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.
Hilbert-space fragmentation provides a mechanism for ergodicity breaking in quantum many-body systems even in the absence of disorder, leading to dynamically disconnected sectors and strong memory of initial conditions. However, identifying such structures is often challenging and typically relies on prior knowledge of conservation laws or model-specific analytical insight. Here we introduce an unbiased approach based on spectral graph theory and, within this framework, formulate the concept of nearly fragmented systems, in which perturbative processes couple otherwise fragmented sectors while preserving their dynamical imprint. By representing basis states as vertices and Hamiltonian matrix elements as edges, we map the connectivity structure of the many-body Hilbert space onto a graph and analyze it using tools such as the Laplacian spectrum, Fiedler vectors, and modularity. Exact fragmentation corresponds to disconnected graph components, while nearly fragmented systems manifest as weakly connected communities whose structure can still be resolved spectrally. Applying this framework to the one-dimensional
t-
J model and its perturbations, we demonstrate that graph-theoretic diagnostics reliably identify both fragmented and nearly fragmented Hilbert-space structures and capture the hierarchy of dynamical time scales that governs the system's evolution. We further show that the method extends beyond kinetically constrained models by applying it directly to the Hubbard chain, where it reveals the emergence of nearly decoupled subspaces associated with doublon dynamics and spin configurations. Our results establish the spectral graph analysis as a general and scalable tool for diagnosing fragmentation and approximate dynamical constraints in complex quantum many-body systems.
Introduction: Variational quantum algorithms can show unstable trainability and uneven generalisation that are not fully explained by circuit depth or raw parameter count alone. We study the data quantum Fisher information matrix (DQFIM) as a data-dependent effective-capacity diagnostic for controlled supervised unitary-learning tasks. Materials and methods: We used exact-state, classically simulated, 4-qubit, supervised unitary-learning benchmarks with two ansatz families: a hardware-efficient ansatz (HEA) for the baseline trainability study and a symmetry-preserving ansatz for symmetry-controlled comparisons. The analysis covered four linked experiments: E0 trainability phase boundaries, E1 symmetry-controlled data-regime comparisons, E2 trainability versus generalisation, and E3 pre-training prediction of optimisation success. For the main analyses we used the support-basis DQFIM, 30 random seeds per main configuration, grouped cross-validation in the predictive benchmark, and additional diagnostics for threshold sensitivity, phase indeterminacy, Hamming-sector generalisation, and test set size sensitivity. Results: In E0, the empirical trainability boundary increased from M c = 32 at L = 1 to M c = 192 at L = 8 , and the support-basis DQFIM-predicted boundary matched the empirical boundary on the resolved scanned grid under the main analysis setting. In E1, the sector-preserving and sector-broken conditions showed no resolved empirical boundary shift on the scanned grid, with only mild low-L asymmetry in the DQFIM-predicted boundary. In E2, trainability and generalisation separated clearly: some regimes generalised well, while others reached near-zero training loss but retained high test loss. The phase diagnostic supported relative phase indeterminacy as a mechanism for failure on in-sector superpositions after basis-state training. In E3, parameter count alone was a weak predictor of optimisation success, with receiver operating characteristic area under the curve (ROC AUC) = 0.648 ; the structural baseline was stronger, with ROC AUC = 0.958 ; and the DQFIM-enhanced model performed best, with ROC AUC = 0.989 . Conclusions: In these small, idealised, classically simulated matched-family tasks, the support-basis DQFIM provides a useful data-dependent pre-training diagnostic of effective capacity on the retained data support. It tracks resolved trainability boundaries and contributes predictive information beyond raw parameter count and structural metadata. The conclusions remain scoped to controlled small-system simulations; larger-qubit, noisy, hardware-executed, and less-matched settings require further validation before claiming practical scalability.
