Quantum-Efficient Kernel Target Alignment

Andreas Rosskopf; Georg Kruse; Rodrigo Coelho

REVIEW 3 major objections 5 minor 2 cited by

Quantum-Efficient Kernel Target Alignment

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The paper claims that using the Nyström low-rank approximation after Kernel Target Alignment yields a quantum kernel SVM pipeline whose circuit executions scale linearly with training set size, with comparable accuracy on four datasets.

desk verdict A useful, honest empirical adaptation of Nyström to quantum kernel SVMs, but the headline linear-scaling claim rests on an untested assumption about M(N). read the letter →

arxiv 2502.08225 v1 pith:QMUXTP4U submitted 2025-02-12 quant-ph

Rodrigo Coelho , Georg Kruse , Andreas Rosskopf This is my paper · ORCID

classification quant-ph PACS 03.67.Lx

keywords quantumembeddingkernelskerneltargetalignmentNyströmapproximationsupportvectormachinesvariationalcircuitsmachinelearningnoisescalability

verification ladder T0 review T1 audit T2 compute T3 formal

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 claims that the main scalability bottleneck of quantum embedding kernel SVMs—the need to run a quantum circuit for each pair of training points—can be bypassed by using the Nyström low-rank matrix approximation after training. Instead of building the full N×N kernel matrix pairwise, it builds a small M×M landmark kernel and an N×M cross-kernel, reconstructing the full matrix approximately. With random landmarks M≪N, quantum circuit executions drop from O($N^{2}$) to O($NM^{2}$), and the testing matrix drops from O(PN) to O(PM). On four small synthetic datasets the SVM accuracy is comparable to the standard approach, and the authors claim this is the first quantum kernel pipeline whose circuit-execution count scales linearly with training set size at every step.

What carries the argument

The central object is the Nyström approximation of the quantum kernel Gram matrix. Given M randomly chosen landmark points, the quantum circuit computes the landmark-landmark fidelity matrix $K_{MM}$ and the training-landmark cross matrix $K_{NM}$; the full training matrix is reconstructed as $\tilde{K} \approx K_{NM} K_{MM}^{-1} K_{NM}^T$. This replaces $N^2$ quantum circuit executions with $NM + M^2$ executions. The training step uses a separate mini-batch KTA cost on $D$ points, making each iteration cost $O(D^2)$ and independent of $N$, and the same reconstruction formula gives an $O(PM)$ inference cost for $P$ test points.

What would settle it

Compute the eigenspectrum of a trained quantum kernel Gram matrix on a dataset with complex structure; if the relative Nyström error $\|K - K_{NM} K_{MM}^{-1} K_{NM}^T\|_F / \|K\|_F$ does not fall sharply before $M$ becomes a substantial fraction of $N$, the linear-scaling pipeline fails on exactly the regime where it is needed.

Watch

Extended reading notes

Core claim

On its own terms, the paper's discovery is that the KTA-optimized quantum kernel matrix behaves like a low-rank Gram matrix well enough that Nyström reconstruction from M randomly chosen landmarks preserves SVM classification accuracy while removing the quadratic circuit bottleneck. The authors combine this with mini-batch KTA training, which subsamples D points per training step, so that the whole pipeline—training, final kernel construction, and inference—has quantum complexity linear in N. They demonstrate on checkers, corners, donuts, and spirals that training and test accuracies are close to, and often identical with, the standard full-kernel method, and that under coherent and depolarizing noise the approximation remains usable, though less robust than the standard method.

Load-bearing premise

The whole speedup rests on the assumption that an optimized quantum kernel Gram matrix is well captured by a low-rank reconstruction from M randomly chosen landmarks; if the kernel's useful information is spread across many eigenvalues, accuracy collapses.

