Quantum Walk Search on Johnson Graphs

TL;DR

This paper demonstrates that Johnson graphs J(n,3) support fast quantum walk search, extending known results from simpler Johnson graphs and introducing a basis change technique for accurate analysis.

Contribution

It proves that Johnson graphs J(n,3) enable efficient quantum search and develops a basis change method for analyzing such quantum walks.

Findings

J(n,3) supports fast quantum walk search.

A basis change is necessary for accurate perturbation analysis.

Method applicable to general J(n,k) graphs.

Abstract

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 $k$.