We established a continuous-time framework for entangled quantum Markov chains derived from classical continuous-time Markov processes. Unlike previous studies focused on discrete-time dynamics, we demonstrated that a robust mixing property emerges directly from the structural architecture of the underlying continuous-time process. Our analysis showed that inhomogeneous entangled chains exhibited path-independent convergence to a limiting distribution determined by asymptotic temporal parameters. The results motivated future study of these entangled Markov chains in connection with open quantum system dynamics and quantum master equations.
We established a continuous-time framework for entangled quantum Markov chains derived from classical continuous-time Markov processes. Unlike previous studies focused on discrete-time dynamics, we demonstrated that a robust mixing property emerges directly from the structural architecture of the underlying continuous-time process. Our analysis showed that inhomogeneous entangled chains exhibited path-independent convergence to a limiting distribution determined by asymptotic temporal parameters. The results motivated future study of these entangled Markov chains in connection with open quantum system dynamics and quantum master equations.
The problem of sampling from the stationary distribution of a Markov chain finds widespread applications in a variety of fields. The time required for a Markov chain to converge to its stationary distribution is known as the classical mixing time. In this article, we deal with analog quantum algorithms for mixing. First, we provide an analog quantum algorithm that, given a Markov chain, allows us to sample from its stationary distribution in a time that scales as the sum of the square root of the classical mixing time and the square root of the classical hitting time. Our algorithm makes use of the framework of interpolated quantum walks and relies on Hamiltonian evolution in conjunction with von Neumann measurements. There also exists a different notion for quantum mixing: the problem of sampling from the limiting distribution of quantum walks, defined in a time-averaged sense. In this scenario, the quantum mixing time is defined as the time required to sample from a distribution that is close to this limiting distribution. Recently, we provided an upper bound on the quantum mixing time for Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi random graphs [Phys. Rev. Lett. 124, 050501 (2020)]. Here, we also extend and expand upon our findings therein. Namely, we provide an intuitive understanding of the state-of-the-art random matrix theory tools used to derive our results. In particular, for our analysis we require information about macroscopic, mesoscopic, and microscopic statistics of eigenvalues of random matrices which we highlight here. Furthermore, we provide numerical simulations that corroborate our analytical findings and extend this notion of mixing from simple graphs to any ergodic, reversible, Markov chain.
We study special quantum Markov chains on a Fock space related to iterated beam splittings as introduced in [23]. Besides a characterization of the position distributions of the chain, we show some kind of weak convergence of such discrete time quantum Markov chains to a kind of continuous time quantum Markov process. Furthermore, we provide existence and uniqueness for the solution of a quantum stochastic differential equation related to this quantum Markov process both on an exponential domain and, on a larger domain, in a pointwise sense.
In quantum computation theory, quantum Markov chains and quantum walks have been utilized by many quantum search algorithms which provide improved performance over their classical counterparts. More recently, due to the importance of the quantum decoherence phenomenon, decoherent quantum walks and their applications have been studied on a wide variety of structures. We study time-inhomogeneous quantum Markov chains with decoherence on finite spaces and discrete infinite spaces and their large scale equilibrium properties. In this thesis, we prove the convergence of decoherent time-inhomogeneous quantum Markov chain on finite state spaces, and a representation theorem for time-inhomogeneous quantum walk on discrete infinite state space. Additionally, the convergence of the distributions of the decoherent quantum walks are numerically estimated as an application of the representation theorem, and the convergence in distribution of the quantum analogues of Bernoulli, uniform, arcsine and semicircle laws are statistically analyzed.
We continue the analysis of nontrivial examples of quantum Markov processes. This is done by applying the construction of entangled Markov chains obtained from classical Markov chains with infinite state–space. The formula giving the joint correlations arises from the corresponding classical formula by replacing the usual matrix multiplication by the Schur multiplication. In this way, we provide nontrivial examples of entangled Markov chains on , F being any infinite dimensional type I factor, J a finite interval of , and the bar the von Neumann tensor product between von Neumann algebras. We then have new nontrivial examples of quantum random walks which could play a rôle in quantum information theory. In view of applications to quantum statistical mechanics too, we see that the ergodic type of an entangled Markov chain is completely determined by the corresponding ergodic type of the underlying classical chain, provided that the latter admits an invariant probability distribution. This result parallels the corresponding one relative to the finite dimensional case. Finally, starting from random walks on discrete ICC groups, we exhibit examples of quantum Markov processes based on type II 1 von Neumann factors.
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.
We continue the analysis of nontrivial examples of quantum Markov processes. This is done by applying the construction of entangled Markov chains obtained from classical Markov chains with infinite state–space. The formula giving the joint correlations arises from the corresponding classical formula by replacing the usual matrix multiplication by the Schur multiplication. In this way, we provide nontrivial examples of entangled Markov chains on , F being any infinite dimensional type I factor, J a finite interval of , and the bar the von Neumann tensor product between von Neumann algebras. We then have new nontrivial examples of quantum random walks which could play a rôle in quantum information theory. In view of applications to quantum statistical mechanics too, we see that the ergodic type of an entangled Markov chain is completely determined by the corresponding ergodic type of the underlying classical chain, provided that the latter admits an invariant probability distribution. This result parallels the corresponding one relative to the finite dimensional case. Finally, starting from random walks on discrete ICC groups, we exhibit examples of quantum Markov processes based on type II 1 von Neumann factors.