{"id":"712cd02a-8cb7-421b-95b5-77a35ff1dffb","arxiv_id":"2411.16560","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Training parameterized quantum circuits by gradually adding reuploader blocks, rather than using fixed depth from the start, reduces loss and variance on the tested regression and PDE tasks.","lead":"This paper introduces three ways to make quantum machine learning circuits grow during training, adding small pieces of the circuit only when needed instead of starting with a full deep circuit. In small regression tasks and a physics simulation, the growing circuits trained more reliably and generalized better than fixed-depth circuits.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 3.1 reports best-epoch test loss, while the Abstract claims lower final losses; the central comparison is not on the stated metric.","rationale":"The paper is a competent empirical study with a plausible Fourier-frequency motivation, clear experimental presentations, and released code. My stress-test focuses on whether the evidence supports the central claim as stated in the Abstract. The strongest controlled evidence, Section 3.1, measures the best test loss over training rather than the final loss. Algorithm 1 evaluates test loss every epoch; 'best loss' is therefore selected using the test set. This systematically lowers reported error and can favor methods whose loss dips and then rises. Because the Abstract claims 'lower final losses,' the student-teacher results are not directly on the claimed metric. The noisy and Laplace experiments do report end-of-training errors, which is why I do not call the entire claim unsubstantiated; however, the student-teacher task is the cleanest comparison of identical architecture and depth, so its metric choice is load-bearing. The reader's weakest assumption concerns undisclosed growth hyperparameters; that is a real reproducibility concern, but the code link makes it checkable, whereas the best-epoch issue is a property of the reported metric that no code inspection can resolve. A single re-analysis with final losses would settle whether the main comparison holds. I keep the reader's CONDITIONAL verdict: the paper needs this re-analysis before the superiority claim is accepted.","tokens_in":11109,"tokens_out":7106,"duration_ms":70770,"concrete_test":"Use the public GitHub code to rerun the 1-qubit and 2-qubit student-teacher experiments with the same 50 seeds and teacher depths, and report the test MSE of the final epoch (epoch 1000) for every seed instead of the best test MSE over each run. Recompute Table 1, Table 2, and Figures 5a-5b with these final losses. If all three growth methods still have lower mean and standard deviation than the matched-depth CDL-I models, the central claim survives; if any growth method's advantage disappears or reverses, the paper must restrict its claim to best-epoch behavior or provide a different controlled experiment.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The Abstract claims that growth-based PQCs achieve lower final losses and reduced variance than fixed-depth CDL, but the principal controlled experiments in Section 3.1 do not measure final losses. The text states: 'The best loss achieved during each run is reported' (Section 3.1), and Algorithm 1 evaluates test loss every epoch. Thus Tables 1-2 and Figures 5a-5b compare the best test loss selected over 1000 epochs, not the loss of the model produced at the end of training. Selecting the best epoch on the test set is a form of test-set model selection; it lowers reported errors and can change the ranking of methods with different loss trajectories. A growing method may reach a low loss immediately after a growth step and then worsen, while a CDL model may converge more slowly but end lower; the reported metric would hide this. The variance columns also describe variance of best losses, not final losses. The noisy experiment (Section 3.2) and Laplace experiment (Section 3.3) do report end-of-training errors, so those parts of the claim are unaffected, but the generality of the central claim rests heavily on Section 3.1. This is an internal mismatch with the Abstract's wording and is more load-bearing than the undisclosed growth schedule, because no amount of hyperparameter disclosure fixes the metric.