# Faster Search by Lackadaisical Quantum Walk

## TL;DR

This paper demonstrates that adding self-loops to a quantum walk on a 2D grid significantly speeds up the search for a marked vertex, achieving a constant success probability in fewer steps than traditional methods.

## Contribution

Introducing lackadaisical quantum walks with self-loops to improve search efficiency on 2D grids, surpassing previous algorithms in runtime.

## Key findings

- Success probability reaches near 1 in O(√N log N) steps.
- Achieves an O(√log N) speedup over loopless quantum search.
- Matches the best known quantum search algorithms for 2D grids.

## Abstract

In the typical model, a discrete-time coined quantum walk searching the 2D grid for a marked vertex achieves a success probability of $O(1/\log N)$ in $O(\sqrt{N \log N})$ steps, which with amplitude amplification yields an overall runtime of $O(\sqrt{N} \log N)$. We show that making the quantum walk lackadaisical or lazy by adding a self-loop of weight $4/N$ to each vertex speeds up the search, causing the success probability to reach a constant near $1$ in $O(\sqrt{N \log N})$ steps, thus yielding an $O(\sqrt{\log N})$ improvement over the typical, loopless algorithm. This improved runtime matches the best known quantum algorithms for this search problem. Our results are based on numerical simulations since the algorithm is not an instance of the abstract search algorithm.