{"id":"86a88a4f-0bf9-48de-b2f5-1cb309c843ee","arxiv_id":"2505.19906","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A quantum-classical resolvent method with a new fermionic block-encoding input scheme computes the spectrum and J values of 20O in a truncated sd-shell space, matching classical diagonalization.","lead":"This paper introduces a hybrid quantum-classical algorithm that computes the full bound-state spectrum and the total angular momentum of each state of a many-fermion system, demonstrated for the nucleus 20O on a noiseless quantum simulator. It couples a new fermionic Hamiltonian input scheme to a two-stage scan, and the results match classical shell-model calculations in a truncated model space.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The single-qubit validity ancilla described in the Hamiltonian input scheme cannot reject invalid multi-fermion products; the block-encoding proof in the SM assumes an unverified circuit invariant.","rationale":"Good-faith reading: the paper's algorithmic skeleton, Chebyshev expansion of the resolvent with a block-encoded second-quantized Hamiltonian and a cascading-M scan, is plausible, and the authors' classical cross-checks are positive evidence. However, the central computation depends on the ancilla registers e_P and e_Q correctly rejecting invalid fermionic products. The manuscript's own description of the single-operator circuits does not scale to multi-operator products, and the SM proof jumps from the circuit description to the invariant without proving it. If a referee implements the text literally, the block-encoded operator is not H/(B Xi). This is more fundamental than the pivot-coverage issue the reader identified: pivot coverage affects the completeness of the resolved spectrum, while the validity ancilla affects every matrix element entering the Chebyshev moments. I also agree with the reader that the abstract's 'complete bound-state spectrum' overstates what a single-configuration pivot can deliver, given the admitted missing odd-J states in the SM 'Pivot choices' section. No ad hominem is intended; this is a request for the missing circuit specification and proof, which could be supplied in a revision. The reader's CONDITIONAL verdict is appropriate, but the conditions should include an explicit validity proof for the fermionic input circuits, not only pivot coverage and error analysis.","tokens_in":18699,"tokens_out":20727,"duration_ms":236474,"concrete_test":"Take the state with b_p=0, b_q=1 and all other modes empty. Assemble the two-annihilation operator a_q a_p exactly as the elementary circuits in the main text are described (controlled flip of e_P from |1> to |0> on b_p/b_q=1, X on the target, Zs on lower indices), and read e_P after both steps. If the final state of e_P is |0>, the validity invariant behind SM Eq. (11) is violated. As a second verification, recompute the reported 20O Chebyshev moments with any spurious terms this circuit would introduce removed, and compare with classical diagonalization; agreement would show the published text does not describe the implemented circuit.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The input scheme's block-encoding proof (SM Eqs. 18-21) requires that y^P_{F,j}=y^Q_{F,j}=0 exactly when every annihilation in b_P and every creation in b^dagger_Q acted on a valid occupancy, and only then does the term contribute. The textual circuit for a_p flips the single qubit e_P from |1> to |0> controlled by b_p=|1>, then applies X to b_p and Z gates. This cannot implement the required invariant for products. For a two-annihilation term b_P=a_q a_p (p<q) acting on a state with b_p=0, b_q=1, the a_p step leaves e_P=|1> but changes b_p to 1; the a_q step then sets e_P=|0>. The term is marked physical even though a_q a_p|...0_p...1_q...