{"id":"a0e35877-53cc-458b-b2ed-99ea33b77d22","arxiv_id":"1908.10439","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A quantum-computer algorithm is proposed for computing parton distribution functions and hadronic tensors, with a Thirring-model demonstration and QCD resource estimates that favor extracting PDFs by fitting the hadronic tensor.","lead":"This paper explains how a future quantum computer could compute parton distribution functions and hadronic tensors from first principles, illustrating the algorithms on a simple 1+1 dimensional toy model. It argues that in QCD the practical route is to compute the hadronic tensor and fit the parton distributions from it, and it gives rough qubit and gate-count estimates.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed impracticality of direct QCD PDFs rests on an unproven finite-differencing requirement; a controlled-unitary or LCU implementation of Eq. (11) may avoid it, so the main QCD recommendation is not secure.","rationale":"Good-faith reading: this is a proposals paper, the Thirring model algorithm is internally consistent, and the resource scaling is derived from stated assumptions. The load-bearing point is not the hadronic-tensor method itself but the comparative claim against direct QCD PDFs. The reader's weakest_assumption is exactly the finite-differencing statement after Eq. (11), and I agree. Because Eq. (11) is a sequence of unitary factors up to the standard non-unitarity of link components, and because those components can be expanded in unitary shift operators for finite gauge groups, the assertion that direct PDF computation necessarily requires high-order finite differencing is not established. This does not falsify the paper: if direct PDFs become feasible, the hadronic-tensor route remains a valid alternative, and the main claim that parton physics can be computed on a quantum computer may still hold. But the 'best obtained by fitting' conclusion, which is part of the abstract, would lose its stated basis. The paper also explicitly notes that no circuits have been produced for any concrete SU(3) simulation and that several spectrum assumptions are untested, which independently supports a conditional acceptance. No ad hominem; the critique targets the strength of an obstacle argument. A concrete small-group simulation would settle the issue.","tokens_in":11155,"tokens_out":30627,"duration_ms":345202,"concrete_test":"Classically simulate a small non-Abelian lattice gauge theory (e.g., S3 in 1+1D with staggered fermions) and compute the PDF correlator of Eq. (10) two ways: (i) exact diagonalization; (ii) a Hadamard-test circuit implementing Eq. (11), where each link component is realized through its unitary group-algebra expansion and the time evolutions are Trotterized. If (ii) reproduces (i) without any numerical finite differencing and with LCU 1-norm and gate count scaling polynomially in the number of time slices, the paper's dismissal of direct QCD PDFs is not supported; if (ii) requires an exponential overhead, the paper's concern lands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract's QCD-specific conclusion—that the PDF is best obtained by fitting the hadronic tensor rather than direct calculation—is supported only by the assertion after Eq. (11) that a light-like Wilson line requires finite differencing of order equal to the number of affected time slices, which is 'not practical even in the absence of quantum noise.' This is an obstacle claim tied to the source-insertion measurement method of [41], not a proven lower bound. Eq. (11) is a product of unitary time evolutions and spatial link operators; for a finite subgroup such as S(1080), the non-unitary link components can be decomposed into sums of unitary group-algebra shift operators and implemented via linear combinations of unitaries, exactly the strategy the paper uses for the non-Hermitian fermion operator in Eqs. (6)-(8). The paper provides no analysis of such direct implementations, no resource estimate for the alleged finite differencing, and no argument that the LCU overhead is exponential in the number of time slices. Since this unquantified obstacle is the sole stated reason to avoid direct QCD PDF computation, the central recommendation is underdetermined even if the hadronic-tensor algorithm itself is sound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11352,"tokens_out":15096,"duration_ms":147318,"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":[{"comment":"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":"Section II, after Eq. (11)"},{"comment":"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.","section":"Section V"}],"minor_comments":[{"comment":"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.","section":"Eq. (9)"},{"comment":"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":"Eq. (4)"},{"comment":"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":"Section V"},{"comment":"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":"Section