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes three growth-based training strategies for reuploader parameterized quantum circuits (Block Growth, Sequential Feature Map Growth, and Interleave Feature Map Growth), in which identity-initialized reuploader blocks are added during training to expand the accessible frequency spectrum. The methods are compared with fixed-depth complete-depth learning (CDL) circuits, with both random and identity initialization, on three tasks: student-teacher regression for 1- and 2-qubit circuits, a noisy student-teacher regression, and a 2D Laplace equation benchmark with an analytic solution. The paper reports that growth-based methods achieve lower losses, reduced variance across 50 random seeds, and better generalization than fixed-depth approaches, and it provides a pseudocode training procedure and a link to a code repository.","tokens_in":11362,"tokens_out":4616,"duration_ms":59526,"significance":"The central idea of growing a PQC during training to balance expressivity and trainability is timely and practically relevant for quantum scientific machine learning. The paper has several strengths: the use of 50 random seeds, clear boxplot visualizations, an analytic Laplace solution for a physically meaningful benchmark, a noisy regression setting that directly tests overfitting, and a publicly available code repository. If the reported advantages hold under a correct evaluation protocol, the work would be a useful contribution to adaptive variational quantum circuit training. However, the central claim is currently overstated because the main student-teacher experiments report best-epoch test losses while the abstract claims lower final losses, and the growth schedule hyperparameters are not disclosed. These issues undermine the current evidence for the paper's headline conclusions, although they appear fixable with additional reporting and analysis.","major_comments":[{"comment":"The abstract claims that growth-based PQCs achieve 'lower final losses and reduced variance between runs,' but the primary student-teacher experiments in Section 3.1 report 'the best loss achieved during each run,' and Algorithm 1 evaluates the test loss every epoch without storing or reporting the final-epoch loss. Tables 1-2 and Figures 5a-5b therefore summarize distributions of best test losses selected over the training trajectory, not final losses. Selecting the best epoch on the test set is a form of test-set model selection and can change the relative ranking of methods with different loss trajectories; for example, a growing method that peaks early and then degrades would look better than a fixed-depth method that converges slowly but ends lower. Because the abstract's central claim is about final losses and Section 3.1 is the main controlled comparison, this mismatch is load-bearing. The authors should either report final-epoch losses (or use a validation-based early-stopping rule) or explicitly revise the claim to refer to best-attained losses rather than final losses.","section":"Section 3.1 and Algorithm 1; Abstract"},{"comment":"The growth hyperparameters are never specified. Section 2 says circuits grow 'after a predefined number of epochs' and that the number of blocks added per stage, ℓ, is 'a user-defined number,' while Section 3.2 says growth continues 'until no further improvement is observed.' No values for the growth interval, ℓ, patience, or stopping threshold are given anywhere in the paper. Without these details the experiments are not reproducible, and the reader cannot rule out that the reported advantage of growth over CDL depends on task-specific tuning of the growth schedule, including possible selection based on test performance. The authors should disclose the exact schedule, all growth-related hyperparameters, and the criterion used to decide when to stop growing, ideally with a sensitivity analysis.","section":"Section 2 and Section 3.2"},{"comment":"In the 2-qubit student-teacher task, the reported mean best losses for the growth methods are 3.19e-4, 3.21e-4, and 3.75e-4, while the 9-layer identity-initialized CDL model has mean 3.97e-4 with standard errors of roughly 1.4e-4 to 3.0e-4 for all methods. The differences between the growth methods and the best identity-initialized CDL baseline are therefore small relative to the reported standard errors, and no statistical significance tests are provided. The text states that the growing models 'outperformed both the identity-initialized CDL circuits,' but the reported summary statistics do not clearly establish this. The authors should add appropriate hypothesis tests or confidence intervals, or temper the claim in light of the overlapping uncertainties.","section":"Appendix A.2 and Section 3.1.2"}],"minor_comments":[{"comment":"There is a typo in 'generator Hamiltonian’s' and the notation Ġ_m is used inconsistently: Eq. (1) writes Ġ_m(γ_m), while Eq. (3) and the following paragraph use Ġ_m(γ_m, ψ).","section":"Section 2, Eq. (1)"},{"comment":"The circuit diagrams are not self-contained; the captions should define whether U_i and F_i represent ansatz blocks, feature map blocks, or both, and how they correspond to the growth stages described in the text.","section":"Figures 1-3"},{"comment":"The labels '5-layer CDL ()' and '20-layer CDL ()' appear to have a missing symbol, likely the identity-initialization marker; please fix the rendering.","section":"Figure 5"},{"comment":"The x-axis label and caption