dorsal/arxiv
View SchemaA New Type of Limit Theorems for the One-Dimensional Quantum Random Walk
| Authors | Norio Konno |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0206103 |
| URL | https://arxiv.org/abs/quant-ph/0206103 |
| Journal | Journal of the Mathematical Society of Japan, Vol.57, No.4, pp.1179-1195 (2005) |
Abstract
In this paper we consider the one-dimensional quantum random walk X^{varphi} _n at time n starting from initial qubit state varphi determined by 2 times 2 unitary matrix U. We give a combinatorial expression for the characteristic function of X^{varphi}_n. The expression clarifies the dependence of it on components of unitary matrix U and initial qubit state varphi. As a consequence of the above results, we present a new type of limit theorems for the quantum random walk. In contrast with the de Moivre-Laplace limit theorem, our symmetric case implies that X^{varphi}_n /n converges in distribution to a limit Z^{varphi} as n to infty where Z^{varphi} has a density 1 / pi (1-x^2) sqrt{1-2x^2} for x in (- 1/sqrt{2}, 1/sqrt{2}). Moreover we discuss some known simulation results based on our limit theorems.
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"abstract": "In this paper we consider the one-dimensional quantum random walk X^{varphi}\n_n at time n starting from initial qubit state varphi determined by 2 times 2\nunitary matrix U. We give a combinatorial expression for the characteristic\nfunction of X^{varphi}_n. The expression clarifies the dependence of it on\ncomponents of unitary matrix U and initial qubit state varphi. As a consequence\nof the above results, we present a new type of limit theorems for the quantum\nrandom walk. In contrast with the de Moivre-Laplace limit theorem, our\nsymmetric case implies that X^{varphi}_n /n converges in distribution to a\nlimit Z^{varphi} as n to infty where Z^{varphi} has a density 1 / pi (1-x^2)\nsqrt{1-2x^2} for x in (- 1/sqrt{2}, 1/sqrt{2}). Moreover we discuss some known\nsimulation results based on our limit theorems.",
"arxiv_id": "quant-ph/0206103",
"authors": [
"Norio Konno"
],
"categories": [
"quant-ph"
],
"journal_ref": "Journal of the Mathematical Society of Japan, Vol.57, No.4,\n pp.1179-1195 (2005)",
"title": "A New Type of Limit Theorems for the One-Dimensional Quantum Random Walk",
"url": "https://arxiv.org/abs/quant-ph/0206103"
},
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