Ebook: Quantum walks for computer scientists

Quantum computation, one of the latest joint ventures between physics and the theory of computation, is a scientific field whose main goals include the development of hardware and algorithms based on the quantum mechanical properties of those physical systems used to implement such algorithms. Solving difficult tasks (for example, the Satisfiability Problem and other NP-complete problems) requires the development of sophisticated algorithms, many of which employ stochastic processes as their mathematical basis. Discrete random walks are a popular choice among those stochastic processes. Inspired on the success of discrete random walks in algorithm development, quantum walks, an emerging field of quantum computation, is a generalization of random walks into the quantum mechanical world. The purpose of this lecture is to provide a concise yet comprehensive introduction to quantum walks.;Introduction -- Quantum mechanics -- Mathematical preliminaries -- Postulates of quantum mechanics -- Entanglement -- Theory of computation -- What is the theory of computation -- The birth of the theory of computation: Alan Turing and his machines -- Deterministic and nondeterministic computation -- A quick tour on algorithmic complexity and NP-completeness -- Physics and the theory of computation -- Classical random walks -- Probability theory and stochastic processes -- Classical discrete random walks: results and applications -- Stochastic algorithms based on classical discrete random walks -- Classical continuous random walks -- Quantum walks -- Quantum walk on a line -- Quantum walks on graphs -- More considerations on classical and quantum walks -- Continuous quantum walks -- Whether discrete or continuous: is it quantum random walks or just quantum walks -- How are continuous and discrete quantum walks connected -- Computer science and quantum walks -- Algorithmic applications of quantum walks -- Universality of quantum walks.