{"id":"cf23b911-07cb-4e0c-bc25-5b858a06532f","arxiv_id":"1908.05238","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A method that projects out the ground state of a Hamiltonian and truncates the result back to the original operator basis yields an effective Hamiltonian whose ground state approximates the first excited state, demonstrated for H2 and partially for LiH.","lead":"This paper proposes a way to compute excited-state energies of molecules on near-term quantum computers by repeatedly removing the ground state and reusing the same quantum measurements. A smart generalist would read it because it tries to stretch today's quantum hardware from ground-state chemistry toward excited-state chemistry, which matters for spectroscopy and photochemistry.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central claim rests on an uncontrolled truncation: the covariance assertion has no error bound outside an exactly-solvable Abelian stabilizer class, and the paper's own LiH run fails once the truncation error accumulates.","rationale":"The reader's verdict identifies the covariance assertion as the weakest assumption, and I agree. My stress-test sharpens that concern: the replacement in Eqs. (5)-(6) is not a controlled low-energy approximation in the perturbed-Cz class. The only formal justification in S1 requires the Hamiltonian to be a complete weighted sum over a finite Abelian group and the ground state to stabilize that group's generators. H2 and LiH lie outside this class, and the numerical support consists of one molecule plus random perturbed-Cz models averaged over realizations, with LiH failing at the third iteration. The exact-versus-truncated comparison would settle whether the discarded Pauli components of P matter for the low-lying spectrum; if they do, the claimed validity for H2 is tied to the specific coefficient values rather than to a stable property of the method. I do not reject the paper: the H2 demonstration is genuine evidence for a restricted regime, and the authors disclose the energy-shift and fidelity limitations in the supplement. However, the abstract overstates the method's generality, so the reader's CONDITIONAL verdict is appropriate and no adjustment is needed.","tokens_in":11002,"tokens_out":9841,"duration_ms":111926,"concrete_test":"For the H2 Hamiltonian of Eq. (8) at a fixed internuclear distance, compute the exact projected operator H' = (I-P)H(I-P) in the full 16-element two-qubit Pauli basis and compare its low-lying spectrum and eigenstates with those of the covariance-asserted effective Hamiltonian sum_j (λ_j - E_g f_j) h_j from Eq. (6). If the spectral norm of the discarded Pauli components of P is comparable to the relevant gap, or if the first-excited eigenvector overlap is below the paper's claimed fidelity threshold, then the covariance assertion is not a controlled approximation for H2 even at the point claimed to be fully valid. Repeating the comparison with the X1X2 and Y1Y2 coefficients scaled by 0.5 and 2.0 would make the dependence on the unquantified perturbative assumption explicit.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract's strongest claim is that the ground state of the covariance-asserted effective Hamiltonian is an excited state of the original problem. The chain of equalities in Eqs. (5)-(6) is exact only if the projector P in Eq. (3) is both the true ground-state projector and supported entirely on the original string set {h_j}. The supplement's S1 justification requires the Hamiltonian to be a weighted sum over a finite Abelian group whose generators the ground state stabilizes; only then is P exactly of the form (S4) and contained in {h_j}. H2 and LiH are not in this class: H2 has non-commuting X1X2 and Y1Y2 terms treated as perturbations, and LiH is explicitly identified as perturbed-Cz. For such Hamiltonians, the replacement H^(i+1) = sum_j (λ_j^(i) - E_g^(i) f_j^(i)) h_j discards all Pauli components of P that are not already in the original Hamiltonian, with no estimate of the discarded operator's norm or of its effect on the low-lying spectrum. The consequence is not merely a small energy error: the truncated operator need not be of the form (I-P')H(I-P'), so the variational guarantee that its ground state is an eigenstate of H^(i) is lost, and errors can accumulate. The LiH section confirms this: the second iteration gives a vector with almost zero overlap with the true second excited state, and the S4 sweep over the identity coefficient cannot satisfy E_g^(3) >= E_g^(2). Separately, S1 concedes that when E_1^(i) >= 0 the VQE minimization of the projected Hamiltonian