  Spatial search on graphs is one of the most important algorithmic
applications of quantum walks. To show that a quantum-walk-based search is more
efficient than a random-walk-based search is a difficult problem, which has
been addressed in several ways. Usually, graph symmetries aid in the
calculation of the algorithm's computational complexity, and Johnson graphs are
an interesting class regarding symmetries because they are regular,
Hamilton-connected, vertex- and distance-transitive. In this work, we show that
spatial search on Johnson graphs by continuous-time quantum walk achieves the
Grover lower bound $\pi\sqrt{N}/2$ with success probability $1$ asymptotically
for every fixed diameter, where $N$ is the number of vertices. The proof is
mathematically rigorous and can be used for other graph classes.
