{"id":"415f1ca5-5b94-4218-bfa2-2bf6af2bb0e9","arxiv_id":"2506.07430","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Neural quantum states plus the kernel polynomial method yield ab initio absorption spectra for molecules up to 52 electrons, matching FCI where an exact comparison is possible.","lead":"A new computational method pairs neural-network ground states with the kernel polynomial method to compute molecular absorption spectra without enumerating excited states. It reports full-configuration-interaction-level spectra for molecules up to 52 electrons, beyond what exact diagonalization can reach.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"KPM moment evaluation (Eqs. 30-31) has no specified tractable estimator for |α_n>=T_n(H~)μ|0> from an autoregressive NQS; the central claim therefore lacks a demonstrated algorithm, independent of ground-state accuracy.","rationale":"The central claim is that sNQS-KPM computes FCI-level absorption spectra for up to 52 electrons from the ground state alone. I agree with the reader that ground-state fidelity for CNa2O3 and C6H8 is unverified; however, the reader's weakest assumption is not the first obstruction. The method section's moment recursion (Eqs. 30-31) is a linear-algebra statement. To turn it into an algorithm one must evaluate or sample |α_n> = T_n(H~)μ|0>, a state that is not parameterized by the autoregressive network and for which no sampler is provided. The only option described in the paper is the local-energy-style summation over Pauli terms; applying that recursively to a single configuration branches over K terms per Chebyshev order. For CNa2O3, K≈1.6M and N_Ω=200k, so the reported linear scaling in Fig. 4b cannot arise from the displayed equations. This is an internal gap, not a disagreement with community consensus. The small-molecule energy matches in Table 1 are real evidence for the ground-state optimizer, but they do not exercise the KPM moment path; H2O and NH3 spectra in Fig. 3 use a Lanczos FCI baseline and thus do not validate the recursive estimator. The concrete test of instrumenting the released code will decide whether the authors have a hidden but valid estimator. Because that test has not been described in the paper, I would not accept or reject outright; I would mark the article UNVERDICTED until the moment algorithm is specified and verified.","tokens_in":11680,"tokens_out":9464,"duration_ms":128950,"concrete_test":"Check the public code at https://github.com/Weitheskmt/sNQS-KPM for the routine implementing Eqs. 30-31. Instrument it for H2O in STO-3G and count the number of NQS forward passes and the storage used per Chebyshev moment. If computing 60,000 moments requires storing or exactly applying H~ to vectors of dimension 2^N, or if per-moment cost grows with n, the scalability and 'ground-state-only' claims fail. If an auxiliary sampler or truncation is used, it must be documented and benchmarked; until then the central claim is unverified.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"Section 3, Eqs. 30-31, defines the spectrum through Ω_n = <α|T_n(H~)|α> with |α>=μ|0>, and the recursion requires the overlaps <α_n|α_n> and <α_{n+1}|α_n>. The paper never states how these overlaps are obtained from the NQS ansatz. |α_n> is not the ground-state network; it is a sequence of states produced by repeated application of H~ to μ|0>. To estimate <α_n|α_n> by sampling one would need either a generator for |α_n|^2 or pointwise values α_n(x) for sampled x, and the Chebyshev recursion makes each α_n(x) depend on α_{n-1} at K Pauli-flipped configurations (K=1,625,991 for CNa2O3), so a direct estimator costs ~K^n evaluations for the largest molecule. The reported 200,000 moments and near-linear runtime in Fig. 4b are not consistent with this unless some additional representation (truncated basis, auxiliary network, or exact vector storage) is used silently. This is logically prior to the reader's NQS-expressiveness concern: even a perfect ground state does not yield spectra unless the moments are computable. The text's claim that 'only the ground state is needed' is therefore unsupported by the algorithmic description.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a scalable neural quantum state (NQS) approach combined with the kernel polynomial method (KPM) to compute molecular absorption spectra. An autoregressive NQS is used to approximate the ground state of the ab initio electronic Hamiltonian in second quantization, and the dipole autocorrelation function is expressed through Chebyshev moments Ω_n = <0|μ T_n(H~) μ|0> (Eq. 26). The authors report ground-state energies for molecules with up to 52 electrons (CNa2O3) and claim FCI-level precision for the resulting spectra, with spectral comparisons for H2O and NH3 against truncated FCI baselines. The paper emphasizes parallel batch autoregressive sampling, GPU-based local energy computation, and a recursive moment evaluation that avoids storing excited states.","tokens_in":11986,"tokens_out":9260,"duration_ms":93436,"significance":"If correct, the method would extend NQS-based quantum chemistry from ground states to optical spectra for systems of roughly 50 electrons, beyond the reach of FCI and competitive with approximate excited-state methods. The paper's contributions include a