A new type of limit theorems for the one-dimensional quantum random walk
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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 × 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, 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 weakly to a limit Z^{\varphi} as n → ∞, where Z^{\varphi} has a density 1/π (1-x^2) √{1-2x^2} for x ∈ (- 1/√{2}, 1/√{2}). Moreover we discuss some known simulation results based on our limit theorems.
- 社団法人 日本数学会の論文
- 2005-10-01
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