{"id":"6d5e4d72-999e-4bce-9bd3-98941a33bf7f","arxiv_id":"2504.16950","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A comment identifying that West et al.'s dynamical Lie algebra for a rotationally equivariant quantum circuit improperly includes fixed CZ gates as parameterized generators.","lead":"This comment argues that a published analysis of trainable rotationally equivariant quantum circuits makes a technical error: it treats fixed entangling gates as if they were adjustable gates when building the circuit's dynamical Lie algebra. The authors illustrate the pitfall with a CNOT decomposition and ask the original authors to redo the calculation; if correct, the published trainability guarantees could change.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The comment never computes the corrected DLA for the CZ-based circuit, so its claim that West et al.'s DLA is inflated remains an unverified analogy rather than an established inconsistency.","rationale":"The reader's verdict of CONDITIONAL already hinges on the fragility of the convention assumption and the CNOT-to-CZ transfer. My stress-test identifies the same soft spot but sharpens it: the comment's argument is an analogy, not a proof, because it never computes the DLA of the actual circuit. This is the most load-bearing concern because it directly affects whether the central claim lands. If the corrected DLA for the original circuit turns out to coincide with West et al.'s, the comment's accusation becomes a pedantic point about phrasing rather than a substantive error. If it differs, the comment is correct. Since the comment does not provide the necessary computation, the claim is plausible but unverified. The reader already conditions acceptance on addressing this, so the verdict does not need to change. I partly agree with the reader: I agree that the CNOT-to-CZ transfer is a fragile premise, but I do not think the definitional convention is fragile—standard references support the comment's reading. Therefore agreement is partial.","tokens_in":1770,"tokens_out":17961,"duration_ms":170543,"concrete_test":"Compute the DLA for the West et al. circuit using the standard convention: take S as the set of generators of all parameterized gates on the radical qubits; form the Lie closure of S together with all conjugates CZ H CZ (and iterates thereof) for each fixed CZ gate, and compute the resulting algebra's dimension and basis. Compare this with the DLA obtained by starting from S ∪ {H_CZ}, where H_CZ is the generator of each fixed CZ gate, as West et al. apparently did. If the two algebras are identical, the comment's central claim fails; if the latter is strictly larger, the comment is substantiated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The comment's central claim is that West et al. err by including fixed CZ gates as generators in the DLA. The only concrete demonstration is a CNOT-based decomposition of a ZZ rotation, where including the CNOT generator enlarges the Lie algebra beyond the true one-parameter dynamics. That example is not transferred to the original circuit. West et al.'s circuit has parameterized gates on radical qubits and fixed CZ gates on nearest-neighbor qubits. The correct DLA under the standard convention is the Lie closure of the parameterized generators together with their conjugates under the fixed CZ gates (i.e., CZ H CZ). The comment does not compute this closure, nor does it compare it to the DLA reported in Appendix A of West et al. It is possible that the parameterized generators and their CZ-conjugates already span the same algebra as the set that also includes the CZ generator itself, in which case the alleged inconsistency has no effect on the DLA dimension or the trainability guarantees. The comment itself suggests a strategy for such an analysis but stops short of executing it. Thus the load-bearing premise—that the published DLA is actually larger than the true dynamics—is not established; the CNOT example shows a potential pitfall, not a proven error in the specific circuit.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript is a comment on West et al., 'Provably Trainable Rotationally Equivariant Quantum Machine Learning' (PRX Quantum 5, 030320 (2024)). The authors argue that West et al. incorrectly include fixed controlled-Z (CZ) gates as generators of the dynamical Lie algebra (DLA), which is inconsistent with the standard definition in which the DLA is generated by parameterized gates only. They illustrate the issue with a CNOT-based decomposition of a ZZ rotation and suggest that the effect of fixed CZ gates should instead be handled by conjugating the parameterized generators. The comment concludes by encouraging the authors to revisit their DLA computation.","tokens_in":1939,"tokens_out":5986,"duration_ms":58270,"significance":"If