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On a reduced SDSS dataset, the quantum kernel SVM outperforms the HHL-based least-squares SVM in most accuracy, precision, recall, and F1 comparisons, while classical SVMs stay slightly ahead.

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On a reduced SDSS dataset, quantum-kernel QSVM outperforms HHL LS-SVM, classical SVMs are slightly ahead, and the HHL method's constant scaling stems from using only two class-average representatives.

T0 review reviewed 2026-08-04 challenge →

load-bearing objection The HHL baseline is reduced to two class-average centroids, so the headline QSVM-vs-HHL comparison is not about the method itself; the paper still has useful scaling and multi-class decomposition data, but the claims need rescoping. the 3 major comments →

arxiv 2509.10190 v1 pith:ZON5TDZY submitted 2025-09-12 quant-ph

Quantum kernel and HHL-based support vector machines for multi-class classification

Gabriela Pinheiro , Donovan Slabbert , Luis Kowada , Francesco Petruccione This is my paper
classification quant-ph
keywords quantum kernelHHL algorithmleast-squares SVMmulti-class classificationSDSS datasetquantum machine learningclass imbalancescaling analysis
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 reading

This paper compares two quantum SVM approaches for three-class classification of astronomical objects—stars, galaxies, and quasars—from a reduced Sloan Digital Sky Survey dataset. It implements a quantum kernel SVM (QSVM) using angle encoding and a projection measurement, and an HHL-based least-squares SVM (HHL LS-SVM) that solves a small linear system built from per-class feature averages. Across one-vs-rest and two-step hierarchical schemes, the QSVM achieves higher accuracy, precision, recall, and macro-F1 than the HHL LS-SVM in most cases, while classical SVMs perform comparably or slightly better. The paper also reports a scaling trade-off: QSVM cost grows quadratically with the number of samples, whereas HHL LS-SVM circuit cost stays essentially constant, making the latter potentially attractive for very large datasets if its representation and noise limitations can be overcome.

Core claim

The central claim is that on this ten-feature, three-class SDSS subset, the quantum kernel approach yields better classification performance than an HHL-based least-squares SVM. The QSVM typically achieves accuracy around 0.96-0.97 and macro-F1 above 0.93, whereas the HHL LS-SVM's best macro-F1 is about 0.86 for galaxy isolation and much lower for quasars (0.55-0.70). The HHL method's weakness is traced to its reliance on only two representative elements—the class-averaged feature vectors—which cannot adequately capture the minority QSO class, while the QSVM benefits from explicit point-by-point encoding and class-weight balancing. Classical SVMs remain slightly ahead of both, indicating tha

What carries the argument

The QSVM's quantum kernel is a fidelity between angle-encoded states: each sample's ten features are mapped to y-rotations on ten qubits, and the kernel value for a pair of samples is the squared overlap obtained by projecting the composite state onto the all-zero state; this kernel matrix is then fed to a classical SVM. The HHL LS-SVM reformulates training as solving the linear system (K + gamma^-1 I)a = y, with kernel entries computed from the average feature vector of each class (two representative elements), and uses the HHL quantum algorithm to obtain a normalized solution. The contrast between explicit sample encoding and compressed class-average representation is what drives the obser

Load-bearing premise

The load-bearing premise is that a 2x2 linear system built from the per-class average features adequately represents the SDSS classes for classification; if the class means do not capture the decision boundaries, especially for minority quasars, the HHL LS-SVM comparison and the constant-scaling conclusion apply only to this compressed toy representation.

What would settle it

Train the HHL LS-SVM with substantially more representative elements, say ten or fifty per-class centroids derived from clustering rather than a single mean, on the same SDSS data and check whether the QSO macro-F1 rises above about 0.7. If it does, the reported performance gap is an artifact of the two-element compression, not an inherent limitation of the HHL algorithm. Conversely, running the same comparison on a dataset with more than three classes or where class means are nearly identical would test whether the representation is the true bottleneck.

