Spectral Phase Encoding for Quantum Kernel Methods

Pablo Herrero Gómez; Antonio Jimeno Morenilla; David Muñoz-Hernández; Higinio Mora Mora

cs.LGFeb 23, 2026

Spectral Phase Encoding for Quantum Kernel Methods

Authors: Pablo Herrero Gómez, Antonio Jimeno Morenilla, David Muñoz-Hernández, Higinio Mora Mora

Organizations: Department of Computer Science Technology and Computation, University of Alicante, Alicante, Spain

Abstract

Quantum kernel methods are promising for near-term quantum ma- chine learning, yet their behavior under data corruption remains insuf- ficiently understood. We analyze how quantum feature constructions degrade under controlled additive noise. We introduce Spectral Phase Encoding (SPE), a hybrid construc- tion combining a discrete Fourier transform (DFT) front-end with a diagonal phase-only embedding aligned with the geometry of diagonal quantum maps. Within a unified framework, we compare QK-DFT against alternative quantum variants (QK-PCA, QK-RP) and classi- cal SVM baselines under identical clean-data hyperparameter selection, quantifying robustness via dataset fixed-effects regression with wild cluster bootstrap inference across heterogeneous real-world datasets. Across the quantum family, DFT-based preprocessing yields the smallest degradation rate as noise increases, with statistically sup- ported slope differences relative to PCA and RP. Compared to classical baselines, QK-DFT shows degradation comparable to linear SVM and more stable than RBF SVM under matched tuning. Hardware exper- iments confirm that SPE remains executable and numerically stable for overlap estimation. These results indicate that robustness in quan- tum kernels depends critically on structure-aligned preprocessing and its interaction with diagonal embeddings, supporting a robustness-first perspective for NISQ-era quantum machine learning.

Figures & tables

Figure 1: Conceptual pipeline of Spectral Phase Encoding. Classical data are transformed to the frequency domain, mapped to a vector of relative phases, and embedded into a quantum state using a diagonal gate acting on a uniform superposition. Kernel values are obtained via overlap estimation.

Panel of Figure 2 — Figure 2: Accuracy vs noise level $\sigma$ for four representative datasets under sigma 0 -frozen configuration selection.

Figure 3: Number of datasets (out of 20) for which each method achieves the highest accuracy at each noise level $\sigma$ , after clean-data configuration selection and freezing.

Parameter	Estimate	CI 95% Low	CI 95% High	$\Pr(\beta<0)$
$\beta_{\text{QK-DFT}}$	-0.214	-0.645	0.222	0.801
$\beta_{\text{QK-PCA}}$	-0.568	-1.063	-0.075	0.995
$\Delta(\text{QK-PCA}-\text{QK-DFT})$	-0.354	-0.524	-0.188	1.000
$\beta_{\text{QK-RP}}$	-1.367	-1.962	-0.760	1.000
$\Delta(\text{QK-RP}-\text{QK-DFT})$	-1.153	-1.592	-0.707	1.000
$\beta_{\text{SVM-Linear}}$	-0.253	-0.630	0.118	0.867

Table 1: Noise robustness slopes estimated via dataset fixed-effects regression with wild cluster bootstrap. Slopes correspond to accuracy change per unit increase in noise

\sigma

. QK-DFT is the reference category.

Panel of Figure 4 — Figure 4: Global robustness analysis. (a) Degradation slopes with bootstrap confidence intervals. (b) Aggregated accuracy curves with 95% bootstrap bands.

Figure 5: Hardware overlap-estimation error across qubit sizes. Mean absolute error (MAE) of

\lvert\Delta p_{0}\rvert=\lvert p_{0}^{\mathrm{hw}}-p_{0}^{\mathrm{exp}}\rvert

for SPE (DFT + DiagonalGate) and a low-depth angle-encoding baseline, averaged over two IBM Quantum backends ( ibm_fez , ibm_marrakesh ), similarity values, and repeated runs. Error bars indicate variability across measurement instances.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Classes ( $C$ )	Domain	Source
BreastMNIST	2	Medical imaging	MedMNIST
CIFAR-10	10	Natural images	CIFAR
DermaMNIST	7	Dermatology	MedMNIST
Digits	10	Handwritten	scikit-learn
DTD	20	Texture	Oxford VGG
EMNIST (ByClass)	20	Characters	NIST

Appendix

Table 2: Overview of the 20 real-world 2D datasets used in the experiments. All datasets are converted to grayscale (when applicable), resized to

32\times 32

, and subsampled to a balanced subset as described in Sec. 2.3 .

Figure 6: Representative samples under increasing Gaussian noise $\sigma$ .

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Quantum machine learning is often motivated by the idea that quantum systems can expose useful high-dimensional structure that is difficult to access with classical models. We isolate one central component of this claim: the fixed data-encoding map. Amplitude, angle, and basis encoding are evaluated as deterministic feature maps for classical supervised learning under matched output dimensionality and strong classical controls. The benchmark compares these encodings against raw linear models, random Fourier features, polynomial features, PCA, RBF SVMs, and shallow neural networks across diverse classical datasets. Rather than treating performance as a single endpoint, we analyze the geometry of each representation through effective rank, condition number, centered kernel alignment, predictive performance, and practical overhead. The resulting picture is mechanistic: amplitude encoding can remove magnitude information through unit-sphere normalization, angle encoding can become geometrically redundant with raw linear features, and basis encoding can impose a binary Hamming geometry that is poorly aligned with smooth decision structure. These findings do not argue against quantum computation, however, they show that fixed quantum-inspired encoding geometry alone is not a reliable source of machine-learning advantage on classical data.

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