for Figure 8 are ambiguous for growth methods: 'began with or grew to' does not clarify whether the reported layer count is the initial depth, the final depth, or a target depth for each method.","section":"Section 3.3, Figure 8"},{"comment":"Reference [6] is incomplete (no title or journal details), Reference [8] is a bare Semantic Scholar URL rather than a full citation, and References [9] and [21] cite the same paper by Grant et al. with different levels of detail; these should be unified.","section":"References"},{"comment":"Algorithm 1 computes the test loss each epoch but does not state how the 'best loss' reported in Section 3.1 is obtained; please make explicit the selection procedure and clarify whether it uses the test set or a validation set.","section":"Algorithm 1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits the journal's scope and the growth-based training idea is worth publishing if the central claims are properly supported. The metric mismatch in Section 3.1 and the undisclosed growth hyperparameters are the main technical obstacles; both are fixable with additional experiments or revised claims. I do not see evidence of misconduct, and the code link is a positive feature. Please ensure the authors address the final-loss versus best-loss issue directly, since it affects the abstract's main promise."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a competent empirical paper on growing PQCs during training. It proposes three identity-initialized growth strategies (block growth, sequential feature-map growth, interleaved feature-map growth) and tests them on student-teacher regression, a noisy regression, and a 2D Laplace PDE. The experiments are clearly presented, 50 seeds, and they ship code. Credit where due: the Laplace setup with an analytic solution is a nice stress test, and the noisy-student experiment shows a real regularization effect with final test losses separated from training losses.\n\nThe soft spots are real, and one is load-bearing. The abstract claims growing methods achieve 'lower final losses,' but Section 3.1 and Tables 1-2 report the best test loss achieved during each 1000-epoch run, selected on the test set. That is test-set model selection, not final loss. A method that spikes down after a growth step and then worsens would look great under this metric. The noisy and Laplace experiments do report end-of-training errors, so the regularization claim survives, but the central student-teacher comparison is not on the stated metric.\n\nSecond, the growth schedule is underspecified. The paper says growth happens after 'a predefined number of epochs,' with a 'user-defined' number of blocks per stage, and the noisy experiment grows 'until no further improvement is observed.' No values, no patience, no thresholds. If those were tuned per task with test-set feedback, the advantage could be partly scheduling artifact. This is fixable by disclosure, but right now it is a fairness gap.\n\nThird, the closest prior work—layerwise learning (Ref. [18])—is cited but not compared. Block growth especially is close to that method (identity-initialized additions, incremental depth), so the incremental-advantage claim needs that baseline.\n\nThe Fourier motivation is sound and the citation pattern is fair; no self-citation loop. I don't think the central direction is wrong. The paper is useful for anyone working on QML training dynamics or QSciML, and with a cleaned-up metric and disclosed hyperparameters it would be a solid contribution. As is, it deserves peer review but with major revision. I would not cite it in its current form.","headline":"Useful empirical study of growing PQCs, but the 'final loss' claim is undercut by best-epoch test-set selection and undisclosed growth schedules.","tokens_in":11882,"tokens_out":1848,"would_cite":false,"duration_ms":17168,"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 paper claims that growing a parameterized quantum circuit during training—adding identity-initialized reuploader or feature-map blocks when needed—yields lower losses, less variance, and better generalization than fixed-depth training.","keywords":["parameterized quantum circuits","growth-based architectures","reuploader circuits","frequency spectrum","quantum machine learning","barren plateaus","overfitting","2D Laplace equation"],"falsifier":"Rerun the student-teacher and Laplace experiments with a sweep over growth schedules (epochs between growth, number of blocks $\\ell$ per stage, stopping patience), selecting schedules by validation loss for each method; if fixed-depth CDL with the same final depth and identical initialization then matches or beats the best growth model, the reported advantage is a scheduling artifact rather than a property of growth.","tokens_in":10914,"feed_emoji":"🌱","tokens_out":12948,"duration_ms":109256,"temperature":0.7,"pith_summary":"Training