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

Abstract

In this paper we consider the one-dimensional quantum random walk X ϕ n 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 ϕ n . 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 ϕ n /n converges weakly to a limit Z ϕ as n → ∞, where Z ϕ 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.

Decoherence in quantum walks – a review

“…We will give examples using these methods in Section 5. Path counting (path integrals) was further refined in Carteret et al (2003), and a third method using the algebra of the matrix operators was presented in Konno (2002;2005a). A solution using the tools of classical (wave) optics can be found in Knight et al (2003;.…”

The generator and quantum Markov semigroup for quantum walks

“…We first introduce the definition of 1-dimensional quantum walks following [2,7,11,12,18]. A quantum particle has an intrinsic degree of freedom, called "chirality".…”

“…One is so called the path integral approach, in which the explicit probability amplitude is computed by using a great deal of combinatorics. This method has been extensively developed by Konno [11,12]. In particular, Konno obtained the scaled limit distribution of the QW very concretely.…”

“…After it was initiated by Meyer [17], it attracted many interests and there are many works developing it in mathematically rigorous way on the one hand and explaining possible practical applications, e.g., in quantum computation (see [2,7,9,11,12,14,17], and references therein for more details).…”

“…By it we will recover the limit distributions for QW's which was concretely studied by Konno [11,12] via path integral approach. This paper is organized as follows.…”