Localization and discrete probability function of Szegedy's quantum search one-dimensional cycle with self-loops

Abstract

We study the localization and the discrete probability function of a quantum search on the one-dimensional (1D) cycle with self-loops for n vertices and m marked vertices. First, unmarked vertices have no localization since the quantum search on unmarked vertices behaves like the 1D three-state quantum walk (3QW) and localization does not occur with nonlocal initial states on a 3QW, according to residue calculations and the Riemann-Lebesgue theorem. Second, we show that localization does occur on the marked vertices and derive an analytic expression for localization by the degenerate 1eigenvalues contributing to marked vertices. Therefore localization can contribute to a quantum search. Furthermore, we emphasize that localization comes from the self-loops. Third, using the localization of a quantum search, the asymptotic average probability distribution (AAPD) and the discrete probability function (DPF) of a quantum search are obtained. The DPF shows that Szegedys quantum search on the 1D cycle with self-loops spreads ballistically.