Parton Physics on a Quantum Computer

Henry Lamm; Scott Lawrence; Yukari Yamauchi (for the NuQS Collaboration)

REVIEW 2 major objections 7 minor 6 cited by

Parton Physics on a Quantum Computer

T0 review · 2 major / 7 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper argues that parton distribution functions and hadronic tensors can be computed on a universal quantum computer, and that for QCD the practical route is to fit PDFs from the hadronic tensor rather than to measure them directly.

desk verdict The paper is a solid Thirring-model proposal with transparent resource estimates, but the central QCD recommendation—PDFs via hadronic-tensor fitting—rests on an unquantified finite-differencing obstacle that likely can be avoided by direct unitary implementation. read the letter →

arxiv 1908.10439 v2 pith:ZZTZUJAN submitted 2019-08-27 hep-lat hep-phnucl-thquant-ph

Henry Lamm , Scott Lawrence , Yukari Yamauchi (for the NuQS Collaboration) This is my paper · ORCID

classification hep-lathep-phnucl-thquant-ph

keywords partondistributionfunctionshadronictensorquantumcomputationHamiltonianlatticegaugetheoryThirringmodeldeepinelasticscatteringadiabaticstatepreparationWilsonline

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

The paper proposes a way to compute parton distribution functions (PDFs) and hadronic tensors—the nonperturbative objects behind deep inelastic scattering—directly in real time on a universal quantum computer, sidestepping the sign problem and other complications of Euclidean lattice QCD. It demonstrates the algorithms in the 1+1-dimensional Thirring model, and argues that in QCD the PDF is best extracted not by direct measurement but by fitting the hadronic tensor, exactly as experiments do. If correct, a large fault-tolerant quantum computer could produce first-principles hadron-structure predictions, including lepton-hadron cross sections, at an estimated cost of about 400,000 qubits for a $20^{3}$ lattice at a=0.1 fm with O($V^{2}$) scaling.

What carries the argument

The central machinery is Hamiltonian lattice field theory evolved in real time. Fermions are mapped to qubits via the Jordan-Wigner transformation, time evolution is Trotterized, and non-Hermitian correlator operators are decomposed into unitary pieces whose expectation values are read out with an ancillary qubit. The hadronic tensor is evaluated by the linear-response trick: a second derivative of an expectation value of a product of time-evolved currents gives the current–current correlator, so no Wilson line and no ancillary decomposition are needed. The final Fourier transform is regulated with a Gaussian window to suppress oscillatory artifacts. The QCD resource estimate uses the S(1080) discrete-subgroup approximation to SU(3), giving roughly 50 qubits per lattice site, and adiabatic state preparation in the zero-momentum proton sector, whose gap at weak coupling is O(1/L) and therefore demands O($L^{2}$) evolution steps.

What would settle it

Construct and cost an explicit circuit that evaluates the light-like Wilson-line correlator of Eq. (10) on the S(1080) Hamiltonian lattice with finite-difference order independent of the number of time slices; if such a circuit is practical, the paper's conclusion that QCD PDFs must be fitted from the hadronic tensor is wrong.

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Extended reading notes

Core claim

The paper claims that parton distribution functions and hadronic tensors, the nonperturbative ingredients of deep inelastic scattering, can be computed on a universal quantum computer from Hamiltonian lattice field theory, avoiding the sign problem and other complications of Euclidean lattice calculations. It demonstrates the algorithms explicitly in the 1+1-dimensional staggered Thirring model, computing the quark distribution of the lowest fermion state by exact diagonalization as a proof of principle. For QCD, the paper argues that directly measuring the PDF is impractical because the gauge-invariant definition contains a light-like Wilson line that would require high-order finite differencing over many time slices; instead, it recommends computing the gauge-invariant hadronic tensor and fitting the PDF from it using the same collinear-factorization framework that experiments use. A side benefit is that lepton-hadron cross sections follow at leading order from the same hadronic tensor, without preparing two asymptotic scattering states.

Load-bearing premise

The load-bearing premise is that a light-like Wilson line—the gauge-invariance connector needed for direct QCD parton distributions—cannot be implemented practically, since it would require finite differencing of order equal to the number of affected time slices, an assertion the paper leaves unquantified.