the criticism is correct, it affects the validity of the trainability guarantees in a published PRX Quantum paper, making this a potentially important comment for the field of symmetry-informed variational quantum algorithms. The comment also highlights a general methodological pitfall: treating fixed gates as Hamiltonian generators in DLA computations can overestimate the accessible algebra. However, the significance is currently limited because the comment does not establish that the specific DLA in West et al. is actually inflated; it only presents an analogy and a proposal for a corrected analysis. The strength of the paper is its clear identification of a subtle definitional issue and a pedagogically useful example, but the load-bearing technical verification is missing.","major_comments":[{"comment":"The central claim of the comment is that West et al.'s DLA is inflated by including fixed CZ gates, but the comment never computes the corrected DLA for the actual circuit. The final paragraph correctly proposes that the effect of fixed gates should be analyzed by conjugating parameterized generators by the CZ gates, but it stops at 'one potential strategy' without executing it. Without showing that the Lie algebra generated by the parameterized generators together with their CZ-conjugates is strictly smaller than the algebra generated when the CZ generator itself is included, the alleged inflation is not established. The authors should add this computation, or at least a dimension comparison, for the specific equivariant ansatz of West et al.","section":"Final paragraph, 'One potential strategy'"},{"comment":"The CNOT example demonstrates a pitfall in a decomposition of a ZZ rotation, but it is not directly applicable to West et al.'s circuit. In the example, the parameterized gate is the ZZ rotation and the CNOT gates appear only in a decomposition; they are not fixed gates in the circuit whose DLA is being computed. In West et al., CZ gates are applied as fixed gates in the actual circuit, so the relevant question is whether the CZ generator is already contained in the Lie closure of the parameterized generators under conjugation by products of CZ gates. The comment should either explain explicitly how the CNOT example transfers to this setting or provide a direct analysis of the CZ-based circuit.","section":"Fig. 1, CNOT decomposition example"},{"comment":"The comment's premise that only parameterized gates may serve as DLA generators is consistent with the definitions in Refs. [2,3], but the comment does not engage with the possibility that West et al. intentionally define a DLA for the full circuit including fixed gates. In that reading, the DLA with the CZ generator included is an upper bound on the true dynamics rather than an inconsistency. The comment should either show that this upper bound is strictly too large and that the trainability guarantees fail as a consequence, or soften the claim from 'key inconsistency' to 'potentially overly loose guarantee.' As written, the unqualified assertion that the construction is 'inconsistent' overstates the force of the definitional disagreement.","section":"Intro, Appendix A quote"}],"minor_comments":[{"comment":"There are two typos in the third sentence: 'But In their analysis' has an unnecessary capital 'I' and should be 'But in their analysis,' and the phrase 'only parameterized gates are explicitly defined' is awkward because the preceding sentence already says this.","section":"p. 1, first paragraph"},{"comment":"The text references 'FIG. 1. Decomposition of ZZ-rotation gates' but the figure is not included in the manuscript; either include the figure or remove the reference.","section":"Fig. 1"},{"comment":"Reference [1] lists 'PRX Quantum 4, 010001 (2024)', but the abstract and header of the comment cite 'PRX Quantum 5, 030320 (2024)'; the reference entry should be corrected to match the actual publication details.","section":"References"},{"comment":"The sentence 'This interaction is analogous to how a CNOT gate impacts the generator (e.g., ) of a Z-rotation gate' contains an empty parenthesis; the generator expression is missing and should be filled in (presumably the Z generator).","section":"p. 2, 'One potential strategy'"}],"recommendation":"major_revision","confidential_remarks":"This comment identifies a potentially important issue, but as written it is more of a research note than a fully supported critique. The central claim is defensible but unproven for the specific circuit of West et al. The authors should be encouraged to supply the missing DLA computation or the comment should be reframed as a suggestion for revisiting the analysis rather than a demonstration of an error. This is a fit for the comment format if the missing technical analysis is added."