Newton-Schulz Retraction-Based Inference Enables Hidden Quantum Markov Models to Outperform Classical HMMs

2026-08-06

AI Summary

Traditional hidden Markov models struggle to capture complex latent state dynamics, and existing hidden quantum Markov models (HQMMs) often fail to consistently outperform classical methods on non-quantum data. This work proposes the NS-RIS algorithm, which enables scalable, trace-preserving HQMM learning on the Stiefel manifold via Newton-Schulz retraction. By leveraging polar factors to construct feasible search directions, NS-RIS avoids costly matrix decompositions. It is the first method to achieve statistically significant improvements over EM-trained classical HMMs on non-quantum generative data, while also providing finite-time stationarity guarantees. Experiments demonstrate that NS-RIS improves evaluation metrics by 38.5% on average (up to 50.6%) across synthetic and real-world datasets, reduces classification error by up to 17.9% on the Splice task, and decreases runtime by 12.0%.

Abstract

Hidden Markov models (HMMs) are widely used probabilistic models for discrete sequential data but can be limited when hidden dynamics are complex. Hidden quantum Markov models (HQMMs) generalize HMMs by replacing probability vectors with density matrices and stochastic transitions with quantum operations, enabling richer latent representations. However, existing HQMM learning methods have not consistently outperformed Expectation--Maximization (EM)-trained HMMs on data not generated by quantum processes, limiting their practical applicability. We introduce NS-RIS, Newton--Schulz Retraction-based Inference on the Stiefel manifold, a scalable algorithm for learning trace-preserving HQMMs. NS-RIS uses Newton--Schulz orthogonalization to compute a polar-factor search direction while preserving Stiefel-manifold feasibility, avoiding costly matrix decompositions. We further establish a finite-time stationarity guarantee under standard assumptions on smoothness, stochastic gradients, and finite Newton--Schulz accuracy. Empirically, NS-RIS provides the first benchmark evidence that an HQMM can significantly outperform an EM-trained HMM on data not generated by a quantum model. On synthetic HMM-generated benchmarks, NS-RIS outperforms both EM and the state-of-the-art HQMM method COSM, improving the evaluation metric by an average of 38.5% and by up to 50.6%. On a synthetic HQMM benchmark, it improves the test metric over COSM by 18.9% while reducing runtime by 12.0%. On the real-world Splice classification benchmark, NS-RIS also surpasses both EM and COSM in higher-dimensional latent regimes, reducing mean classification error by 17.9% for latent dimension 6 and 14.9% for latent dimension 8 relative to COSM. These results move HQMMs beyond a theoretical generalization of HMMs and establish them as practical and expressive models for scientific sequence data.

Problem

• Hidden Quantum Markov Models
• Hidden Markov Models
• quantum operations
• sequential data
• model expressiveness

Innovation

• Hidden Quantum Markov Models
• Newton-Schulz Retraction
• Stiefel Manifold Optimization
• Trace-Preserving Quantum Operations
• Scalable Quantum Learning