Simulating Quantum-Inspired Kernel Complexity via Analytic Geometric and Spectral Feature Maps

Aimé T Shangula

Aimé T Shangula

Yudai Suzuki, Hiroshi Yano, Qi Gao +4 more

Toheeb Ogunade, Taofeek Kassim, Etinosa Osaro

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.

Zhang Wei

Semi-supervised learning (SSL) has emerged as a critical paradigm for leveraging abundant unlabeled data alongside scarce labeled samples, yet classical approaches often struggle in sparse label regimes where the ratio of labeled to unlabeled data is extremely low. Meanwhile, quantum machine learning (QML) has demonstrated that quantum feature spaces can induce rich kernel structures with provable representational advantages, but these methods typically require quantum hardware that remains inaccessible for practical deployment. In this work, we propose a quantum-inspired semi-supervised learning framework that harnesses the geometric properties of quantum Hilbert space embeddings while remaining entirely classical and hardware-independent. Our approach, which we term Quantum-Inspired SemiSupervised Learning (QISSL), constructs feature maps that mimic the action of quantum encoding circuits via classically simulable kernel functions derived from randomized trigonometric expansions. We provide rigorous theoretical analysis establishing: (1) a bounded approximation guarantee between our classical embeddings and their quantum counterparts; (2) a generalization bound for QISSL under low-label regimes via Rademacher complexity analysis; and (3) a convergence proof for the alternating optimization procedure that jointly trains the embedding and classifier modules. Extensive experiments on benchmark datasets demonstrate that QISSL achieves state-of-the-art performance in extreme low-label settings (0.1%–1% labeled data), outperforming strong baselines including FixMatch, MixMatch, and FlexMatch by margins of 2–7% in accuracy. Our results suggest that the inductive biases of quantum feature spaces—when properly classically approximated—offer a powerful and practical tool for semi-supervised learning without quantum hardware.

M Usman Mughal