{"id":"a5e31c35-a66e-4f2b-ab54-7b42ddcb4c55","arxiv_id":"2506.08594","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A determinant-based neural network ansatz computes multiple low-lying excited states of long-range interacting spin systems with up to 300 ions, with benchmarks on exactly solvable models.","lead":"A new neural-network algorithm computes several low-energy excited states of quantum spin systems at once, using a determinant trick that keeps the states automatically orthogonal. This makes it possible to study long-range interacting systems with hundreds of spins, including trapped-ion simulators, and to understand why imperfect preparation still reproduces ground-state correlations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Large-N excited states are validated only by variational energy decrease; without local-energy variance or an independent reference, the claimed accuracy and gap/correlation conclusions for 100-300 ions are not established.","rationale":"The paper's determinant-based construction is built on a sound trace variational principle, and the small-system benchmarks (transverse-field Ising at N=20 and Haldane-Shastry at N=20 and N=128) provide credible evidence that the method can work when a reference is available. The Haldane-Shastry curriculum-learning results, with sub-10^-3 relative energy errors and reproduced long-range correlations, are the strongest support for the central claim. However, the headline results for trapped ions at N=100 and N=300 are offered without any internal eigenstate diagnostic. The reader's weakest assumption focuses on the experimental interpretation (whether the quasi-adiabatic ramp populates the computed low-lying states); that concern is real but secondary to the algorithmic core. If the NQES states are not themselves close to eigenstates at these sizes, then both the experimental 'explanation' and the gap/correlation conclusions fail. The requested variance check is a standard VMC diagnostic that does not require an exact solution and would settle whether the large-N states are genuinely eigenstates. I agree with the reader that a conditional verdict is appropriate: the method is promising and well motivated, but the large-N claims need either an internal eigenstate diagnostic, error bars, or an independent reference before they should be treated as benchmark-grade results. No fatal flaw is identified, so I keep the verdict unchanged rather than moving to reject or accept.","tokens_in":22667,"tokens_out":10261,"duration_ms":129150,"concrete_test":"Recompute the trained N=100 single-phonon-mode and N=300 power-law-model states and evaluate the Monte Carlo variance of the diagonal local-energy matrix elements Var[E_loc,ii] for each of the K states (equivalently, the squared residual norm ||H|psi_i> - E_i|psi_i>||^2). If the per-site variance is not at least an order of magnitude smaller than the squared gaps reported in Fig. 4 (for example, variance per site < 0.1 x gap^2), then the optimized states are not eigenstates and the claimed energies, gap scaling, and correlation features are unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the trapped-ion sections (Figs. 3c-g and 4a-d), the evidence that NQES has actually found the low-lying eigenstates is only a decrease of the total variational energy as the hidden-neuron density M rises from N to 3N (Fig. 3c), plus qualitative similarity of correlation maps. No local-energy variance, no overlap with a reference, and no independent computation at N>=100 are reported; the small-N=20 exact-diagonalization checks in Supp. Figs. S5-S6 do not control for expressivity or optimization failures at N=100 and N=300. For the central claim of 'accurate and efficient' excited states to hold in the headline regime, each of the K diagonalized states must be close to a true eigenstate of the model Hamiltonian. A variational energy decrease is necessary but not sufficient: the ansatz can be trapped in a local minimum, or the Monte Carlo estimates of the local energy matrix can be biased by the determinant sampling, while the total energy still decreases with M. Hence the gap scaling in Fig. 4 and the excited-state correlation patterns in Figs. 3d-g and 4c-d are not yet certified as physical, and the claim that NQES 'successfully uncovers gap scaling and correlation features' out to 300 ions rests on an unverified premise.