Lackadaisical quantum walks on triangular and honeycomb 2D grids

TL;DR

This paper extends the lackadaisical quantum walk technique to triangular and honeycomb 2D grids, achieving improved search times comparable to rectangular grids by adding specific self-loops.

Contribution

It demonstrates that adding tailored self-loops to triangular and honeycomb grids enables faster quantum search algorithms similar to those on rectangular grids.

Findings

Achieves $O(\sqrt{N \log{N}})$ search time on triangular grids

Achieves $O(\sqrt{N \\log{N}})$ search time on honeycomb grids

Shows effectiveness of lackadaisical approach on different 2D lattice types

Abstract

In the typical model, a discrete-time coined quantum walk search has the same running time of $O(\sqrt{N} \log{N})$ for 2D rectangular, triangular and honeycomb grids. It is known that for 2D rectangular grid the running time can be improved to $O(\sqrt{N \log{N}})$ using several different techniques. One of such techniques is adding a self-loop of weight $4/N$ to each vertex (i.e. making the walk lackadaisical). In this paper we apply lackadaisical approach to quantum walk search on triangular and honeycomb 2D grids. We show that for both types of grids adding a self-loop of weight $6/N$ and $3/N$ for triangular and honeycomb grids, respectively, results in $O(\sqrt{N \log{N}})$ running time.