returns the old ground state with eigenvalue zero, and the required energy shift has no generic selection rule under covariance assertion. Both limitations are acknowledged in the manuscript but are absent from the abstract's unconditional claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an iterative method for computing excited states of a Hamiltonian by projecting out the already-found ground state and constructing an effective Hamiltonian whose ground state supposedly coincides with the next excited state of the original problem. To avoid an exponential growth in the number of Pauli terms, the authors introduce a \"covariance assertion\": only the string operators already present in the current Hamiltonian are retained in the projected Hamiltonian, with coefficients updated as lambda'_j = lambda_j - E_g f_j. The method is demonstrated numerically for H2 and LiH, and a class of Hamiltonians (perturbed Cz-type) is discussed in the supplemental material. The central claim is that this gives a resource-efficient way to extend ground-state VQE to excited states.","tokens_in":11419,"tokens_out":4678,"duration_ms":49896,"significance":"If the covariance assertion could be justified with controlled error, the method would be a simple and attractive extension of ground-state VQE, requiring only expectation values that are already measured during the ground-state calculation. The paper gives a clean derivation for the exactly solvable Abelian-stabilizer class and produces clean H2 binding curves. It is also to the authors' credit that the LiH failure and the unresolved energy-shift problem are disclosed explicitly. However, as it stands, the central claim is not established for general or even realistic Hamiltonians: the main numerical example uses exact diagonalization rather than a hybrid quantum-classical procedure, and the covariance assertion is applied without any quantitative error bound.","major_comments":[{"comment":"The covariance assertion is exact only when the Hamiltonian is a weighted sum over all elements of a finite Abelian group and the ground state stabilizes the generators of that group. For H2 (Eq. (8), where X1X2 and Y1Y2 do not commute with the Z terms) and for LiH (explicitly treated as perturbed-Cz in S4), this condition is not met. S1 states that the projection operator is \"replaced by the projection to the stabilizing subspace of a set of commuting independent operators without further assumptions.\" No norm bound or spectral error estimate is given for the discarded Pauli components of the projector, so the equality in Eq. (5) is uncontrolled. This truncation is load-bearing: without it, the updated Hamiltonian need not be of the form (I-P)H(I-P), and the variational guarantee that its ground state is an eigenstate of the previous Hamiltonian is lost.","section":"Eqs. (5)-(6) and S1"},{"comment":"The H2 numerical results are obtained by exact diagonalization, as S2 explicitly states, not by a variational ansatz or on quantum hardware. Therefore the abstract's claim that low-lying excited states can be calculated \"using existing hybrid-quantum classical techniques\" is not tested by the paper's main numerical example. The noise model in S2 perturbs the coefficients of the exact state and is not a simulation of VQE optimization, so it does not close this gap.","section":"S2"},{"comment":"The energy-shift problem is admitted in S1 as \"not addressed in the main text.\" Equation (S2) shows that when the first excited energy E1 is nonnegative, a variational minimization of the projected Hamiltonian returns the ground state with eigenvalue zero rather than the first excited state. The LiH procedure in S4 sweeps the identity coefficient lambda_I over [-10,10] and imposes monotonicity constraints, and S4 reports that the condition E_g^(3) >= E_g^(2) cannot be satisfied. This unresolved obstacle directly affects the iterative use of the method.","section":"S1"},{"comment":"The LiH demonstration fails beyond the first excited state: the main text states that in the second round of iteration the excited-state vector has almost zero overlap with the true second excited state, and S4 reports that the third iteration deviates from the exact value. This is evidence that the covariance assertion breaks down for a realistic molecular Hamiltonian after only one successful iteration, which contradicts the broader claim that the method \"determines the excited-state vector with high fidelity\" and can be used to iteratively extract eigenstates and eigenenergies.","section":"S4 and LiH discussion in main text"}],"minor_comments":[{"comment":"There are