Complexity analysis of quantum walk based search algorithms

TL;DR

This paper introduces specific graph families suitable for efficient quantum walk search algorithms, providing explicit quantum circuit implementations and analyzing their practical complexity for real-world applications.

Contribution

It constructs explicit quantum circuits for quantum walk search algorithms on certain graphs, moving beyond black-box models and analyzing their gate complexity.

Findings

Quantum circuits for search algorithms are explicitly constructed.

Search on twisted toroid graphs requires O(√n log n) gates.

The approach enables practical exploration of quantum walk algorithms.

Abstract

We present several families of graphs that allow both efficient quantum walk implementations and efficient quantum walk based search algorithms. For these graphs, we construct quantum circuits that explicitly implement the full quantum walk search algorithm, without reference to a `black box' oracle. These circuits provide a practically implementable method to explore quantum walk based search algorithms with the aim of eventual real-world applications. We also provide a numerical analysis of a quantum walk based search along a twisted toroid family of graphs, which requires O($\sqrt{n}$ log($n$)) elementary 2-qubit quantum gate operations to find a marked node.