{"id":"3dfafe36-5ff6-4a37-93b7-351e9d041baf","arxiv_id":"2506.22275","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Excited-state energies are obtained by diagonalizing the Hamiltonian in the subspace spanned by the intermediate states of an ADAPT-VQE ground-state calculation.","lead":"This paper extracts low-lying excited states from the intermediate steps of an ADAPT-VQE ground-state calculation, by diagonalizing the Hamiltonian in the subspace spanned by those steps. Applications to nuclear pairing models and the H4 molecule show good energy accuracy in noiseless simulation, though symmetry sectors can be missed and extra strategies are then needed.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central premise that ADAPT-VQE path states span low-lying states is heuristic; no proof or criterion, and H4 shows missed symmetry sectors; generality is overstated.","rationale":"The reader's weakest_assumption is exactly the load-bearing point: the subspace-richness of the ADAPT-VQE convergence path is asserted but not proved. The paper's own H4 results show that the plain algorithm misses entire symmetry sectors, and the state-averaged workaround is presented as an add-on rather than as a condition in the central claim. The concern is not that the numerical demonstrations are wrong; they are plausible and internally consistent for the tested systems. The issue is the logical gap between energy convergence and spectral richness of the path, and the absence of any quantitative criterion for when the path is a good trial space. A random-subspace control would separate the hypothesis 'ADAPT path is a good trial space' from the weaker statement 'any reasonably spread set of states in that sector would do as well.' If the latter were true, the method's value would reduce to cost rather than subspace generation. Consequently, the existing CONDITIONAL verdict is appropriate, and this stress-test does not change it.","tokens_in":14734,"tokens_out":7438,"duration_ms":101109,"concrete_test":"Classically simulate ADAPT-VQE+QSD on an ensemble of 8-12 qubit test Hamiltonians (e.g., random pairing/Hubbard instances with known FCI spectra). For each instance, in a fixed symmetry sector reached by the chosen initial state, compare the mean relative error of the 5 lowest QSD energies from the ADAPT path against the same calculation using N random states of that sector, and record any low-lying exact states whose overlap with the path subspace is below, say, 10^-3. If the ADAPT path does not systematically outperform the random control, or if it misses an in-sector state, the subspace-richness premise fails and the general claim must be restricted; if it consistently wins, the concern is substantially mitigated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Sec. 3.2 argues that because energy and variance decrease, the ADAPT path 'depopulates high-energy spectrum in favor of low-energy part' and therefore the intermediate states form a subspace rich in low-lying spectral information. The Rayleigh-Ritz bound quoted there only lower-bounds the ground-state fidelity; it says nothing about the amplitude of the first (or any) excited state in the intermediate states. A trajectory could approach the ground state while remaining in a subspace (a symmetry sector, or the dynamical Lie algebra of a truncated operator pool) whose intersection with a given low-lying eigenspace is zero. The H4 calculation is an explicit instance: with a single Ag initial state (Fig. 7), entire spectral sectors are missed, and the paper must add state-averaged optimization with K=4 initial states (Fig. 8) to recover them. Thus the abstract's unqualified claim that low-lying excited states are obtained from the ground-state convergence path with small overhead is not supported in general; at present the method is a successful heuristic for the demonstrated sectors/systems, without a criterion for when the path subspace is spectrally complete. This is a correctness/scope risk, not a matter of consensus.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes using the intermediate states generated along the ADAPT-VQE convergence path toward the ground state as a subspace for quantum subspace diagonalization (QSD), solving the generalized eigenvalue problem in Eq. (7) to approximate low-lying excited states. The authors argue that the simultaneous decrease of energy and energy variance along the path implies that the subspace is rich in low-energy spectral information. They demonstrate the approach on like-particle pairing, neutron-proton pairing, and H4 dissociation, using noiseless statevector simulations and comparing against exact FCI spectra. They also introduce a single-parameter-optimization variant and augment the H4 calculation with a state-averaged strategy to recover symmetry sectors missed by the ground-state-only path.","tokens_in":14898,"tokens_out":5781,"duration_ms":66279,"significance":"If the central premise held, the method would be