Quantum Markov Models Outperform Classical HMMs with New Inference.

Hidden Markov Models (HMMs) are a cornerstone for modeling sequential data, but their ability to capture complex hidden dynamics can be limited. Hidden Quantum Markov Models (HQMMs) offer a more expressive generalization by replacing classical probabilities with quantum density matrices and stochastic transitions with quantum operations. However, a major hurdle has been HQMMs' inability to consistently outperform classical HMMs on data not inherently quantum-generated, limiting their practical appeal. This research introduces NS-RIS (Newton-Schulz Retraction-based Inference on the Stiefel manifold), a novel and scalable algorithm for learning trace-preserving HQMMs. NS-RIS efficiently computes search directions while maintaining mathematical feasibility, crucially avoiding the computationally expensive matrix decompositions that plague other HQMM learning methods. The algorithm also comes with a finite-time stationarity guarantee under standard assumptions. Empirically, NS-RIS marks a significant breakthrough. It provides the first evidence that an HQMM can substantially outperform an EM-trained classical HMM on data not originating from quantum processes. Benchmarks show NS-RIS improving evaluation metrics by an average of 38.5% on synthetic HMM data and reducing classification error by up to 17.9% on the real-world Splice classification benchmark, particularly in higher-dimensional latent regimes. These results position HQMMs, with NS-RIS, as powerful and practical models for scientific sequence data beyond theoretical interest.