Quantum Dynamic Time Warping for Multivariate Time Series Classification
Dynamic Time Warping (DTW) is a cornerstone for time series classification, but its reliance on Euclidean distances fails to capture latent cross-channel correlations in complex multivariate data. We propose a hybrid Quantum Dynamic Time Warping (qDTW) architecture, replacing the classical distance metric with the parameterized geometry of a quantum Hilbert space. Through structural ablation on benchmarks up to $C=8$ spatial dimensions, we establish fundamental topological rules for quantum sequence alignment. We introduce a Unified Pre-Embedding Adjoint Ansatz that decouples trainable entanglement from classical data, eliminating the severe phase-scrambling and information bottlenecks inherent to traditional measurements. We demonstrate this decoupled architecture allows untrained quantum kernels to act as highly expressive baselines, while parameterized training effectively untangles deeply overlapping hyper-dimensional data. Furthermore, we identify a strict spatial-temporal expressivity tradeoff: temporal depth (data re-uploading) is necessary for dimensionally restricted univariate circuits, but applying it to wide multi-qubit registers triggers chaotic frequency-spectrum explosions and representation collapse. By navigating these topological hazards, our multivariate quantum architecture outperforms classical baselines, setting a new standard for integrating parameterized quantum circuits with dynamic programming