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis is a comment on the West et al. PRX Quantum paper, and its core point is real: the standard DLA is generated by the parameterized gates' generators, not by fixed gates. Quoting Appendix A, the comment shows West et al. treated the CZ gates as generators, which would indeed overestimate the reachable Lie algebra if taken literally. The CNOT example correctly illustrates the pitfall—including the CNOT generator inflates the DLA relative to the true one-parameter dynamics of a Z-rotation conjugated by a fixed CNOT.\n\nWhat the comment does well is identify a specific, plausible inconsistency in a published derivation and point to a concrete remedy: compute the Lie closure of the parameterized generators together with their CZ-conjugates. The suggested strategy is sensible and aligns with how the field normally handles fixed entanglers.\n\nWhere it falls short is execution. The comment never computes the corrected DLA for the CZ-based circuit. It is entirely possible that the parameterized generators plus their CZ-conjugates already span the same algebra as the set that also includes the CZ generator itself, in which case the alleged inconsistency changes nothing about the DLA dimension or the trainability guarantees. The CNOT example is an analogy, not a proof for the CZ lattice. The comment even gestures at the needed analysis but stops before doing it. That makes the load-bearing claim—that West et al.'s DLA is inflated—unverified.\n\nI also noticed the reference to West et al. has a mismatch: the abstract says PRX Quantum 5, 030320 (2024), while the reference list says PRX Quantum 4, 010001 (2024). That is sloppy in a comment whose entire purpose is correcting another paper's rigor. The \"five elements\" list in the CNOT example is actually five, so no issue there.\n\nOverall, this is a useful flag rather than a refutation. It deserves peer review because the question matters and the resolution—whether the corrected DLA differs—is a concrete calculation that reviewers can check. But it should be sent back to the authors with a request to either perform that calculation or explicitly show that the CZ generator is already in the closure of the parameterized generators. In its current form, the comment is a prompt for clarification, not a definitive correction.\n\nRecommendation: send to peer review, but set the bar at \"complete the analysis,\" not just \"acknowledge the definitional point.\"","headline":"A legitimate definitional flag about treating fixed CZ gates as DLA generators, but the comment stops short of proving the published DLA is actually wrong for West et al.'s circuit.","tokens_in":2474,"tokens_out":5174,"would_cite":false,"duration_ms":48790,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The comment's core claim is that the published dynamical Lie algebra for rotationally equivariant circuits wrongly counts fixed CZ gates as generators, and the trainability proof would need to be redone.","keywords":["dynamical Lie algebra","barren plateaus","rotationally equivariant quantum circuits","controlled-Z gates","quantum machine learning","parameterized quantum circuits","trainability","Lie closure"],"falsifier":"Recompute the DLA for the rotationally equivariant circuit with fixed CZ gates treated as conjugation operations on the parameterized generators, and compare the resulting algebra dimension and trainability bounds with the published ones; if the two algebras coincide for all qubit numbers, the critique fails, and if they differ, the published guarantees must be revised.","tokens_in":1521,"feed_emoji":"⚛️","tokens_out":4795,"duration_ms":39133,"temperature":0.7,"pith_summary":"This comment argues that the dynamical Lie algebra (DLA) computed in the commented paper includes fixed controlled-Z (CZ) gates as generators, even though the standard definition counts only parameterized gates as DLA generators. The authors demonstrate with a CNOT example that including generators of fixed gates changes the algebra that results from Lie closure. They conclude that the published DLA may be larger than the true dynamics, so the absence-of-barren-plateaus guarantees would need to be recomputed with a corrected generator set. A reader should care because that trainability guarantee is the main practical payoff of the original work.","feed_headline":"Fixed CZ gates may break the trainability guarantee","feed_subtitle":"A comment says the published algebra counts fixed CZ gates as generators, so the no-barren-plateau proof needs rechecking.","key_machinery":"The dynamical Lie algebra (DLA): the Lie closure (repeated commutators) of the Hermitian generators of the circuit's parameterized gates. The paper's argument turns on which operators are allowed into the generating set. The illustrative machinery