Dirac dynamics produce half-domain kinetic-energy (or related) observables that track classical references with mean absolute errors between 5.9×10^{-3} and 2.4×10^{-2} for encoded state spaces up to 4096, at fixed retained Fourier bandwidth.

What carries the argument

Fourier-block diagonalization via the quantum Fourier transform, plus an exact bright–dark (Morris–Shore) reduction of each 2D acoustic Fourier block that isolates the longitudinal velocity coupled to pressure and removes directional product-formula error; Dirac kinetics and mass are combined by Strang splitting.

Load-bearing premise

The inputs stay on a fixed low Fourier bandwidth so mode angles can be precomputed classically on a small retained set rather than evaluated coherently for broadband fields.

What would settle it

Repeat the same H2-2 kinetic-energy protocol on a broadband or unstructured initial field (or full coherent 2D mode arithmetic without retained-set precomputation) and check whether mean absolute error stays in the reported 10^{-2} range as grid qubits grow.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • At fixed retained bandwidth, compiled acoustic gate counts scale roughly quadratically in grid qubits while depth is near-linear in 1D and more quadratic in 2D, and acoustic depth is essentially independent of evolution time.
  • Observable-level wave dynamics, not full-state tomography, are a practical NISQ output model for hyperbolic PDEs on trapped-ion hardware.
  • The bright–dark block reduction gives an exact alternative to directional Trotterization for constant-coefficient periodic 2D acoustics within each retained Fourier mode.
  • Heterogeneous Dirac-type models remain accessible via position–Fourier Strang splitting, at the cost of depth that grows with product-formula steps.
  • These benchmarks do not claim quantum advantage over classical PDE solvers; they bound what structured wave observables present hardware can resolve.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If low-bandwidth structure is the real enabler, hybrid pipelines that compress classical fields into few Fourier modes before quantum evolution may matter more than raw grid size.
  • The same bright–dark idea may transfer to other rank-one pressure–velocity or flux couplings (e.g., simplified elastic or Maxwell blocks) before general sparse encodings are needed.
  • Emulator error sometimes exceeding hardware error suggests device-specific calibration, not just gate-count models, should drive next mitigation choices for QFT-heavy wave circuits.
  • Scaling beyond retained-mode precomputation will likely be gated first by reversible arithmetic for dispersion and Givens angles, not by the ion-trap connectivity that already suits nonlocal QFTs.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 7 minor

Summary. The manuscript constructs and benchmarks structure-preserving Fourier-based quantum circuits for the 1D and 2D acoustic wave equations and a 1D Dirac equation with stepwise mass on Quantinuum H2-2. A Hermitian first-order finite-difference discretization plus QFT reduces 1D acoustics to mode-controlled rotations with a low-mode linearization error bound; 2D acoustics uses an exact bright–dark (Morris–Shore) reduction of each Fourier block that removes directional product-formula error; Dirac dynamics uses Strang splitting between Fourier kinetic and position-space mass terms. Hardware runs reach Nh=1024 (1D) and 32×32 (2D), encoded dimension up to 4096, and report half-domain kinetic-energy (or spinor-population) observables rather than full fields. Across tested cases, H2-2 tracks classical references with MAE between 5.9×10^{-3} and 2.4×10^{-2}. Resource scans show roughly quadratic compiled gate count in nh at fixed retained bandwidth, with acoustic depth essentially t-independent and Dirac cost growing with product-formula steps.

Significance. If the reported hardware agreement and resource scalings hold as stated, this is a meaningful near-term benchmark: it extends prior QFT wave-equation hardware work (notably Wright et al. on H1-1 at nh=6) to nh=10 and to 2D and inhomogeneous Dirac dynamics, introduces an exact bright–dark digital reduction for the constant-coefficient periodic 2D acoustic block, and supplies logical vs H2-2-native gate/depth fits under independent t and nh scans. Strengths include careful structure preservation (Hermitian semi-discrete generators), an explicit low-mode linearization bound O(t K^3/N_h^2), a Strang commutator proposition with bulk/interface split, and transparent classical/Aer/H2-2E/H2-2 comparisons on a physically motivated marginal observable. The authors correctly disclaim quantum advantage and restrict scope to periodic, low-bandwidth inputs and coarse observables. Within that scope the work is a solid hardware-level contribution to quantum PDE simulation.