Editorial extensions

If this is right

A large fault-tolerant quantum computer could produce first-principles PDFs and hadronic tensors for the proton, with systematics different from Euclidean lattice methods.
Lepton–hadron cross sections at leading order in QED can be obtained from the hadronic tensor alone, removing the need to prepare two asymptotic states and avoiding long-range QED complications.
The same procedure extends to generalized parton distributions and related observables by choosing different hadron momenta and current insertions, something not straightforward on a Euclidean lattice.
The estimated cost—about $4\times10^5$ qubits for a $20^3$ lattice at $a=0.1$ fm, with $O(V^2)$ total scaling—puts practical QCD parton-physics calculations beyond the NISQ era but within plausible future fault-tolerant machines.
In lower dimensions the Thirring-model version could be a NISQ-era target for demonstrating a classically hard hadronic-structure calculation.

Reading between the lines

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

The paper leaves open the possibility that a better implementation of light-like Wilson lines—for example one whose finite-difference order does not grow with the number of time slices—could make direct PDF computation practical; that would change the QCD recommendation without undermining the hadronic-tensor route.
The adiabatic state-preparation step dominates the resource estimate through its $O(L^2)$ evolution steps; a state-preparation method that avoids the weak-coupling $O(1/L)$ gap, such as spectral combing, could substantially reduce the total cost.
Because the PDF extraction assumes collinear factorization and parameterized fitting, the method's uncertainties inherit those assumptions; a cleaner demonstration of advantage might target Minkowski-signature observables that Euclidean methods cannot access at all.
A small-scale test on a noisy device—measuring the Thirring-model hadronic tensor via the linear-response circuit and comparing the extracted PDF to exact diagonalization—would be a natural near-term validation.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague —

This is a proposals paper, not a completed calculation, and it should be read that way. The Thirring-model part is concrete and internally consistent: Trotterized time evolution, Jordan-Wigner fermions, a controlled-unitary decomposition for the non-Hermitian PDF correlator, and a 10-site exact-diagonalization plot. I was able to follow the steps without finding errors. The resource estimates for QCD — about 4×10^5 qubits for a 20^3 lattice at a=0.1 fm, with O(V²) scaling — are transparently built from stated assumptions, and the authors are explicit that SU(3) circuits do not yet exist and that the adiabatic-preparation spectrum assumptions are untested.

The soft spot is the paper's central QCD recommendation: that PDFs are best obtained by fitting the hadronic tensor rather than by direct computation. The only stated reason is the assertion after Eq. (11) that a light-like Wilson line requires finite differencing of order equal to the number of affected time slices, and that this is 'not practical even in the absence of quantum noise.' That is an obstacle claim tied to the source-insertion measurement scheme of the companion paper [41], not a proven lower bound. Eq. (11) is a product of unitary time evolutions and spatial link operators; for a finite subgroup like S(1080) the links are unitary operators on the register, and the whole product is a unitary that could in principle be implemented directly, with the non-Hermitian fermion pieces handled by the same linear-combination-of-unitaries decomposition the paper already uses in Eqs. (6)–(8). The authors provide no resource estimate for the alleged finite differencing and no argument that a direct implementation is exponential. So the abstract's 'best obtained by fitting' claim is underdetermined. The fix is straightforward: either prove the obstacle is fundamental or restrict the conclusion to the [41] scheme. As written, the main QCD-facing message is not secure.

The rest is proportionate. The hadronic-tensor algorithm is genuinely cheaper than the earlier two-state scattering proposals, and the paper correctly notes that flavor-selective currents avoid the experimental fitting problem. The Gaussian-window Fourier processing is standard; the PDF extraction by fitting W_{μν} is a legitimate analog of experimental extractions. Citation pattern is appropriate.

Who this is for: people working on quantum simulation of gauge theories or Hamiltonian lattice methods for hadron structure. It's a useful roadmap with a clean toy-model benchmark. It deserves a serious referee; I would send it out, but I would ask the authors to substantiate or soften the direct-PDF impracticality claim before publication. My own verdict on the QCD headline is skeptical; the Thirring-model content stands on its own.

Recommendation: engage with it, use it as a baseline, but don't let the 'fit the hadronic tensor' conclusion become the takeaway without the Eq. (11) analysis.