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper introduces NQES, a variational neural-network method for simultaneously computing several low-lying excited states of quantum spin Hamiltonians. K restricted Boltzmann machine ansätze are combined into a determinant Ψ(S)=det[ψ_i(S_j)]; the total energy of the expanded Hamiltonian is minimized, which yields the sum of the K lowest eigenvalues, and the local energy matrix is diagonalized to extract individual energies and state-resolved observables. The method is benchmarked on a 20-site transverse-field Ising chain against exact diagonalization and on 20/128-site Haldane-Shastry chains against analytic spectra; the latter requires a curriculum-learning initialization from the XXZ model. The paper then applies NQES to two trapped-ion Ising models: a 100-ion single-phonon-mode model with alternating-sign all-to-all couplings, and a power-law antiferromagnetic model up to 300 ions, reporting gap scaling and correlation maps. The authors argue that the similar correlation patterns of low-lying excited states and the ground state explain the surprising robustness of the experimental ground-state correlations to nonadiabatic ramps.","tokens_in":22917,"tokens_out":7065,"duration_ms":80821,"significance":"If correct, the paper establishes a scalable classical tool for excited-state spectra and correlations in long-range spin systems, a regime where tensor-network methods are limited and where existing neural excited-state methods often require explicit orthogonalization penalties. The determinant construction is a clean way to enforce linear independence without penalty terms, and the benchmarks at N=20 (ED) and N=20/128 (analytic Haldane-Shastry) are meaningful and appear correctly implemented. The memory-efficient Krylov stochastic-reconfiguration solver and the bit-encoding Monte Carlo updates are useful engineering contributions. The main caveat is that the headline trapped-ion results at N=100 and N=300 are not certified: the only convergence evidence is a decrease of the variational energy with hidden-neuron density, which is necessary but not sufficient for the claim that individual excited states are accurately represented. The physical explanation of the experiment also relies on an unverified assumption about which excited states are populated during the ramp.","major_comments":[{"comment":"For the N=100 and N=300 systems, the only evidence that NQES has found the true low-lying eigenstates is the decrease of the variational total energy as M/N increases from 1 to 3 (Fig. 3c) and the qualitative similarity of the correlation maps. A variational energy decrease is necessary but not sufficient: the optimization could be trapped in a local minimum, or the Monte Carlo estimate of the local-energy matrix could be biased by the determinant sampling, while the total energy still declines with M. The small-N=20 exact-diagonalization checks in Supplementary Figs. S5-S6 do not control for expressivity or optimization failures at N=100/300. The nonmonotonic behavior at M=5N further shows that the M-dependence is not a simple convergence certificate. Please report per-state local-energy variances (or equivalently the variance of Tr[Eloc] and of the diagonalized eigenvalues) and, where possible, cross-check at intermediate sizes (e.g., N=32/64) against ED or high-quality tensor-network results. Without this, the claims in the abstract and the statement that NQES 'successfully uncovers gap scaling and correlation features' for up to 300 ions are not established.","section":"Trapped ions, Fig. 3c and Fig. 4"},{"comment":"The explanation of the experimental robustness rests on the premise that the quasi-adiabatically prepared state is dominated by the first few low-lying eigenstates computed by NQES. The paper states that truncated ramps inevitably populate excited states, but it never simulates the ramp dynamics or quantifies the overlap of the time-evolved state with the computed eigenstates. For N=20 (or another size accessible to ED), please compute the actual time-dependent Schrödinger evolution under the experimental ramp and compare the instantaneous state's projection onto the NQES eigenstates. Only then can one conclude that these specific eigenstates, rather than a broad superposition of higher states, are responsible for the measured correlation patterns.","section":"Trapped ions, first scenario"},{"comment":"At N=128, the paper reports that both ground and first excited state energies agree with the analytic spectrum to relative error 10^-3, but the only state-resolved correlation function shown (Fig. 2e) is for the ground state. Since the abstract claims faithful reproduction of long-range spin correlations for multiple excited states, please also show excited-state correlation functions at N=128 (or at least at N=20 where ED is available) to verify that the state-resolved observables, not only the energies, are accurate at scale.","section":"Haldane-Shastry results, Fig. 2e"}],"minor_comments":[{"comment":"The text contains a typo: 'Y et' should be 'Yet' in the opening sentence of the abstract/full text.","section":"Abstract"},{"comment":"The x-axis is labeled 'Hidden neuron density' with tick values extending to 6, but the text describes M/N = 1,...,5; please clarify the axis range and tick labels to match the described protocol.","section":"Fig. 3c"},{"comment":"The sentence 'For