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arXiv:quant-ph/0206103 · Journal of the Mathematical Society of Japan

A new type of limit theorems for the one-dimensional quantum random walk

Norio Konno

Published Oct 1, 2005• 229 citations
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Abstract

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.

DOI
10.2969/jmsj/1150287309

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In this paper we consider limit theorems, symmetry of distribution, and absorption problems for two types of one-dimensional quantum random walks determined by 2×22 \times 22×2 unitary matrices using our PQRS method. The one type was introduced by Gudder in 1988, and the other type was studied intensively by Ambainis et al. in 2001. The difference between both types of quantum random walks is also clarified.
The discrete-time quantum walk (QW) is determined by a unitary matrix whose components are complex numbers. Konno (2015) extended the QW to the quaternionic quantum walk (QQW) whose components are quaternions and presented some properties of the QQW. Furthermore, Konno (2015) presented the question of whether or not the dynamics of a QQW is exactly the same as that of the corresponding QW. We give an answer to the problem by calculating the probability distribution and the weak limit density function of some classes of the QQW.
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