  The spatial search problem aims to find a marked vertex of a finite graph
using a dynamic with two constraints: (1) The walker has no compass and (2) the
walker can check whether a vertex is marked only after reaching it. This
problem is a generalization of unsorted database search and has many
applications to algorithms. Classical algorithms that solve the spatial search
problem are based on random walks and the computational complexity is
determined by the hitting time. On the other hand, quantum algorithms are based
on quantum walks and the computational complexity is determined not only by the
number of steps to reach a marked vertex, but also by the success probability,
since we need to perform a measurement at the end of the algorithm to determine
the walker's position. In this work, we address the spatial search problem on
Johnson graphs using the coined quantum walk model. Since Johnson graphs are
vertex- and distance-transitive, we have found an invariant subspace of the
Hilbert space, which aids in the calculation of the computational complexity.
We have shown that, for every fixed diameter, the asymptotic success
probability is $1/2$ after taking $\pi\sqrt N/(2\sqrt 2)$ steps, where $N$ is
the number of vertices of the Johnson graph.
