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Chaos Solitons & Fractals

Degree of entanglement in Entangled Hidden Markov Models

Luigi Accardi, Abdessatar Souissi, El Gheteb Soueidi, Mohamed Rhaima

Published Apr 11, 2025• 2 citations
Publisher
DOI
10.1016/j.chaos.2025.116389

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A solution of the identifiability problem (IP) for hidden Markov models (HMMs), based on a novel algebraic theory for random sources, is presented. It gives rise to an efficient and practical algorithm that can be easily implemented. Extant approaches are exponential in the number of hidden states and therefore only applicable to a limited degree. The algorithm can be equally applied to solve the IP for quantum random walks (QRWs) that have recently been presented as an analogon of Markov chains in quantum information theory. Moreover, the algorithm can be used to efficiently test HMMs and QRWs for ergodicity, which had remained an open problem so far.
Hidden Quantum Markov Models (HQMMs) can be thought of as quantum probabilistic graphical models that can model sequential data. We extend previous work on HQMMs with three contributions: (1) we show how classical hidden Markov models (HMMs) can be simulated on a quantum circuit, (2) we reformulate HQMMs by relaxing the constraints for modeling HMMs on quantum circuits, and (3) we present a learning algorithm to estimate the parameters of an HQMM from data. While our algorithm requires further optimization to handle larger datasets, we are able to evaluate our algorithm using several synthetic datasets. We show that on HQMM generated data, our algorithm learns HQMMs with the same number of hidden states and predictive accuracy as the true HQMMs, while HMMs learned with the Baum-Welch algorithm require more states to match the predictive accuracy.