Open quantum walk: Probability distribution and central limit theorem

As a basic tool for researching quantum algorithms, quantum walk has been an important aspect of the quantum computation. In an open quantum environment, homogenous quantum walk has all been studied, including its limit probability distribution and its central limit theorem. However, the evolution formula, the probability distribution and the central limit theorem have not been studied about open quantum walk with higher dimensional lattices and non-homogenous environment. Based on these works, we propose an open quantum walk on higher dimensional lattices and give its evolution formula, especially for one under non-homogenous environment, focus on its probability distribution and their central limit theorem with a different type of quantum operators. Firstly, we give an evolution formula of open quantum walk on higher dimensional lattices, show that it not only adapts to homogenous quantum walk, but also adapts to non-homogenous quantum walk. Compared with the existing evolution formulas, this result is more general. Secondly, by using the fourier transform and the fourier inverse transform, we give a computational formula for the probability distribution of open quantum walk, study its applicability for homogenous quantum walk and non-homogenous quantum walk. To present by examples, we show a computational process about their probability distributions of non-homogenous quantum walks on 1-dimensional lattice and 2-dimensional lattice. Finally, according to the central theorem for maringale difference sequence, we prove a central limit theorem for non-homogenous open quantum walk on higher dimensional lattices under certain condition, show that a central limit theorem is its special case about a one-dimensional lattice homogenous open quantum walk, illustrate a specific process of calculating the limit distribution. © 2016, Science Press. All right reserved.