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REVIEW 4 major objections 6 minor 37 references

Proton-proton scattering on a quantum computer

T0 review · 4 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A hybrid quantum-classical calculation on a noisy processor reproduces the low-energy proton-proton s-wave phase shift from a Coulomb-regulated lattice effective field theory, using one quantum energy measurement followed by classical relax

desk verdict A serious first demonstration of Coulomb phase shifts on real QPU hardware, but the hybrid claim rests on an unshown convergence step and the figure is largely a re-fit of the input. read the letter →

arxiv 2508.07986 v1 pith:S2KV7ESI submitted 2025-08-11 nucl-th quant-ph

Ratna Khadka , Gautam Rupak This is my paper · ORCID
classification nucl-thquant-ph
keywords proton-protonscatteringCoulombphaseshiftquantumcomputingeffectivefieldtheorylatticeEFThard-wallboundaryconditionhybridquantum-classicalalgorithmSchmidtdecomposition
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 tries to establish that a near-term quantum computer can compute a charged-particle scattering observable, specifically the low-energy proton-proton s-wave phase shift, by a hybrid scheme in which only one energy expectation value is measured on the quantum device and all subsequent relaxation iterations run classically. The authors combine a leading-order effective field theory with a short-range contact interaction and the long-range Coulomb force, discretize the radial Schrödinger equation on a lattice, and convert the ground-state energy in a hard spherical wall into a phase shift through the ratio of regular and irregular Coulomb wave functions. They report that the resulting phase shifts track the analytical modified effective range expansion over momenta 10--70 MeV/c and across several lattice spacings and volumes. If this holds, quantum computation becomes a viable route to the proton-proton scattering input needed for solar fusion rates, extending earlier quantum-computer scattering calculations from neutral to charged particles.

What carries the argument

The load-bearing identity is the hard-wall relation $\delta_0(p)=\tan^{-1}(-F_0(\eta_p,R_{\mathrm{wall}} p)/G_0(\eta_p,R_{\mathrm{wall}} p))$, which turns a single ground-state energy into a scattering phase shift: the wall fixes $u(R_{\mathrm{wall}})=0$, so the ratio of regular to irregular Coulomb functions at the wall determines $\delta_0$. Around this sit the regulator prescription $V(0)=c_0$, $V(b)=(c_0+\alpha_{\mathrm{EM}}/(2b))/2$, $V(r>b)=\alpha_{\mathrm{EM}}/r$, which renders the Coulomb singularity finite at the origin while leaving the exterior potential unchanged, and the Gauss-Seidel relaxation update $u^{(\mathrm{new})}(r)=2t\,\bar{u}(r)/(2t+b^2[V(r)-E])$, with $E$ measured onc

What would settle it

Compare the phase shift from Eq. (9) obtained with exact classical ground-state energies to Eq. (8) for several $b$ values at fixed physical volume $(L-1)b$; if the residuals do not shrink as $b\to 0$ or if they depend on $R_{\mathrm{wall}}$ at fixed $b$, the uniformly-charged-sphere regulator is altering the momentum dependence and the central claim fails.

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Extended reading notes

Core claim

The central claim is that the Coulomb-modified s-wave phase shift $\delta_0(p)$ for proton-proton scattering can be recovered from the ground-state energy of a two-proton lattice Hamiltonian with a hard wall, even when that energy comes from one noisy quantum computation. The phase shift is fixed by $\delta_0(p)=\tan^{-1}(-F_0(\eta_p,R_{\mathrm{wall}} p)/G_0(\eta_p,R_{\mathrm{wall}} p))$, where $F_0$ and $G_0$ are the regular and irregular Coulomb wave functions, and the momentum $p$ is obtained from the spectrum. The authors regulate the combined short-range and Coulomb potential as $V(0)=c_0$, $V(b)=(c_0+\alpha_{\mathrm{EM}}/(2b))/2$, $V(r>b)=\alpha_{\mathrm{EM}}/r$, tune $c_0$ at each lat

Load-bearing premise

The regulated Coulomb potential (a uniformly charged sphere of radius $2b$ plus a retuned contact coupling) leaves the wave function unchanged outside $r=2b$; if its finite size shifts the momentum dependence of the phase shift in the 10--70 MeV/c window, the reproduced curve would deviate from Eq. (8) even with perfect energies.

