Grover algorithm with zero theoretical failure rate

Abstract

In a standard Grover's algorithm for quantum searching, the probability of finding the marked item is not exactly 1. In this paper we present a modified version of Grover's algorithm that searches a marked state with full successful rate. The modification is done by replacing the phase inversion by phase rotation through angle $\ensuremath{\varphi}.$ The rotation angle is given analytically to be $\ensuremath{\varphi}=2\mathrm{arcsin}(\mathrm{sin}[\ensuremath{\pi}/(4J+6)]/\mathrm{sin}\ensuremath{\beta}),$ where $\mathrm{sin}\ensuremath{\beta}=1/\sqrt{N},$ N is the number of items in the database, and J is any integer equal to or greater than the integer part of $[(\ensuremath{\pi}/2)\ensuremath{-}\ensuremath{\beta}]/(2\ensuremath{\beta}).$ Upon measurement at the $(J+1)\mathrm{th}$ iteration, the marked state is obtained with certainty.