major comments (3)
  1. [Abstract, §I, §VII] Abstract and §I/§VII: the headline claim that accurate observable dynamics remain resolvable for structured wave problems with “thousands of encoded degrees of freedom” is only supported under fixed retained Fourier bandwidth (1D linearization §V A 2, Eqs. 35–54; 2D K1={0,±1,±2}, |K|=25, Boolean synthesis §V B 4 and App. B 1, Eqs. B7–B19). The body states this, but the abstract’s resource sentence (“At fixed retained bandwidth…”) is easy to miss relative to the dimension claim. Please elevate one explicit sentence in the abstract (and the concluding claim in §VII) that the nontrivial dynamical support and non-QFT interaction cost are those of a fixed mode set, so the resolvability evidence does not transfer to broadband or unstructured fields without coherent angle evaluation and different depth scaling.
  2. [§VI C 1, Eq. (128); Prop. V.1] §VI C 1, Eq. (128): the Dirac Strang step schedule r(t) is a hand-tuned staircase (1,2,3,6,7) chosen by classical fidelity sweeps, not derived from the commutator bound in Prop. V.1 / Eq. (115). Because the central Dirac hardware claim (MAE 2.4×10^{-2}, largest in the paper) and the linear-in-t resource fits are tied to this schedule, either (i) report the classical fidelity or ||U−S2^r|| versus r used to select the breakpoints, or (ii) replace/augment with a bound-driven r(t) from CS=∥XA∥+2∥XB∥ for the step mass so readers can separate Trotter error from device error. As written, the t-resource slope mixes algorithmic step growth with compilation in a non-reproducible way.
  3. [§VI A 2, §VI B 2, §VI C 2; Figs. 3–4, 8–12] §VI A 2 / VI B 2 / VI C 2 and Figs. 3–4, 8–10, 12: hardware MAE is compared to classical references and Aer (8192 shots) while H2-2 uses 1024 shots. For several Gaussian/nonseparable points the H2-2E mean error exceeds H2-2, which the text notes but does not quantify via binomial/shot-noise error bars on ⟨KE⟩. Because the strongest claim is that device errors remain O(10^{-2}) and “track” classical dynamics, add shot-noise uncertainty (or bootstrap intervals) on all hardware and Aer markers and, where MAE is within a few σ of sampling noise, state that explicitly so device noise is not over-attributed.
minor comments (7)
  1. [Table I] Table I lists this work’s depth as O(log^2 Nh) per step; for Dirac the depth also scales with r(t), and for 2D retained-mode synthesis the prefactor depends on |K|. A footnote clarifying “acoustic, fixed K” versus “Dirac per Strang step” would align the table with §VI.
  2. [§III B] Eq. (10) and nearby text: “displacenentu” is a typo for “displacement u”.
  3. [§VI A 1, Fig. 1] Fig. 1 caption and §VI A 1: fits are written with missing signs/spacing in places (e.g. “1.3n2_h + 19.7nh 163.1”); use explicit ± or − for intercepts consistently with the annotated figures.
  4. [§II] §II: “Suauet al.” and similar missing spaces/italics before “et al.” appear repeatedly; normalize author–et al. spacing.
  5. [§VI A 2] §VI A 2: “16×16 increase in grid resolution over Wright et al.” for nh=6→10 is a 2^4=16× increase in Nh, not 16×16; rephrase to avoid reading as two-dimensional.
  6. [§V, Appendix C] Appendix circuit diagrams (Figs. 14–24) are valuable; a single sentence in §V pointing to which figure matches which hardware experiment (cosine/Gaussian/2D/Dirac) would improve navigability.
  7. [§V B 2, Appendix D] Glossary (Table II) defines N as “system size” and n as total qubits; in the main text N is sometimes used where Nh is meant (e.g. around Eq. 80). Unify Nh versus N.

