REVIEW 3 major objections 4 minor 3 references
Comment on "Provably Trainable Rotationally Equivariant Quantum Machine Learning"
T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash
Pith's one-line read 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.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (3)
- [Final paragraph, 'One potential strategy'] 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.
- [Fig. 1, CNOT decomposition example] 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.
- [Intro, Appendix A quote] 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.
minor comments (4)
- [p. 1, first paragraph] 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.
- [Fig. 1] 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.
- [References] 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.
- [p. 2, 'One potential strategy'] 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).
Circularity Check
No circularity: the comment's critique rests on externally cited DLA definitions and a direct worked example, not on any fitted or self-referential input.
full rationale
This comment does not derive a new prediction from fitted parameters or from the authors' own prior results. Its load-bearing premise is that the standard DLA is the Lie closure of parameterized generators only, and that premise is explicitly attributed to external references (Ragone et al. and Fontana et al.), not to a self-citation or to the target paper's own conclusions. The CNOT/ZZ decomposition is an explicit mathematical identity followed by a direct computation of the Lie closure; it is not a fitted quantity renamed as a prediction, and it does not presuppose that the CZ-based DLA in West et al. is wrong. The comment openly stops short of computing the corrected DLA for the actual nearest-neighbour CZ circuit, instead suggesting a strategy for doing so. That incompleteness may weaken the force of the critique, but an uncompleted or analogical argument is not circular. No uniqueness theorem is imported from the authors, no ansatz is smuggled in via self-citation, and no known result is merely renamed. Even if the target authors adopted a broader but internally consistent DLA definition, that would make the dispute definitional rather than circular. Under the given rules, the absence of any fitted input, self-citation chain, or definitional identification of the conclusion with the premises yields a circularity score of 0.
Assumptions & free parameters
assumptions (3)
- domain assumption The DLA of a parameterized circuit is the Lie closure of the generators of the parameterized gates only.
- standard math The CNOT gate admits the exponential representation CNOT = exp(i pi / 4 (I1 - Z1) tensor (I2 - X2)).
- domain assumption Fixed gates should enter the DLA through their adjoint action on parameterized generators, not as generators themselves.
Cite this review
Pith. "Pith review of Comment on "Provably Trainable Rotationally Equivariant Quantum Machine Learning"." pith.science (2026). https://pith.science/paper/YPFYX63W
@misc{pith2026250416950,
author = {Pith},
title = {Pith review of: Comment on "Provably Trainable Rotationally Equivariant Quantum Machine Learning"},
year = {2026},
howpublished = {\url{https://pith.science/paper/YPFYX63W}},
note = {Machine review of arXiv:2504.16950}
}
read the original abstract
We comment on the article by West {et al.}, ``Provably Trainable Rotationally Equivariant Quantum Machine Learning'' [PRX Quantum , 030320 (2024)]. While the general framework is insightful, we identify a key inconsistency in the construction of the dynamical Lie algebra (DLA). Specifically, the fixed controlled-Z (CZ) gates applied to all nearest-neighbor qubits are treated as if they were parameterized gates, with generators expressed in terms of combinations of Pauli operators. We discuss the implications of this inclusion and encourage the authors to revisit their analysis using a corrected DLA formulation.
Figures
Reference graph
Works this paper leans on
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[1]
M. T. West, J. Heredge, M. Sevior, and M. Usman, Prov- ably Trainable Rotationally Equivariant Quantum Ma- chine Learning, PRX Quantum 4, 010001 (2024)
work page 2024
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[2]
M. Ragone et al., A Unified Theory of Barren Plateaus for Deep Parameterized Quantum Circuits , arXiv:2309.09342 (2023)
arXiv 2023
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[3]
E. Fontana et al. , The Adjoint Is All You Need: Characterizing Barren Plateaus in Quantum Ans¨ atze , arXiv:2309.07902 (2023)
arXiv 2023
Reviewed August 16, 2026 · model on record in the stance chip above.
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