Spatial Search via Memoryless Walk with Selfloop
Peter H{\o}yer, Janet Leahy

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 . Our walk uses minimal memory, 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…
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Taxonomy
TopicsQuantum Computing Algorithms and Architecture · Quantum-Dot Cellular Automata
