The analysis of a potential quantum computing device is a difficult subject to investigate, since it involves the study of computational methods from the theoretical viewpoint of quantum physics modeling. Such results provide a theoretical foundation for the development of quantum computational algorithms, and are highly important. This paper focuses on the analysis of quantum random walks in one and higher dimensions, in analogy to nonquantum random walks and their application in established computer science. In particular, new results are presented for absorption probabilities for systems involving one and two absorbing walls. Results for the one-dimensional case are obtained by means of generating functions and eigenfunction analysis. The eigenfunction approach is also applicable to higher dimensions (D) that involve a (D-1) dimensional wall.

The authors present a thorough and detailed analysis of the development of these results, after first providing a solid background for the analogy of these quantum results with the ordinary case, so the reader understands their significance.