parallel sampling scheme, a memory-efficient local energy tensor representation, and public code links. However, the manuscript's central algorithmic gap - the missing specification of how the KPM moments are computed from the NQS - currently prevents verification of the headline claim, and the spectral validation against exact references is limited to two small molecules with truncated FCI baselines.","major_comments":[{"comment":"The manuscript does not specify how the states |α_n> = T_n(H~)|α> are represented or how the overlaps <α_n|α_n> and <α_{n+1}|α_n> in Eq. (31) are computed. The only wavefunction representation described in the paper is the autoregressive ground-state NQS; |α_n> is not a ground state and the sampling machinery in Section 3 (Variational Monte Carlo optimization) applies only to the probability distribution |Ψ(x,θ)|^2 of the ground-state ansatz. Evaluating <α_n|α_n> by sampling would require either a generator for |α_n|^2 or pointwise values α_n(x) for sampled x, and the recurrence (30) makes each α_n depend on α_{n-1} evaluated at K Pauli-flipped configurations (K=1,625,991 for CNa2O3). A direct implementation would therefore have cost growing rapidly with n, inconsistent with the near-linear scaling in Fig. 4b. The statement in the Introduction that 'only the ground state is needed' is unsupported by the algorithmic description; an additional representation or estimator for the |α_n> states is required. This gap is load-bearing for the central claim of computing spectra via Eq. (26).","section":"3 Methods, KPM for absorption spectra (Eqs. 30-31)"},{"comment":"The H2S row in Table 1 lists an HF ground-state energy of -349.3114 Hartree, about 45 Hartree above the CCSD and FCI values of -394.3546 Hartree. For the STO-3G basis, the HF energy should be within a few tenths of a Hartree of the correlated result; the tabulated value is unphysical and likely a data-entry error. Because Table 1 is the primary evidence for the ground-state accuracy of sNQS-KPM, the authors should correct this entry and clarify how the mean absolute errors in the last row are computed (which molecules are included for each method).","section":"1 Results, Table 1"},{"comment":"The only spectral baselines shown are for H2O and NH3, and those FCI references are obtained by diagonalizing only the lowest 1000 eigenstates of the full FCI matrix. The authors acknowledge minor discrepancies in relative peak heights arising from this truncation, so the comparison is indicative rather than exact. For the larger molecules in Fig. 3 (C2H4O, Li2CO3, C6H8, CNa2O3), no spectral reference is provided, so the claim of FCI-level precision for absorption spectra is not directly validated. The ground-state energy agreement in Table 1 is necessary but not sufficient: the KPM spectrum depends on the moments Ω_n via Eq. (26), and the Jackson kernel and truncation at N_Ω introduce additional approximations. The authors should either provide a spectral baseline for at least one molecule beyond H2O and NH3 using a high-level method (e.g., DMRG or FCI in a reduced space) or qualify the claim.","section":"1 Results, 'Absorption spectra of different molecules' and Fig. 3"}],"minor_comments":[{"comment":"The Jackson kernel formula appears to be misprinted: the standard expression is (N-n+1) cos(π n/(N+1)) + sin(π n/(N+1)) cot(π/(N+1)) divided by (N+1), whereas the text has (N-n-1). Please check whether this is a typo in the formula or in the implementation.","section":"3 Methods, Eq. (21)"},{"comment":"The text states that the computation time for the moments is approximately linear in the number of moments 'as shown in Fig. 3b'; the correct reference appears to be Fig. 4b, which displays the time versus N_Ω relationship.","section":"1 Results, 'Resolution of absorption spectra'"},{"comment":"The empirical formula N_Ω = 400|e_g| mixes a dimensionless count with an energy in Hartree; the authors should clarify its domain of validity (e.g., basis set, energy range) or present it as a practical guideline rather than a universal rule.","section":"1 Results, 'Resolution of absorption spectra', Eq. (11)"},{"comment":"The caption says 'The black dotted line represents the absorption spectra of H2O and NH3, obtained by diagonalizing the smallest 1000 eigenvalues...' but the figure contains separate panels for H2O and NH3; please clarify whether the black dotted line is the FCI reference for each panel.","section":"1 Results, Fig. 3 caption"}],"recommendation":"major_revision","confidential_remarks":"The central issue is that the KPM moment evaluation is not described. The paper claims a recursive computation but does not state how |α_n> is represented or how the overlaps are sampled. This is a reproducibility-blocking gap, not a mere presentation issue. The H2S HF value in Table 1 also suggests a data error. The authors should be asked to provide a complete algorithmic description of the moment computation, including the cost scaling, before resubmission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nYou should know two things about this paper: it's an original combination of neural quantum states and the kernel polynomial method for molecular absorption spectra, and it has a gap in its algorithmic description that is much more serious than the reader's report suggests. The key issue is that the paper never states how the KPM moments are computed from the NQS ground state.