Limit theorem for continuous-time quantum walk on the line
Norio Konno · Physical Review E · 2005
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: ${X}_{t}^{(d)}∕t\ensuremath{\rightarrow}dx∕\ensuremath{\pi}(1\ensuremath{-}{x}^{2})\sqrt{1\ensuremath{-}2{x}^{2}}$ as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. The present paper shows that a similar type of weak limit theorem is satisfied for a continuous-time quantum walk ${X}_{t}^{(c)}$ on the line as follows: ${X}_{t}^{(c)}∕t\ensuremath{\rightarrow}dx∕\ensuremath{\pi}\sqrt{1\ensuremath{-}{x}^{2}}$ as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. These results for quantum walks form a striking contrast to the central limit theorem for symmetric discrete- and continuous-time classical random walks: ${Y}_{t}∕\sqrt{t}\ensuremath{\rightarrow}{e}^{\ensuremath{-}{x}^{2}∕2}dx∕\sqrt{2\ensuremath{\pi}}$ as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. 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.