  Perfect state transfer (PST) has great significance due to its applications
in quantum information processing and quantum computation. In this paper we
present a characterization on connected simple Cayley graph $\Gamma={\rm
Cay}(G,S)$ having PST. We show that many previous results on periodicity and
existence of PST of circulant graphs (where the underlying group $G$ is cyclic)
and cubelike graphs ($G=(\mathbb{F}_2^n,+)$) can be derived or generalized to
arbitrary abelian case in unified and more simple ways from our
characterization. We also get several new results including answers on some
problems raised before.
