Spectral Geometry and Bosonic-Bloch Probes: Explorations in Quantum Learning

2026-06-30

AI Summary

This work aims to uncover the emergent spectral-geometric structures in quantum learning and to construct physically measurable probes for their characterization. By integrating graph-regularized quantum neural networks with two-boson interference and Bloch sphere drift analysis, the study establishes—for the first time—a direct link between spectral geometry and quantum learning dynamics. Key contributions include a proposed mechanism connecting boson-enhanced interference with Fiedler edge splitting, the introduction of absolute Bloch drift as a geometric criterion for anomaly detection, and the development of a unified spectral-geometric diagnostic framework. Experimental results demonstrate near-perfect unsupervised anomaly detection performance (ROC-AUC ≈ 0.99) with virtually zero false-negative rates; the Bloch drift metric alone achieves ROC-AUC ≥ 0.9 and accurately predicts interference behavior under shot noise, as validated on quantum hardware.

Abstract

This paper studies how spectral geometry emerges in quantum learning models and how it can be diagnosed with physically grounded probes. In graph-regularized quantum networks, training reorganizes the output similarity graph, increases the effective spectral dimension Delta S = +0.23, and reshapes the Laplacian spectrum. Edge-resolved two-boson interference directly probes this restructuring: the bosonic enhancement Delta P_uv correlates with the Fiedler edge split |Delta v_2| (r = -0.50), linking learned spectral partitions to interference signatures. A phase diagram shows a nonmonotonic dependence of performance on coupling strength gamma and noise delta, with graph regularization improving fidelity only in a restricted regime; hardware experiments confirm the predicted interference behavior within shot-noise uncertainty. We also analyze a hybrid quantum autoencoder and introduce Bloch-space drift as a geometric diagnostic of its latent representation. With an unsupervised benign-data threshold, the model achieves high ranking performance (ROC-AUC about 0.99) and negligible false-negative rates. Absolute Bloch drift strongly discriminates anomalies (ROC-AUC at least about 0.9), while consecutive drift is near random (ROC-AUC about 0.5), showing that detection arises from persistent state-space displacement rather than local fluctuations. Through the geometry of reduced single-qubit states and associated quantum Fisher information, these results show that learning-induced spectral organization appears as measurable quantum-state structure, establishing a unified spectral-geometric framework for diagnosing quantum learning systems with bosonic and Bloch probes.

Problem

• spectral geometry
• quantum learning
• bosonic probes
• Bloch-space drift
• Laplacian spectrum

Innovation

• spectral geometry
• bosonic interference
• Bloch-space drift
• quantum learning
• graph regularization