a parameterized quantum circuit (PQC) usually means fixing its architecture and optimizing all parameters from the start. This paper argues that starting small and letting the circuit grow during training is better. Three growth strategies—Block Growth, Sequential Feature Map Growth, and Interleave Feature Map Growth—insert reuploader blocks or feature-map gates one stage at a time, initialized as identities so the function already learned is preserved. Because a reuploader circuit—one that alternates data-encoding feature maps with trainable ansatz blocks—is a truncated Fourier series, each added feature map widens the accessible frequency spectrum, so capacity tracks what the data actually needs. Across student-teacher regressions, a noisy regression, and the 2D Laplace equation, the growing circuits achieve lower final losses, lower variance across random seeds, and better generalization than fixed-depth complete-depth learning, including a regime where the fixed circuit appears to hit a barren plateau.","feed_headline":"Growing quantum circuits as they train beats fixed-depth models","feed_subtitle":"Adding capacity only when needed cuts losses, variance, and overfitting on noisy and PDE tasks.","key_machinery":"The load-bearing object is the reuploader PQC viewed as a truncated Fourier series, $f(x,\\theta,\\psi) = \\sum_{\\omega_j \\in \\Omega} c_j(\\theta,\\hat{C}) e^{i \\omega_j(\\psi)\\cdot \\phi(x)}$. The accessible frequencies $\\Omega$ come from the eigenspectrum of the feature-map generator $\\hat{G} = \\sum_m \\psi_m \\hat{X}_m/2$; repeating the feature map more times enlarges $|\\Omega|$ roughly linearly. Growth methods insert identity-initialized blocks: each added gate is paired with a cancellation gate so the circuit still implements the identity at insertion, preserving the learned function while giving the new parameters nonzero gradients from the start. Block Growth appends whole reuploader blocks (feature map plus ansatz), increasing both the frequency range and the Fourier-coefficient expressivity; Sequential FM Growth inserts feature maps from left to right between existing ansatz blocks; Interleave FM Growth inserts them from the center outward. Identity insertion is what makes mid-training growth safe, and the Fourier-spectrum view turns growth into a controlled expansion of capacity rather than a random widening.","core_discovery":"The paper's central discovery is that dynamically grown PQCs dominate static ones in every setting tested. In student-teacher tasks where a randomly initialized teacher circuit with $L$ reuploader layers generates the data, all three growth methods reach the same final depth as the teacher but end with lower mean and worst-case mean squared error than complete-depth learning: the best 1-qubit mean is $(1.029 \\pm 3.52)\\times 10^{-6}$ for Interleave FM Growth versus $(8.13 \\pm 1.42)\\times 10^{-5}$ for the best randomly initialized CDL baseline. On a noisy 20-point regression, block growth finishes with train/test losses of $0.035/0.063$, while a 20-layer CDL model overfits to $0.005/0.158$. Solving the 2D Laplace equation with 5 to 13 reuploader layers, all three growth methods keep low $L^2$ relative error with small variance, while identity-initialized CDL degrades sharply at 13 layers—behavior the paper attributes to the onset of a barren plateau. The common mechanism is spectral growth: expanding the model's Fourier frequencies gradually acts as a regularizer and keeps the circuit shallow until extra capacity is justified.","pith_inferences":["Extension: if the spectral explanation is right, growth should act as a curriculum over Fourier modes—low frequencies are learned first and higher frequencies added only when the target demands them—so the gains should be largest on targets with decaying spectral power, such as natural images or PDE solutions.","Extension: the paper triggers growth after a fixed number of epochs; an adaptive trigger based on validation-loss plateaus would make the method self-terminating and would likely enlarge its advantage on tasks of unknown complexity.","Extension: a decisive comparison not reported here is against a fixed-depth CDL model matched in total final parameters; if growth still wins at equal final capacity, the benefit comes from the training trajectory itself rather than from having fewer parameters.","Extension: because added feature maps enlarge the frequency spectrum roughly linearly in repetitions, the approach could be combined with trainable frequency generators (the paper's Equation 3 setting) to give the optimizer continuous control over both coefficients and spectrum."],"forward_implications":["Practitioners can start with a compact PQC and grow it only as the task demands, keeping circuits shallower on noisy hardware without sacrificing final accuracy.","Growth acts as an implicit regularizer: because high-frequency