models which in general does not possess the ground-state correlation robustness' has a grammatical error; please rephrase for clarity.","section":"Trapped ions, second scenario"},{"comment":"The statement that the ground states are configured with σ_i^z = ±sign(b_ik) should specify that the sign choice is global up to the Z2 symmetry; as written, it could be misread as allowing independent per-site signs.","section":"Supplementary Section IV C"},{"comment":"For a computational methods paper, consider making the code and data available at submission time or providing a reviewer access link, rather than promising public release only upon publication.","section":"Code and data availability"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely to be a useful contribution after the convergence and ramp-dynamics validation are added. The small-system benchmarks are sound and the determinant construction is elegant, but the large-N trapped-ion results are the headline and currently rest on variational energy decrease alone. I recommend major revision rather than rejection because the missing checks appear feasible within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The bottom line: this is a real method paper, not an incremental tweak. The determinant-based orthogonalization comes from Pfau et al., but the adaptation to spin RBMs with long-range, sign-problematic Hamiltonians, the curriculum pipeline, and the Krylov SR memory reduction are genuinely new and practically useful. The small-system benchmarks are clean: 20-site transverse-field Ising against ED, 20- and 128-site Haldane-Shastry against exact solutions, and 20-site ED checks for both trapped-ion models. The 128-site HS result at sub-10^-3 relative error is the strongest piece of evidence that the method works where tensor networks struggle.\n\nThe soft spots are mostly concentrated in the large-N trapped-ion sections. For the 100-ion and 300-ion results, the evidence that NQES has actually converged to the low-lying eigenstates is a variational energy decrease as M grows, plus qualitative similarity of correlation maps. No local-energy variance, no overlap with a reference, no independent check at N>=100. The stress-test note is right: a variational energy decrease is necessary but not sufficient. That doesn't sink the paper, but it means the gap scaling in Fig. 4 and the correlation patterns in Figs. 3-4 are not yet certified as physical. The claim that these states 'closely match' the experiment is also qualitative; the paper doesn't simulate the ramp dynamics or verify that the populated states are the computed ones. That assumption is stated but not tested.\n\nNone of this is fatal. The methods section is solid, the algorithm is clearly explained, and the authors are honest about the sign problem and the undecidability caveat. The main missing piece is code and data, which they say will be on Zenodo upon publication; that's the right move, but it makes current verification harder.\n\nWho this is for: anyone working on neural quantum states, long-range spin models, or classical simulation of trapped-ion experiments. It deserves a serious referee. I'd send it out, and I'd expect the referee to push for variance estimates and an independent convergence check at the largest sizes before the headline claims are taken as benchmark-grade.\n\nRecommendation: accept for peer review, with the understanding that it will need revision.","headline":"A genuine extension of Pfau's excited-state method to spin systems, with clean small-system benchmarks but unverified 100-300 ion claims.","tokens_in":23510,"tokens_out":1644,"would_cite":true,"duration_ms":18294,"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":"The NQES algorithm trains K restricted-Boltzmann neural states inside a determinant-shaped wavefunction so that K low-lying excited states of a long-range spin system are solved together, with demonstrated accuracy up to 300 ions.","keywords":["neural quantum excited states","restricted Boltzmann machine","long-range Ising model","trapped-ion quantum simulator","Haldane-Shastry model","stochastic reconfiguration","sign problem","variational Monte Carlo"],"falsifier":"Measure the actual excited-state populations after the truncated ramp in the 100-ion alternating-sign system, or simulate the ramp time evolution classically for small $N$ and compare the resulting correlation matrix with NQES predictions; if the prepared state has substantial weight outside the low-lying manifold, or its correlations differ from the computed excited-state correlations, the explanation fails.","tokens_in":22438,"feed_emoji":"⚛️","tokens_out":9360,"duration_ms":102547,"temperature":0.7,"pith_summary":"The paper introduces the neural quantum excited-state (NQES) algorithm, which combines K restricted Boltzmann machine wavefunctions into a Slater-determinant-like composite ansatz so that