Adjacent vertices can be hard to find by quantum walks

TL;DR

This paper investigates how the placement of multiple marked vertices affects the efficiency of quantum walk-based search algorithms, revealing configurations where quantum speed-up is lost, especially with adjacent marked vertices.

Contribution

It demonstrates that certain arrangements of multiple marked vertices, particularly adjacent ones, can negate quantum speed-up in search algorithms across various graphs.

Findings

Quantum walk search can require (N) steps with specific marked vertex configurations.

Adjacent marked vertices can significantly impair quantum search efficiency.

The results apply to general graphs, including grids and hypercubes.

Abstract

Quantum walks have been useful for designing quantum algorithms that outperform their classical versions for a variety of search problems. Most of the papers, however, consider a search space containing a single marked element only. We show that if the search space contains more than one marked element, their placement may drastically affect the performance of the search. More specifically, we study search by quantum walks on general graphs and show a wide class of configurations of marked vertices, for which search by quantum walk needs $\Omega(N)$ steps, that is, it has no speed-up over the classical exhaustive search. The demonstrated configurations occur for certain placements of two or more adjacent marked vertices. The analysis is done for the two-dimensional grid and hypercube, and then is generalized for any graph.