\n\nThe good parts: integrating autoregressive NQS with KPM is new for ab initio spectra, and the engineering around parallel sampling and GPU local energy is credible. The ground-state energies for small molecules match FCI to five decimal places, and they ship code and data. That is real evidence.\n\nThe bad part: Eqs. 30-31 define the moments as overlaps of the states |α_n> = T_n(H~) μ|0>, generated by repeated application of the Hamiltonian. The NQS only represents the ground state |0>. It does not represent |α_n>, and applying H~ n times to μ|0> is not tractable for a 52-electron molecule. No estimator for these overlaps is provided. This is not a minor omission; it is the central algorithm. The claim that 'only the ground state is needed' is true in principle, but the moments still require manipulating excited-type states, and the manuscript doesn't say how. The reported linear scaling of the moment computation time is hard to square with this.\n\nOther soft spots are secondary: the H2S HF entry in Table 1 is about 45 Hartree off and must be a typo; Eq. 17 has an unexplained factor of 1/2 that would halve the energy if actually used; there are no statistical error bars on the spectra; and the FCI baselines for H2O/NH3 use only 1000 eigenstates, which is fine for peaks but not heights.\n\nAll that said, the code is on GitHub, so a referee can check whether the moment computation has an implementation trick that the text leaves out. That makes the paper worth sending to peer review, but only with the expectation that the methods section gets a major rewrite. If the gap is real, the headline claim is unsupported; if it's not, the authors need to say how it's not.\n\nFor you: interesting to read, but don't cite it yet.\n\nCheers","headline":"Original NQS+KPM combination for optical spectra, but the manuscript never explains how the KPM moments are computed from the neural ground state, which is a load-bearing gap; the code may help, but the text is incomplete.","tokens_in":12509,"tokens_out":4098,"would_cite":false,"duration_ms":51291,"reading_group":"maybe","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 claims that combining autoregressive neural-network ground states with the kernel polynomial method reproduces exact absorption spectra for molecules with up to 52 electrons, using only the ground state.","keywords":["neural quantum states","kernel polynomial method","absorption spectra","variational Monte Carlo","autoregressive sampling","second quantization","Chebyshev moments","full configuration interaction"],"falsifier":"A converged DMRG or selected CI calculation of the absorption spectrum of CNa2O3 in the STO-3G basis, compared peak-by-peak with the sNQS-KPM spectrum; disagreement in peak positions beyond linewidth would disprove the FCI-level claim for large molecules.","tokens_in":11481,"feed_emoji":"⚛️","tokens_out":6176,"duration_ms":58286,"temperature":0.7,"pith_summary":"The paper aims to make optical absorption spectra of molecules with dozens of electrons computationally accessible at full configuration interaction (FCI) accuracy. It does so by computing the spectrum from the ground state alone, using the kernel polynomial method to build the dipole autocorrelation function from Chebyshev moments, and it represents the ground state with an autoregressive neural-network quantum state trained by variational Monte Carlo. The reported results cover molecules from 8 to 52 electrons, with spectra that match FCI for water and ammonia and, the authors argue, FCI-level ground states for larger systems where no exact reference exists. If correct, this means spectra for strongly correlated molecules no longer require storing or diagonalizing excited states, reducing both runtime and memory.","feed_headline":"Absorption spectra for 52-electron molecules from a neural state","feed_subtitle":"Only the ground state is needed: moments of the dipole operator rebuild the whole spectrum.","key_machinery":"The load-bearing object is the Chebyshev moment sequence $\\Omega_n = \\langle 0|\\hat{\\mu}T_n(\\tilde{H})\\hat{\\mu}|0\\rangle$ evaluated recursively from the ground state, so that the spectrum is a polynomial expansion of the dipole autocorrelation function. The ground state is provided by an autoregressive neural-network quantum state whose amplitude is a product of conditional probabilities, enabling exact parallel sampling with a number-conserving constraint, and whose local energy is computed on GPUs from a tensorized Hamiltonian. A Jackson kernel multiplies the moments to control truncation error, and the empirically chosen number of moments is $N_\\Omega = 400|e_g|$.","core_discovery":"The central claim is that the pair (neural quantum state, kernel polynomial method) computes absorption spectra at FCI-level precision using only the ground state. The spectrum follows from moments $\\Omega_n = \\langle 0|\\hat{\\mu}T_n(\\tilde{H})\\hat{\\mu}|0\\rangle$, obtained by the Chebyshev recursion $|\\alpha_{n+1}\\rangle = 2\\tilde{H}|\\alpha_n\\rangle - |\\alpha_{n-1}\\rangle$ starting from the dipole-excited state $|\\alpha\\rangle = \\hat{\\mu}|0\\rangle$. Because the method never needs eigenstates, it avoids the