Time-Inhomogeneous Quantum Markov Chains

Abstract

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.