Spectral fluctuations of quantum graphs

Akihito Hora, Obata, Nobuaki

Priyanga Ganesan

Santanu Ganguly, Xing Liang, Dimitrios Makris

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.

Clive H. Elphick, Paweł Wocjan

Daniel Volya, Prabhat Mishra

Spectral graph partitioning is a well known technique to estimate clusters in undirected graphs. Recent approaches explored efficient spectral algorithms for directed and mixed graphs utilizing various matrix representations. Despite its success in clustering tasks, classical spectral algorithms suffer from a cubic growth in runtime. In this paper, we propose a quantum spectral clustering algorithm for discovering clusters and properties of mixed graphs. Our experimental results based on numerical simulations demonstrate that our quantum spectral clustering outperforms classical spectral clustering techniques. Specifically, our approach leads to a linear growth in complexity, while state-of-the-art classical counterpart leads to cubic growth. In a case study, we apply our proposed algorithm to preform unsupervised machine learning using both real and simulated quantum computers. This work opens an avenue for efficient implementation of machine learning algorithms on directed as well as mixed graphs by making use of the inherent potential quantum speedup.