exponential memory of configuration interaction. The authors validate against FCI for small molecules, report good peak placement, and apply the pipeline to molecules with up to 52 electrons and Hamiltonians with over one million terms, claiming ground-state energies that stay within about $10^{-4}$ Hartree of available references.","pith_inferences":["The spectral accuracy is only as good as the dipole-weighted part of the approximate ground state; agreement in total energy does not guarantee that the low-lying transition moments are correct, so spectra for the largest molecules should be treated as predictions pending independent checks.","Because the KPM moments are correlation functions, the same pipeline could yield conductivities, spin correlations, or higher-order response functions by swapping the dipole operator, provided the operator is cheap to apply in the autoregressive basis.","The number-conserving sampling constraint and the unique-sample pruning are what make 52 electrons feasible; pushing to larger systems will likely require the same two tricks rather than a larger network alone."],"forward_implications":["With an FCI-quality ground state, absorption spectra come without storing any excited state; the memory cost drops to the Hamiltonian and the moment recursion.","The $O(N^4)$ scaling measured on H2 chains implies the method's advantage over exact diagonalization grows with molecule size.","Any two-point correlation function that can be written as $\\langle 0| A T_n(\\tilde H) B |0\\rangle$ becomes accessible with the same machinery.","Hamiltonians with millions of Pauli terms are no longer a barrier, since local energy is computed term-wise in parallel across GPUs.","The empirical moment count $N_\\Omega = 400|e_g|$ offers a practical, size-consistent recipe for choosing the resolution."],"supporting_citations":[{"why":"Supplies the Chebyshev expansion, moments, and Jackson kernel that turn ground-state correlation functions into spectra.","marker":"[20]"},{"why":"Introduces the batch autoregressive sampling and the NAQS ansatz on which the ground-state solver is built.","marker":"[34]"},{"why":"Provides the MADE parallel sampling scheme and the memory-efficiency baseline that this work improves on.","marker":"[37]"},{"why":"Supplies the variational Monte Carlo training formalism and local-energy estimator used for the ground state.","marker":"[32]"},{"why":"Contributes the number-conserving constrained sampling and pruning ideas applied in the parallel sampler.","marker":"[35]"}],"fun_headline_variants":["Neural state + kernel polynomial yields spectra from ground state","Ground-state-only absorption spectra at FCI accuracy","FCI-level optical spectra without excited-state calculations","Kernel polynomial on neural state: spectra for 52 electrons","Scalable spectra: neural quantum state and KPM"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire pipeline inherits the accuracy of the neural ground state: for the largest molecules (44 and 52 electrons), the paper asserts FCI-level accuracy without an FCI calculation to test against.","fun_headline_variants_meta":{"raw":{"variants":["Neural state + kernel polynomial yields spectra from ground state","Ground-state-only absorption spectra at FCI accuracy","FCI-level optical spectra without excited-state calculations","Kernel polynomial on neural state: spectra for 52 electrons","Scalable spectra: neural quantum state and KPM"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000671,"raw_usage":{"total_tokens":3002,"prompt_tokens":835,"completion_tokens":2167,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":451,"completion_tokens_details":{"reasoning_tokens":2090}},"tokens_in":451,"tokens_out":2167,"duration_ms":16817,"temperature":1.0,"reasoning_tokens":2090,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:34:41.612856+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A converged DMRG or selected CI calculation of the absorption spectrum of CNa2O3 in the STO-3G basis, compared peak-by-peak with the sNQS-KPM spectrum; disagreement in peak positions beyond linewidth would disprove the FCI-level claim for large molecules.","supporting_citations":[{"cited_title":"Reviews of modern physics78(1), 275–306 (2006)","cited_arxiv_id":null,"evidence_quote":"Supplies the Chebyshev expansion, moments, and Jackson kernel that turn ground-state correlation functions into spectra."},{"cited_title":"Nature Machine Intelligence4(4), 351–358 (2022)","cited_arxiv_id":null,"evidence_quote":"Introduces the batch autoregressive sampling and the NAQS ansatz on which the ground-state solver is built."},{"cited_title":"Machine Learning: Science and Technology4(2), 025034 (2023)","cited_arxiv_id":null,"evidence_quote":"Provides the MADE parallel sampling scheme and the memory-efficiency baseline that this work improves on."},{"cited_title":"Nature communications11(1), 2368 (2020)","cited_arxiv_id":null,"evidence_quote":"Supplies the variational Monte Carlo training formalism and local-energy estimator used for the ground state."},{"cited_title":"In: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, pp","cited_arxiv_id":null,"evidence_quote":"Contributes the number-conserving constrained sampling and pruning ideas applied in the parallel sampler."}],"review_version":1}