{"id":"ecd66bc3-af12-4ece-bc8f-1d12d838e4b9","arxiv_id":"2506.06063","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Truncated linear response excitation energies depend on redundant wave function parameters even at the FCI ground state, and constrained state-averaged UCC optimization can reduce this dependence.","lead":"This paper shows that excitation energies from truncated linear response or equation of motion calculations can depend on arbitrary orbital rotations even when the ground state is exact. It proposes a constrained trace optimization of the response matrix, called CSA-UCC, to reduce this sensitivity for small expansions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The redundant-parameter dependence of truncated LR is rigorously supported, but the trace-optimization remedy (Eq. 49) is validated only on He/6-31G and LiH/STO-3G, and the energy-only constraint does not provably select the intended ground-state branch for truncated ansätze.","rationale":"The reader's verdict of CONDITIONAL is appropriate. The core mathematical observation — that truncated ST/sc* LR excitation energies inherit dependence on redundant parameters even when the reference is an exact ground state — follows from trace conservation of the full unitarily rotated Hamiltonian together with non-conservation of its subspace traces. The He/6-31G and LiH/STO-3G numerical demonstrations are consistent with this argument. I do not find an internal inconsistency in that central derivation. The concern that matters for the paper's forward-looking remedy is that Eq. (49) is a heuristic: minimizing a subspace trace subject to an energy penalty does not carry a proven guarantee of improving individual excitation energies, and the energy constraint only fixes the reference state when the ansatz is exact and the ground state is nondegenerate. For truncated ansätze or degenerate ground states, isoenergetic parameter directions can move the reference state while preserving the penalty, so the optimized response matrix may be built from a different wavefunction than intended. This does not refute the paper's negative result, but it does mean the 'alleviated' claim rests on limited numerical evidence. The proposed overlap and third-molecule checks would settle whether the remedy is general or an artifact of the two test systems. Since the reader already flagged essentially this weakness and issued a conditional verdict, the appropriate action is to leave the verdict unchanged.","tokens_in":15747,"tokens_out":15178,"duration_ms":178629,"concrete_test":"For LiH/STO-3G at each truncation level (oo-UCCSD, oo-UCCSDT, oo-UCCSDTQ), after solving Eq. (49), compute the overlap |<Psi(θopt,κopt)|FCI>|^2 and the st-LRSD eigenvalue spectrum. If the overlap deviates from 1 by more than 1e-4 for an ansatz capable of representing FCI (or, for truncated ansätze, deviates from the best variational ground-state overlap), the energy-only constraint is not selecting the intended ground-state branch. Additionally, repeat the trace-optimized st-LRSD calculation on a third molecule, e.g., H2O/STO-3G or N2/STO-3G, and compare optimized, bad-parameter, and FCI low-energy spectra; if the improvement is not systematic, the remedy lacks generality.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim — Eqs. (42)–(46), that for state-transfer/sc* LR with an exact ground state the truncated excitation-energy sum equals the subspace trace of a unitarily rotated Hamiltonian and therefore varies with redundant orbital/cluster parameters — is internally consistent and numerically supported. The load-bearing weakness is the proposed fix. Eq. (49) minimizes a subspace trace subject only to a penalty on the ground-state energy. For an exact FCI-level ansatz with a nondegenerate ground state, the energy constraint does pin the state to FCI, because a positive-semidefinite shifted Hamiltonian has zero variance only at its ground state. For truncated ansätze (oo-UCCSD, oo-UCCSDT) the optimized energy is only the best variational energy of that ansatz; flat directions in the energy surface can connect different reference states with the same energy, and the penalty does not select among them. The trace objective is a proxy for spectral quality, not an error bound on individual excitation energies, and the evidence is limited to two small molecules with no shipped code or data. Thus the claim that the problem is 'alleviated' remains conditional, even though the negative dependence result itself is credible.