an attractive way to obtain excited states almost as a by-product of a ground-state ADAPT-VQE run, with no fitted constants in the energy predictions and with clear benchmarking against exact FCI in small systems. These are real strengths: the numerical demonstrations are internally consistent, the comparison to exact spectra is transparent, and the authors explicitly identify when symmetry sectors are missed. However, the central premise is only illustrated heuristically, not proved, and the H4 example shows that the pure ground-state path can fail entirely for some symmetry sectors. The resource-overhead claim in the abstract is unquantified. As a numerical study of a promising heuristic, the paper is valuable, but the generality and overhead claims need substantial qualification.","major_comments":[{"comment":"The central premise that the ADAPT-VQE path states span the low-lying spectrum is not established. The Rayleigh-Ritz argument quoted in Sec. 3.2 only lower-bounds the ground-state fidelity; it says nothing about the presence of excited-state amplitudes in the intermediate states. A trajectory can approach the ground state while remaining in a symmetry-restricted subspace that is orthogonal to a given low-lying excited eigenspace. The H4 calculation in Fig. 7 is an explicit counterexample: starting from an Ag-symmetric state, the B1g, B2u, and B3u sectors are missed entirely, and the paper must switch to a state-averaged ADAPT-VQE with K=4 initial states (Fig. 8) to recover them. Thus the abstract's unqualified claim that low-lying excited states are obtained from the ground-state convergence path with small overhead is not supported in general. The paper should either provide a criterion for when the path subspace is spectrally complete or explicitly restrict the claim to the demonstrated sectors.","section":"Sec. 3.2, Fig. 2; Sec. 4.3, Figs. 7-8"},{"comment":"The claim of 'small overhead in terms of quantum resources' is unquantified. Computing H_lk and O_lk via the Hadamard-test circuits in Fig. 3 requires, for each pair (l,k) and each Pauli string P_beta, an additional circuit evaluation; with n_f+1 path states this is O(n_f^2 * N_Pauli) extra circuits relative to ground-state ADAPT-VQE. In the state-averaged H4 application, the paper itself states at the end of Sec. 4.3 that the subspace cost is K^2 times higher (with K=4), which is not obviously 'small'. No comparison is made with the measurement cost of the ground-state calculation or with other QSD approaches. The resource claim should be either quantitatively supported or removed/qualified.","section":"Sec. 3.3; Sec. 4.3"},{"comment":"It is unclear whether the neutron-proton pairing results in Fig. 6 use SPO or FPO. This matters because Fig. 1 shows that, for the full pairing Hamiltonian, SPO does not converge to the ground state within 40 iterations, while FPO does. Since Sec. 4.2 is the main demonstration of the method for neutron-proton pairing, the reader cannot tell whether the successful QSD spectra there rely on full optimization (which weakens the paper's SPO recommendation) or on single optimization (in which case QSD is compensating for non-convergence and that effect deserves discussion). Please state the optimization strategy used in each figure and discuss the implications for the SPO proposal.","section":"Sec. 4.2, Fig. 6; Sec. 3.1, Fig. 1"}],"minor_comments":[{"comment":"There is a typo: 'descrease' should be 'decrease' in the sentence about MRE1.","section":"Sec. 4.1"},{"comment":"The MRE formula 'MRE_Lambda = 1/Lambda Delta-epsilon ...' is ambiguous; a parenthesized denominator, e.g. 1/(Lambda Delta-epsilon), would be clearer, and the choice of Delta-epsilon as the energy scale should be stated explicitly.","section":"Sec. 4.1"},{"comment":"The product of exponentials in Eq. (3) and in the definition of U_lk in Sec. 3.3 should specify the multiplication order (left-to-right or right-to-left), since the operators do not commute.","section":"Sec. 3.1, Eq. (3)"},{"comment":"The caption of Fig. 7 states that about 12 ADAPT-VQE iterations per r value were used, but Fig. 8 does not specify the number of iterations or the K=4 details; please add these parameters for reproducibility.","section":"Figs. 7-8"},{"comment":"The state-averaged weights c_i are all set to 1 without discussion; some justification or a reference to the sensitivity of the results to these weights would strengthen the presentation.","section":"Sec. 4.3"},{"comment":"All numerical results are obtained with a noiseless statevector simulator; a brief comment on the expected sensitivity to measurement noise and on the additional measurement overhead would help readers assess NISQ applicability.