Spatial Search on Johnson Graphs by Continuous-Time Quantum Walk

TL;DR

This paper demonstrates that continuous-time quantum walks can perform spatial search on Johnson graphs with optimal efficiency, matching the Grover lower bound and achieving success probability one asymptotically.

Contribution

It provides a rigorous proof that quantum spatial search on Johnson graphs reaches the optimal Grover bound for fixed diameter, expanding understanding of quantum algorithms on symmetric graphs.

Findings

Achieves Grover lower bound on Johnson graphs

Success probability approaches 1 asymptotically

Proof applicable to other graph classes

Abstract

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.