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

If this is right

  • If the paper is correct, quantum kernel SVMs can be applied to moderately sized multi-class datasets and match classical performance while operating in a quantum feature space.
  • The HHL LS-SVM's essentially constant circuit scaling means it could in principle handle datasets with many more samples without adding qubits or circuit depth, if the number of representative elements is increased beyond two.
  • The two-step hierarchical scheme reduces the number of quantum classifiers from three to two with only a small accuracy loss, offering a resource-efficient structure for multi-class problems.
  • The minority-class failures of HHL LS-SVM indicate that class imbalance must be addressed directly, through weights, more representative elements, or a different decomposition, before the method becomes practically useful.
  • The reported noise sensitivity of HHL suggests that error-corrected quantum hardware is a prerequisite for real-device deployment of this approach.

Where Pith is reading between the lines

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

  • The two-centroid representation used for HHL LS-SVM means the comparison effectively tests whether per-class means can separate the SDSS classes; a dataset with heavily overlapping class distributions would likely degrade the HHL results even further than observed here.
  • A natural extension is to increase the number of representative elements, for instance using several centroids per class from clustering, and check whether HHL LS-SVM approaches QSVM accuracy while preserving its scaling advantage.
  • The constant-scaling claim applies to circuit size and gate count; the classical preprocessing and kernel computation would still grow with dataset size, so end-to-end computational cost is not constant.
  • The QSVM's need for pairwise kernel evaluations makes it impractical for very large surveys; randomized or low-rank kernel approximations could close that gap without changing the core methodology.
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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 / 5 minor

Summary. This paper compares two quantum SVM pipelines for three-class SDSS classification (stars, galaxies, QSOs): a quantum-kernel QSVM with angle encoding and a classical SVM backend, and an HHL-based least-squares SVM in which the kernel matrix is built from two representative elements defined as the per-class average feature vectors. Both one-vs-rest and two-step hierarchical decompositions are evaluated, with accuracy, macro precision/recall/F1, and a scaling analysis reported. The authors state that QSVM outperforms the HHL implementation in most settings, that classical SVMs are competitive or slightly better, and that the HHL implementation has essentially constant resource scaling. All reported results are simulations; real-device HHL runs were abandoned because of noise.

Significance. If the claims were taken at face value, the paper would provide a useful benchmark of two quantum classification pipelines on a realistic astronomical dataset. Its most reliable contribution is the direct QSVM-vs-CSVM comparison, which again shows no quantum advantage, and the transparent account of HHL's noise sensitivity. However, the headline comparison is weakened by the fact that the HHL "LS-SVM" is not a sample-based LS-SVM but a two-centroid classifier, so the performance and scaling results are largely consequences of the chosen n_rep=2 rather than empirical discoveries about HHL LS-SVM. The paper does not ship code, and several hyperparameters needed for reproduction are omitted. The manuscript is potentially useful as a scoped benchmark, but the title and abstract overstate the generality of the QSVM-vs-HHL comparison.

