Markov chain decision-making with quantum-assisted entropy kernels

Sequential anomaly detection requires summary statistics that remain stable under nominal dynamics yet react sharply to distributional shifts. We introduce QEK-DTMC, a hybrid quantum–classical sequential detector in which quantum circuits are used solely to estimate overlap-based kernel similarities, while entropy estimation, density smoothing, and decision-making are performed classically via a Bayesian two-state DTMC filter. Quantum overlaps derived from a parameterized ZZ feature map are smoothed by a boundary-aware kernel density estimator, yielding a Shannon entropy statistic that feeds a Bayesian posterior update on a two-state DTMC. This design explicitly accounts for finite-shot (quantum) measurement noise and kernel concentration, aligning representation, estimation, and decision within a unified sequential rule. We establish finite-sample guarantees for kernel estimation and entropy evaluation and prove the correctness of the DTMC recursion under calibrated thresholds. Empirical results on controlled drift scenarios show strong performance: AUROC of 0.90, AUPRC of 0.28, false-alarm rate < 0.01, and mean detection delay of 14 steps. Robustness analyses confirm graceful degradation under noise, missingness, and covariate shift, and debouncing reduces spurious triggers without undermining sensitivity. These findings demonstrate that entropy-aware quantum kernels can be systematically integrated into sequential Bayesian detectors, offering a statistically principled and compatible pathway for real-time anomaly detection.