II"},{"comment":"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.","section":"Section II"},{"comment":"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":"Eq. (15)"},{"comment":"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.","section":"Section III"}],"recommendation":"major_revision","confidential_remarks":"The paper is a well-written methods proposal from the NuQS collaboration, and the Thirring-model part is sound. The central QCD recommendation is not yet secure because of the unquantified finite-differencing claim after Eq. (11); the authors should either prove the obstruction or reposition the paper as a proposal for hadronic-tensor computations with PDF extraction as one application. The equal-time commutator issue in Sec. V is a technical error that may be fixable in revision. I would not reject, but the revision should address both points and recalibrate the resource estimates if needed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague —\n\nThis 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.\n\nThe 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.\n\nThe 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.\n\nWho 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.\n\nRecommendation: 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.","headline":"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.","tokens_in":11904,"tokens_out":8380,"would_cite":true,"duration_ms":82576,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["parton distribution functions","hadronic tensor","quantum computation","Hamiltonian lattice gauge theory","Thirring model","deep inelastic scattering","adiabatic state preparation","Wilson line"],"falsifier":"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.","tokens_in":10942,"feed_emoji":"⚛️","tokens_out":7570,"duration_ms":73223,"temperature":0.7,"pith_summary":"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.","feed_headline":"Parton distributions could be computed on a quantum computer","feed_subtitle":"Direct PDFs in QCD are impractical, so the algorithm fits them from the hadronic tensor at a cost of about 400,000 qubits.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Hamiltonian-lattice gauge-theory simulation scheme and the Wilson-loop argument from which the paper infers that direct PDF Wilson lines require high-order finite differencing.","marker":"[41]"},{"why":"Provides the S(1080) discrete-subgroup approximation to SU(3) that sets the qubit cost and lattice-spacing range for the QCD resource estimate.","marker":"[42]"},{"why":"Gives the ancillary-qubit measurement technique used to evaluate the non-Hermitian quark-correlator operators in the PDF algorithm.","marker":"[52]"},{"why":"Supplies the linear-response second-derivative method used to compute the hadronic tensor without ancillary decomposition.","marker":"[55]"},{"why":"Underpins the adiabatic state-preparation cost model and the alternative direct-scattering approach that the hadronic-tensor method is designed to improve upon.","marker":"[45–47]"},{"why":"Defines the gauge-invariant PDF with a light-like Wilson line, the object that motivates the switch to hadronic-tensor fitting in QCD.","marker":"[50]"},{"why":"Provides the adiabatic theorem used to relate the spectral gap along the deformation trajectory to the required state-preparation time.","marker":"[60]"},{"why":"Provides the deep-inelastic-scattering cross-section formula and the standard experimental PDF-extraction fitting framework that the quantum procedure mimics.","marker":"[1]"}],"fun_headline_variants":["Quantum algorithm computes parton physics via hadronic tensor","Parton PDFs on a quantum computer: fit from hadronic tensor","Quantum computing for parton physics: fit PDFs from tensor","Hadronic tensor on quantum computer yields parton PDFs","Parton physics with qubits: fit PDFs from hadronic tensor"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Quantum algorithm computes parton physics via hadronic tensor","Parton PDFs on a quantum computer: fit from hadronic tensor","Quantum computing for parton physics: fit PDFs from tensor","Hadronic tensor on quantum computer yields parton PDFs","Parton physics with qubits: fit PDFs from hadronic tensor"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000873,"raw_usage":{"total_tokens":3724,"prompt_tokens":836,"completion_tokens":2888,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":452,"completion_tokens_details":{"reasoning_tokens":2811}},"tokens_in":452,"tokens_out":2888,"duration_ms":21763,"temperature":1.0,"reasoning_tokens":2811,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:43:31.240971+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Messiah, Quantum mechanics, volume II (North- Holland Publishing Company, 1969)","cited_arxiv_id":null,"evidence_quote":"Provides the adiabatic theorem used to relate the spectral gap along the deformation trajectory to the required state-preparation time."}],"review_version":1}