>=0, and the system register has been corrupted. If 'flip' is read as a CNOT toggle, the all-occupied valid case ends with e_P=|1> and is rejected. A single reset-on-hit bit records only that at least one annihilation succeeded, not that all of them did. Since the 20O Hamiltonian contains two-body terms (Table S3), these spurious contributions would alter the Chebyshev moments unless the circuits in Fig. 1 implement a different, unspecified validation mechanism. The paper must specify the multi-operator circuits explicitly and prove the ancilla condition before the spectral results can be regarded as a correct quantum computation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a hybrid quantum-classical framework for configuration-interaction (CI) many-fermion structure calculations and demonstrates it on the 20O nucleus. The approach evaluates the retarded Green's function (resolvent) of the second-quantized Hamiltonian from Chebyshev polynomial moments computed on a quantum computer through a new 'direct basis encoding' Hamiltonian input scheme: fermionic operator monomials are realized by circuits on occupation qubits with auxiliary validity ancillae, and the Hamiltonian is block-encoded via forward/backward walk states with a claimed gate cost of O~(N_sp^{2k+1}) for Hamiltonians containing up to k-body terms. The spectral function is assembled classically from the moments, and a two-fold scan, consisting of a cascading-M scan and a resolution scan, assigns total angular momentum J to the resolved energy levels. The demonstration uses 20O in a truncated sd-shell space (0d5/2 and 1s1/2 orbits, four valence neutrons) with an effective interaction from Ref. [68]; the Chebyshev moments are evaluated with the Qiskit Statevector simulator in noiseless mode and cross-checked against classical calculations, yielding an excitation spectrum with J assignments.","tokens_in":18993,"tokens_out":49512,"duration_ms":429172,"significance":"If the Hamiltonian input scheme is correct as described, the paper contributes a concrete, oracle-free circuit construction for second-quantized fermionic Hamiltonians with up to k-body interactions that avoids Pauli-string compilation, a stated gate-cost bound, and a spectral method that can in principle return complete bound-state spectra with angular momentum assignments. The manuscript's strengths include explicitly drawn circuit building blocks, a self-contained walk-state construction, the use of standard qubitization for Chebyshev moment evaluation, and computed moments that are machine-checked against classical results, which is a commendable validation step. The physics demonstration itself is modest: an 8-mode, four-particle model space computed on a noiseless simulator with a phenomenological sd-shell interaction, so the contribution is methodological. The claimed applicability to BLFQ hadron spectra is plausible but not demonstrated in this work.","major_comments":[{"comment":"The validity-ancilla mechanism described in the main text and Fig. 1 does not, on its face, implement the condition y^P_{F,j}=y^Q_{F,j}=0 if and only if the monomial action b†_{Qj}b_{Pj}|F⟩ is physical, which is the invariant on which the block-encoding identity, SM Eq. (21), and Eq. (8) of the main text rest. For a single-qubit register e_P initialized in |1⟩ and 'flipped' controlled on each annihilated mode's occupation, the final state of e_P records the parity of the occupancies rather than their conjunction. For a two-annihilation term a_q a_p (p<q), a Fock state with b_p=0 and b_q=1 ends with e_P=|0⟩ and is accepted as physical even though a_q a_p|F⟩=0; read as a CNOT toggle, the valid all-occupied case ends with e_P=|1⟩ and is rejected, while read as a flip-only-while-|1⟩ gate, the exactly-one-occupied cases are still accepted because the failure of the first annihilation leaves e_P=|1⟩ and the second flip then toggles it to |0⟩. In the wrongly accepted cases the X gates also corrupt the system register, so the spurious terms contribute to the block-encoded matrix rather than being projected out. Because the 20O Hamiltonian contains two-body terms (Table S3), these spurious contributions would alter the Chebyshev moments ⟨ψ|T_n(H')|ψ⟩ unless the implemented circuits differ from the text. The authors must specify the multi-operator circuits explicitly and prove the ancilla invariant before the input scheme claim can be regarded as established.","section":"Hamiltonian input scheme (Fig. 1); SM Eqs. (18)-(21)"},{"comment":"The abstract claims the method is 'capable of resolving the complete bound-state spectrum' and of providing 'the total angular momentum J associated with each eigenstate,' but the results reported in the manuscript do not support the completeness claim. The cascading-M assignment logic requires the pivot |ψ_M⟩ to have nonzero overlap with every eigenstate with J≥M; with single-configuration pivots this condition is not guaranteed and in fact fails in