Quantum Walks

Quantum walks can be considered as a generalized version of the classical random walk. There are two classes of quantum walks, that is, the discrete-time (or coined) and the continuous-time quantum walks. This manuscript treats the discrete case in Part I and continuous case in Part II, respectively. Most of the contents are based on our results. Furthermore, papers on quantum walks are listed in References. Studies of discrete-time walks appeared from the late 1980s from Gudder (1988), for example. Meyer (1996) investigated the model as a quantum lattice gas automaton. Nayak and Vishwanath (2000) and Ambainis et al. (2001) studied intensively the behaviour of discrete-time walks, in particular, the Hadamard walk. In contrast with the central limit theorem for the classical random walks, Konno (2002a, 2005a) showed a new type of weak limit theorems for the one-dimensional lattice. Grimmett, Janson, and Scudo (2004) extended the limit theorem to a wider range of the walks. On the other hand, the continuous-time quantum walk was introduced and studied by Childs, Farhi, and Gutmann (2002). Excellent reviews on quantum walks are found in Kempe (2003), Tregenna et al. (2003), Ambainis (2003), Kendon (2007).