Circularity Check

0 steps flagged

No significant circularity: hardware KE comparisons and resource counts are empirical, with classical references and circuit reductions independently derived.

full rationale

The paper’s load-bearing claims are (i) structure-preserving QFT/bright–dark/Strang circuit constructions and (ii) H2-2 measurements of subdomain kinetic energy versus classical references. Classical trajectories are obtained by independent NumPy/SciPy exponentiation of the same semi-discrete Hermitian generators, not by fitting to hardware samples. The 1D low-mode linearization error bound follows from the cubic remainder of sin θ_k; the 2D bright–dark reduction is an algebraic Morris–Shore factorization of each Fourier block (external 1983 structure), removing directional Trotter error by construction of the model rather than by normalizing to measured KE. Resource scalings are compiled gate/depth counts under fixed retained bandwidth—an explicit scope choice, not a fitted parameter renamed as a prediction. Citations to Wright et al. and Lubasch et al. supply prior QFT context; the hardware instances, bright–dark compilation, and Dirac Strang runs are self-contained experiments. Nothing in the derivation chain reduces a claimed prediction to its own inputs by definition or self-citation uniqueness.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 1 invented entities

The work rests on standard unitary quantum circuit model and circulant FD + QFT facts, plus domain choices that define the experimental regime: periodic BCs, constant coefficients (except stepwise mass), amplitude encoding, and fixed low retained bandwidth so angles are classically precomputed. No new physical entities. Free parameters are experimental/compilation knobs (retained modes, σ, mass step, Trotter schedule, shots) that set which instances run, not fitted “predictions” of a theory constant.

free parameters (5)
  • Retained Fourier set K / K1 = K1={0,1,2,Nh-2,Nh-1}; cosine often single mode k0=1
    Hand-chosen low-mode support (e.g. five 1D Gaussian modes; K1={0,±1,±2} per 2D direction, |K|=25 fixed) on which angles are precomputed; central resource and hardware claims are at fixed retained bandwidth.
  • Dirac Strang step schedule r(t) = r=1,2,3,6,7 on intervals up to t=1
    Coarse piecewise r(t) in Eq. (128) chosen by classical fidelity simulations, not a unique error-optimal schedule; directly sets depth vs t for Dirac hardware.
  • Gaussian width σ and nonseparable (κ,γ) = σ=0.2; κ=1.0, γ=0.4
    Initial-condition shape parameters selecting spectral width and entanglement of prep circuits.
  • Mass step (m-, m+) = m-=0, m+=2 on half-domains
    Defines heterogeneous Dirac benchmark; aligned to MSB for a single controlled rotation.
  • Hardware shot count = 1024 (device/emulator), 8192 (Aer)
    1024 shots on H2-2/H2-2E (8192 ideal) chosen for HQC cost; affects KE estimator variance.
axioms (6)
  • domain assumption Periodic finite-difference first-derivative blocks paired with adjoints yield a Hermitian semi-discrete generator, so evolution is unitary on the encoded space.
    Section III; structure-preserving discretization premise for all Schrödinger-form circuits.
  • standard math Under periodic BCs, circulant difference operators are diagonalized by the QFT, giving independent Fourier-mode blocks.
    Section III E; standard circulant diagonalization used throughout.
  • domain assumption Smooth initial data are effectively supported on a fixed low-wavenumber band so sin-linearization / retained-mode synthesis errors remain small (bounds Eqs. 51–54, 105–106).
    Sections V A 2 and V B 4; required for shallow angle synthesis and stated infidelity scaling.
  • ad hoc to paper Subdomain kinetic-energy projectors estimated from computational-basis samples are the figure of merit for “accurate observable dynamics.”
    Sections VI A 2, VI B 2, VI C 2; deliberately avoids full-field reconstruction; claim scope is tied to this observable.
  • domain assumption Quantinuum H2-2 native compilation via pytket/qnexus and the device noise process are adequately represented by reported gate counts and H2-2 / H2-2E runs.
    Section IV experimental pipeline; hardware conclusions depend on this stack.
  • domain assumption For noncommuting wave and mass terms, Strang splitting with the chosen r controls the relevant error for the plotted KE trajectories.
    Section V C and Proposition V.1; Dirac results depend on product-formula adequacy at selected r(t).
invented entities (1)
  • Bright–dark (Morris–Shore) digital reduction of 2D acoustic Fourier blocks independent evidence
    purpose: Exact mode-wise isolation of longitudinal velocity–pressure coupling to remove directional Trotter error and compile a two-level bright-flux rotation plus Givens/phase gates.
    Not a new particle or force; a circuit-design identification of existing block structure (Section V B 3). Independent mathematical content is the explicit factorization and retained-mode synthesis; no beyond-paper physical entity.