Editorial extensions

If this is right

The dominant final-kernel cost becomes linear in $N$, so quantum kernel SVMs can be applied to larger training sets than the quadratic pairwise approach allows.
The hyperparameter $M$ is a direct resource-accuracy knob: smaller $M$ means fewer quantum runs, while larger $M$ restores accuracy, as the experiments show.
The testing kernel matrix costs only $O(PM)$ executions, so classifying unseen points is independent of the training set size $N$.
Mini-batch KTA training plus Nyström reconstruction forms a complete recipe for end-to-end linear-scaling quantum kernel classification.
Under depolarizing and coherent noise the approximation error and device errors compound, so realistic deployments need larger $M$ or additional error mitigation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A practical diagnostic before adopting this pipeline is to inspect the eigenspectrum of the trained quantum kernel Gram matrix: Nyström reconstruction is faithful when its top $M$ eigenvalues dominate, and choosing $M$ by spectral energy would be more reliable than random selection.
The landmark selection strategy is an obvious lever; k-means-style landmarks could outperform random sampling on datasets with cluster structure, which the authors note but do not test.
The same Nyström reconstruction can be applied to other quantum kernel tasks, such as regression, clustering, or kernel PCA, whenever a Gram matrix must be materialized.
The noise results suggest that noisy landmark entries are amplified by the inversion $K_{MM}^{-1}$, which would explain why the Nyström method degrades faster than the full kernel under hardware noise.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this paper does something real and modest. It takes the classical Nyström low-rank approximation and applies it to the kernel matrix of a KTA-trained quantum embedding kernel, after training, so that the final matrix fed to the SVM requires O(NM) circuit executions instead of O(N^2). Combined with mini-batch KTA training à la Sahin et al., you get a pipeline where every stage's circuit count is linear in N if M and D stay fixed. The authors test this on four small synthetic datasets and report accuracy comparable to the exact kernel in the noiseless case. They also run coherent and depolarizing noise experiments, which honestly show the approximation becomes more fragile under noise.

What's good: the problem is real; the final kernel matrix construction is indeed a quadratic bottleneck in quantum kernel SVMs, and nobody else seems to have applied Nyström there. The presentation is clear, the code is on GitHub, and the authors explicitly flag that M and D may need to grow with dataset complexity, rather than hiding that caveat.

The soft spots are real but not fatal. First, the complexity count is off: they state O(NM^2) circuit executions, but the actual number is O(NM + M^2), since you compute an N-by-M cross matrix and an M-by-M block. It's an easy fix, but as written it's wrong. Second, the central linear-scaling claim is conditional on M not growing with N. The paper gives no spectral decay or error-bound evidence for the quantum kernel Gram matrix, and no scaling law for M(N). Their experiments use N up to 100 and M in {2,4,8}, so the linear-in-N claim is an assumption about the low-rank structure, not a demonstrated property. The noise results strengthen this concern: the Nyström approximation degrades faster than the full kernel under both coherent and depolarizing noise, so in realistic settings the accuracy-comparability claim is fragile. Third, the priority claim over prior centroid-based work is asserted rather than investigated.

Whom is this for? Anyone working on practical quantum kernel methods on NISQ hardware. It's a useful empirical starting point, not a theoretical breakthrough. I'd send it to review — the core idea is valid and the issues are addressable. A revision should fix the circuit count, add at least a basic spectral/error-bound discussion, and test larger N or varying M to show what happens to M(N). If that scaling turns out to be superlinear, the headline claim falls, but the paper would still be a decent contribution.

Referee Report

3 major / 5 minor

Summary. The paper proposes using the Nyström low-rank approximation to construct the quantum embedding kernel matrix for an SVM after Kernel Target Alignment (KTA) training, with the goal of reducing the number of quantum circuit executions needed for the final training and testing kernel matrices. The training phase uses mini-batch KTA from prior work, and the final kernel is approximated via M randomly chosen landmarks. The authors report noiseless experiments on four synthetic 2D datasets showing SVM accuracy comparable to the standard full-kernel approach, plus noise experiments under coherent and depolarizing noise. The paper claims that this is the first quantum-kernel pipeline whose quantum circuit execution count scales linearly with training set size N in all steps.

Significance. If the central claim holds, the paper is a useful engineering contribution: it combines a scalable KTA training procedure with a standard classical low-rank approximation to remove the O(N^2) quantum circuit bottleneck from the final kernel-matrix construction, and it makes the code publicly available. The noiseless empirical comparison on four datasets supports the accuracy statement for the tested small-scale cases, and the authors are transparent about the dependence of the method on hyperparameters M and D. The significance is limited, however, by the lack of any spectral-decay or error-bound evidence for the optimized quantum kernels and by the fact that the linear-scaling claim is conditional on M and D remaining bounded as N grows; the paper does not currently establish that condition.

major comments (3)

