Quantum Optimal Transport for Railway Track Degradation Monitoring

Railway track geometry is monitored by comparing measurements against regulatory thresholds, but this approach does not capture how the distribution of geometry parameters evolves over time or across track segments. Optimal transport provides a natural framework for such distributional comparisons. This study investigates whether quantum optimal transport (QOT) can provide additional diagnostic information. Track geometry measurements are encoded as quantum states via angle encoding on three qubits, and transport distances are computed through semidefinite programming under four quantum cost functions alongside classical OT, maximum mean discrepancy, energy distance, principal component analysis, and Mahalanobis distance. The framework is applied to three rail datasets. In a temporal analysis of monthly recordings, the quantum swap cost nearly matches classical squared Euclidean (𝑟 = 0.998), while the Hamming cost ranks transitions differently (Spearman 𝜌 = 0.45). Comparison against a monthly Track Quality Index reveals that Hamming QOT is the only metric with a statistically significant correlation after Bonferroni correction (Pearson 𝑟 = 0.701, 𝑝 < 0.001). In a cross-sectional comparison of healthy and degraded track sections, the eigenvalue spectra of density matrices reveal that degraded sections produce higher-rank quantum states (rank 3–4 versus rank 2 for healthy). Cross-validation on a second Class 3 track replicates this pattern. Applied as a per-section anomaly score, Hamming QOT achieves an AUC of 0.756 for classifying section health. Sensitivity analyses confirm that these findings are largely preserved across parameter choices. These results suggest QOT offers a distinct but complementary diagnostic lens for infrastructure condition assessment.