the present calculation, as the SM ('Pivot choices') concedes that some states are absent from the spectral function (the odd-J states in Fig. 3(f)). The Summary itself more cautiously claims only that 'most eigenstates' are resolved. The assignment-by-emergence procedure will mislabel a state if the pivot at the higher M has zero overlap with it, so the claimed completeness requires either a proven overlap condition for the chosen pivots or an explicit method for selecting pivots that guarantees completeness. The paper should reconcile the abstract with the Summary and state precisely which eigenstates the demonstrated procedure actually resolves.","section":"Abstract; SM 'Pivot choices'; Summary"},{"comment":"The J-assignment protocol that produces the spectrum in Fig. 3(b) is not documented in a reproducible way. The spectral functions shown are computed only with an M=4 pivot (Fig. 3(c,d)) and an M=0 pivot (Fig. 3(e,f)); the cascading-M scan used to assign the individual J values in Fig. 3(b) is not specified: which M values were scanned, which single-configuration pivots were used at each M, which resolutions N were used, and how the criterion 'additional states emerge' was applied when comparing F_{ψ_M} with F_{ψ_{M−1}} are all absent. No peak-matching threshold or error estimate for the fitted peak positions E_x is given. Since the J assignment is a central claimed output of the method, the protocol should be described precisely, or the relevant cascading spectra should be made available.","section":"Fig. 3; 'Hybrid method' section"}],"minor_comments":[{"comment":"The integral representation in Eq. (1) integrates over all t with H→H−iϵ; the exponential e^{i(E−H)t} then diverges for t<0, so the integration range should be [0,∞) for the retarded Green's function. The discretized version in Eq. (3) correctly uses τ=0,...,N−1.","section":"Formalism, Eq. (1)"},{"comment":"The demonstration is performed on the Qiskit Statevector simulator in noiseless mode in a truncated model space with the phenomenological single-particle energies of Ref. [68]; the abstract's 'on a quantum computer' and the title's 'Ab initio' should be qualified to reflect this, with 'ab initio' reserved for the method's intended target rather than the reported calculation.","section":"Title and Abstract"},{"comment":"The phrase 'the lowest rigorous upper bound on the gate cost' is not supported, since no lower bound is proven; the sentence should state simply that the scheme achieves a gate cost of O~(N_sp^{2k+1}).","section":"Eq. (8) discussion"},{"comment":"The peak-extraction procedure should be documented: the fit model f(x) is stated, but the fitting algorithm, the statistical error on E_x, and the criterion for declaring a resolved peak are not; the use of Re⟨ψ|G(E)|ψ⟩ rather than −Im⟨ψ|G(E)|ψ⟩ also deserves a comment, since Re G has a dispersive (zero-crossing) shape near a pole.","section":"Fig. 3 caption"},{"comment":"The column header 'n l2j2m' is garbled and should read 'n, l, 2j, 2m'.","section":"SM Table S1"},{"comment":"Ref. [104] (NNDC) should be given a proper database citation rather than a bare URL and access date.","section":"References"},{"comment":"The sentence 'this input scheme preserves Hamiltonian symmetries' should specify which symmetries (presumably the M-projection) are preserved and how, since the direct-basis encoding with M-conserving matrix elements is what carries this property.","section":"Hamiltonian input scheme"},{"comment":"The 'for the first time' claim should specify that the spectrum is computed for the first time by this hybrid quantum algorithm on a simulator; classical shell-model diagonalization in this small model space is routine.","section":"Introduction and Abstract"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the manuscript builds directly on the authors' prior work (Refs. [29] and [30], arXiv:2402.08969 and PRD 111, 016013), and the revision should state explicitly what is new relative to those papers. The most serious technical issue is the validity-ancilla circuit description, which needs a concrete fix and a proof; the abstract's completeness claim should also be reconciled with the Summary's 'most eigenstates.' The paper is within the scope of the journal and, once the circuit details and claims are corrected, would be a plausible contribution to quantum algorithms for nuclear spectroscopy."