typos: \"reresentation\" in S1 should be \"representation\", and \"Hamiltotnian\" in the main text should be \"Hamiltonian\".","section":"S1 and main text"},{"comment":"The normalization 1/2Ng should be written 1/2^{N_g} to avoid ambiguity between 1/(2 N_g) and 2^{-N_g}.","section":"Eq. (9) and Eq. (S4)"},{"comment":"The averages in Fig. S2 are over thirty random realizations, but no error bars or standard deviations are shown, making the claimed convergence to the thermodynamic limit difficult to assess.","section":"Fig. S2"},{"comment":"The noise-corrupted states are not renormalized before computing the expectation values f_j; the authors state this is intentional, but the resulting quantity called \"fidelity\" is then not a standard state overlap and should be discussed as such.","section":"S2 noise model"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is honest about its limitations, which is commendable. The main issue is that the abstract and conclusion overstate what has been demonstrated: the H2 calculation is classical exact diagonalization, and the LiH example fails precisely where the covariance assertion is most needed. A revision that narrows the claims, adds a quantitative error analysis or an actual VQE-based demonstration, and resolves or clearly circumscribes the energy-shift problem would materially improve the paper. I do not see grounds for outright rejection, but the current form is not yet suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nQuick read: the paper is a real algorithmic proposal, not a repackaged trick. The deflation step (I-P)H(I-P) is textbook, but the covariance assertion — keep the operator basis fixed and update coefficients as λ_j - E_g f_j — is a concrete, testable heuristic. For Hamiltonians that are weighted sums over an Abelian Pauli group with a stabilizer ground state, it is exact. The paper is honest that H2 and LiH are only perturbed-Cz instances of that class.\n\nWhat it does well: the H2 binding-curve calculation is clean and reproducible, and the added noise model (corrupting the exact state vector) gives some evidence of robustness, at least for small W. The LiH test is also honest: 97% fidelity for the first excited state, good second energy but zero overlap with the exact second state, and failure of the third iteration to satisfy E_g^(3) >= E_g^(2). That is exactly the signature of an uncontrolled truncation error. The supplement also openly admits the energy-shift problem: without a shift, the VQE minimization of the projected Hamiltonian can return the old ground state with eigenvalue zero, and the shift is nontrivial once covariance is imposed.\n\nSoft spots, in proportion: the abstract overstates the result. The ground state of the effective Hamiltonian coincides with an excited state only when the covariance assertion holds, and the paper gives no error bound for the truncation outside the exact Abelian group case. The numerical demonstration is exact diagonalization, not VQE on hardware or a simulator, so the claim about 'existing hybrid-quantum classical techniques' is not actually tested. The noise model is a crude proxy for optimizer noise and ansatz error. The LiH λI sweep in S4 is ad hoc — they tune a constant to satisfy ordering constraints, which works for the first two iterations but not the third. None of these flaws are fatal for the paper as a heuristic, but they should be stated in the main text, not only in the supplement.\n\nThe citation pattern looks fine: QSE, VQE, and related excited-state work are all covered. No self-citation issues.\n\nBottom line: this is a useful, honestly-grounded heuristic for a limited class of Hamiltonians, with a clean H2 demonstration and an instructive LiH failure. It deserves peer review, with referees asked to probe the covariance assertion for a practical error estimate and to push the authors to address the energy-shift issue. I would not cite it in my own work yet, but I'd recommend a careful referee for the archive version.","headline":"A useful iterative projection heuristic for excited-state VQE, honestly caveated in the supplement, but with an uncontrolled truncation that limits it to near-Cz Hamiltonians; H2 works, LiH partially fails.","tokens_in":11907,"tokens_out":3196,"would_cite":false,"duration_ms":32596,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.67.Ac","31.15.