In networks, the Markov clustering (MCL) algorithm is one of the most efficient approaches in detecting clustered structures. The MCL algorithm takes as input a stochastic matrix, which depends on the adjacency matrix of the graph network under consideration. Quantum clustering algorithms are proven to be superefficient over the classical ones. Motivated by the idea of a potential clustering algorithm based on quantum Markov chains, we prove a clustering property for quantum Markov chains (QMCs) on Cayley trees associated with open quantum random walks (OQRW).

This paper extended the framework of quantum Markovianity by introducing backward and inverse backward quantum Markov chains (QMCs). We established the existence of these models under general conditions, demonstrating their applicability to a wide range of quantum systems. Our findings revealed distinct structural properties within these models, providing new insights into their dynamics and relationships to finitely correlated states. These advancements contributed to a deeper understanding of quantum processes and have potential implications for various quantum applications, including hidden quantum Markov processes.