Quantum walk search on Johnson graphs

Abstract

The Johnson graph 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, is the complete graph Kn, and 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 , 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 with fixed k.