  The quantum-walk-based spatial search problem aims to find a marked vertex
using a quantum walk on a graph with marked vertices. We describe a framework
for determining the computational complexity of spatial search by
continuous-time quantum walk on arbitrary graphs by providing a recipe for
finding the optimal running time and the success probability of the algorithm.
The quantum walk is driven by a Hamiltonian derived from the adjacency matrix
of the graph modified by the presence of the marked vertices. The success of
our framework depends on the knowledge of the eigenvalues and eigenvectors of
the adjacency matrix. The spectrum of the Hamiltonian is subsequently obtained
from the roots of the determinant of a real symmetric matrix $M$, the
dimensions of which depend on the number of marked vertices. The eigenvectors
are determined from a basis of the kernel of $M$. We show each step of the
framework by solving the spatial searching problem on the Johnson graphs with a
fixed diameter and with two marked vertices. Our calculations show that the
optimal running time is $O(\sqrt{N})$ with an asymptotic probability of
$1+o(1)$, where $N$ is the number of vertices.