pith-pipeline@v1.2.0-daily-grok45 · 41440 in / 4374 out tokens · 88291 ms · 2026-07-31T05:25:30.366016+00:00 · methodology

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read the original abstract

Wave equations provide a natural testbed for near-term quantum simulation of partial differential equations, but hardware demonstrations have remained limited in spatial dimension, equation class, system size, and physically meaningful output. We develop and benchmark structure-preserving, Fourier-based quantum circuits for the one- and two dimensional acoustic wave equations and Dirac dynamics with variable mass on the Quantinuum H2-2 trapped-ion processor. The experiments include one-dimensional grids with up to \(1024\) points and \(32\times32\) two-dimensional grids, corresponding to an encoded state-space dimension of up to \(4096\). Rather than reconstructing the full fields, we estimate subdomain kinetic energies directly from measurement samples. Across all tested acoustic and Dirac dynamics problems, the H2-2 results track the classical kinetic-energy dynamics with mean absolute errors between \(5.9\times10^{-3}\) and \(2.4\times10^{-2}\). At fixed retained bandwidth, the compiled gate counts grow approximately quadratically with the number of grid qubits; the acoustic circuit sizes are essentially independent of evolution time, whereas the cost also grows with the number of product-formula steps. These results provide hardware-level evidence that accurate observable dynamics can remain resolvable for structured wave problems with thousands of encoded degrees of freedom on a present-day trapped-ion processor.

Figures

Figures reproduced from arXiv: 2607.28499 by Abhishek Shringi, Ahmed Shokry, Hsuan-Cheng Wu, Mahmut Taylan Kandemir, Xiantao Li.

Figure 1
Figure 1. Figure 1: FIG. 1. Circuit resources for the one-dimensional cosine pressure initial state with [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Circuit resources for the one-dimensional Gaussian pressure initial state with [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Half-domain kinetic energy for the one-dimensional cosine pressure initial state with [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Half-domain kinetic energy for the one-dimensional Gaussian pressure initial state with [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Circuit resources for the two-dimensional separable cosine pressure initial state with [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Circuit resources for the two-dimensional separable Gaussian pressure initial state with [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Circuit resources for the two-dimensional nonseparable pressure initial state in Eq. (126), with [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Kinetic energy of the [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Kinetic energy of the [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Kinetic energy of the [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Circuit resources for the one-dimensional linear Dirac dynamics with cosine pressure initial data ( [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Half-domain kinetic energy for the one-dimensional linear Dirac dynamics equation with cosine pressure [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Overall circuit for the one-dimensional acoustic-wave evolution in Eq. (59). The Fourier-space flux state is [PITH_FULL_IMAGE:figures/full_fig_p038_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Gate-level decomposition of the approximate one-dimensional Fourier-space propagator [PITH_FULL_IMAGE:figures/full_fig_p039_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Fourier-space state-preparation circuit for the one-dimensional cosine pressure initial condition with [PITH_FULL_IMAGE:figures/full_fig_p039_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Fourier-space state-preparation circuit for the one-dimensional Gaussian pressure initial condition. The [PITH_FULL_IMAGE:figures/full_fig_p039_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Overall pressure-input circuit for the two-dimensional acoustic wave equation. The two-qubit field register [PITH_FULL_IMAGE:figures/full_fig_p040_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. Pressure-specific bright–dark propagator in Eq. (B19). The reversible circuit [PITH_FULL_IMAGE:figures/full_fig_p040_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Gate-level decomposition of the two-dimensional phase transformation [PITH_FULL_IMAGE:figures/full_fig_p040_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. Gate-level decomposition of the Givens transformation [PITH_FULL_IMAGE:figures/full_fig_p041_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. Gate-level decomposition of the bright–flux rotation [PITH_FULL_IMAGE:figures/full_fig_p041_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. Rank-two preparation of a nonseparable two-dimensional Fourier state. The rotation [PITH_FULL_IMAGE:figures/full_fig_p041_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23. Circuit for the Strang-split evolution of the one-dimensional linear Dirac dynamics equation. After prepa [PITH_FULL_IMAGE:figures/full_fig_p042_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24. Mass-evolution circuit for the stepwise profile [PITH_FULL_IMAGE:figures/full_fig_p042_24.png] view at source ↗