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper shows that truncated linear response (LR) / equation-of-motion (EOM) excitation energies depend on orbital rotation and UCC cluster parameters that are redundant with respect to the ground-state energy. For the state-transfer (st) and self-consistent (sc*) parametrizations, the authors derive a trace identity: the sum of truncated excitation energies equals the subspace trace of a unitarily transformed Hamiltonian, which is not conserved under redundant unitary rotations. They demonstrate this dependence numerically for He/6-31G (with naive, projected, self-consistent, and state-transfer parametrizations) and for LiH/STO-3G with st-LRSD and st-LRS. As a remedy, they propose a ground-state-constrained trace optimization of the Hessian (Eq. 49), including a state-averaged variant (CSA-UCC), and report improved spectra for LiH/STO-3G.","tokens_in":16065,"tokens_out":6409,"duration_ms":64314,"significance":"If the central claim holds, it is an important and non-obvious result: truncated LR/EOM spectra are not well-defined until redundant parameters are fixed, with direct consequences for orbital-optimized and quantum-computing formulations (oo-qLR, qEOM, sc-qEOM). The trace identity for the st/sc* parametrizations is derived cleanly and is a useful diagnostic. The paper also ships no code or data, but the analytical argument is self-contained. The proposed trace-optimization remedy is, however, only a heuristic proxy, validated on two small systems, and the numerical evidence contains some ambiguities; this weakens the otherwise credible contribution.","major_comments":[{"comment":"The description of the post-optimization energy minimization is methodologically unclear and potentially load-bearing. After the penalty-function optimization with K=10^12, the text states that 'an energy minimization was performed with respect to the ground state, to ensure that the ground-state wave function was correct.' If this energy minimization moves along flat directions of the ground-state energy surface (which are precisely the redundant directions identified in Section II.C), the final parameters may no longer be the trace-optimized ones, and the traces in Table I and the spectra in Figs. 3-5 might not correspond to the reported θ_opt/κ_opt. The paper does not report whether the trace is preserved after this step, nor does it justify why this additional minimization is needed given that the penalty already enforces the energy to 1e-8 Hartree. The authors should either remove this step, demonstrate that it does not change the trace, or explain how the final wave function is unambiguously defined.","section":"Section III and Section IV.B"},{"comment":"The claim that trace optimization 'alleviates' the redundant-parameter problem rests on a heuristic objective that is not theoretically justified. Minimizing the subspace trace of the transformed Hamiltonian is equivalent to minimizing the sum of the truncated excitation energies for the st/sc* parametrizations, but there is no argument that this moves individual excitation energies toward the FCI values. The numerical support is limited to LiH/STO-3G, and for oo-UCCSD the reported difference between bad and opt traces is only 2.27e-3 Hartree (Table I), so the improvement is negligible in that case. The abstract's statement that the problem 'can be alleviated' is therefore only empirically demonstrated for two small molecules and a specific proxy; the paper should be more cautious in its wording or provide additional evidence that the trace objective correlates with spectral accuracy beyond these examples.","section":"Section II.D, Eq. (49), and Section IV.B"},{"comment":"Table I reports the subspace Hamiltonian and Hessian traces to only two decimal places, which makes it impossible for the reader to verify the quantitative differences quoted in the text. For oo-UCCSD, the text states that the difference between the bad-parameter and opt-parameter traces is 2.27e-3 Hartree, but the table lists both values as -274.60 and 113.93. For oo-UCCSDT and oo-UCCSDTQ the differences (0.71 and 28.25 Hartree) are visible, but the oo-UCCSD result is not. The table should include enough significant digits to substantiate all quoted differences.","section":"Table I"}],"minor_comments":[{"comment":"The notation (θ_red, κ_red) is used in Eq. (41) but the subscript 'red' is not explicitly defined; it should be stated that these denote redundant parameters as defined in Eqs. (14)-(15).","section":"Section II.C, Eq. (41)"},{"comment":"The notation min_{θ,κ\\κ_pq} is somewhat awkward; consider using a clearer formulation such as minimization over all parameters except κ_pq, with κ_pq free.","section":"Section II.C, Eqs. (14)-(15)"},{"comment":"The y-axis labels '1 [Hartree]' and 'LRS_1 + LRD_1 [Hartree]' are informal; using ε_1 and ε_1^S + ε_1^D (or a textual description) would improve readability.","section":"Figures 1 and 2"},{"comment":"The symbol 'sc*' is introduced in Eq. (35) without an explicit definition of the asterisk; the text should state that sc* refers to the self-consistent parametrization evaluated for a wave function that is an eigenfunction of the Hamiltonian.","section":"Section II.B, Eq. (35)"},{"comment":"Reference [61] (Grimsley and Evangelista) is a duplicate of Reference [28]; this should be merged or cross-referenced.