Searches on star graphs and equivalent oracle problems

TL;DR

This paper investigates quantum search algorithms on star graphs with non-uniform unmarked vertices, showing success probabilities decrease with more vertex types and relating graph searches to oracle problems.

Contribution

It introduces a framework for analyzing quantum searches on non-uniform graphs and establishes their equivalence to oracle problems.

Findings

Success probability decreases as the number of unmarked vertex types increases.

Graph searches can be reformulated as equivalent oracle problems.

Quantum search remains feasible despite non-uniform background vertices.

Abstract

We examine a search on a graph among a number of different kinds of objects (vertices), one of which we want to find. In a standard graph search, all of the vertices are the same, except for one, the marked vertex, and that is the one we wish to find. We examine the case in which the unmarked vertices can be of different types, so the background against which the search is done is not uniform. We find that the search can still be successful, but the probability of success is lower than in the uniform background case, and that probability decreases with the number of types of unmarked vertices. We also show how the graph searches can be rephrased as equivalent oracle problems.