K low-lying excited states of a spin Hamiltonian are obtained in one variational run, with no explicit orthogonalization. Its central demonstration is that this works for long-range and all-to-all interactions where tensor-network methods struggle: a 20-site transverse-field Ising chain, a 128-site Haldane-Shastry chain with sub-$10^{-3}$ relative energy errors, a 100-ion 2D Wigner-crystal Ising model, and a power-law antiferromagnetic model with up to 300 ions. The paper also uses NQES to solve an experimental puzzle: a 300-ion simulator's quasi-adiabatic ramp is too short to be adiabatic, yet its measured correlations match ground-state predictions, and the computed low-lying excited states carry essentially the same correlation pattern. If correct, NQES offers a scalable classical route to excitation spectra, state-resolved correlations, and gap scaling in long-range interacting quantum systems, with direct use in benchmarking quantum simulators.","feed_headline":"Multiple excited states in one shot for a 300-ion simulator","feed_subtitle":"Computes several eigenstates in one pass and explains why fast ion ramps still mimic ground-state correlations.","key_machinery":"The load-bearing object is the expanded wavefunction $\\Psi(\\mathcal{S}) = \\det[\\psi_k(S_{k'})]$, built from $K$ independently parameterized restricted Boltzmann machines—neural-network state ansätze in which visible spin units couple to hidden binary units. The determinant enforces mutual orthogonality automatically and prevents spectral collapse, while the cost function is the trace of a sampled local-energy matrix $E_{\\mathrm{loc}} = \\mathbb{E}[\\,\\Psi^{-1} H \\Psi\\,]$; diagonalizing it after training gives the eigenvalues and, through the same rotation, the correlation functions of each state. Three engineering pieces carry the scalability: stochastic reconfiguration with a matrix-free MINRES-QLP Krylov solver so the covariance matrix is never stored, a bit-packed spin encoding that reduces Pauli actions to shifts, XORs, and population counts, and a curriculum-learning schedule with a local unitary transformation that seeds the sign structure needed for the Haldane-Shastry model.","core_discovery":"The central claim, stated on the paper's own terms, is that multiple low-lying eigenstates can be learned simultaneously by minimizing the total energy of the expanded determinant ansatz, and that orthogonality comes for free because the determinant vanishes whenever the single states become linearly dependent. At convergence the sampled $K\\times K$ local-energy matrix approaches the exact eigenvalues up to a basis rotation, and diagonalizing that matrix yields the individual excitation energies plus state-resolved expectation values via the same rotation. The numerical evidence spans the transverse-field Ising chain (all four lowest states with relative errors below $10^{-3}$ for $M\\ge 4N$ hidden units), the Haldane-Shastry chain at $N=128$ after a curriculum-learning pass through an XXZ model, the 100-ion alternating-sign all-to-all Ising model in both ferromagnetic and paramagnetic regimes, and the power-law antiferromagnetic model up to 300 ions where the gap is size-independent at $h=2$ and closes near $h=0.5$.","pith_inferences":["Editorial extension: the paper does not simulate the ramp dynamics, so a natural next test is to evolve the actual time-dependent Hamiltonian for $N=20$–$100$ and check whether the population-weighted correlation matrix matches NQES's static low-lying eigenstates.","Editorial extension: the correlation robustness in the alternating-sign model is likely tied to the single-phonon-mode structure of that Hamiltonian; in broader classes of long-range models, excited states should be expected to differ from the ground state, as the paper's antiferromagnetic example itself shows.","Editorial extension: a direct experimental probe would be to selectively drive or measure the first excited state of the power-law model at $N=300$; NQES predicts long-range correlations in that state, which would distinguish it sharply from the short-range-ordered ground state."],"forward_implications":["System sizes of 100–300 sites with all-to-all or power-law long-range couplings become accessible for excited-state spectra, correlations, and gaps, beyond exact diagonalization and area-law tensor-network methods.","The fast-ramp experiment on the 100-ion alternating-sign model is explained: the first and second excited states show correlation maps nearly indistinguishable from the ground state, in both ferromagnetic and paramagnetic phases.","For the power-law antiferromagnetic model the first excited state has a different, longer-range correlation pattern, so ground-state correlations are not robust to nonadiabatic excitations; NQES-computed gaps indicate how slow adiabatic ramps must be.","Because the determinant construction is agnostic to the underlying