Spatial Search via Memoryless Walk with Selfloop

TL;DR

This paper introduces a memoryless quantum walk with a selfloop technique that efficiently finds a marked vertex on a 2D grid, achieving near-certain success with minimal memory and applications.

Contribution

It presents a novel memoryless quantum walk on a grid using selfloops to ensure high success probability, advancing quantum search algorithms with minimal memory requirements.

Findings

Achieves near-perfect success probability in locating a marked vertex.

Uses minimal memory, $O( oot{N ext{log} N})$, applications.

Proves the walk's asymptotic behavior into a single rotational space.

Abstract

The defining feature of memoryless quantum walks is that they operate on the vertex space of a graph, and therefore can be used to produce search algorithms with minimal memory. We present a memoryless walk that can find a unique marked vertex on a two-dimensional grid. Our walk is based on the construction proposed by Falk, which tessellates the grid with squares of size $2 \times 2$. Our walk uses minimal memory, $O(\sqrt{N \log N})$ applications of the walk operator, and outputs the marked vertex with vanishing error probability. To accomplish this, we apply a selfloop to the marked vertex - a technique we adapt from interpolated walks. We prove that with our explicit choice of selfloop weight, this forces the action of the walk asymptotically into a single rotational space. We characterize this space and as a result, show that our memoryless walk produces the marked vertex with a success probability asymptotically approaching one.