major comments (3)
  1. [3.3 and Eq. (12)] The HHL implementation uses exactly two representative elements, defined as the per-class averages of the selected features. Equation (12) is therefore a 2x2 linear system, and the decision rule in Eq. (10) is a linear function of similarities to two class centroids. This is not an LS-SVM trained on the data, nor is it an HHL solution of the general LS-SVM formulation in Eq. (9) with a kernel matrix over training samples. The abstract's statement that 'QSVM outperforms HHL LS-SVM in most cases' is therefore not established for HHL LS-SVM as such; it is established only for this capacity-limited, two-centroid variant. The conclusion's sentence 'The point was to bias the comparison in favour of the well-established QSVM' confirms this. The authors should either (a) run HHL with more representative elements and report how performance changes, or (b) consistently and prominently scope all cl
  2. [4 and Fig. 3] The 'essentially constant scaling' of HHL LS-SVM is a consequence of the construction, not an empirical finding: the circuit width and depth depend on n_rep=2 and log kappa, not on the number of dataset samples. Counting gate operations for a fixed 2x2 linear system cannot reveal any sample-size scaling. This statement should be presented as an identity of the chosen reduction, and the resource analysis should also account for the classical O(N d) cost of constructing the class-average representatives and for how resources grow with n_rep.
  3. [3.3 and 4] Reproducibility is hampered by missing hyperparameters. The value of the regularization parameter gamma in Eq. (12) is never stated, despite directly defining the linear system solved by HHL. Similarly, Section 3.2 says the QSVM was repeated 'a number of times' without giving the number of repetitions or the exact standard-error formula. These omissions prevent an independent check of the reported accuracies and uncertainties.
minor comments (5)
  1. [Table 1] The HHL LS-QSVM Star row reports recall as '0.67.12', which is not a valid number. Table 2 uses a decimal comma ('0,704') in the Star-vs-All precision. These formatting errors should be corrected.
  2. [Eq. (10)] The index structure in Eq. (10) is unclear: the summand is written as a_i k(x_i, x_j) while the sum runs over j. It should likely be a_j k(x_j, x_i) (or similar) with the test sample index kept separate.
  3. [Figures] The text refers to the HHL circuit as 'illustrated in 3' (Section 2.2), but the figure is numbered 2. Please fix the cross-reference.
  4. [Tables 1 and 2] The notation 'HHL LS-QSVM' and 'HHL LS-CSVM' is used without explicit definition. Since 'LS-QSVM' could be read as a quantum SVM, and 'LS-CSVM' as a classical SVM, the authors should define these terms clearly (e.g., quantum HHL vs. classical HHL solution of the same reduced system).
  5. [3.2] The preprocessing of the ten SDSS features to [0, pi] is mentioned only in passing; the exact scaling or normalization procedure should be specified for reproducibility.

Circularity Check

0 steps flagged

No circular derivation: the QSVM-vs-HHL comparison is empirical and self-contained; self-citations are contextual and not load-bearing.

full rationale

After walking the derivation chain, I find no circular step that warrants a nonzero score on the circularity scale. The QSVM/HHL comparison is an empirical benchmark: kernel values are computed from fixed angle-encoding circuits (Eqs. 3-5), the HHL LS-SVM solves the fixed linear system (K+γ^{-1}I)a = y (Eq. 12) with K built from class-averaged representative elements, and all reported accuracies/precisions/recalls/F1 are evaluated on held-out samples. No parameter is fitted to the reported metrics, and no 'prediction' is defined in terms of the conclusion. The self-citations ([27], [34], [35], [51]) are contextual or prior applications; the central equations are attributed to Rebentrost et al. [14] and Yang et al. [46], and the comparison does not rest on the authority of these self-citations. The 'essentially constant' HHL scaling is a disclosed design consequence of fixing n_rep=2 and processing data classically, not a hidden equivalence or a fitted result. The manuscript itself flags the main validity limitation—'the number of representative elements was fixed to two, which limits the amount of information fed into the circuit' and 'The point was to bias the comparison in favour of the well-established QSVM'—which is a fairness/scoping issue rather than circular reasoning. Score 2 reflects only the presence of minor, non-load-bearing self-citations; the central claim has independent empirical content.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

The numerical claims rest on choices not derived from the data: gamma for the HHL regularized system, a fixed two-element class-average representation, a 300-sample cap for QSVM, inverse-frequency class weights, and [0, pi] angle-encoding scaling. The HHL scaling behavior is a logical consequence of the two-element representation rather than an emergent property. No new physical entities are introduced; the main trusted inputs are prior ML labels, the bias-free LS-SVM reduction, and ideal simulator behavior.