ansatz, the same simultaneous-excited-state scheme can be combined with other network architectures.","The method provides a quantitative tool for benchmarking quantum simulators: compute spectra and correlations classically, then compare against device measurements."],"supporting_citations":[{"why":"It supplies the 300-ion 2D Wigner-crystal experiment, its Ising Hamiltonian parameters, and the quasi-adiabatic ground-state correlation measurements that NQES is used to interpret.","marker":"[13]"},{"why":"It supplies the determinant-based simultaneous excited-state construction that NQES extends from continuous real space to many-body spin systems.","marker":"[32]"},{"why":"It supplies the restricted Boltzmann machine wavefunction ansatz and its variational Monte Carlo evaluation, which serve as the single-state building blocks.","marker":"[17]"},{"why":"It supplies stochastic reconfiguration, the optimization scheme the paper generalizes to the composite determinant ansatz.","marker":"[47]"},{"why":"It supplies the MINRES-QLP Krylov solver used to apply the inverse covariance matrix without storing it, enabling optimization with millions of parameters.","marker":"[48]"},{"why":"They supply the exactly solvable Haldane-Shastry spectrum used as the benchmark for the computed energies.","marker":"[35, 36]"},{"why":"It supplies the result that no single-qubit local unitary makes the Haldane-Shastry Hamiltonian stoquastic, identifying the sign problem that motivates the curriculum initialization.","marker":"[37]"},{"why":"It supplies curriculum learning, the staged training strategy used to overcome the Haldane-Shastry sign problem.","marker":"[38]"}],"fun_headline_variants":["Neural network computes multiple excited states for trapped-ion systems","Excited states without orthogonalization: neural algorithm scales to 300 ions","One pass, multiple eigenstates: NQES tackles long-range ion crystals","NQES: simultaneous eigenstates for 300-ion antiferromagnetic crystals","Neural excited states: from Haldane-Shastry to 300-ion crystals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The trapped-ion explanation assumes the fast experimental ramp mainly excites only the first few low-lying eigenstates that NQES computes; the paper never simulates the ramp dynamics or verifies those populations.","fun_headline_variants_meta":{"raw":{"variants":["Neural network computes multiple excited states for trapped-ion systems","Excited states without orthogonalization: neural algorithm scales to 300 ions","One pass, multiple eigenstates: NQES tackles long-range ion crystals","NQES: simultaneous eigenstates for 300-ion antiferromagnetic crystals","Neural excited states: from Haldane-Shastry to 300-ion crystals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000507,"raw_usage":{"total_tokens":2505,"prompt_tokens":1013,"completion_tokens":1492,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":629,"completion_tokens_details":{"reasoning_tokens":1394}},"tokens_in":629,"tokens_out":1492,"duration_ms":14311,"temperature":1.0,"reasoning_tokens":1394,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:07:19.942864+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the actual excited-state populations after the truncated ramp in the 100-ion alternating-sign system, or simulate the ramp time evolution classically for small $N$ and compare the resulting correlation matrix with NQES predictions; if the prepared state has substantial weight outside the low-lying manifold, or its correlations differ from the computed excited-state correlations, the explanation fails.","supporting_citations":[{"cited_title":"Guo, Y .-K","cited_arxiv_id":null,"evidence_quote":"It supplies the 300-ion 2D Wigner-crystal experiment, its Ising Hamiltonian parameters, and the quasi-adiabatic ground-state correlation measurements that NQES is used to interpret."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the determinant-based simultaneous excited-state construction that NQES extends from continuous real space to many-body spin systems."},{"cited_title":"Sorella, M","cited_arxiv_id":null,"evidence_quote":"It supplies stochastic reconfiguration, the optimization scheme the paper generalizes to the composite determinant ansatz."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the MINRES-QLP Krylov solver used to apply the inverse covariance matrix without storing it, enabling optimization with millions of parameters."},{"cited_title":"Bravyi, G","cited_arxiv_id":null,"evidence_quote":"It supplies the result that no single-qubit local unitary makes the Haldane-Shastry Hamiltonian stoquastic, identifying the sign problem that motivates the curriculum initialization."},{"cited_title":"Bengio, J","cited_arxiv_id":null,"evidence_quote":"It supplies curriculum learning, the staged training strategy used to overcome the Haldane-Shastry sign problem."}],"review_version":1}