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The numerical work is sound and the paper addresses a relevant problem, but the abstract and conclusion overstate both the generality and the resource efficiency of the method. The H4 example, in particular, shows that the pure ground-state path is insufficient without a state-averaged augmentation, so the central claim needs to be reframed. I recommend major revision rather than rejection, since the demonstrated cases are valid and the method may still be a useful heuristic with clearer scope limitations."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid, honest demonstration that the states picked up along an ADAPT-VQE run can be reused as a QSD subspace to get low-lying excited states in the particular systems tested. The nuclear pairing applications and the single-parameter optimization (SPO) variant are the genuinely new pieces. But the abstract overreaches: \"small overhead\" and \"low-lying states from the convergence path\" are only true within a symmetry sector, and the H4 results show whole sectors are missed unless you bolt on a state-averaged strategy from Ref [61]. Treat it as a heuristic with demonstrated sector-wise success, not a general method.\n\nWhat I liked: The paper is transparent about the limitations. Sec. 3.2 is honest that the subspace-richness claim is an intuition; Fig. 2 is schematic. The numerical comparisons against full CI are clean; the MRE metric and forward/backward iteration selection are sensible. SPO is a practical twist: if you don't re-optimize old parameters, the states along the path are related by simple products of exponentials, which makes the overlap and Hamiltonian measurements cheaper and more consistent. The like-particle pairing results are convincing, with n_f ~ 30 states enough to get ten low-lying states to good precision. And the authors explicitly flag the symmetry-sector failures in both Fig. 6 and Fig. 7, then show the state-averaged fix works.\n\nSoft spots, in proportion: The load-bearing premise in Sec. 3.2 is not proven. The Rayleigh-Ritz bound they quote lower-bounds the ground-state fidelity; it says nothing about the overlap of the first excited state with the path subspace. The claim that the descent \"depopulates high-energy states\" is an interpretation, not a consequence. H4 is the concrete counterexample: with an initial Ag state, the B2u and B3u sectors are entirely absent until you add a state-averaged optimization with K=4 initial states. So the abstract's unqualified statement is wrong as written. The \"small overhead\" claim is also qualitative; the paper gives no resource estimate beyond the K^2 cost factor for state averaging. Finally, no code or data is shipped, and everything is a noiseless statevector simulation, so practicality on real hardware is untested. These are real constraints but they don't undo the demonstrated results.\n\nWho this is for: people working on quantum subspace methods, especially in nuclear structure and quantum chemistry on small systems. It's an incremental but useful extension of Ref [61], and the pairing examples give the nuclear community a concrete benchmark. The citation pattern looks appropriate; the authors are honest that the subspace-expansion idea overlaps with Ref [61] and they are positioning this as an application plus the SPO variant.\n\nRecommendation: Send it to peer review. A competent referee should push for a corrected abstract, a quantitative resource comparison, and ideally a diagnostic or criterion for when the path subspace is spectrally complete. With those revisions it would be a solid contribution to the QSD literature.","headline":"Useful, honest demonstration that ADAPT-VQE path states can seed a QSD subspace for low-lying states in the tested sectors, but the abstract overclaims; the key premise is unproven and H4 shows symmetry-sector failures without a state-averaged add-on.","tokens_in":15481,"tokens_out":3874,"would_cite":true,"duration_ms":42006,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P68"],"pacs":["03.67.Ac","03.67.Lx"],"model":"deepseek-v4-flash","headline":"The paper claims that the intermediate states generated during ADAPT-VQE's descent to the ground state span a subspace whose diagonalization gives accurate low-lying excited states, with only a small overhead beyond the ground-state…","keywords":["ADAPT-VQE","quantum subspace diagonalization","excited states","nuclear pairing","neutron-proton pairing","H4 dissociation","many-body quantum systems","low-lying spectra"],"falsifier":"Take a small Hamiltonian that is exactly diagonalizable and whose first excited state lies in a symmetry sector orthogonal to the initial ADAPT-VQE state. If after a large number of iterations the generalized eigenvalue problem still yields no eigenvalue within numerical tolerance of that excited state, the generic claim that the convergence path contains the low-lying spectrum fails; the paper itself exhibits this behavior for H4 before the state-averaged strategy is added.","tokens_in":14457,"feed_emoji":"⚛️","tokens_out":8547,"duration_ms":88755,"temperature":0.7,"pith_summary":"The paper's claim is that the intermediate states produced while ADAPT-VQE, an adaptive variational quantum eigensolver