Lackadaisical quantum walk in the hypercube to search for multiple marked vertices

TL;DR

This paper explores how adding self-loops to hypercube quantum walks enhances the search for multiple marked vertices, revealing optimal parameters and success probabilities for different configurations.

Contribution

It introduces a new self-loop parameter for hypercube quantum walks to improve multi-vertex search efficiency and analyzes success probabilities for various marked vertex arrangements.

Findings

Optimal self-loop value for multiple vertices is l = (n/N) * k.

High success probability for non-adjacent marked vertices.

Success probability increases with marked neighbors and the presence of non-adjacent marked vertices.

Abstract

Adding self-loops at each vertex of a graph improves the performance of quantum walks algorithms over loopless algorithms. Many works approach quantum walks to search for a single marked vertex. In this article, we experimentally address several problems related to quantum walk in the hypercube with self-loops to search for multiple marked vertices. We first investigate the quantum walk in the loopless hypercube. We saw that neighbor vertices are also amplified and that approximately $1/2$ of the system energy is concentrated in them. We show that the optimal value of $l$ for a single marked vertex is not optimal for multiple marked vertices. We define a new value of $l = (n/N)\cdot k$ to search multiple marked vertices. Next, we use this new value of $l$ found to analyze the search for multiple marked vertices non-adjacent and show that the probability of success is close to $1$. We also use the new value of $l$ found to analyze the search for several marked vertices that are adjacent and show that the probability of success is directly proportional to the density of marked vertices in the neighborhood. We also show that, in the case where neighbors are marked, if there is at least one non-adjacent marked vertex, the probability of success increases to close to $1$. The results found show that the self-loop value for the quantum walk in the hypercube to search for several marked vertices is $l = (n / N) \cdot k $.