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REVIEW 3 major objections 6 minor 51 references

The paper claims that replacing a quantum kernel's full Gram matrix with a trainable sample-to-class-centroid fidelity matrix yields an intrinsically multiclass quantum kernel method with linear runtime O(n_train·M), and that the model is t

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2026-08-01 11:43 UTC pith:JKAKIFBG

load-bearing objection A clean, honest multiclass extension of Quack with a sound linear-scaling argument, undercut by a benchmark comparison to Bowles et al. baselines that are taken as reported rather than re-run under matched conditions. the 3 major comments →

arxiv 2607.19782 v1 pith:JKAKIFBG submitted 2026-07-22 quant-ph cs.LG

A Multiclass Quantum Aligned Centroid Kernel

Kilian Tscharke , Pascal Debus This is my paper
classification quant-ph cs.LG
keywords quantum kernel methodsmulticlass classificationkernel-target alignmenttrainable embeddingssample-to-centroid fidelitylinear runtimevariational classifiersbarren plateaus
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to show that three standard limitations of kernel methods—quadratic scaling with training-set size, fixed non-trainable kernels, and the absence of a native multiclass formulation—can be removed simultaneously in a quantum setting. Its vehicle is McQuack, a quantum kernel built from a trainable sample-to-class-centroid fidelity matrix instead of a full Gram matrix. The authors claim that training and inference run in time O(n_train·M), that the trained model is the strongest 'pure' quantum baseline across a 133-dataset benchmark, and that even an untrained hardware version on 124 qubits performs comparably to an RBF kernel. If the linear-scaling claim holds, quantum kernel methods could be applied to substantially larger training sets than Gram-matrix approaches allow.

Core claim

McQuack replaces the n_train×n_train Gram matrix with an n_train×M matrix whose entry K_{i,m} is the fidelity between the encoded data point x_i and a trainable class centroid c_m. The embedding parameters (w,b) and the centroids are optimized alternately, both against the same kernel-target alignment objective, so the model learns both a data embedding and a set of class prototypes in Hilbert space. The paper reports that in simulation on 133 synthetic and 19 real-world datasets, the model outperforms existing pure quantum baselines; on hardware, an untrained instance on a 124-qubit subgraph of two 156-qubit devices reaches accuracy in the range of an RBF kernel. The authors also report no

What carries the argument

The central object is the sample-to-centroid fidelity matrix K ∈ R^{n_train×M} with K_{i,m} = |⟨0^{⊗n}|U†(c_m,w,b)U(x_i;w,b)|0^{⊗n}⟩|². It defines a kernel between each sample and each class centroid, rather than between pairs of samples. Training alternates kernel-target alignment on the shared embedding parameters and on the centroid parameters, giving an intrinsically multiclass model whose cost per epoch is O(n_train M) shots via the parameter-shift rule. A secondary object is the mapping of this kernel to a variational-classifier form, which shows that the shared embedding parameters are the key structural difference from an ensemble of independently trained variational classifiers.

Load-bearing premise

The claim that McQuack is the strongest pure quantum model depends on comparing its benchmark rankings to baseline rankings produced in a separate study with possibly different splits, seeds, and hyperparameter choices; if those protocols are not interchangeable, the comparison is not apples-to-apples.

