  We investigate the effect of stochastic control errors on the Hamiltonian
that controls a closed quantum system. Quantum information technologies require
careful control for preparing a desired state used as an information resource.
However, because the stochastic control errors inevitably appear in realistic
situation, it is difficult to completely implement the control Hamiltonian.
Under this error, the actual performance of quantum control is far away from
the ideal one, and thus it is of great importance to evaluate the effect of the
control errors. In this paper, we derive a lower bound of the fidelity between
two closed quantum systems obeying the dynamics with and without errors. This
bound reveals a reachable and unreachable set of the controlled quantum system
under stochastic noises. Also, it is easily computable without considering the
stochastic process and needing the full dynamics of the states. We demonstrate
the actual performance of this bound via a simple control example. Furthermore,
based on this result, we quantitatively evaluate the probability of obtaining
the target state in the presence of control errors.
