Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Volume 57, Number 4 (2005), 1179-1195.
A new type of limit theorems for the one-dimensional quantum random walk
In this paper we consider the one-dimensional quantum random walk at time starting from initial qubit state determined by unitary matrix . We give a combinatorial expression for the characteristic function of . The expression clarifies the dependence of it on components of unitary matrix and initial qubit state . 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 converges weakly to a limit as , where has a density for . Moreover we discuss some known simulation results based on our limit theorems.
J. Math. Soc. Japan, Volume 57, Number 4 (2005), 1179-1195.
First available in Project Euclid: 14 June 2006
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Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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. https://projecteuclid.org/euclid.jmsj/1150287309