What would settle it

Re-run the 133-dataset benchmark with identical train/test splits, seeds, and hyperparameter optimization budgets for McQuack and all baseline quantum models; if McQuack's average normalized rank is not lower than the other pure quantum models, the paper's headline comparative claim fails.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Training and inference costs scale linearly with the number of training samples and the number of classes, removing the O(n_train²) Gram-matrix bottleneck of standard kernel methods.
  • Because the kernel is trainable through kernel-target alignment, the embedding adapts to the dataset instead of being fixed in advance.
  • The multiclass formulation is intrinsic: a single model handles M classes through M trainable centroids, without one-vs-rest or one-vs-one decomposition.
  • Untrained inference on more than 100 qubits is feasible with error mitigation, suggesting that large fidelity-based quantum kernels are not out of reach on current hardware.
  • Parameter initialization—not regularization—dominates trainability in the tested regime, giving concrete guidance for practitioners.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the centroid representation is the real source of the gain, a purely classical analogue—optimizing RBF centers directly against kernel-target alignment—should reproduce much of McQuack's behavior; this is testable and would help isolate whether a quantum advantage is present.
  • The O(n_train M) scaling shifts the practical bottleneck toward the shot count and the number of classes, so for very large M the method may still be expensive; gradient-free or hybrid optimizers could change that trade-off.
  • The 'no barren plateaus' result is confined to 13 qubits; extending the trainability study to larger system sizes with hardware noise would tell whether the observed stability survives scale-up.
  • If linear-time trainable quantum kernels prove robust, a natural next step is to apply them in online or streaming settings where Gram matrices cannot be maintained.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper introduces McQuack, a multiclass quantum kernel method that replaces the full n_train×n_train Gram matrix with an n_train×M fidelity matrix between data samples and trainable class-centroid parameters. The shared embedding parameters (w,b) and the centroid parameters (c_m) are optimized alternately via kernel-target alignment (KTA) losses (Eqs. 22–23), and the method is claimed to have O(n_train·M) training cost (Eq. 10). The authors evaluate an untrained version on 124-qubit IBM hardware, benchmark the trained model in simulation on the QML Benchmark Suite of Bowles et al. and on 19 real-world datasets, and present a trainability study for up to 13 qubits. The central claims are the linear-scaling formulation, the empirical claim that McQuack is the strongest "pure" quantum model in the QML Benchmark, and the absence of observed barren plateaus in the tested range.

Significance. If the empirical claims are supported, this is a useful contribution to quantum kernel methods: it directly addresses the quadratic scaling of full-Gram kernels, provides a native multiclass formulation, and includes a rare hardware demonstration on 100+ qubits. The O(n_train·M) derivation in Eq. (10) is explicit and appears sound. The hardware experiment is honestly reported as a feasibility demonstration, and the trainability study, though limited, is practically relevant. However, the headline benchmark comparison rests on a cross-paper ranking against baselines "as reported in [4]" without a matched protocol, and this is not yet auditable. I do not see a circularity problem: KTA training on labeled data and test evaluation are standard supervised learning.