modes are added late, models trained on small noisy datasets are less likely to fit noise, as shown by the noisy student-teacher train/test gap.","At depths where fixed-depth training degrades (the 13-layer Laplace case), growth-based training maintains performance, which the authors interpret as resilience to barren-plateau onset.","The three growth variants are largely interchangeable in accuracy on the Laplace task, so the choice can be driven by hardware layout or implementation convenience rather than performance.","Identity initialization of added blocks preserves the current function, so growth can be inserted into an ongoing optimization without resetting progress."],"supporting_citations":[{"why":"Establishes that reuploader PQC outputs are truncated Fourier series with frequencies fixed by the encoding gate spectrum; supplies the spectral-growth rationale.","marker":"[17]"},{"why":"Introduces trainable feature-map generator frequencies, letting the accessible spectrum change during training; underpins the paper's Equation 3 and the trainable $\\psi$ parameters.","marker":"[20]"},{"why":"Identity-pair initialization for added gates preserves the current function while providing nonzero gradients from the start of optimization.","marker":"[21]"},{"why":"Defines barren plateaus in quantum neural network training landscapes; motivates the depth-related trainability claims.","marker":"[8]"},{"why":"Prior layerwise-growth method that builds PQCs layer-by-layer with zero-initialized blocks; the paper distinguishes its identity-initialized growth from this approach.","marker":"[18]"},{"why":"Supplies the differentiable-quantum-circuit method used for the 2D Laplace equation application.","marker":"[6]"}],"fun_headline_variants":["Grow-as-you-go quantum circuits beat fixed-depth models","Add reuploader blocks mid-training to curb overfitting","Dynamic circuit growth delivers stable quantum convergence","Adaptive depth training wins on noisy quantum regression","Spectral growth regularizes quantum circuit training"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central comparison assumes that the growth schedule—how many epochs pass before growing, how many blocks are added per stage, and when growth stops—was fixed in advance and not tuned on the test set to favor the growing methods.","fun_headline_variants_meta":{"raw":{"variants":["Grow-as-you-go quantum circuits beat fixed-depth models","Add reuploader blocks mid-training to curb overfitting","Dynamic circuit growth delivers stable quantum convergence","Adaptive depth training wins on noisy quantum regression","Spectral growth regularizes quantum circuit training"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000385,"raw_usage":{"total_tokens":2027,"prompt_tokens":931,"completion_tokens":1096,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":547,"completion_tokens_details":{"reasoning_tokens":1024}},"tokens_in":547,"tokens_out":1096,"duration_ms":12233,"temperature":1.0,"reasoning_tokens":1024,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:58:42.060991+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Rerun the student-teacher and Laplace experiments with a sweep over growth schedules (epochs between growth, number of blocks $\\ell$ per stage, stopping patience), selecting schedules by validation loss for each method; if fixed-depth CDL with the same final depth and identical initialization then matches or beats the best growth model, the reported advantage is a scheduling artifact rather than a property of growth.","supporting_citations":[{"cited_title":"Ef- fect of data encoding on the expressive power of variational quantum-machine-learning mod- els,","cited_arxiv_id":null,"evidence_quote":"Establishes that reuploader PQC outputs are truncated Fourier series with frequencies fixed by the encoding gate spectrum; supplies the spectral-growth rationale."},{"cited_title":"An initialization strategy for ad- dressing barren plateaus in parametrized quan- tum circuits,","cited_arxiv_id":null,"evidence_quote":"Identity-pair initialization for added gates preserves the current function while providing nonzero gradients from the start of optimization."},{"cited_title":"Barren plateaus in quantum neural network training landscapes,","cited_arxiv_id":null,"evidence_quote":"Defines barren plateaus in quantum neural network training landscapes; motivates the depth-related trainability claims."},{"cited_title":"Layerwise learning for quantum neural networks,","cited_arxiv_id":null,"evidence_quote":"Prior layerwise-growth method that builds PQCs layer-by-layer with zero-initialized blocks; the paper distinguishes its identity-initialized growth from this approach."},{"cited_title":"Kyriienko, A","cited_arxiv_id":null,"evidence_quote":"Supplies the differentiable-quantum-circuit method used for the 2D Laplace equation application."}],"review_version":1}