that builds its ansatz step by step from an operator pool, descends toward the ground state are not just a computational by-product: their span is a compact trial space for low-lying excited states. Solving a generalized eigenvalue problem for the Hamiltonian in this span reproduces accurate ground- and excited-state energies for like-particle and neutron-proton pairing and for H4 dissociation, with a small additional quantum cost beyond the ground-state calculation. This matters because excited-state spectra are normally a separate and more expensive task, while here they come almost for free from the convergence history. The paper also shows that the same subspace diagonalization improves the ground-state estimate obtained from single-parameter optimization, and that symmetry sectors missed by the descent can be recovered by additional starting states or a state-averaged variant.","feed_headline":"ADAPT-VQE's ground-state path also yields excited spectra","feed_subtitle":"Diagonalizing in the space of intermediate states reproduces pairing and H4 spectra with little added quantum cost.","key_machinery":"The machinery is a quantum subspace diagonalization built from the path states: one forms the matrices $H_{lk}=\\langle l|H|k\\rangle$ and $O_{lk}=\\langle l|k\\rangle$ and solves $H c = E O c$. The matrix elements are measured with Hadamard-test circuits that evaluate $\\langle l|P_\\beta|k\\rangle$ for the Pauli string $P_\\beta$ in the Hamiltonian decomposition, using the unitary that maps state $|k\\rangle$ to $|l\\rangle$ along the ADAPT-VQE path. The paper uses the qubit-excitation and Qubit operator pools, compares full versus single-parameter optimization, thresholds the overlap eigenvalues to remove nearly dependent states, and invokes a state-averaged cost function when different symmetry sectors coexist.","core_discovery":"The central discovery is that the ADAPT-VQE convergence path carries low-lying spectroscopy. During iterations the energy fluctuation $\\sigma_n^2$ decreases and the occupation of low-energy eigenstates increases while high-energy components are depopulated, so the states $\\{|k\\rangle\\}$ along the path form a subspace with substantial overlap on low-lying eigenstates. The paper diagonalizes the Hamiltonian in this subspace through the generalized eigenvalue equation $H_{lk} c_k = E\\, O_{lk} c_k$ and shows numerically that the ten lowest states of a like-particle pairing problem are reproduced with about thirty path states out of a 252-dimensional seniority-zero space. In the full neutron-proton pairing case, the method works within each seniority sector, and different symmetry sectors are obtained by restarting from a different initial state. For H4 dissociation, the lowest Ag states are captured directly, while states of other symmetries require restarting from states of those sectors, and the B2u-B3u level crossing requires the state-averaged strategy; after that, QSD again recovers the missing states.","pith_inferences":["If the path states' low-energy overlap grows systematically with iteration count, then the required subspace size may scale with the number of target low-lying states rather than with Hilbert-space dimension; a size-scaling study on larger pairing or molecular models would test this implied scaling.","The overlap matrix's condition number and the threshold epsilon will likely control error propagation; a per-state error bound in terms of the overlap of the exact excited state with the path subspace would turn the mean-relative-error benchmarks into a certificate, which the paper does not provide.","A practical extension is to use the path states of a symmetry-breaking ADAPT-VQE run as generator coordinates for a quantum generator-coordinate-method calculation, connecting the technique to nuclear potential-energy-surface studies without any new hardware requirement.","On noisy hardware the dominant cost will be the O(n_f^2) Hamiltonian matrix elements among path states; using randomized sampling of path states or classically post-processing subsets of them, as the paper notes but does not explore, could reduce that overhead."],"forward_implications":["Low-lying excited-state energies become a nearly free by-product of a ground-state ADAPT-VQE run, needing only the measurement of H and overlap matrix elements among the path states.","Single-parameter optimization, which is much cheaper than full re-optimization, becomes competitive because the later QSD step improves the ground-state energy by an order of magnitude in the like-particle pairing test.","For Hamiltonians that are block-diagonal by symmetry, the method should be run once per symmetry sector, starting from a state of that sector, to cover all low-lying states.","In the H4 example, state-averaged ADAPT-VQE plus QSD solves the B2u-B3u level-crossing problem and reproduces spectra that a single descent from an Ag seed misses.","The subspace dimension used in the pairing benchmark, about thirty path