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Reference graph

Works this paper leans on

61 extracted references · 17 linked inside Pith

  1. [1]

    Withh= 1/N h and using sinθ k ≥0 for 0≤k < N h, |λk|= 2N h sinθ k, θ k = πk Nh .(29) Thus each block in Eq

    Fourier-mode propagator It is convenient to record the modulus ofλ k. Withh= 1/N h and using sinθ k ≥0 for 0≤k < N h, |λk|= 2N h sinθ k, θ k = πk Nh .(29) Thus each block in Eq. (27) simply consists of Pauli operators, Hk =−|λ k| cosθ k σx + sinθ k σy =−|λ k|R z(θk)σ x Rz(−θk).(30) Equivalently, usingσ x = Hσ z H (with H the Hadamard gate), Hk = Rz(θk) H ...

  2. [2]

    , nh −1}and anyϕ∈R, CR (r) Z (ϕ) = Nh−1X k=0 |k⟩⟨k| ⊗Rz(ϕkr),(39) wherek r ∈ {0,1}is ther-th bit in the binary expansionk= Pnh−1 r=0 kr2r, withN h = 2nh

    Low-mode linearization For low Fourier modesk=O(1), we can apply a linear approximation, sin πk Nh ≈ π Nh min(k, Nh −k) = π Nh ks, k s ≡min(k, Nh −k).(35) Substituting this into the diagonal generator of (31) replaces−2N h sinθ k by−2N h · π Nh ks =−2πk s, giving the approximate diagonal generator eHZ = Nh−1X k=0 |k⟩⟨k| ⊗ −2πks σz .(36) For evenN h = 2nh ...

  3. [3]

    Complete evolution circuit The complete construction for the evolution of the wave equation is given by |Ψ(t)⟩=e −iHt |Ψ(0)⟩(55) H= (F ⊗I)† bH(F ⊗I) =⇒e −iHt = (F † ⊗I) bU(t) (F ⊗I), bU(t) =e −i bHt (56) For the initial conditions considered here, the velocity field vanishes,v(x j,0) = 0, while the flux field is initialized with the non-uniform profilep(x...

  4. [4]

    (24) to the Fourier domain, we apply the two-dimensional QFT acting on the two spatial registers

    Fourier block structure To reduce the Hamiltonian in Eq. (24) to the Fourier domain, we apply the two-dimensional QFT acting on the two spatial registers. F2 =F ⊗ F(60) The transform is applied directly and unconditionally to both registers; no ancillary qubit is needed to choose anx- ory-directional QFT. Specifically, by using Eqs. (23) and (26), we have...

  5. [5]

    Each of the Hamiltonians is acting in one direction, for which the circuit implemented has been outlined in Section V A

    Directional operator splitting A natural approach is to decompose the block into directional couplings, H k =H x(kx) +H y(ky),(67) where Hx =λ x |vx⟩⟨p|+λ ∗ x |p⟩⟨vx|, H y =λ y |vy⟩⟨p|+λ ∗ y |p⟩⟨vy|,(68) and implement them one at a time. Each of the Hamiltonians is acting in one direction, for which the circuit implemented has been outlined in Section V A...

  6. [6]

    A more efficient construction follows by first identifying the velocity superposition that actually couples to pressure

    Bright–dark reduction The directional decomposition treats the two pressure–velocity couplings separately, even though they belong to the same three-state Fourier block. A more efficient construction follows by first identifying the velocity superposition that actually couples to pressure. This is the same algebraic structure that appears in the Morris–Sh...

  7. [7]

    In the hardware implementation reported here, we do not evaluate the square root or atan2 coherently

    Compilation of the nonlinear mode dependence The bright–dark reduction leaves two nonlinear, mode-dependent rotation angles, αk = 2ηk = 2 atan2(Ay, Ax), β k = 2tΩk = 2t q A2x +A 2y.(101) The phasesϕ x andϕ y inP k are simpler because each depends on only one Fourier index. In the hardware implementation reported here, we do not evaluate the square root or...