Referee Report

2 major / 7 minor

Summary. This paper proposes quantum algorithms for computing parton distribution functions (PDFs) and hadronic tensors on a universal quantum computer. The authors illustrate the method in the 1+1-dimensional staggered Thirring model, presenting an exact-diagonalization calculation of the quark distribution function on 10 sites, and they describe how the hadronic tensor can be obtained via linear-response circuits without Wilson lines. For QCD, they argue that the PDF is best extracted by fitting the hadronic tensor rather than by direct computation, and they estimate that a 20^3 lattice at a=0.1 fm would require about 4e10^5 qubits with O(V^2) total scaling, including adiabatic state preparation. The paper is a methods proposal rather than a full calculation, and its correctness depends on a few technical assertions that need tightening.

Significance. If the algorithms and resource estimates are correct, this work would be an important step toward first-principles hadronic structure calculations on fault-tolerant quantum computers, avoiding some Euclidean lattice complications such as the inverse problem for real-time correlators. The Thirring-model illustration is a useful proof of principle, and the linear-response treatment of the hadronic tensor is explicit and well-suited to quantum simulation. However, the QCD-specific recommendation is not yet secure because it rests on an unsupported claim about the difficulty of direct PDF computation, and the cost scaling contains a technical issue with the simultaneous measurement of noncommuting currents. These points need to be resolved before the central claims can be accepted.

major comments (2)

[Section II, after Eq. (11)] The statement that direct evaluation of Eq. (10) requires finite differencing of order equal to the number of affected time slices, and that this is 'not practical even in the absence of quantum noise', is not substantiated. Equation (11) is a product of unitary time evolutions and spatial link operators; in the S(1080) scheme these are unitary gates, so the correlator could in principle be measured directly (e.g., via a Hadamard test) or with a linear-combination-of-unitaries decomposition, exactly as the authors do for the non-Hermitian fermion operator in Eqs. (6)-(8). No resource estimate for the alleged finite differencing is provided, and no argument is given that the LCU overhead is exponential in the number of time slices. Since this assertion is the sole basis for the abstract's QCD-specific recommendation that the PDF is 'best obtained by fitting the hadronic tensor', the recommendation is underdetermined. The authors should either provide a concrete obstruction with a complexity analysis or qualify the claim as specific to the source-insertion method of Ref. [41].
[Section V] The claim that 'the J_mu(x) at a single time are mutually commuting' is not correct for the staggered currents defined in Eqs. (18)-(19). For example, J^1(x) and J^1(x+1) have overlapping support at site x+1, and their commutator contains non-vanishing hopping terms such as chi^dagger(x) chi(x+2) - chi^dagger(x+2) chi(x). Therefore the L^3 spatial measurements for a fixed time cannot all be performed with a single measurement basis, and the stated O(V^2) total scaling is not established. Please clarify the intended simultaneous measurement procedure and recalculate the scaling if necessary.

minor comments (7)

[Eq. (9)] The Gaussian window width epsilon is a free parameter; the paper does not discuss how the limit epsilon -> 0 is taken in practice or the associated systematic uncertainty. The figure uses epsilon = 3 with no sensitivity study.
[Eq. (4)] The notation |y| is defined in words but would be clearer as an explicit piecewise definition or a different symbol (e.g., y mod 2).
[Section V] The resource estimate neglects error-correction overhead and does not specify the Trotter step size or error tolerance; a statement that these are not included would strengthen the presentation.
[Section II] The claim that classical algorithms 'struggle to obtain this observable' for the Thirring model is not quantified; since the continuum Thirring model is exactly solvable, the authors should cite or briefly justify the classical difficulty for the lattice version.
[Section II] The discussion of the speed of light could note that the value c = 1 in the continuum limit of the lattice Thirring model is a numerical observation, not an exact statement for finite lattice spacing.
[Eq. (15)] The perturbing operators should be specified as applied at fixed times; the text is slightly ambiguous about the time ordering in the linear-response circuit.
[Section III] The paper would benefit from a brief discussion of renormalization of the lattice currents and the hadronic tensor, since this is a standard concern for any lattice calculation but is not mentioned.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction: the Thirring-model algorithm is self-contained, and the QCD recommendation rests on an unquantified practical obstacle rather than a circular fit.