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The one thing to know: this paper has a real new idea—computing complete bound-state spectra and total angular momenta from a block-encoded Hamiltonian using a resolvent plus Chebyshev moments on a quantum computer. If the input scheme holds up, it goes beyond the usual ground-state-focused algorithms and gives you J quantum numbers as a bonus. But the abstract says 'complete bound-state spectrum,' and the paper's own SM admits some odd-J states are missed with the chosen pivots, so treat the claim as 'most states until you scan more pivots.'\n\nWhat it does well: the block-encoding algebra in the SM is internally consistent, and the simulated Chebyshev moments are cross-checked against classical moments—so the circuits they actually ran on Qiskit do something correct. The cascading-M scan is a clever way to assign J by comparing spectral functions from different M-projections. The authors are also honest about limitations: missing states, the excluded 0d3/2 orbit, simulator-only execution. The gate-count claim O~(N_sp^{2k+1}) is plausible, though stated rather than proved.\n\nNow the soft spots, in proportion. The serious one is the validation ancilla. The text describes a single qubit e_P that 'flips' from 1 to 0 when an annihilation acts on an occupied mode. For a product like a_q a_p, that cannot enforce that every annihilation succeeded. If 'flip' means reset-to-0, a subsequent failure on a vacant mode is ignored and invalid products get marked valid; if it means a CNOT toggle, a fully valid product leaves e_P=1 and gets rejected. Either way, the SM proof (Eqs. 18–21) relies on the invariant y^P=y^Q=0 iff the fermionic product is valid, and the textual description does not establish that invariant for multi-operator terms. The fact that the physics cross-checks pass suggests the actual Fig. 1 circuits may implement a different, correct mechanism—but the paper as written doesn't show it. That is a fixable but load-bearing omission.\n\nLesser issues: the 'complete' wording, no code or exhaustive circuit listings, no quantitative resolution/truncation error analysis, and the pivot-coverage problem (acknowledged in the SM, but it does undercut the headline claim).\n\nWho is this for: people working on quantum algorithms for nuclear or hadronic structure, and anyone interested in Hamiltonian input schemes that dodge Pauli-string overhead. I would send it to peer review, but with major revisions demanded: spell out the multi-operator validation circuits, prove the ancilla invariant, ship code or full circuit specifications, and soften 'complete' to 'most' or 'scannable.' The core idea is worth engaging seriously.","headline":"A genuinely new hybrid approach to full nuclear spectra plus J values, but the abstract overreaches with 'complete' and the validation-ancilla description is under-specified in a load-bearing way.","tokens_in":19579,"tokens_out":4943,"would_cite":false,"duration_ms":51878,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["21.60.Cs","03.67.Ac","21.10.Dr"],"model":"deepseek-v4-flash","headline":"A quantum-classical algorithm now resolves complete bound-state spectra of many-fermion systems, assigning each eigenstate its total angular momentum.","keywords":["quantum computation","nuclear structure","configuration interaction","bound-state spectrum","angular momentum","block encoding","Chebyshev polynomial","resolvent"],"falsifier":"Take a small configuration-interaction Hamiltonian whose exact eigenstates are known, run the cascading-$M$ scan with a pivot that is a single Slater determinant deliberately orthogonal to one eigenstate of angular momentum $J \\ge M$, and observe that this state never appears in any spectral function; this directly contradicts the claim that the method resolves the complete bound-state spectrum for arbitrary pivots.","tokens_in":18452,"feed_emoji":"⚛️","tokens_out":7091,"duration_ms":60836,"temperature":0.7,"pith_summary":"This paper introduces a hybrid quantum-classical algorithm that aims to extract the complete bound-state spectrum of a self-bound many-fermion system, including the total angular momentum $J$ of every eigenstate, from a quantum computer. The key move is to block-encode the second-quantized Hamiltonian through a new fermionic circuit representation that avoids the overhead of converting operators into