states, is far below the 252-dimensional seniority-zero full configuration-interaction space for the same problem, indicating the method extracts spectra from a compact trial space."],"supporting_citations":[{"why":"supplies the ADAPT-VQE algorithm whose intermediate states are reused as the subspace.","marker":"[52]"},{"why":"supplies the neutron-proton pairing Hamiltonian, the QEB operator pool, and the qubit encoding used in the pairing tests.","marker":"[53]"},{"why":"supplies the pair-to-qubit encoding used for the like-particle pairing benchmark.","marker":"[50]"},{"why":"supplies the state-averaged ADAPT-VQE strategy and the analysis of the H4 B2u-B3u level crossing used to recover missing symmetry sectors.","marker":"[61]"},{"why":"supplies the review of quantum subspace diagonalization that frames the method and its alternatives.","marker":"[66]"},{"why":"supplies the original subspace-expansion idea against which the convergence-path subspace is contrasted.","marker":"[67]"},{"why":"supplies the qubit-excitation-based operator pool used for the pairing problems.","marker":"[80]"},{"why":"supplies the Qubit pool used in the H4 calculations to allow symmetry breaking in the ansatz.","marker":"[81]"},{"why":"supplies the generalized-eigenvalue and overlap-diagonalization treatment borrowed from the generator coordinate method.","marker":"[44]"}],"fun_headline_variants":["Excited states come cheaply with ADAPT-VQE's ground-state path","Low-cost excited spectra from ADAPT-VQE's convergence path","Mine ADAPT-VQE's path for excited states with minimal extra circuits","ADAPT-VQE path diagonalization reveals excited spectra cheaply","ADAPT-VQE's path to ground state also hands you excited spectra"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the ADAPT-VQE descent enriches low-energy eigenstate components fast enough that the span of the intermediate states is a good trial space for low-lying excited states; for symmetry sectors not represented in the initial state this premise is false unless extra starting states or state-averaging are added.","fun_headline_variants_meta":{"raw":{"variants":["Excited states come cheaply with ADAPT-VQE's ground-state path","Low-cost excited spectra from ADAPT-VQE's convergence path","Mine ADAPT-VQE's path for excited states with minimal extra circuits","ADAPT-VQE path diagonalization reveals excited spectra cheaply","ADAPT-VQE's path to ground state also hands you excited spectra"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001084,"raw_usage":{"total_tokens":4520,"prompt_tokens":918,"completion_tokens":3602,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":534,"completion_tokens_details":{"reasoning_tokens":3508}},"tokens_in":534,"tokens_out":3602,"duration_ms":28451,"temperature":1.0,"reasoning_tokens":3508,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:07:35.209229+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a small Hamiltonian that is exactly diagonalizable and whose first excited state lies in a symmetry sector orthogonal to the initial ADAPT-VQE state. If after a large number of iterations the generalized eigenvalue problem still yields no eigenvalue within numerical tolerance of that excited state, the generic claim that the convergence path contains the low-lying spectrum fails; the paper itself exhibits this behavior for H4 before the state-averaged strategy is added.","supporting_citations":[{"cited_title":"Grimsley, Sophia E","cited_arxiv_id":null,"evidence_quote":"supplies the ADAPT-VQE algorithm whose intermediate states are reused as the subspace."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the neutron-proton pairing Hamiltonian, the QEB operator pool, and the qubit encoding used in the pairing tests."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the pair-to-qubit encoding used for the like-particle pairing benchmark."},{"cited_title":"state-averaged strategies, Quantum Sci","cited_arxiv_id":null,"evidence_quote":"supplies the state-averaged ADAPT-VQE strategy and the analysis of the H4 B2u-B3u level crossing used to recover missing symmetry sectors."},{"cited_title":"Struct., 6, 013001, (2024)","cited_arxiv_id":null,"evidence_quote":"supplies the review of quantum subspace diagonalization that frames the method and its alternatives."},{"cited_title":"McClean, Mollie E","cited_arxiv_id":null,"evidence_quote":"supplies the original subspace-expansion idea against which the convergence-path subspace is contrasted."},{"cited_title":"Yordanov, V","cited_arxiv_id":null,"evidence_quote":"supplies the qubit-excitation-based operator pool used for the pairing problems."},{"cited_title":"Shkolnikov, George S","cited_arxiv_id":null,"evidence_quote":"supplies the Qubit pool used in the H4 calculations to allow symmetry breaking in the ansatz."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the generalized-eigenvalue and overlap-diagonalization treatment borrowed from the generator coordinate method."}],"review_version":1}