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The central trace-dependence result appears sound and is a useful contribution to the quantum-chemistry/quantum-computing literature. However, the remedy (trace optimization) is the weakest part of the paper and is presented in the abstract as a solution. The post-optimization energy-minimization step is a potential source of inconsistency in the reported numerical data. I would like the editor to ensure the authors clarify this step and either strengthen or temper the claim that the problem is 'alleviated.' The paper is likely acceptable after major revision if these issues are resolved, but it is not yet ready for publication in its current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should read this one. The central observation is solid and important: truncated linear response excitation energies can depend on parameters that are redundant with respect to the ground-state energy, even when the ground state is FCI. The authors show this for four parametrizations on He/6-31G, and the trace identity they derive for state-transfer and self-consistent parametrizations is clean and convincing. I checked the math in Eqs. (42)-(46); it goes through, and the numerical behavior in Figs. 1-2 matches the argument. The He and LiH results are small models, but they are exactly the right models to expose the problem. This is a real, previously underappreciated source of non-uniqueness in truncated EOM/LR methods, and it matters for qEOM/qLR work that builds on arbitrary variational states.\n\nThe soft spot is the remedy, not the diagnosis. The trace optimization in Eq. (49) is a heuristic objective, and the evidence that it improves spectra is two small molecules. The stress-test note gets this right: for a truncated ansatz like oo-UCCSD, the penalty on the ground-state energy does not provably select a unique wave function, because flat directions in the energy surface can connect different reference states. The authors acknowledge the scope implicitly, but the abstract's claim that the problem \"can be alleviated\" is stronger than the data support. I would like to see at least one larger molecule or a more systematic scan over starting points before trusting the remedy as a general prescription. The CSA-UCC variant is interesting but again rests on one example. Also, no code or data are shipped; the availability statement points to \"reasonable request,\" which is a reproducibility hit.\n\nBottom line: the negative result—that truncated LR spectra are not well-defined until redundant parameters are fixed or optimized—is well established here. The positive prescription needs more work. The paper deserves serious refereeing because the core observation will affect how people think about qEOM/qLR and truncated classical response theory. I would send it to review, and I would push the authors on the remedy, not on the central theorem. If you work on excited-state methods, cite it. I'd bring it to reading group, mostly to argue about whether the trace objective can be grounded more rigorously.","headline":"A clean, credible demonstration that truncated LR/EOM excitation energies depend on redundant orbital and cluster parameters, with a remedy that is plausible but only lightly validated.","tokens_in":16533,"tokens_out":1316,"would_cite":true,"duration_ms":16129,"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":"Truncated linear-response and equation-of-motion excitation energies depend on wave-function parameters that are redundant for the ground-state energy, and the paper proposes a constrained trace-optimization of the response Hessian to…","keywords":["linear response theory","equation of motion","redundant parameters","excitation energies","unitary coupled cluster","orbital rotation","trace optimization","quantum linear response"],"falsifier":"A computation showing that a truncated LR excitation energy of an FCI ground state is invariant under a redundant orbital