[§4, Eq. (13) and §5] The paper repeatedly states that the Nyström method reduces the number of quantum circuit executions for the training kernel matrix to O(NM^2). This is not the quantum circuit-execution count: constructing K_MM requires M^2 circuit evaluations (or M(M+1)/2 unique entries) and constructing K_NM requires N·M circuit evaluations, for a total of O(NM + M^2). The O(NM^2) figure is the classical arithmetic cost of forming the product K_NM K_MM^{-1} K_NM^T in Eq. (13). Because the manuscript's central claim is about quantum circuit executions, this count should be corrected. The linear scaling in N remains valid for fixed M, but the prefactor and the quantitative comparison in Fig. 2 should reflect NM + M^2 rather than NM^2.
[§6.1 and §7] The end-to-end linear-in-N claim is conditional on M and D not growing with N. The manuscript gives no spectral-decay information for the optimized quantum kernels, no rank estimate, and no scaling law for M(N); the experiments use N between 30 and 100 and M in {2,4,8}. The authors themselves note in §6.1 that larger or more complex datasets may require increasing M and D. If M grows as N^α, the quantum circuit count becomes superlinear, so the advertised "first pipeline that depends only linearly on the training dataset size" is an assumption rather than a demonstrated property. Please add evidence such as spectral decay curves or approximation-error curves as N grows at fixed M, or explicitly restate the claim as conditional on constant M and D.
[§4, Eq. (14)] There is a dimensional inconsistency in Eq. (14). If K_PM denotes the (P,M) cross-kernel matrix between P test points and M landmarks, then K_PM^T K_MM^{-1} K_NM is an undefined product with dimensions (M,P) × (M,M) × (N,M). The correct expression is K_PM K_MM^{-1} K_MN, where K_MN = K_NM^T (or equivalently one can define K_PM as having shape (M,P)). Please fix the indices, since this formula underlies the claimed O(PM) test-time scaling.

minor comments (5)

[§2.1, Eq. (8)] The text says "Trace(ρ^2) = 1, where ρ = ⟨ϕ|ϕ⟩"; the density operator should be ρ = |ϕ⟩⟨ϕ|, and the inner-product notation used for ρ is incorrect.
[§6 and Appendix] The "spirals" dataset is said to be generated using the make_moons method from scikit-learn; make_moons produces two interleaving moon-shaped clusters rather than a spiral, and the library is scikit-learn rather than scipy. Please correct the name or the attribution.
[§6] The statement that the input scaling weights are "initialized as 1s, ensuring they initially have no effect on the output of the quantum kernel" is unclear; a scaling weight of 1 still applies the encoded feature, so the sense in which there is no effect should be clarified.
[§6.2, §6.3, and §7] The conclusion that "both methods perform well under realistic levels of noise" is broader than the evidence in Fig. 5, where the Nyström accuracies decrease sharply for σ > 0.2; please qualify the noise-robustness statement to the tested ranges.
[§4, Eq. (13)] The text does not account for the symmetry of K_MM and K_NM when counting unique circuit executions; stating this explicitly would make the reported execution counts reproducible.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claims are empirically evaluated with fixed hyperparameters, and the Nyström approximation is applied as a standard post-training low-rank reconstruction rather than as a fitted input disguised as a prediction.

full rationale

The paper's derivation chain is self-contained and non-circular. The Nyström approximation (Eq. 13) is a standard linear-algebra identity applied to the optimized quantum kernel matrix; its complexity reduction from O(N^2) to O(N M^2) follows directly from the number of computed entries and is not a re-labeling of a fitted quantity. KTA is used both as the training cost (on mini-batches of size D) and as a reporting metric on the full training kernel matrix, but the reported SVM accuracies are measured only after training is complete, so the evaluation is not statistically forced by the training objective. The hyperparameters M and D are fixed across datasets (M in {2,4,8}, D = 8), not tuned to the test results, and the authors explicitly disclaim that these values are guaranteed for other datasets. No load-bearing self-citation appears: the cited prior work on KTA (Hubregtsen et al., 2022), mini-batch training (Sahin et al., 2024), and Nyström methods (Drineas et al., 2005) is external, and the paper's own claims do not reduce to a self-citation chain. The acknowledged limitation that larger datasets may require larger M and D is a validity concern about an unproven low-rank assumption, not circularity: it does not make the empirical comparison equivalent to its inputs. The paper honestly reports that the standard approach achieves higher KTA and that the Nyström method degrades faster under noise, which is consistent with a genuine empirical study rather than a constructed confirmation. Therefore, the appropriate circularity score is 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central efficiency claim rests on two hand-chosen hyperparameters, M and D, plus the untested assumption that the optimized quantum kernels are effectively low-rank under random landmark selection. The paper provides no rank bounds and no approximation error analysis, so the approximation quality is assumed rather than established.

free parameters (2)

M (number of Nyström landmarks) = M in {2, 4, 8}
Chosen by hand for the experiments. Controls approximation quality and the quantum circuit count, so the central resource-scaling claim depends on it.
D (KTA mini-batch size) = 8
Fixed mini-batch size used during all training iterations. It is a hyperparameter, not derived, and affects training cost and stability.