major comments (3)
  1. [§4.2.1, Fig. 2a, App. B.2.2] The central empirical claim that McQuack is "the strongest pure quantum model in this benchmark" is based on normalized ranks in which the seven QNN baselines are taken "as reported in [4]" rather than re-run under the protocol used for McQuack. McQuack's hyperparameters were chosen by its own grid search (Table 7), accuracies are averaged over 5 seeds, and for downscaled MNIST the training set was reduced to 500 samples (B.2.2), while the baselines come from a separate pipeline with their own splits, seeds, and search budgets. Moreover, Fig. 2a and Fig. 10 show that some baselines (DressedQuantumCircuitClassifier, QuantumBoltzmannMachine) were evaluated on only a subset of the 133 datasets. Averaging normalized ranks over heterogeneous evaluation sets can bias the comparison. To support the claim, the authors should either rerun all baselines under the identical data pipeline and search
  2. [§4.1, Table 1, §1.2 Contribution 2] The hardware experiment consists of three runs per model and is explicitly non-significant (Sec. 4.1). The abstract and Contribution 2 nevertheless state that McQuack "demonstrate[s] performance comparable to an RBF kernel" on 124 qubits. Table 1 shows 0.83±0.02 (Pittsburgh) and 0.79±0.02 (Aachen) vs. 0.80 for SimpleKernel; with n=3 these differences are within noise. This is a legitimate feasibility demonstration, but the word "demonstrate" overstates the evidence. Please align the abstract/conclusion wording with the statistical caveat stated in Sec. 4.1.
  3. [§4.3, Figs. 4–5] The trainability claim "no evidence of barren plateaus up to 13 qubits" is based on gradient statistics and loss curves for a few system sizes, with no quantitative scaling analysis of gradient variance vs. qubit number. The paper itself notes the result may not generalize (Sec. 6). I do not block on this, but if the claim is kept as a contribution, report the variance of gradients as a function of n (e.g., a table or fit) so readers can judge whether the data actually constrain barren-plateau behavior. At minimum, the abstract should say "no evidence in the tested range," matching the Limitations section.
minor comments (6)
  1. [Abstract] Typo: "formulticlass" should be "for multiclass".
  2. [§2.3, Eq. (6)] The stated upper bound 1/sqrt(M) assumes nonnegative kernel entries; please state this assumption explicitly in the main text and clarify the affine shift in LKAO introduced in footnote 5.
  3. [§3.2, Eq. (10)] The O(ntrain·M) statement suppresses D and d, which are fixed for a chosen architecture and data dimension; this is fine, but please say so explicitly so readers do not misread the complexity as independent of feature dimension or qubit count.
  4. [§4.3, Fig. 3] The fANOVA method is not defined or referenced. Add a citation or a sentence describing how the relative importance values are computed.
  5. [Table 6] Several entries contain formatting artifacts such as "0 .75" (e.g., Census, McQuack F1). Please fix the spacing throughout the table.
  6. [§7] Code availability is pending export control clearance; this is understandable, but in the meantime the QML benchmark comparison cannot be reproduced. Consider releasing at least the benchmark-evaluation scripts and seeds, or provide a detailed protocol document.

Circularity Check

0 steps flagged

No circularity found: McQuack's derivation is self-contained; cross-paper benchmark comparability is an external-validity concern, not circularity.

full rationale

The core model is defined explicitly in Section 3: K is a fidelity matrix between samples and trainable class centroids (eqs. 7-9), K* is the label-derived ideal matrix (eq. 4), and training minimizes 1 - A(K,K*) (eqs. 6, 22, 23). Test labels are predicted by argmax over fidelities (eqs. 11, 25). No equation is defined in terms of its own output, and no fitted parameter is renamed as a prediction. The linear-runtime claim (eq. 10) is a direct shot-count of the parameter-shift-rule training loop, not an assertion derived from the model's performance. The 'no barren plateaus' statement is an empirical observation over 13 qubits, explicitly acknowledged as possibly not generalizing (Section 6). The only self-citation to the binary predecessor Quack [7] is used for provenance ('builds upon'), while the multiclass construction and all equations are given in the paper; no load-bearing result is imported solely from [7]. The benchmark claim that McQuack is the strongest 'pure' quantum model does rely on baseline rankings 'as reported in [4]' with McQuack run separately, but this is a question of experimental comparability and external validity, not circularity: it does not make the model's output equal to its input by construction. Code availability is pending, which hampers audit but does not constitute circular reasoning.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

The paper's empirical claims rest on a large number of fitted trainable parameters and grid-searched hyperparameters, and on the assumption that benchmark baselines from [4] are comparable. No new physical entities are introduced.