is the identity $\\mathrm{CNOT} = \\exp[i\\pi/4 (I_1-Z_1)\\otimes(I_2-X_2)]$, which expresses a fixed CNOT as a Hamiltonian evolution and produces a Pauli generator $(I_1-Z_1)\\otimes(I_2-X_2)$; including that generator changes the closure, and the same logic is applied to fixed CZ gates in the target circuit.","core_discovery":"The central claim is that the DLA construction in the target paper is inconsistent with the standard definition: the DLA should be the Lie closure of the generators of parameterized gates only, but the target paper also inserts fixed CZ gates between nearest neighbours as generators written as Pauli combinations. To show why this matters, the comment decomposes a ZZ rotation into CNOT plus Z rotations and computes the algebra if the CNOT generator is included. The resulting five-element algebra differs from what the parameterized circuit alone generates, illustrating that including fixed-gate generators can inflate the algebra and misrepresent the true dynamics of the parameterized circuit.","pith_inferences":["The same issue may affect any parameterized quantum circuit paper that includes fixed entangling layers as DLA generators; the convention of what counts as a generator deserves explicit statement in every ansatz.","A direct numerical test would be to compute the DLA dimension for small qubit numbers both ways (fixed CZ as a generated direction versus fixed CZ as a conjugation) and compare the resulting loss landscape curvature; the comment's critique predicts these differ.","If the published DLA is indeed inflated, the flaw may not change the qualitative conclusion of trainability but would change the quantitative bounds; that distinction is testable."],"forward_implications":["If the corrected DLA is smaller than the published one, the dimension bounds used to argue trainability may not hold, and the absence of barren plateaus is not established by the current proof.","The expressibility analysis of rotationally equivariant circuits would need to account for fixed CZ gates as transformations that conjugate the parameterized generators rather than as extra algebra directions.","Comparisons to other equivariant architectures and numerical benchmarks would need to be rerun with the corrected DLA to see whether the guarantees and performance claims survive.","The comment's suggested strategy, analysing how the collective action of CZ gates transforms the DLA generated by the parameterized gates, provides a concrete route to a corrected calculation."],"supporting_citations":[{"why":"Defines the rotationally equivariant circuit and its DLA construction; it is the target of the comment.","marker":"[1]"},{"why":"Cited as the reference for the standard definition of the DLA as the Lie closure of parameterized gate generators.","marker":"[2]"},{"why":"Cited alongside [2] as the standard Lie-algebraic framework for DLA and barren-plateau analysis.","marker":"[3]"}],"fun_headline_variants":["Fixed CZ gates wrongly treated as parameterized generators","DLA construction flaw voids trainability guarantee","No-barren-plateau proof relies on fixed gate error","Rotational equivariance analysis counts fixed gates as trainable","Comment: Fixed CZ gates break the DLA logic"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The critique assumes that only parameterized gates may serve as DLA generators; if the original authors' broader but internally consistent definition is accepted, the disagreement could be purely definitional.","fun_headline_variants_meta":{"raw":{"variants":["Fixed CZ gates wrongly treated as parameterized generators","DLA construction flaw voids trainability guarantee","No-barren-plateau proof relies on fixed gate error","Rotational equivariance analysis counts fixed gates as trainable","Comment: Fixed CZ gates break the DLA logic"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000483,"raw_usage":{"total_tokens":2299,"prompt_tokens":769,"completion_tokens":1530,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":385,"completion_tokens_details":{"reasoning_tokens":1452}},"tokens_in":385,"tokens_out":1530,"duration_ms":10523,"temperature":1.0,"reasoning_tokens":1452,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:12:39.214532+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the DLA for the rotationally equivariant circuit with fixed CZ gates treated as conjugation operations on the parameterized generators, and compare the resulting algebra dimension and trainability bounds with the published ones; if the two algebras coincide for all qubit numbers, the critique fails, and if they differ, the published guarantees must be revised.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the rotationally equivariant circuit and its DLA construction; it is the target of the comment."}],"review_version":1}