Quantum walk approach to search on fractal structures

TL;DR

This paper explores how the topology of fractal structures influences the efficiency of continuous-time quantum walks used for search algorithms, highlighting the importance of structural properties in quantum search performance.

Contribution

It introduces a quantum walk framework on fractal structures and analyzes the impact of topology on search efficiency through analytical and numerical methods.

Findings

Topological structure significantly affects quantum search success probabilities.

Fractal structures exhibit unique spectral properties influencing quantum walk dynamics.

Transition states are sensitive to the underlying topology of the database.

Abstract

We study continuous-time quantum walks mimicking the quantum search based on Grover's procedure. This allows us to consider structures, that is, databases, with arbitrary topological arrangements of their entries. We show that the topological structure of the database plays a crucial role by analyzing, both analytically and numerically, the transition from the ground to the first excited state of the Hamiltonian associated with different (fractal) structures. Additionally, we use the probability of successfully finding a specific target as another indicator of the importance of the topological structure.