Multi-target quantum walk search on Johnson graph

Abstract

The discrete-time quantum walk on the Johnson graph $J(n,k)$ is a useful tool for performing target vertex searches with high success probability. This graph is defined by $n$ distinct elements, with vertices being all the \(\binom{n}{k}\) $k$-element subsets and two vertices are connected by an edge if they differ exactly by one element. However, most works in the literature focus solely on the search for a single target vertex on the Johnson graph. In this article, we utilize lackadaisical quantum walk--a form of discrete-time coined quantum walk with a wighted self-loop at each vertex of the graph--along with our recently proposed modified coin operator, $\mathcal{C}_g$, to find multiple target vertices on the Johnson graph $J(n,k)$ for various values of $k$. Additionally, a comparison based on the numerical analysis of the performance of the $\mathcal{C}_g$ coin operator in searching for multiple target vertices on the Johnson graph, against various other frequently used coin operators by the discrete-time quantum walk search algorithms, shows that only $\mathcal{C}_g$ coin can search for multiple target vertices with a very high success probability in all the scenarios discussed in this article, outperforming other widely used coin operators in the literature.