Spectral Phase Encoding for Quantum Kernel Methods

TL;DR

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.

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 (6)

• 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.
• 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.
• 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.

Theorems & Definitions (1)

• Remark 1: Phase-only encodings and complex-valued features