Editorial extensions

If this is right

  • Charged-particle scattering is no longer out of reach for near-term quantum hardware; only one noisy energy measurement is required, and the rest of the relaxation is classical.
  • Because the Schmidt decomposition reduces every Pauli-string measurement to one- or two-qubit circuits, the method sidesteps the deep state-preparation circuits that otherwise destroy the signal on current processors.
  • The agreement across $b=2$, 1, and 0.75 fm indicates the extracted phase shift is insensitive to the ultraviolet regulator once $c_0$ is retuned, so the effective field theory power counting is doing the work expected of it.
  • The same hard-wall phase-shift technique that has already been applied classically to proton fusion, proton-deuteron scattering, and alpha-alpha scattering is now demonstrated in a quantum-classical hybrid form for the two-proton case.

Reading between the lines

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

  • The paper uses only the leading-order modified effective range expansion, with the effective-range term omitted; a natural next test is to include the next-to-leading-order $r_0$ term and ask whether the same one-measurement relaxation loop still reproduces the phase shift, which would probe the method's sensitivity to momentum-dependent short-range physics.
  • Because the regulator is varied independently of lattice size, the residuals at fixed $b$ could be extrapolated to $b\to 0$; the deviation near $p\approx 60$ MeV at $b=1$ fm looks like a discretization artifact, so a quantitative continuum extrapolation would turn this demonstration into a precision statement.
  • The Gauss-Seidel variant requires only one quantum energy evaluation, but the Jacobi variant would need one per iteration; if gate errors continue to fall, a fully quantum iteration loop might become feasible and would remove the classical relaxation step entirely.
  • The same wall-plus-EFT pipeline could be applied to other charged-light systems where the classical wall method already works, such as proton-deuteron and alpha-alpha scattering, making the quantum resource cost a state-preparation question rather than a scattering-dynamics question.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The manuscript proposes a hybrid quantum-classical computation of the low-energy proton-proton s-wave phase shift. The authors reduce the two-body problem to a 1D radial Schrödinger equation with a short-range regulated Coulomb potential, discretize it on a spatial lattice, and impose a hard spherical wall. The phase shift δ0(p) is obtained from Eq. (9) using the ground-state energy of the lattice Hamiltonian. The ground-state energy is measured on the IBMQ-Brisbane QPU for a particle-in-a-box trial state, using Schmidt decomposition to reduce measurements to one- and two-qubit circuits, and the energy is used as the seed for a classical Gauss-Seidel relaxation. The short-distance coupling c0 is tuned at each lattice spacing so that the classical lattice phase shift matches the leading-order Coulomb-modified effective range expansion Eq. (8). Results from noisy QPU measurements at b = 2, 1, 0.75 fm and several lattice sizes L are reported in Table I and Fig. 1, showing phase shifts close to the analytic curve.

Significance. If the pipeline performs as advertised, this is a useful step toward real-device quantum computation of nuclear scattering observables. The Schmidt-decomposition reduction to one- and two-qubit circuits is a practical error-reduction technique, and using a single QPU energy measurement to seed a classical relaxation is a plausible division of labor for near-term hardware. The manuscript also connects the lattice EFT wall method to a 1D radial formulation, which is convenient for quantum simulation. However, the central quantitative claim is currently underdetermined: the good agreement in Fig. 1 largely follows from tuning c0 to Eq. (8), and the specific role of the QPU measurement is not verified. With additional convergence and uncertainty analyses the method could be compelling, but the present evidence does not yet establish the hybrid pipeline as the source of the accurate phase shifts.