Pauli strings, then evaluate Chebyshev moments of the resolvent $G(E)=1/(E-H)$ and assemble the spectral function classically. A two-fold scan—varying the $M$-projection of the pivot and increasing the resolution—assigns $J$ values to each resolved level. The authors apply the method for the first time to the oxygen-20 nucleus with a realistic strong-interaction Hamiltonian and find agreement with classical shell-model results, with direct extension to hadron spectra in basis light-front quantization.","feed_headline":"Quantum-classical method yields full nuclear spectrum with spins","feed_subtitle":"First demonstration on oxygen-20 resolves bound-state energies and angular momenta from a realistic strong-force interaction.","key_machinery":"The load-bearing object is the resolvent $G(E) = (E - H)^{-1}$, expressed through the integral identity $G(E) = -i \\int_{-\\infty}^{\\infty} e^{iEt} e^{-iHt} dt$ and evaluated by a truncated Chebyshev polynomial expansion of $e^{-iHt}$, so that only moments $\\langle \\psi_\\mathrm{out} | T_n(H') | \\psi_\\mathrm{in} \\rangle$ must be produced on the quantum device. The Hamiltonian is block-encoded by a fermionic circuit representation built from direct occupation-basis encoding with validation registers that mark physical versus nonphysical actions of creation and annihilation operators, eliminating the Pauli-string compilation of standard mappings and the oracle/uncomputation overhead of other block encodings. A quantum-walk-style construction of forward and backward walk states turns the scaled Hamiltonian matrix element $\\langle G | H | F \\rangle / (B\\Xi)$ into an inner product, and Chebyshev polynomials are block-encoded by alternating the walk operator with a reflection on ancilla qubits. The cascading-$M$ scan then reads off $J$ from the emergence pattern of peaks in the spectral function.","core_discovery":"The central claim is that the full bound-state spectrum and the total angular momentum of each eigenstate can be obtained by combining a quantum-computed set of Chebyshev moments $\\langle \\psi | T_n(H') | \\psi \\rangle$ with a classical reconstruction of the spectral function $F_\\psi(E) = \\Re \\langle \\psi | G(E) | \\psi \\rangle$. The $J$-assignment is made by a cascading-$M$ scan: a state first appearing when the pivot projection is lowered from $M$ to $M-1$ carries $J = M-1$, provided the pivot has nonzero overlap with it. The paper reports the first such computation for $^{20}\\mathrm{O}$ using a realistic strong interaction, resolving the energies and $J$ values of most bound eigenstates and matching classical calculations.","pith_inferences":["Editorially, the completeness of the spectrum hinges on pivot coverage: a single-determinant pivot that is orthogonal to some eigenstate with $J \\ge M$ will simply not show that state, and the paper itself notes that odd-$J$ states can be missing from one pivot's spectral function; a practical recipe must therefore use several pivots or a superposition.","Editorially, the $\\tilde{O}(N_\\mathrm{sp}^{2k+1})$ gate count is the lowest rigorous upper bound for general $k$-body Hamiltonians, yet it still grows rapidly with the single-particle basis, so the near-term niche is small valence-space problems rather than full no-core shell-model scales.","Editorially, the symmetry-preserving block encoding could be combined with quantum subspace diagonalization to target a fixed $J$ sector directly, potentially removing the need for the cascading-$M$ scan.","Editorially, a natural stress-test is to apply the method to a nucleus with closely spaced levels of different angular momenta and compare the $J$ assignment against exact diagonalization for several randomly chosen single-configuration pivots."],"forward_implications":["For the first time, a full bound-state spectrum with $J$ assignments is produced for $^{20}\\mathrm{O}$ from a realistic strong-interaction Hamiltonian on a quantum computer.","The method carries a gate cost $\\tilde{O}(N_\\mathrm{sp}^{2k+1})$ for Hamiltonians with up to $k$-body interactions, the lowest rigorous upper bound for a general $k$-body input, making larger configuration-interaction spaces accessible than with Pauli-string-based encodings.","The same block-encoding and scan framework applies directly to hadron spectra computed in basis light-front quantization, where a 2D harmonic-oscillator basis