Quantum search of matching on signed graphs

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