dorsal/arxiv
View SchemaLimit Theorem for Continuous-Time Quantum Walk on the Line
| Authors | Norio Konno |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0408140 |
| URL | https://arxiv.org/abs/quant-ph/0408140 |
| DOI | 10.1103/PhysRevE.72.026113 |
| Journal | Physical Review E, Vol.72, 026113 (2005) |
Abstract
Concerning a discrete-time quantum walk X^{(d)}_t 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^{(d)}_t /t \to dx / \pi (1-x^2) \sqrt{1 - 2 x^2} as t \to \infty. The present paper shows that a similar type of weak limit theorems is satisfied for a {\it continuous-time} quantum walk X^{(c)}_t on the line as follows: X^{(c)}_t /t \to dx / \pi \sqrt{1 - x^2} as t \to \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} \to e^{-x^2/2} dx / \sqrt{2 \pi} as t \to \infty. The work deals also with 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.
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"abstract": "Concerning a discrete-time quantum walk X^{(d)}_t with a symmetric\ndistribution on the line, whose evolution is described by the Hadamard\ntransformation, it was proved by the author that the following weak limit\ntheorem holds: X^{(d)}_t /t \\to dx / \\pi (1-x^2) \\sqrt{1 - 2 x^2} as t \\to\n\\infty. The present paper shows that a similar type of weak limit theorems is\nsatisfied for a {\\it continuous-time} quantum walk X^{(c)}_t on the line as\nfollows: X^{(c)}_t /t \\to dx / \\pi \\sqrt{1 - x^2} as t \\to \\infty. These\nresults for quantum walks form a striking contrast to the central limit theorem\nfor symmetric discrete- and continuous-time classical random walks: Y_{t}/\n\\sqrt{t} \\to e^{-x^2/2} dx / \\sqrt{2 \\pi} as t \\to \\infty. The work deals also\nwith issue of the relationship between discrete and continuous-time quantum\nwalks. This topic, subject of a long debate in the previous literature, is\ntreated within the formalism of matrix representation and the limit\ndistributions are exhaustively compared in the two cases.",
"arxiv_id": "quant-ph/0408140",
"authors": [
"Norio Konno"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevE.72.026113",
"journal_ref": "Physical Review E, Vol.72, 026113 (2005)",
"title": "Limit Theorem for Continuous-Time Quantum Walk on the Line",
"url": "https://arxiv.org/abs/quant-ph/0408140"
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