-p"],"model":"deepseek-v4-flash","headline":"Projecting out the ground state turns excited-state energies into ground-state problems.","keywords":["excited states","effective Hamiltonian","covariance assertion","variational quantum eigensolver","Pauli string operators","perturbed-Cz class","quantum chemistry"],"falsifier":"Take a molecule beyond H2 or LiH (such as water, or LiH at a non-equilibrium bond length), compute the exact projected Hamiltonian $\\hat{H}^{(1)} = (\\hat{I}-\\hat{P})\\hat{H}(\\hat{I}-\\hat{P})$ by full diagonalization, expand it in Pauli string operators, and compare the ground-state energy of this exact projected Hamiltonian with that of the covariance-asserted effective Hamiltonian $\\sum_j(\\lambda_j - E_g f_j)\\hat{h}_j$; if the difference exceeds chemical accuracy while the discarded operator weights are non-negligible, the covariance assertion is falsified for that system.","tokens_in":10783,"feed_emoji":"⚛️","tokens_out":6808,"duration_ms":64178,"temperature":0.7,"pith_summary":"The paper proposes a way to compute excited-state energies of a molecular Hamiltonian using the same hybrid quantum-classical machinery that already computes ground-state energies, such as the variational quantum eigensolver. The idea is to construct, from a known ground state, a projected Hamiltonian whose lowest eigenvalue is the first excited-state energy of the original problem, and then iterate. Because the exact projection would contain exponentially many Pauli-string terms, the paper introduces a covariance assertion: keep only the string operators already present in the original Hamiltonian, renormalizing their coefficients using ground-state expectation values. The method is shown to reproduce the full hydrogen-molecule spectrum, to obtain the first excited state of LiH with high fidelity, and to become exact in the thermodynamic limit for a perturbed classical (Cz) class of Hamiltonians.","feed_headline":"Project out the ground state, and excited states become ground states","feed_subtitle":"By renormalizing the same Pauli terms, the method lets VQE-style hardware reach molecular spectra without extra qubits.","key_machinery":"The load-bearing mechanism is the covariance assertion: the assumption that the ground-state projection can be represented using only the Pauli string operators that already appear in the Hamiltonian, with renormalized coefficients $\\lambda_j' = \\lambda_j - E_g f_j$, where $f_j$ is the ground-state expectation value of the $j$-th string operator. This keeps the number of terms constant across iterations, avoiding the exponential growth that a general projection operator would incur. Its justification is group-theoretic: if the Hamiltonian is a weighted sum over all elements of a finite Abelian group of commuting string operators and the ground state stabilizes the group's generators, then the projection operator has the product form $\\hat{P} = \\frac{1}{2^{N_g}}\\prod_g (\\hat{I} - \\hat{h}_g)$, making the covariance assertion exact.","core_discovery":"The central discovery is the identity $\\hat{H}^{(1)} = (\\hat{I} - \\hat{P}^{(0)})\\hat{H}^{(0)}(\\hat{I} - \\hat{P}^{(0)}) = \\hat{H}^{(0)} - E_g^{(0)}\\hat{P}^{(0)}$, where $\\hat{P}^{(0)}$ projects onto the ground state of $\\hat{H}^{(0)}$. In a basis where $\\hat{H}^{(0)}$ is diagonal, this projected Hamiltonian has eigenvalue zero for the ground state, and its remaining spectrum is exactly the spectrum of $\\hat{H}^{(0)}$ above the ground state. The paper's contribution is to make this projection practical by the covariance assertion: the projection operator is replaced by a stabilizer projection of a set of commuting generators, so that the effective Hamiltonian at each iteration is $\\hat{H}^{(i+1)} = \\sum_j (\\lambda_j^{(i)} - E_g^{(i)} f_j^{(i)})\\,\\hat{h}_j$, with $f_j^{(i)} = \\langle\\varphi_{\\mathrm{int}}| (\\hat{U}^{(i)})^{\\dagger} \\hat{h}_j \\hat{U}^{(i)} |\\varphi_{\\mathrm{int}}\\rangle$ directly measurable on a quantum device. Iterating this update yields successive low-lying excited states, and the paper demonstrates the procedure on H$_2$ and partially on LiH.","pith_inferences":["For generic molecular Hamiltonians, the covariance assertion is unlikely to hold beyond low-lying states, as the LiH example already shows at the second excited state; a quantitative measure of the discarded part of the projection operator would be a useful diagnostic.","The method effectively treats the molecule as a member of the fixed-point class of a renormalization-group flow, which suggests that combining it with symmetry-adapted ansatze that enforce stabilizer structure could push the covariance assertion closer to exactness.","Because the $f_j$ coefficients are standard VQE expectation values, the approach has almost no additional quantum overhead; the main cost is classical, and the authors' heuristic for choosing