replaces the 3D basis used here.","Because the input scheme preserves Hamiltonian symmetries, targeted $M$-projection calculations can prune the Hilbert space and isolate states by angular momentum.","The resolvent-based machinery extends to response functions and reaction observables, not just bound-state energies."],"supporting_citations":[{"why":"Supplies the realistic sd-shell effective interaction used for the 20O calculation.","marker":"[68]"},{"why":"Provides the resolvent/strength-function formalism that converts Chebyshev moments into the spectral function.","marker":"[88]"},{"why":"Establishes the Chebyshev polynomial expansion of the time-evolution operator used to approximate the resolvent.","marker":"[89]"},{"why":"The kernel polynomial method, the standard background for expanding functions of H in Chebyshev polynomials.","marker":"[90]"},{"why":"The quantum-walk formalism that the Hamiltonian block encoding builds on.","marker":"[95]"},{"why":"Standard protocol for block-encoding Chebyshev polynomials via alternating applications of the walk operators and reflections.","marker":"[103]"},{"why":"Prior work providing the circuit construction for block-encoding Chebyshev polynomials, adapted in the supplementary material.","marker":"[29]"},{"why":"Experimental nuclear data used to compare the computed 20O spectrum.","marker":"[104]"}],"fun_headline_variants":["Quantum method resolves full oxygen-20 spectrum and spins","First full nuclear spectrum with spins from a quantum computer","Quantum scan yields complete spectrum and spins for oxygen-20","Hybrid quantum algorithm maps full bound-state spectrum and J"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The angular-momentum assignment assumes that the chosen single-configuration pivot has nonzero overlap with every eigenstate whose total angular momentum is at least the pivot's $M$-projection, so that a missing peak is evidence of absence rather than of pivot invisibility.","fun_headline_variants_meta":{"raw":{"variants":["Quantum method resolves full oxygen-20 spectrum and spins","First full nuclear spectrum with spins from a quantum computer","Quantum scan yields complete spectrum and spins for oxygen-20","Hybrid quantum algorithm maps full bound-state spectrum and J"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001038,"raw_usage":{"total_tokens":4324,"prompt_tokens":857,"completion_tokens":3467,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":473,"completion_tokens_details":{"reasoning_tokens":3402}},"tokens_in":473,"tokens_out":3467,"duration_ms":23966,"temperature":1.0,"reasoning_tokens":3402,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:06:54.457602+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a small configuration-interaction Hamiltonian whose exact eigenstates are known, run the cascading-$M$ scan with a pivot that is a single Slater determinant deliberately orthogonal to one eigenstate of angular momentum $J \\ge M$, and observe that this state never appears in any spectral function; this directly contradicts the claim that the method resolves the complete bound-state spectrum for arbitrary pivots.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the resolvent/strength-function formalism that converts Chebyshev moments into the spectral function."},{"cited_title":"Volya, Time-dependent approach to the continuum shell model, Phys","cited_arxiv_id":null,"evidence_quote":"Establishes the Chebyshev polynomial expansion of the time-evolution operator used to approximate the resolvent."},{"cited_title":"Tal-Ezer and R","cited_arxiv_id":null,"evidence_quote":"The kernel polynomial method, the standard background for expanding functions of H in Chebyshev polynomials."},{"cited_title":"Helgaker, P","cited_arxiv_id":null,"evidence_quote":"The quantum-walk formalism that the Hamiltonian block encoding builds on."},{"cited_title":"Gily´ en, Y","cited_arxiv_id":null,"evidence_quote":"Standard protocol for block-encoding Chebyshev polynomials via alternating applications of the walk operators and reflections."},{"cited_title":"Systematic input scheme of many-boson Hamiltonians with applications to the two-dimensional $\\phi ^4$ theory","cited_arxiv_id":"2407.13672","evidence_quote":"Prior work providing the circuit construction for block-encoding Chebyshev polynomials, adapted in the supplementary material."}],"review_version":1}