Exact results on Quantum search algorithm

TL;DR

This paper derives exact analytical expressions for the success probability of a generalized quantum search algorithm with arbitrary phases, analyzing how initial coherence and noise affect its performance.

Contribution

It extends Grover's algorithm to include arbitrary phases and provides exact success probability formulas, also studying the impact of coherence and noise.

Findings

Success probability can reach >= 0.8 with specific phase matching and knowledge of lower bound of bb.

Derived exact formulas for success probability after any number of iterations.

Quantified how initial coherence and noise influence the algorithm's success.

Abstract

We generalize Grover algorithm with two arbitrary phases in a density matrix set up. We give exact analytic expressions for the success probability after arbitrary number of iteration of the generalized Grover operator as a function of number of iterations, two phase angles ({\alpha}, \{beta}) and parameter {\xi} introduced in the off diagonal terms of the density matrix in a sense to capture the coherence present in the initial quantum register. We extend Li and Li's idea and show for the phase matching condition {\alpha} = -\{beta} = 0.35{\pi} with two iterations and {\xi} = 1, we can achieve success probability >= 0.8 only with a knowledge about the lower bound of {\lambda} = 0.166 where {\lambda} is the ratio of marked to total number states in the database. Finally we quantify success probability of the algorithm with decrease in coherence of the initial quantum state against modest noise in this simple model.