Quantum Search Algorithms on Hierarchical Networks

TL;DR

This paper investigates quantum search algorithms on hierarchical networks, revealing limitations of standard methods and exploring alternative quantum walk strategies with potential for improved efficiency.

Contribution

It introduces a new analysis of quantum search on hierarchical networks using non-Groverian quantum walks, highlighting cases where traditional algorithms are suboptimal.

Findings

Standard quantum search fails to be optimal on certain hierarchical networks.

Hierarchical structures enable analytical insights into quantum walk behavior.

Numerical simulations support the potential for alternative quantum algorithms.

Abstract

The "abstract search algorithm" is a well known quantum method to find a marked vertex in a graph. It has been applied with success to searching algorithms for the hypercube and the two-dimensional grid. In this work we provide an example for which that method fails to provide the best algorithm in terms of time complexity. We analyze search algorithms in degree-3 hierarchical networks using quantum walks driven by non-groverian coins. Our conclusions are based on numerical simulations, but the hierarchical structures of the graphs seems to allow analytical results.