  In this work, we consider the spatial search for a general marked state on
graphs by continuous time quantum walks. As a simplest case, we compute the
amplitude expression of the search for the multi-vertex uniform superposition
state on hypercube, and find that the spatial search algorithm is optimal for
the two-vertex uniform state. However, on general graphs, a common formula
can't be obtained for searching a general non-uniform superposition state.
Fortunately, a Laplacian spectrum condition which determines whether the
associated graph could be appropriate for performing the optimal spatial search
is presented. The condition implies that if the proportion of the maximum and
the non-zero minimum Laplacian eigenvalues is less or equal to 1+sqrt(1/2),
then the spatial search is optimal for any general state. At last, we apply
this condition to three kind graphs, the induced complete graph, the strongly
regular graph and the regular complete multi-partite graphs. By the condition,
one can conclude that these graphs will become suitable for optimal search with
properly setting their graph parameters.
