Exponential quantum advantage for learning signals with a single qubit

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.

One-bit post-training quantization represents each weight using only its sign, requiring all deployment contexts to share the same binary weight matrix even when their activation statistics favor different sign patterns. We study this shared-sign constraint and introduce Quantum Random Access Quantization (QRAQ). This framework encodes context-dependent signs in a quantum random-access code and retrieves them via context-matched Pauli measurements. Under an explicit fresh-copy logical readout model, QRAQ produces an unbiased, context-specific binary surrogate with a tractable shot-noise penalty. We prove a row-wise separation from shared-sign one-bit PTQ with signed per-row scales. When the optimal context-wise signs are incompatible, QRAQ achieves a strictly lower ideal reconstruction risk. We also derive finite-shot and calibrated-noise conditions under which this separation is retained. Fixed-readout quantum schemes are classically simulable, so the relevant resource in this model is measurement incompatibility rather than quantization alone. Finally, we characterize the role of scale granularity, provide finite-sample certificates, and evaluate the predicted ideal, finite-shot, noisy, and multi-context regimes in simulator experiments.

We develop theoretical foundations for a practical quantum-advantage mechanism in quantum-informed machine learning for chaotic dynamical systems. A family of $k$-indexed higher-order quantum statistical priors (Q-Priors) hosts the $k$-point marginal of the invariant measure on $n_q = kq$ qubits, extending the single-site construction of prior work. We prove a two-stage advantage. In the representation stage, superposition and entanglement compactly store non-factorisable spatial correlations of the invariant measure on $n_q$ qubits. In the extraction stage, joint Bell measurements on two copies estimate any \emph{post hoc} Pauli functional with a copy-pair count independent of $n_q$, whereas any adaptive single-copy protocol for the corresponding full-Pauli read-out requires $Ω(2^{n_q})$ copies; this is a provable quantum-classical separation in copy-measurement complexity. The two-copy read-out is realised in simulation and on IQM superconducting processors. Two case studies instantiate the mechanism in workflows of independent scientific value. In a turbulent channel-flow study, the two-copy read-out yields the velocity-direction coherence as a named non-diagonal correlator of the invariant measure, and the multi-site $k = 2$ Q-Prior recovers DNS-level invariant-measure statistics that the unregularised baseline loses. In a medium-range weather forecasting workflow on the European Centre for Medium-Range Weather Forecasts ERA5 reanalysis, the diagonal $k \leq 2$ Q-Prior steers a Koopman rollout, improves anomaly correlation skill by 10% to 39% across 48 to 240h lead times, and stabilises long-horizon rollouts against collapse onto a static mean field. Together, the mechanism and these two case studies satisfy our practical-advantage definition, identifying a candidate route to practical quantum advantage before fault-tolerant hardware.