We consider an extension of discrete time Markov chain queueing model to the quantum domain by use of discrete time quantum Markov chain. We introduce methods for numerical analysis of such models. Using these tools we show that quantum model behaves fundamentally different from the classical one.
A framework for finite-dimensional quantum Markov chains on Hilbert spaces is introduced. Quantum Markov chains generalize both classical Markov chains with possibly hidden states and existing models of quantum walks on finite graphs. Quantum Markov chains are based on Markov operations that may be applied to quantum systems and include quantum measurements, for example. It is proved that quantum Markov chains are asymptotically stationary and hence possess ergodic and entropic properties. With a quantum Markov chain one may associate a quantum Markov process, which is a stochastic process in the classical sense. Generalized Markov chains allow a representation with respect to a generalized Markov source model with definite (but possibly hidden) states relative to which observables give rise to classical stochastic processes. It is demonstrated that this model allows for observables to violate Bell’s inequality.
A quantum chain is analogous to a classical stationary Markov chain except that the probability measure is replaced by a complex amplitude measure and the transition probability matrix is replaced by a transition amplitude matrix. After considering the general situation, we study a particular example of a quantum chain whose transition amplitude matrix has the form of a Dirichlet matrix. Such matrices generate a discrete analog of the usual continuum Feynman amplitude. We then compute the probability distribution for these quantum chains Une chai←ne de Markov quantique est analogue a une chai←ne de Markov stationnaire classique avec la difference que la mesure de probabilite est remplacee par une mesure d'amplitude complexe, et la matrice des probabilites de transition est remplacee par une matrice d'amplitude de transition. Apres avoir considere le cas general, nous etudions le cas particulier d'une chaine quantique dont la matrice de transition est une matrice de Dirichlet. De telles matrices conduisent a un analogue discret des amplitudes de Feynman continues. Nous calculons ensuite les distributions de probabilite de ces chaines quantiques
This paper consists of 3 parts. The first part only considers classical processes and introduces two different extensions of the notion of hidden Markov process. In the second part, the notion of quantum hidden process is introduced. In the third part it is proven that, by restricting various types of quantum Markov chains to appropriate commutative sub-algebras (diagonal sub-algebras) one recovers all the classical hidden process and, in addition, one obtains families of processes which are not usual hidden Markov process, but are included in the above mentioned extensions of these processes. In this paper we only deal with processes with an at most countable state space.
In this paper we discuss the quantum Markov chains of unitary quantum walks.The construction of the quantum Markov chain associated with a unitary quantum walk serves as a good example in the view point of quantum Markov chains.On the other hand, the method of quantum Markov chain provides a tool for the investigation of dynamical properties of the unitary quantum walks.We discuss the reducibility/irreducibility and the recurrence/transience properties.Comparing with the results known in other literature and in classical random walks, we will see some similarity as well as some differences.
Quantum Markov chains are an extension of classical Markov chains which are labelled with super-operators rather than probabilities. They allow to faithfully represent quantum programs and quantum protocols. In this paper, we investigate model checking omega-regular properties, a very general class of properties (including, e.g., LTL properties) of interest, against this model. For classical Markov chains, such properties are usually checked by building the product of the model with a language automaton. Subsequent analysis is then performed on this product. When doing so, one takes into account its graph structure, and for instance performs different analyses per bottom strongly connected component (BSCC). Unfortunately, for quantum Markov chains such an approach does not work directly, because super-operators behave differently from probabilities. To overcome this problem, we transform the product quantum Markov chain into a single super-operator, which induces a decomposition of the state space (the tensor product of classical state space and the quantum one) into a family of BSCC subspaces. Interestingly, we show that this BSCC decomposition provides a solution to the issue of model checking omega-regular properties for quantum Markov chains.