major comments (4)
  1. [Section III, Fig. 1 and Table I] The calibration procedure removes most of the content from the headline agreement. Section III states that at each b, c0 is tuned so that the numerical phase shift from Eq. (9) reproduces Eq. (8); Eq. (8) is exactly the dashed curve in Fig. 1. Thus the classical part of the calculation is guaranteed to lie on the target curve at the tuned points. The QPU contributes only the energy entering p in Eq. (9). Table I shows the SVD QPU trial energies are 16–46% high. To support the hybrid claim, the authors must report the converged energies after relaxation, the corresponding phase shifts, and a comparison of QPU-seeded versus exact-seeded relaxation. Otherwise Fig. 1 tests the classical solver, not the hybrid pipeline.
  2. [Section I A, Eq. (10), Section III] The relaxation algorithm is not fully specified. Eq. (10) updates u using 'the energy E associated with the current wave function', but Section III says that after the initial QPU measurement 'the rest of the relaxation method iterations are done classically'. If E is held fixed at the noisy QPU value, the iteration is solving (H−E)u=0 for a non-eigenvalue E, and convergence to the ground state is not guaranteed. If E is recomputed classically, the QPU result is eventually washed out. The manuscript asserts convergence in 'about 30 iterations' without showing a convergence study, final energy errors, or phase-shift uncertainties. This is load-bearing because the central claim is that a noisy QPU measurement seeds a calculation that accurately reproduces the phase shift.
  3. [Section I, regulator prescription; Fig. 1] The claim that the uniformly charged sphere regulator 'changes nothing beyond r=2b' needs a quantitative demonstration. Retuning c0 removes the leading short-distance effect, but the finite shape of the potential can produce momentum-dependent (effective-range-like) distortions in the 10–70 MeV/c window. The b scan in Fig. 1 is suggestive, but b=1 fm at p≈60 MeV already shows a deviation that the text attributes to discretization. A clean classical test at varying b, L, and regulator shape, comparing against Eq. (8) and including the next-order effective-range term, would isolate regulator artifacts from QPU effects. Without this, the statement that the physics outside 2b is unchanged is an assumption rather than a demonstrated property.
  4. [Eq. (8), Eq. (9), Table I] The reported phase shifts have no error bars, even though the QPU energies in Table I carry statistical errors from 5000 shots and the relaxation seed has a systematic offset of 16–46%. Moreover, Eq. (8) is leading order only; the omitted effective-range term may be non-negligible at the upper end of the momentum range. The authors should provide an uncertainty budget for p and δ0(p), including shot noise, seed offset, lattice discretization, and the truncation of the effective-range expansion.
minor comments (6)
  1. [Eq. (10)] The finite-difference expression contains a repeated u(r+b); presumably it should be u(r+b)+u(r-b).
  2. [Eq. (15)] The notation |2^l⟩ for computational basis states should be defined more carefully; as written it is easy to confuse with the decimal label.
  3. [Table I caption] The 'Exact (MeV)' column values are trial-state energies for a particle in a box of size 5b, but this is stated only in the text, not in the table caption.
  4. [Fig. 1] The figure would benefit from a legend that distinguishes the classical 3D points (open squares) from the hybrid QPU points, and from error bars on the phase-shift data.
  5. [Section III] The text says '10s of qubits' and 'about 30 iterations' — concrete numbers for the qubit counts and iteration counts should be given.
  6. [Eq. (8), Ref. [30]] The scattering length a0 = -7.81 fm is taken from a chiral potential paper; the authors should specify whether this is the Coulomb-subtracted pp scattering length and which convention is used.

Circularity Check

1 steps flagged · score 6.0 of 10

Phase-shift 'reproduction' is the calibration target: c0 is tuned to Eq. (8), so Fig. 1 matches Eq. (8) by construction.

  1. fitted input called prediction [Section III, first paragraph; Section I after Eq. (9)]
    "At a given lattice spacing b, we tune the coupling c0 such that a numerical calculation of the phase shift using Eq. (9) from the ground state energy of the 1st quantized Hamiltonian in Eq. (2) reproduces the analytical result in Eq. (8). ... We find accurate reproduction of the analytical phase shift from the quantum-classical calculation in Fig. 1 over a range of lattice spacings b and sizes L."

    Eq. (8) is the LO modified effective-range curve fixed by the input a0 = -7.81 fm. The only short-range parameter c0 is tuned so that the numerical Eq. (9) phase shift reproduces Eq. (8). Thus the Fig. 1 'reproduction' is the same analytic curve used for calibration; it is not an independent derivation or prediction of delta0(p). If the Gauss-Seidel iterations converge to the ground state, agreement with Eq. (8) is enforced by the fitted c0 plus the literature value of a0. The residual independent content is the robustness of the noisy QPU-seeded relaxation, but the paper provides no convergence study or final-energy errors, so even that is only asserted.