assumptions (4)

domain assumption The optimized quantum Gram matrix is well approximated by a rank-M Nyström reconstruction with M random landmarks.
Used in Sections 4 and 6. No spectral-rank or error-bound evidence is supplied, and the noise experiments show that approximation quality degrades when landmark entries are noisy.
domain assumption Uniformly random sampling of landmarks produces a representative subset for the kernel matrix.
The paper always selects landmarks randomly and notes in Section 6.1 that more sophisticated selection may be needed for harder datasets.
domain assumption Mini-batch KTA with batch size D approximates full-batch KTA well enough to train the quantum kernel.
The training loop in Section 5 relies on the subsampling approach from Sahin et al.; no convergence guarantee is given for the chosen D=8.
domain assumption Fidelity kernels can be evaluated with the adjoint method under the modeled noise, especially depolarizing noise.
Section 6.3 states that the adjoint operation may not be feasible under general device noise and assumes it is feasible for depolarizing channels.

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Cite this review

Pith. "Pith review of Quantum-Efficient Kernel Target Alignment." pith.science (2026). https://pith.science/paper/QMUXTP4U

@misc{pith2026250208225,
  author       = {Pith},
  title        = {Pith review of: Quantum-Efficient Kernel Target Alignment},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QMUXTP4U}},
  note         = {Machine review of arXiv:2502.08225}
}

read the original abstract

In recent years, quantum computers have emerged as promising candidates for implementing kernels. Quantum Embedding Kernels embed data points into quantum states and calculate their inner product in a high-dimensional Hilbert Space by computing the overlap between the resulting quantum states. Variational Quantum Circuits (VQCs) are typically used for this end, with Kernel Target Alignment (KTA) as cost function. The optimized kernels can then be deployed in Support Vector Machines (SVMs) for classification tasks. However, both classical and quantum SVMs scale poorly with increasing dataset sizes. This issue is exacerbated in quantum kernel methods, as each inner product requires a quantum circuit execution. In this paper, we investigate KTA-trained quantum embedding kernels and employ a low-rank matrix approximation, the Nystr\"om method, to reduce the quantum circuit executions needed to construct the Kernel Matrix. We empirically evaluate the performance of our approach across various datasets, focusing on the accuracy of the resulting SVM and the reduction in quantum circuit executions. Additionally, we examine and compare the robustness of our model under different noise types, particularly coherent and depolarizing noise.

Figures

Figures reproduced from arXiv: 2502.08225 by the authors.

**Figure 2.** Quantum circuit executions required for computing the training kernel matrix (Fig.2a) and testing kernel matrix [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Given a training dataset of size N, for T iterations, do: A) sample a random mini-batch of data points of size D. B) Calculate the overlap between the embedded quantum states. C) Once all the pairs have been processed, fill the kernel matrix KDD and compute the cost function (KTA). D) Update the parameters θ of the quantum circuit. Finally, after the T training iterations, E) Compute the approximated training kernel… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Performance of the Standard approach VS the Nystr [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: Performance of the Standard approach VS the Nystr [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: Performance of the Standard approach VS the Nystr [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: Red and blue represent the two classes. Cir [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

Works this paper leans on

22 extracted references · 20 canonical work pages · cited by 2 Pith papers

[1]

write newline

" write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.state := if if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTIO...
[2]

Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., and Woerner, S. (2021). The power of quantum neural networks. Nature Computational Science , 1(6):403--409

work page 2021
[3]

Cai, Z., Xu, X., and Benjamin, S. C. (2020). Mitigating coherent noise using pauli conjugation. npj Quantum Information , 6(1):17

work page 2020
[4]

C., Endo, S., Fujii, K., McClean, J

Cerezo, M., Arrasmith, A., Babbush, R., Benjamin, S. C., Endo, S., Fujii, K., McClean, J. R., Mitarai, K., Yuan, X., Cincio, L., et al. (2021). Variational quantum algorithms. Nature Reviews Physics , 3(9):625--644

work page 2021
[5]

S., Guan, Y., and Kulis, B

Dhillon, I. S., Guan, Y., and Kulis, B. (2004). Kernel k-means: spectral clustering and normalized cuts. In Proceedings of the tenth ACM SIGKDD international conference on Knowledge discovery and data mining , pages 551--556

work page 2004
[6]

W., and Cristianini, N

Drineas, P., Mahoney, M. W., and Cristianini, N. (2005). On the nystr \"o m method for approximating a gram matrix for improved kernel-based learning. journal of machine learning research , 6(12)

work page 2005
[7]

