Limit theorem for continuous-time quantum walk on the line.

Concerning a discrete-time quantum walk X(t)(d) with a symmetric distribution on the line, whose evolution is described by the Hadamard transformation, it was proved by the author that the following weak limit theorem holds: X(t)(d)/t -->dx/pi(1-x2) square root of (1-2x2) as t --> infinity. The present paper shows that a similar type of weak limit theorem is satisfied for a continuous-time quantum walk X((c) )(t ) on the line as follows: X(t)(c)/t --> dx/pi square root of (1-x2) as t --> infinity. These results for quantum walks form a striking contrast to the central limit theorem for symmetric discrete- and continuous-time classical random walks: Y(t)/square root of (t) --> e(-x2/2)dx/square root of (2pi) as t --> infinity. The work deals also with the issue of the relationship between discrete and continuous-time quantum walks. This topic, subject of a long debate in the previous literature, is treated within the formalism of matrix representation and the limit distributions are exhaustively compared in the two cases.