free parameters (5)
  • Trainable feature-map parameters w, b = Learned per dataset via KTA
    Optimized during Kernel Alignment Optimization; all performance results depend on this fit.
  • Trainable centroid parameters c_m = Learned per dataset via centroid optimization
    Centroids are fitted to each class's training data; central to the fidelity kernel.
  • Model hyperparameters (n_qubits, n_repetitions, learning rates, regularization, batch size, patience) = Selected by grid search per dataset (Tables 7-8)
    Performance results are conditional on this hyperparameter selection; grid search is a form of fitting to validation data.
  • Bit flip tolerance b = 45
    Calibrated on the hardware dataset to maximize diagonal/off-diagonal kernel separation (Fig. 9); affects untrained hardware accuracy.
  • Initialization variance gamma^2 = 1/(3 n L) = Set per circuit size
    Chosen to promote trainability; affects convergence in the trainability study.
axioms (5)
  • standard math Fidelity kernel is positive definite (Eq. 3)
    Used to treat K as a valid kernel; proven in Schuld 2021 [15].
  • standard math Parameter-shift rule gives correct analytic gradients
    Relied on for shot-count scaling in Eq. (10) and for hardware gradient estimates.
  • standard math Data re-uploading circuits realize a truncated Fourier series
    Supports the expressivity discussion in Section 2.2, but not load-bearing for empirical results.
  • domain assumption Classes are representable by centroids in the quantum feature space
    Explicitly stated in Limitations (Section 6); if false, the fidelity-based classification rule (Eq. 9) cannot separate classes.
  • domain assumption Benchmark baselines from Bowles et al. [4] are directly comparable to McQuack's runs
    Section 4.2.1/Fig. 2a uses 'as reported in [4]' for baseline rankings without evidence that datasets, splits, seeds, or hyperparameter protocols match the McQuack evaluation.

pith-pipeline@v1.3.0-alltime-deepseek · 26709 in / 16071 out tokens · 143341 ms · 2026-08-01T11:43:36.552380+00:00 · methodology

0 comments
read the original abstract

Kernel methods are powerful tools in machine learning but commonly used full-Gram kernels face three key limitations: (1) quadratic scaling with training set size; (2) the use of fixed, non-trainable kernels; and (3) the absence of an intrinsic formulation for multiclass classification. We present McQuack, a trainable quantum kernel method for multiclass problems that achieves linear scaling in the number of training samples. This is accomplished by replacing the full training-set Gram matrix with a trainable sample-to-(class-centroid) fidelity matrix. We evaluate the model in simulation and on 124 qubits of two IBM devices, across more than 150 datasets. In simulation, McQuack outperforms existing "pure" quantum baselines, while results from hardware inference -- obtained without training -- achieve performance similar to an RBF kernel. Finally, we study the trainability of the model and observe no evidence of barren plateaus in our experiments with up to 13 qubits, and highlight the importance of parameter initialization for successful optimization.

Figures

Figures reproduced from arXiv: 2607.19782 by Kilian Tscharke, Pascal Debus.

Figure 1
Figure 1. Figure 1: Schematic overview of the McQuack training process on a binary dataset. (A–E) Data distribution in two dimensions (ϕ0, ϕ1) of the Hilbert space after sequential steps of model training. (A) Initially, the embedding distributes data almost randomly, and the centroids do not represent the class centers well. (B) During the first KAO iteration, the embedding is trained to cluster data points more tightly arou… view at source ↗
Figure 2
Figure 2. Figure 2: Number of rankings (blue/first to red/last) for [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: fANOVA analysis showing the relative impor [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Mean and standard deviation of the weight gradients computed for the initial batch of the [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Training loss for different system sizes and [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Circuit of a layer unitary Ul consisting of the rotation unitary (green) and the entangling unitary (blue). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |0⟩ U0(xi ; w0, b0) U1(xi ; w1, b1) UL−1(xi ; wL−1, bL−1) U † L−1 (cm; wL−1, bL−1) U † 1 (cm; w1, b1) U † 0 (cm; w0, b0) |0⟩ |0⟩ |0⟩ [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Schematic of the full McQuack circuit. The yellow section encodes the sample xi and the purple section the centroid cm. A single layer unitary Ul is shown in fig. 6. A.3 Centroid Optimization The Centroid Optimization optimizes the centroids C = [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: Bit flip tolerance calibration on ibm_pittsburgh based on kernel estimates between class centroids for varying [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: Map of all 156 qubits and their connectivity on [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Rankings for the Quantum Neural Network family split up by classification tasks. These rankings [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Accuracy scores on the training and test sets are shown as a function of dataset dimensionality. The [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Architecture of the circuit if executed on hardware. The kernel entry [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗

discussion (0)

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