Quantum walk search on Johnson graphs

The Johnson graph J (n, k) is defined by n symbols, where vertices are kelement subsets of the symbols, and vertices are adjacent if they differ in exactly one symbol. In particular, J (n, 1) is the complete graph Kn, and J (n, 2) is the strongly regular triangular graph Tn, 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.