the energy shift could be systematized into a classical optimization.","A direct comparison with quantum subspace expansion on the same H2 and LiH data sets would clarify whether the covariance assertion yields competitive accuracy with a simpler classical post-processing step."],"forward_implications":["Excited-state energies of molecules whose Hamiltonians lie near the perturbed-Cz class can be obtained with the same quantum resources as a single ground-state VQE computation, since the coefficients $f_j$ are already measured during the ground-state optimization.","The iterative procedure produces not only excited-state energies but also the corresponding state vectors, with high fidelity when the covariance assertion holds, such as above 95% fidelity for H2 under noise and roughly 97% for the LiH first excited state.","For the perturbed-Cz class, the error in the predicted first excited-state energy and the state infidelity both go to zero as the number of qubits grows, so the method becomes exact in the thermodynamic limit.","If the covariance assertion fails at some iteration, the method signals its own breakdown: the ground-state energy of the next effective Hamiltonian violates the ordering constraint $E_g \\le E_1 \\le E_2 \\le \\cdots$.","The method can also produce orthogonal degenerate ground states when the target subspace is degenerate, which is useful for computing degenerate spectra."],"supporting_citations":[{"why":"Supplies the LiH Hamiltonian string operators and coefficients used in the LiH test.","marker":"[1]"},{"why":"Provides the fermion-to-qubit mapping that puts the Hamiltonian into the Pauli-string form the method assumes.","marker":"[5]"},{"why":"Defines the VQE ground-state procedure whose measurements supply the $f_j$ coefficients.","marker":"[7]"},{"why":"Provides the H2 Hamiltonian coefficients used in the numerical demonstration.","marker":"[10]"},{"why":"Supplies the group-theoretic framework used to justify the covariance assertion for Hamiltonians built from finite Abelian groups.","marker":"[19]"}],"fun_headline_variants":["Excited states as ground states via projection","Project out ground state, compute excited states","Hamiltonian projection turns excited states into ground states","Effective Hamiltonian: excited states become ground states"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire scheme rests on the covariance assertion: that after the ground state is removed, the remaining Hamiltonian is still faithfully described by the same set of Pauli string operators, with only their coefficients changed.","fun_headline_variants_meta":{"raw":{"variants":["Excited states as ground states via projection","Project out ground state, compute excited states","Hamiltonian projection turns excited states into ground states","Effective Hamiltonian: excited states become ground states"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000466,"raw_usage":{"total_tokens":2305,"prompt_tokens":906,"completion_tokens":1399,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":522,"completion_tokens_details":{"reasoning_tokens":1343}},"tokens_in":522,"tokens_out":1399,"duration_ms":10439,"temperature":1.0,"reasoning_tokens":1343,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:43:30.402820+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a molecule beyond H2 or LiH (such as water, or LiH at a non-equilibrium bond length), compute the exact projected Hamiltonian $\\hat{H}^{(1)} = (\\hat{I}-\\hat{P})\\hat{H}(\\hat{I}-\\hat{P})$ by full diagonalization, expand it in Pauli string operators, and compare the ground-state energy of this exact projected Hamiltonian with that of the covariance-asserted effective Hamiltonian $\\sum_j(\\lambda_j - E_g f_j)\\hat{h}_j$; if the difference exceeds chemical accuracy while the discarded operator weights are non-negligible, the covariance assertion is falsified for that system.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the fermion-to-qubit mapping that puts the Hamiltonian into the Pauli-string form the method assumes."},{"cited_title":"Peruzzo, J","cited_arxiv_id":null,"evidence_quote":"Defines the VQE ground-state procedure whose measurements supply the $f_j$ coefficients."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the H2 Hamiltonian coefficients used in the numerical demonstration."},{"cited_title":"Joshi, Elements of Group Theory for Physicists (1997), ISBN 9788122409758","cited_arxiv_id":null,"evidence_quote":"Supplies the group-theoretic framework used to justify the covariance assertion for Hamiltonians built from finite Abelian groups."}],"review_version":1}