Deterministic quantum search on all Laplacian integral graphs

TL;DR

This paper introduces a deterministic quantum search algorithm that guarantees finding a marked vertex with certainty on any Laplacian integral graph, broadening the class of graphs suitable for reliable quantum search.

Contribution

It presents the first deterministic quantum search method applicable to all Laplacian integral graphs, regardless of the number of marked vertices, unifying and expanding previous approaches.

Findings

Achieves 100% success probability in quantum search on Laplacian integral graphs.

Applicable to graphs with any proportion of marked vertices.

Simplifies the design of deterministic quantum search algorithms.

Abstract

Searching for an unknown marked vertex on a given graph (also known as spatial search) is an extensively discussed topic in the area of quantum algorithms, with a plethora of results based on different quantum walk models and targeting various types of graphs. Most of these algorithms have a non-zero probability of failure. In recent years, there have been some efforts to design quantum spatial search algorithms with $100\%$ success probability. However, these works either only work for very special graphs or only for the case where there is only one marked vertex. In this work, we propose a different and elegant approach to quantum spatial search, obtaining deterministic quantum search algorithms that can find a marked vertex with certainty on any Laplacian integral graph with any predetermined proportion of marked vertices. Thus, this work discovers the largest class of graphs so far that allow deterministic quantum search, making it easy to design deterministic quantum search algorithms for many graphs, including the different graphs discussed in previous works, in a unified framework.