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 1-eigenvalues 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 Szegedy’s quantum search on the 1D cycle with self-loops spreads ballistically.