Continuous-Time Quantum Walk on the Line

Quantum walks have recently been introduced and investigated, with the hope that they may be useful in constructing new efficient quantum algorithms. For reviews of quantum walks, see Refs. [4, 16, 24]. There are two distinct types of the quantum walk: one is a discrete-time case [2, 5, 12, 14, 17, 18, 19, 22, 23], the other is a continuous-time case [1, 3, 7, 8, 10, 15, 20]. The quantum walk can be considered as a quantum analog of the classical random walk. However there are some differences between them. Let σ (d)(t) (resp. σ (c)(t)) be the standard deviation of the probability distribution for a discrete-time (resp. continuous-time) classical random walk on the line starting from the origin at time t. Similarly, σ (d)(t) (resp. σ q (c)(t)) denotes the standard deviation for a discrete-time (resp. continuous-time) quantum walk. Then, it is well known that σ (d)(t), σ c (c)(t) ≍ √ t, where f(t) ≍ g(t) indicates that f(t)/g(t) → c∗( 6= 0) as t → ∞, (related results can be found in [9, 13]). In contrast, σ (d)(t) ≍ t holds, (see [5, 17], for example). That is, the qunatum walk spreads over the line faster than the classical walk in the discrete-time case. However, it is not known that whether or not σ (c)(t) ≍ t also holds for the continuous-time quantum walk. One of the motivations of the present paper is to give an affirmative answer to this question as a consequence of a new limit theorem for the walk. Thus here we focus on a continuous-time qunatum random walk on a line. Let Z be the set of integers. To define the continuous-time qunatum walk on Z, we