full rationale

The circularity is real but partial. The paper's physics target, the s-wave phase shift, is the same object used to fix the short-range coupling c0: Section III states that c0 is tuned so that the lattice phase shift (Eq. 9) reproduces the analytic LO result (Eq. 8), which itself depends on the input scattering length a0 from Ref. [30]. Consequently, the good agreement in Fig. 1 is expected by construction for a correctly converging solver, rather than a first-principles prediction. This is a fitted-input-called-reproduction pattern, though the paper honestly uses the word 'reproduce' rather than 'predict'. The genuinely independent content is the demonstration that a noisy QPU trial energy can seed a classical Gauss-Seidel relaxation without destroying the benchmark, but the paper does not quantify convergence or final-energy errors (it merely asserts 'sufficiently robust' and 'about 30 iterations'), so the hybrid-pipeline claim is not independently evidenced. Self-citation to Ref. [21] for the SVD technique is technical and not a circularity chain, especially because the relaxation method is also credited to independent Ref. [33]. Weighing these, the central physics result reduces to its inputs by construction (score 6), but the algorithmic benchmark prevents a higher score.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central result rests on two external physics inputs (mu, alpha_EM, and the pp scattering length a0 = -7.81 fm from Ref. [30]) and one fitted number, c0, which is tuned to the very phase shift the paper reproduces. The uniform-sphere Coulomb regulator and the wall method are prior-standard or ad hoc choices. No new physical entities are introduced. The quantum component is one energy measurement; the converged phase shifts are classical. Overall correctness risk is medium because the convergence-from-noise assertion is not demonstrated.

free parameters (1)
  • c0 = -0.0449 (b=2 fm), -0.1112 (b=1 fm), -0.1540 (b=0.75 fm)
    Short-range contact coupling at leading order. Tuned at each lattice spacing so the ground state of the 1D lattice Hamiltonian reproduces the analytic phase shift in Eq. (8). This makes the Fig. 1 'reproduction' a calibration check rather than a prediction.
assumptions (5)
  • domain assumption Leading-order EFT: the short-range nuclear interaction is a zero-range contact with a single coupling c0; the modified effective range expansion Eq. (8) keeps only the 1/a0 term and drops r0 and higher terms.
    Adopted from Kong-Ravndal [9,11,12]. The analytic reference curve in Fig. 1 is this leading-order expression; the truncation is stated, not derived.
  • domain assumption Hard-wall quantization: the phase shift is obtained from the ratio of regular and irregular Coulomb wave functions at the wall, delta0 = -arctan(F0/G0), Eq. (9).
    Standard method from Carlson [19] and Borasoy [20], validated classically for pp in [26]; its extension to the 1D radial reduction used here is assumed.
  • ad hoc to paper Coulomb regulation: V(0)=c0, V(b)=(c0+alpha/(2b))/2, V(r>b)=alpha/r describes a uniformly charged sphere of radius 2b, and the physics outside 2b is unchanged.
    Chosen because numerically it 'works more accurately' (Section I). The claim that a single retuned c0 absorbs all short-distance effects is load-bearing for every plotted point.
  • domain assumption s-wave dominance: only the l=0 partial wave contributes, and only the ground state of the radial Hamiltonian determines the phase shift at the momenta considered.
    The partial-wave decomposition is truncated at l=0 and only the lowest eigenvalue is used to fix p. Standard for low-energy scattering, but an assumption at p up to 70 MeV/c.
  • domain assumption Relaxation convergence from a noisy seed: Gauss-Seidel iterations converge to the exact lattice ground state despite initial energy errors of 10-46%.
    Asserted in Sections III-IV ('the relaxation method is sufficiently robust... about 30 iterations'). No convergence plot or seed-error study is provided, and the final phase shifts inherit an unquantified dependence on this assertion.

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Cite this review

Pith. "Pith review of Proton-proton scattering on a quantum computer." pith.science (2026). https://pith.science/paper/S2KV7ESI

@misc{pith2026250807986,
  author       = {Pith},
  title        = {Pith review of: Proton-proton scattering on a quantum computer},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S2KV7ESI}},
  note         = {Machine review of arXiv:2508.07986}
}
abstract

Scattering of charged particles is ubiquitous in nuclear physics. We calculate the proton-proton $s$-wave phase shift at low energy relevant to solar physics. The phase shift is calculated from the ratio of the regular and irregular solutions to the radial Schr\"odinger equation on a hard spherical wall boundary for the ground state. The ground state energy is calculated using a hybrid quantum-classical variational algorithm. A theory with short-ranged nuclear interaction in the presence of the long-ranged Coulomb force is used to describe the scattering. The theory is discretized on a spatial lattice for adaptation to the quantum computer in the second quantized language. The phase shifts at low momenta are accurately reproduced.

Figures

Figures reproduced from arXiv: 2508.07986 by the authors.

Figure 1
Figure 1. FIG. 1. Proton-proton [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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