rotation that preserves the ground-state energy to machine precision would falsify the central claim of parameter dependence.","tokens_in":15594,"feed_emoji":"⚛️","tokens_out":7752,"duration_ms":67765,"temperature":0.7,"pith_summary":"This paper establishes that truncated linear response (LR) and equation-of-motion (EOM) excitation energies are not uniquely defined: they change when the wave function is varied along directions that leave the ground-state energy unchanged. The dependence is shown for a helium atom in a 6-31G basis with four different parametrizations of the excitation operators, and persists for lithium hydride in STO-3G when the response is truncated to singles or singles and doubles, even when the ground state is the full configuration interaction solution. The authors trace the mechanism to the fact that the sum of truncated excitation energies equals the trace of the response matrix on a subspace, which is not invariant under the unitary transformations generated by the redundant parameters, unlike the full-space trace. As a remedy, they minimize this subspace trace subject to keeping the ground-state energy fixed, implemented as a penalty-function optimization, and show it improves the low-lying spectra. They further show that restricting the trace optimization to a few target states (constrained state-averaged UCC) can improve small LR expansions.","feed_headline":"Truncated response spectra change with redundant orbital rotations","feed_subtitle":"Minimizing the subspace trace fixes the orbital dependence of truncated LR/EOM excitation energies.","key_machinery":"The load-bearing object is the subspace trace of the response Hessian, $\\mathrm{tr}\\{[A]_{SS}\\} = \\sum_{I\\in SS} \\langle I| U^\\dagger H U |I\\rangle - E_0$, which for state-transfer and self-consistent (eigenfunction) parametrizations equals the sum of the truncated excitation energies. Under redundant orbital rotations and cluster amplitudes, the full trace is conserved by unitarity but the subspace trace is not, and this is what makes truncated spectra ambiguous. The proposed remedy is a constrained optimization (Eqs. 48-49) that minimizes this subspace trace subject to the ground-state energy staying fixed, implemented with a large penalty parameter $K$; restricting the subspace to a few target states yields the constrained state-averaged UCC (CSA-UCC) variant used to improve small LR expansions.","core_discovery":"The central claim is that for a unitary-parametrized wave function (UCC or a quantum circuit), any truncation of the LR/EOM excitation space makes the excitation energies functions of parameters that are redundant for the ground-state energy. For the state-transfer parametrization, and for the self-consistent parametrization when the wave function is an eigenfunction of the Hamiltonian, the response matrix reduces to a unitary transformation of the Hamiltonian shifted by the ground-state energy. Because a unitary transformation conserves the total trace but not the trace over a truncated subspace, the sum of the truncated excitation energies, $\\sum_i \\varepsilon_i = \\mathrm{tr}\\{[H(U_\\kappa,U_\\theta)]_{SS}\\}$, varies with redundant orbital rotations $\\kappa$ and cluster amplitudes $\\theta$. The authors prove that individual excitation energies must therefore depend on these redundant parameters, and they demonstrate the effect numerically: for He/6-31G the single-excitation energy sweeps over a wide range as one redundant orbital rotation is varied, and for LiH/STO-3G the difference between trace-maximized and trace-minimized parameters grows with wave-function flexibility, reaching tens of Hartrees in the subspace trace for oo-UCCSDTQ.","pith_inferences":["The subspace-trace proxy assumes that lowering the sum of truncated excitation energies (or a selected-state sum) improves the individual states that matter; for general systems the correlation between trace minimization and per-state accuracy is not guaranteed and would need testing on larger molecules.","Because the constraint fixes only the ground-state energy, the optimization can move along near-null directions that leave the energy essentially unchanged but distort the response; the paper's numerical evidence is limited to two small systems, so the robustness of the remedy for realistic molecules remains open.","The subspace-trace viewpoint suggests a diagnostic: one could estimate the sensitivity of any truncated LR spectrum by computing the variance of the subspace trace over redundant parameter directions, without running full optimizations.","The same redundant-parameter ambiguity should