# Quantum search on Hanoi network

## TL;DR

This paper investigates quantum search algorithms on Hanoi networks using regular and lackadaisical quantum walks, demonstrating improved scaling of search times compared to classical methods.

## Contribution

It introduces the application of regular and lackadaisical quantum walks to search on Hanoi networks and analyzes their scaling behavior.

## Key findings

- Regular quantum walks scale as O(N^{0.79}√log N) and O(N^{0.65}√log N) for degrees three and four.
- Lackadaisical quantum walks scale as O(N^{0.57} log N) and O(N^{0.50} log N) for degrees three and four.
- Quantum walks outperform classical exhaustive search on Hanoi networks.

## Abstract

Hanoi network has a one-dimensional periodic lattice as its main structure with additional long-range edges, which allow having efficient quantum walk algorithm that can find a target state on the network faster than the exhaustive classical search. In this article, we use regular quantum walks and lackadaisical quantum walks respectively to search for a target state. From the curve fitting of the numerical results for Hanoi network of degree three and four we find that their running time for the regular quantum walks followed by amplitude amplification scales as $\mathcal{O}\left(N^{0.79} \sqrt{\log N}\right)$ and $\mathcal{O}\left(N^{0.65} \sqrt{\log N}\right)$ respectively. And for the search by lackadaisical quantum walks the running time scales as $\mathcal{O}\left(N^{0.57}\log N\right)$ and $\mathcal{O}\left(N^{0.50}\log N\right)$ respectively.