Implementation and Learning of Quantum Hidden Markov Models

• URL: http://arxiv.org/abs/2212.03796v3
• Date: Fri, 04 Oct 2024 17:23:48 GMT
• Title: Implementation and Learning of Quantum Hidden Markov Models
• Authors: Vanio Markov, Vladimir Rastunkov, Amol Deshmukh, Daniel Fry, Charlee Stefanski,
• Abstract summary: We propose a unitary parameterization and an efficient learning algorithm for Quantum Hidden Markov Models (QHMMs) By leveraging the richer dynamics of quantum channels, we demonstrate the greater efficiency of quantum generators compared to classical ones. We show that any QHMM can be efficiently implemented and simulated using a quantum circuit with mid-circuit measurements.
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• Abstract: In this article, we apply the theory of quantum channels and open-system state evolution to propose a unitary parameterization and an efficient learning algorithm for Quantum Hidden Markov Models (QHMMs). By leveraging the richer dynamics of quantum channels, we demonstrate the greater efficiency of quantum stochastic generators compared to classical ones. Specifically, we prove that a stochastic process can be simulated within a quantum Hilbert space using quadratically fewer dimensions than in a classical stochastic vector space. We show that any QHMM can be efficiently implemented and simulated using a quantum circuit with mid-circuit measurements. A key advantage for feasible QHMM learning in the hypothesis space of unitary circuits lies in the continuity of Stinespring's dilation. Specifically, if the unitary parameterizations of channels are close in the operator norm, the corresponding channels will be close in both diamond norm and Bures distance. This property forms the foundation for defining of efficient learning algorithms with continuous fitness landscapes. By employing the unitary parameterization of QHMMs, we establish a formal generative learning model. This model formalizes the empirical distributions of target stochastic process languages, defines the hypothesis space of quantum circuits, and introduces an empirical stochastic divergence measure-hypothesis fitness-as a criterion for learning success. The smooth mapping between the hypothesis and fitness spaces facilitates the development of efficient heuristic and gradient descent algorithms. We consider four examples of stochastic process languages and train QHMMs with hyperparameter-adaptive evolutionary search and multi-parameter nonlinear optimization technique applied to parameterized quantum ansatz circuits. We confirm our results by running optimal circuits on quantum hardware.

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