Journal of the Mathematical Society of Japan

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 n ϕ at time n starting from initial qubit state ϕ determined by 2 × 2 unitary matrix U . We give a combinatorial expression for the characteristic function of X n ϕ . The expression clarifies the dependence of it on components of unitary matrix U and initial qubit state ϕ . 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 n ϕ / n converges weakly to a limit Z ϕ as n , where Z ϕ has a density 1 / π ( 1 - x 2 ) 1 - 2 x 2 for x ( - 1 / 2 , 1 / 2 ) . Moreover we discuss some known simulation results based on our limit theorems.

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J. Math. Soc. Japan, Volume 57, Number 4 (2005), 1179-1195.

First available in Project Euclid: 14 June 2006

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Primary: 60F05: Central limit and other weak theorems 60G50: Sums of independent random variables; random walks 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 81Q99: None of the above, but in this section

quantum random walk the Hadamard walk limit theorems


KONNO, Norio. A new type of limit theorems for the one-dimensional quantum random walk. J. Math. Soc. Japan 57 (2005), no. 4, 1179--1195. doi:10.2969/jmsj/1150287309.

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