full rationale

The derivation is self-contained. The Thirring-model PDF calculation uses the lattice Hamiltonian of Eq. (1), Jordan-Wigner mapping, Trotterized evolution, and exact diagonalization (Fig. 1); no parameter is fitted and then renamed a prediction. The hadronic-tensor algorithm similarly measures the second derivative of Eq. (15) and classically Fourier transforms the result. The only place where a conclusion is imported from prior work by the same authors is the QCD-specific claim after Eq. (11): 'The order of the finite differencing needed is equal to the number of time slices affected. This high-order finite differencing is not practical even in the absence of quantum noise.' That claim supports the recommendation that the PDF be obtained by fitting the hadronic tensor, but it is an unquantified obstacle argument, not an equation-level reduction: Eq. (11) is not shown to equal a fitted quantity, and the paper does not claim a uniqueness theorem. The extraction of PDFs from the computed hadronic tensor via Eq. (21) is explicitly a numerical fit to parameterized PDFs, analogous to experimental analysis, so it does not disguise an input as a prediction. The S(1080) and adiabatic-cost estimates inherit assumptions from prior work but are not used to define the observables. Overall, no step of the derivation reduces by construction to its inputs; the main correctness risk is the missing resource estimate for direct Wilson-line computation, not circularity.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The algorithms rely on standard quantum simulation tools and on domain assumptions about QCD factorization, discrete subgroup approximations, and the spectrum used for gap estimates. No new physical entities are introduced, and the numerical inputs in the toy figure are regulators or model parameters rather than fitted constants.

free parameters (2)

Gaussian window width epsilon = 3 for Fig. 1
Hand-chosen regulator in Eq. (9) used to stabilize the Fourier transform; the intended limit is epsilon to 0 after L to infinity, but for the finite-lattice illustration it affects the shape of f(x).
Toy-model parameters = N=10 sites; m=1.5, g=0.0 and m=1.4, g=0.4
Chosen by hand for the illustrative exact-diagonalization calculation of Fig. 1; they do not enter the QCD resource estimates.

assumptions (7)

standard math The Jordan-Wigner transformation maps the Thirring fermions to qubit operators exactly.
Used throughout Sec. II to translate the fermionic Hamiltonian and correlators to bosonic qubit operators.
standard math Trotterized time evolution approximates real-time evolution by a product of easily diagonalized terms.
Invoked in Sec. II as the standard technique for implementing e^{-iHt} on a quantum computer.
standard math The hadronic tensor can be obtained by second derivatives of a linear-response unitary evolution, following Ref. [55].
Used in Sec. III, Eqns. (15)-(16), as the measurement strategy for the current-current correlator.
domain assumption Collinear factorization applies to the quantum-computed hadronic tensor, so PDFs can be extracted by fitting parameterized distributions.
Needed in Sec. III, Eq. (21), for the central recommendation to extract QCD PDFs from W_mu_nu.
domain assumption In the zero-momentum, fixed-isospin sector of a finite-volume QCD lattice, the gap above the proton is set by the pion mass, about 135 MeV.
Used in Sec. IV to argue that the end of the adiabatic trajectory can be traversed quickly.
domain assumption At weak coupling the proton state fills the lattice and the gap is O(1/L), leading to O(L^2) adiabatic time steps.
Used in Sec. IV to estimate adiabatic state-preparation cost; this spectral assumption is acknowledged by the authors as unverified.
domain assumption The 1080-element discrete subgroup S(1080) approximates SU(3) down to lattice spacing a=0.08 fm, as reported in Ref. [42].
Adopted in Sec. V for the QCD qubit cost estimate; the paper does not re-derive this validity range.

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Pith. "Pith review of Parton Physics on a Quantum Computer." pith.science (2026). https://pith.science/paper/ZZTZUJAN

@misc{pith2026190810439,
  author       = {Pith},
  title        = {Pith review of: Parton Physics on a Quantum Computer},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZZTZUJAN}},
  note         = {Machine review of arXiv:1908.10439}
}

read the original abstract

Parton distribution functions and hadronic tensors may be computed on a universal quantum computer without many of the complexities that apply to Euclidean lattice calculations. We detail algorithms for computing parton distribution functions and the hadronic tensor in the Thirring model. Their generalization to QCD is discussed, with the conclusion that the parton distribution function is best obtained by fitting the hadronic tensor, rather than direct calculation. As a side effect of this method, we find that lepton-hadron cross sections may be computed relatively cheaply. Finally, we estimate the computational cost of performing such a calculation on a digital quantum computer, including the cost of state preparation, for physically relevant parameters.

Figures

Figures reproduced from arXiv: 1908.10439 by the authors.

Discussion (0). Continue with ORCID to comment.

Forward citations

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