affect other response properties (transition moments, polarizabilities) computed from truncated LR, not just excitation energies, though the paper does not demonstrate this."],"forward_implications":["Truncated LR/EOM spectra are not well-defined quantities until the redundant parameters are fixed or optimized; different but equally valid ground-state parameter sets yield different excitation energies.","Because quantum circuit ansätze are unitary, the same redundant-parameter dependence applies to qLR/qEOM implementations on quantum computers.","The constrained trace optimization of the Hessian, with the ground-state energy held fixed, provides a practical way to remove the ambiguity.","Targeting only a few states in the trace optimization (constrained state-averaged UCC) can improve the accuracy of small LR expansions, reducing the number of excitation operators needed.","Orbital localization or other uses of redundant rotations should be combined with truncated LR only with caution, since the resulting spectra depend on the chosen orbitals."],"supporting_citations":[{"why":"Provides the linear response formalism and working equations used throughout the paper.","marker":"3"},{"why":"Introduces the naive, projected, self-consistent, and state-transfer parametrizations of LR operators for UCC wave functions.","marker":"31"},{"why":"Showed metric singularities in naive parametrizations, used to explain the divergence observed in the helium example.","marker":"36"},{"why":"Establishes that LR and EOM are identical for unitary coupled cluster wave functions, making the conclusions transferable.","marker":"41"},{"why":"Reported instability of sc-qEOM with small expansions, which the paper suggests may be caused by redundant-parameter ambiguity.","marker":"28"}],"fun_headline_variants":["Truncated excitation energies depend on redundant orbital choices","Redundant orbital rotations shift truncated response spectra","Hessian trace optimization fixes orbital dependence in LR","Truncated LR/EOM spectra vary with redundant parameters","Orbital redundancy causes spurious shifts in truncated spectra"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The remedy rests on the hope that minimizing the subspace trace of the response matrix, subject only to keeping the ground-state energy fixed, actually improves the truncated excitation spectrum; this proxy leaves the ground-state wave function itself unconstrained and is only tested on two small molecules.","fun_headline_variants_meta":{"raw":{"variants":["Truncated excitation energies depend on redundant orbital choices","Redundant orbital rotations shift truncated response spectra","Hessian trace optimization fixes orbital dependence in LR","Truncated LR/EOM spectra vary with redundant parameters","Orbital redundancy causes spurious shifts in truncated spectra"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000276,"raw_usage":{"total_tokens":1656,"prompt_tokens":966,"completion_tokens":690,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":582,"completion_tokens_details":{"reasoning_tokens":614}},"tokens_in":582,"tokens_out":690,"duration_ms":6664,"temperature":1.0,"reasoning_tokens":614,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T06:00:48.114320+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A computation showing that a truncated LR excitation energy of an FCI ground state is invariant under a redundant orbital rotation that preserves the ground-state energy to machine precision would falsify the central claim of parameter dependence.","supporting_citations":[{"cited_title":"Linear and nonlinear response functions for an exact state and for an MCSCF state","cited_arxiv_id":null,"evidence_quote":"Provides the linear response formalism and working equations used throughout the paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the naive, projected, self-consistent, and state-transfer parametrizations of LR operators for UCC wave functions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Showed metric singularities in naive parametrizations, used to explain the divergence observed in the helium example."},{"cited_title":"Taube and Rodney J","cited_arxiv_id":null,"evidence_quote":"Establishes that LR and EOM are identical for unitary coupled cluster wave functions, making the conclusions transferable."},{"cited_title":"Challenging excited states from adaptive quantum eigensolvers: subspace expansions vs","cited_arxiv_id":null,"evidence_quote":"Reported instability of sc-qEOM with small expansions, which the paper suggests may be caused by redundant-parameter ambiguity."}],"review_version":1}