  8. [8]

    14 and 15

    Resource scaling For the cosine pressure initial state with a single modek 0 = 1, the explicit circuit schematics are given in Figs. 14 and 15. Firstly we study scaling with evolution time, shown in Fig. 1 ([panels (a) and (c)), after transpilation to the native Quantinuum H2-2 gate set corresponding to grid sizeN h = 1024 (n h = 10), the compiled circuit...

    2000

  9. [9]

    S. A. Moseset al., Physical Review X13, 041052 (2023)

    2023

  10. [10]

    Because the full measurement count distribution is available, the kinetic energy can be computed on any subdomain in post-processing; we report the half-domainX= (0, 1 2 ), i.e

    Hardware results Rather than reconstructing the velocity and flux fieldsv(x), p(x) at every grid pointx∈(0,1), we evaluate a single physically meaningful observable, the kinetic energy on a subdomainX⊂(0,1). Because the full measurement count distribution is available, the kinetic energy can be computed on any subdomain in post-processing; we report the h...

  11. [11]

    The detailed circuit construction is described in Section B 2

    Resource scaling Similar to the one-dimensional case, we first consider the cosine initial condition for the flux withk x = ky = 1. The detailed circuit construction is described in Section B 2. Further, note that there are only 2 Fourier modes retained in each direction, the Hamiltonian circuit is simplified accordingly. The resource estimation here foll...

    2048

  12. [12]

    In the field encoding of Eq

    Hardware results As in Section VI A 2, we evaluate a single physically meaningful observable rather than reconstructing the full field: the kinetic energy carried by thev x component on a spatial subdomainx∈[0,1/2). In the field encoding of Eq. (65),v x corresponds to both field qubit state being|00⟩, so ⟨KEvx ⟩=I y ⊗ |0⟩ ⟨0|M SBx ⊗I nh−1 ⊗ |00⟩⟨00|f ,(12...

  13. [13]

    The simulations have been done for grid-size ofN h = 32(nh = 5) corresponding to total system ofn= 12 qubits. For the cosine intial pressure, the mean absolute error averaged over its sampled points is 2.8×10 −3 for AerSimulator, 9.8×10 −3 for H2-2E, and 7.9×10 −3 for H2-2 - the same ordering, and similar∼3× increase from the shot-noise reference to hardw...

    2000

  14. [14]

    (122) with m− = 0,m + = 2, and, following the same protocol as Section VI A 1

    Resource scaling For the cosine pressure initial state, we takek 0 = 1 and the step-function mass profile of Eq. (122) with m− = 0,m + = 2, and, following the same protocol as Section VI A 1. Because the Trotter step countr=t/τ in Eq. (107) must grow withtto control the localτ 3 error, our implementation uses a coarse step schedule or Trotter number given...

  15. [15]

    (123), since the Dirac-dynamics circuit retains the one-dimensional construction’s velocity–flux field encoding with the mass term added only through the TrotterizedH mass sub-step

    Hardware results We evaluate the same kinetic-energy observable as in Section VI A 2, Eq. (123), since the Dirac-dynamics circuit retains the one-dimensional construction’s velocity–flux field encoding with the mass term added only through the TrotterizedH mass sub-step. Figure 12 shows the resulting kinetic energy as a function of evolution time. 0.0 0.2...

  16. [16]

    It therefore suffices to prepare|ψ p⟩in the spatial register

    Initial-state preparation We restrict attention to initial conditions with vanishing velocity,v(x,0) = 0, so that the complete initial state factorizes as |Ψ(0)⟩=|ψ p⟩ |p⟩,(A1) where|ψ p⟩carries the Fourier coefficients of the initial pressure field and|p⟩is the field register. It therefore suffices to prepare|ψ p⟩in the spatial register. We give two cons...

  17. [17]

    This case is small enough that the nonlinear angles can be synthesized by Boolean expansions in the retained-mode bits, avoiding a general arithmetic circuit

    Explicit synthesis for the retained low-mode set We now specialize to the retained one-dimensional mode set K1 ={0,1,2, N h −2, Nh −1}.(B1) The two-dimensional retained set isK=K 1 × K1. This case is small enough that the nonlinear angles can be synthesized by Boolean expansions in the retained-mode bits, avoiding a general arithmetic circuit. a. Sign-mag...

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