  The Johnson graph $J(n,k)$ is defined by $n$ symbols, where vertices are
$k$-element subsets of the symbols, and vertices are adjacent if they differ in
exactly one symbol. In particular, $J(n,1)$ is the complete graph $K_n$, and
$J(n,2)$ is the strongly regular triangular graph $T_n$, both of which are
known to support fast spatial search by continuous-time quantum walk. In this
paper, we prove that $J(n,3)$, which is the $n$-tetrahedral graph, also
supports fast search. In the process, we show that a change of basis is needed
for degenerate perturbation theory to accurately describe the dynamics. This
method can also be applied to general Johnson graphs $J(n,k)$ with fixed $k$.
