  The graph isomorphism (GI) problem is the computational problem of finding a
permutation of vertices of a given graph $G_1$ that transforms $G_1$ to another
given graph $G_2$ and preserves the adjacency. In this work, we propose a
quantum algorithm to determine whether there exists such a permutation. To find
such a permutation, we introduce isomorphic equivalent graphs of the given
graphs to be tested. We proof that the GI problem of the equivalent graphs is
equivalent to the GI problem of the given graphs. The idea of the algorithm is
to determine whether there exists a permutation can transform the eigenvectors
of the adjacency matrix of the equivalent graphs each other. The cost time of
the algorithm is polynomial.