J., Kaufman, L., Smola, A., and Vapnik, V

Drucker, H., Burges, C. J., Kaufman, L., Smola, A., and Vapnik, V. (1996). Support vector regression machines. Advances in neural information processing systems , 9

work page 1996
[8]

Farhi, E., Goldstone, J., and Gutmann, S. (2014). A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028

arXiv 2014

Show all 22 references

[9]

Grover, L. K. (1996). A fast quantum mechanical algorithm for database search. In Proceedings of the twenty-eighth annual ACM symposium on Theory of computing , pages 212--219

1996
[10]

A., Dumais, S

Hearst, M. A., Dumais, S. T., Osuna, E., Platt, J., and Scholkopf, B. (1998). Support vector machines. IEEE Intelligent Systems and their applications , 13(4):18--28

1998
[11]

H., Faehrmann, P

Hubregtsen, T., Wierichs, D., Gil-Fuster, E., Derks, P.-J. H., Faehrmann, P. K., and Meyer, J. J. (2022). Training quantum embedding kernels on near-term quantum computers. Physical Review A , 106(4):042431

2022
[12]

and Krems, R

J \"a ger, J. and Krems, R. V. (2023). Universal expressiveness of variational quantum classifiers and quantum kernels for support vector machines. Nature Communications , 14(1):576

2023
[13]

M., and Gambetta, J

Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J. M., and Gambetta, J. M. (2017). Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature , 549(7671):242--246

2017
[14]

Preskill, J. (2018). Quantum computing in the nisq era and beyond. Quantum , 2:79

2018
[15]

E., Symons, B

Sahin, M. E., Symons, B. C., Pati, P., Minhas, F., Millar, D., Gabrani, M., Mensa, S., and Robertus, J. L. (2024). Efficient parameter optimisation for quantum kernel alignment: A sub-sampling approach in variational training. Quantum , 8:1502

2024
[16]

o lkopf, B., Smola, A., and M \

Sch \"o lkopf, B., Smola, A., and M \"u ller, K.-R. (1997). Kernel principal component analysis. In International conference on artificial neural networks , pages 583--588. Springer

1997
[17]

Schuld, M. (2018). Supervised learning with quantum computers . Springer

2018
[18]

Shor, P. W. (1999). Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM review , 41(2):303--332

1999
[19]

Skolik, A., Jerbi, S., and Dunjko, V. (2022). Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum , 6:720

2022
[20]

Skolik, A., Mangini, S., B \"a ck, T., Macchiavello, C., and Dunjko, V. (2023). Robustness of quantum reinforcement learning under hardware errors. EPJ Quantum Technology , 10(1):1--43

2023
[21]

Tscharke, K., Issel, S., and Debus, P. (2024). Quack: Quantum aligned centroid kernel. arXiv preprint arXiv:2405.00304

2024 arXiv
[22]

Wang, X., Du, Y., Luo, Y., and Tao, D. (2021). Towards understanding the power of quantum kernels in the nisq era. Quantum , 5:531

2021

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

write newline

" write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.state := if if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTIO...

[2] [2]

Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., and Woerner, S. (2021). The power of quantum neural networks. Nature Computational Science , 1(6):403--409

work page 2021

[3] [3]

Cai, Z., Xu, X., and Benjamin, S. C. (2020). Mitigating coherent noise using pauli conjugation. npj Quantum Information , 6(1):17

work page 2020

[4] [4]

C., Endo, S., Fujii, K., McClean, J

Cerezo, M., Arrasmith, A., Babbush, R., Benjamin, S. C., Endo, S., Fujii, K., McClean, J. R., Mitarai, K., Yuan, X., Cincio, L., et al. (2021). Variational quantum algorithms. Nature Reviews Physics , 3(9):625--644

work page 2021

[5] [5]

S., Guan, Y., and Kulis, B

Dhillon, I. S., Guan, Y., and Kulis, B. (2004). Kernel k-means: spectral clustering and normalized cuts. In Proceedings of the tenth ACM SIGKDD international conference on Knowledge discovery and data mining , pages 551--556

work page 2004

[6] [6]

W., and Cristianini, N

Drineas, P., Mahoney, M. W., and Cristianini, N. (2005). On the nystr \"o m method for approximating a gram matrix for improved kernel-based learning. journal of machine learning research , 6(12)

work page 2005

[7] [7]

J., Kaufman, L., Smola, A., and Vapnik, V

Drucker, H., Burges, C. J., Kaufman, L., Smola, A., and Vapnik, V. (1996). Support vector regression machines. Advances in neural information processing systems , 9

work page 1996

[8] [8]

Farhi, E., Goldstone, J., and Gutmann, S. (2014). A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028

arXiv 2014