Matrix exponential is the key problem to calculate the time evolution of a quantum system. It is usually difficult to obtain the time evolution operator when the system Hamiltonian is of large dimension. Fortunately, it is not necessary to get the time evolution operator but to obtain its action on the initial state. In this talk, a brief introduction to the utilization of Krylov space method to obtain the matrix-exponential-vector multiplication is archived when the dimension of the sparse Hamiltonian is up to 2^22. A 10x speed up of CUDA compared to MKL can be realized with a NVIDIA Tesla GPU.