Quantum search of matching on signed graphs

Abstract

We construct a quantum searching model of a signed edge driven by a quantum walk. The time evolution operator of this quantum walk provides a weighted adjacency matrix induced by the assignment of a sign to each edge. This sign can be regarded as so-called the edge coloring. Then as an application, under an arbitrary edge coloring which gives a matching on a complete graph on $n+1$ vertices we consider a quantum search of a colored edge from the edge set of the complete graph. We show that this quantum walk finds a colored edge within the time complexity of $O(n^{\frac{2-\alpha }{2}})$ with probability $1-o(1)$, while the corresponding random walk on the line graph finds them within the time complexity of $O(n^{2-\alpha })$ if we set the number of the edges of the matching by $t=O(n^{\alpha })$ for $0 \le \alpha \le 1$ red with $t \le \frac{n}{2}$.