Spectral clustering is a powerful unsupervised machine learning algorithm for clustering data with nonconvex or nested structures [A. Y. Ng, M. I. Jordan, and Y. Weiss, On spectral clustering: Analysis and an algorithm, in Advances in Neural Information Processing Systems 14: Proceedings of the 2001 Conference (MIT Press, Cambridge, MA, 2002), pp. 849--856]. With roots in graph theory, it uses the spectral properties of the Laplacian matrix to project the data in a low-dimensional space where clustering is more efficient. Despite its success in clustering tasks, spectral clustering suffers in practice from a fast-growing running time of
O(n3), where
n is the number of points in the data set. In this work we propose an end-to-end quantum algorithm performing spectral clustering, extending a number of works in quantum machine learning. The quantum algorithm is composed of two parts: the first is the efficient creation of the quantum state corresponding to the projected Laplacian matrix, and the second consists of applying the existing quantum analog of the
k-means algorithm [I. Kerenidis, J. Landman, A. Luongo, and A. Prakash,
q-means: A quantum algorithm for unsupervised machine learning, in Advances in Neural Information Processing Systems 32: Proceedings of the 2019 Conference (Curran Associates, Red Hook, NY, 2020), pp. 4136--4146]. Both steps depend polynomially on the number of clusters, as well as precision and data parameters arising from quantum procedures, and polylogarithmically on the dimension of the input vectors. Our numerical simulations show an asymptotic linear growth with
n when all terms are taken into account, significantly better than the classical cubic growth. This work opens the path to other graph-based quantum machine learning algorithms, as it provides routines for efficient computation and quantum access to the incidence, adjacency, and projected Laplacian matrices of a graph.
This work investigates whether classical machine learning can detect statistical signatures associated with entanglement graph structure from partial quantum measurements. Experiments were performed on real IBM Quantum hardware using circuits with controlled entanglement patterns. Only a subset of qubits was observed, while the state of one qubit was hidden. A classical Random Forest classifier was trained to predict the hidden qubit’s measurement outcome based solely on the observed subset. To distinguish structural effects from spatial or hardware-related confounders, a geometry decoupling protocol was introduced. This involved comparing entangled circuits with control circuits of identical spatial layout but without entangling gates. Results show that classification accuracy exceeds chance in entangled configurations and correlates negatively with graph distance, suggesting that classical models can capture statistical signatures consistent with entanglement connectivity. The introduced delta metric partially controls for hardware bias by comparing entangled and non-entangled baselines. The observed signal reflects classical correlations in measurement outcomes generated by entangled states and does not constitute quantum communication or violation of no-signalling principles. This dataset and code provide a reproducible experimental framework for studying the relationship between quantum circuit structure and classical detectability using machine learning.
We study a quantum autoencoder (QAE) for compression-driven anomaly detection in brain MRI data. The approach leverages angle encoding to map image patches into quantum states, followed by a variational encoder-decoder architecture trained to discard information via auxiliary trash qubits. Anomaly scores reflect the degree to which inputs resist compression relative to normal data, with higher scores corresponding to deviations from the learned normal manifold. Evaluated on publicly available brain MRI DICOM datasets, the method achieves a slice-level ROC-AUC of approximately 0.95 and a patch-level ROC-AUC of approximately 0.813, outperforming classical autoencoder and PCA baselines. Analysis of the learned parameters reveals a pronounced encoder-decoder asymmetry, where effective anomaly detection arises from structured information compression within the encoder rather than increased parameter magnitude or decoder expressivity. This results in a controlled compression-reconstruction trade-off with a clear operating regime that supports principled threshold selection. Qualitative evaluation further shows that the QAE produces spatially localized anomaly heatmaps aligned with tumorous regions. The results, supported by promising baseline performances, demonstrate that quantum autoencoders provide an interpretable and controllable mechanism for anomaly detection based on incompressibility with respect to a learned latent representation. This work highlights the potential of quantum autoencoders as a principled tool for studying compression dynamics in quantum machine learning, with promising implications for decision support in medical imaging workflows.