free parameters (5)
  • gamma (regularization) = not reported
    User-defined weight between training error and objective in the HHL LS-SVM linear system (Eqs 9 and 12); affects conditioning and the solution, but no value or sensitivity analysis is given.
  • number of representative elements = 2
    Fixed to two because of simulator qubit limits (Sec 3.3); this determines the 2x2 kernel matrix and makes the scaling constant by construction.
  • QSVM training and test sample size = 300
    Chosen to mitigate quadratic SVM scaling (Sec 3.2); the comparison and all QSVM error bars depend on this arbitrary cap.
  • class weights = inverse frequency normalized by number of classes
    Used for QSVM and CSVM fitting (Eq 13); a modeling choice that affects balanced metrics and is not available in all real-world settings, as the authors note.
  • angle encoding scale = [0, pi]
    Features are scaled to [0, pi] for RY rotations; this preprocessing choice changes the quantum kernel geometry and therefore the QSVM results.
axioms (5)
  • domain assumption SDSS class labels from a prior machine-learning catalogue are treated as ground truth.
    Sec 3.1: labels were assigned using previous machine learning techniques; all accuracy, precision, recall, and F1 metrics assume these labels are correct.
  • domain assumption Setting b' = 0 in the LS-SVM linear system preserves enough information for classification.
    Eq 12 reduces the original system (Eq 9) by removing the bias row and column; the paper relies on the reduction from ref 46 without re-deriving it.
  • ad hoc to paper Class-average feature vectors suffice as representative elements of each class for kernel matrix construction.
    Sec 3.3: kernel entries are computed from the average values of selected features per class, reducing all training data to one point per class.
  • standard math The normalized HHL output vector can be used as the LS-SVM coefficient vector in Eq 10.
    Sec 2.2: HHL returns a normalized version of A^-1|b>, and the paper assumes normalization does not change the classification boundary.
  • domain assumption Quantum phase estimation correctly encodes the matrix F = K + gamma^-1 I into the clock register.
    Sec 2.2 and Fig 2: the circuit assumes the matrix is Hermitian and invertible and that QPE resolution is sufficient for the eigenvalues.

reviewed 2026-08-04 · how reviews work

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Cite this review

Pith. "Pith review of Quantum kernel and HHL-based support vector machines for multi-class classification." pith.science (2026). https://pith.science/paper/ZON5TDZY

@misc{pith2026250910190,
  author       = {Pith},
  title        = {Pith review of: Quantum kernel and HHL-based support vector machines for multi-class classification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZON5TDZY}},
  note         = {Machine review of arXiv:2509.10190}
}
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read the original abstract

We compare two quantum approaches that use support vector machines for multi-class classification on a reduced Sloan Digital Sky Survey (SDSS) dataset: the quantum kernel-based QSVM and the Harrow-Hassidim-Lloyd least-squares SVM (HHL LS-SVM). Both one-vs-rest and two-step hierarchical classification schemes were implemented. The QSVM involves angle encoding of ten features, two unitary operator blocks consisting of rotational operator gates, and a projective measurement that projects the final state to the zero state. The HHL-based method involves solving a system of linear equations using the HHL algorithm and using the solution in a support vector machine approach. The results indicate that the QSVM outperforms HHL LS-SVM in most cases. HHL LS-SVM performs somewhat competitively in selected cases, particularly when isolating galaxies (majority), however, it also performs poorly in others, especially when isolating QSOs (minority). Comparisons with classical SVMs confirm that quantum and classical methods achieve broadly similar performance, with classical models performing slightly ahead overall. Scaling analysis reveals a trade-off: QSVM performance suffers from quadratic scaling with the number of samples and features, but benefits from explicit feature representation during training, while HHL LS-SVM scales essentially constantly, with moderate fluctuations, but suffers from limited representative elements. The HHL-based method is also highly noise-sensitive. These results suggest that QSVM performs better overall and will perform better on current hardware as well, but that the more efficient scaling of HHL LS-SVM makes it a useful option for larger datasets with many samples, especially if we move past the NISQ era.

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This paper was first reviewed by deepseek-v4-flash on August 4, 2026.