Finding Structural Anomalies in Star Graphs Using Quantum Walks: A General Approach

TL;DR

This paper rigorously analyzes quantum walk-based searches on star graphs, identifying conditions for quadratic speedup and demonstrating that such searches cannot surpass order root N time, while establishing a formalism for symmetric graphs.

Contribution

It provides necessary and sufficient conditions for quadratic-speed quantum searches on star graphs and introduces a formalism for analyzing highly symmetric graphs.

Findings

Quantum walks on star graphs are only faster when eigenvalues are degenerate.

Searches cannot be faster than order root N time.

A formalism for analyzing symmetric graphs is developed.

Abstract

In previous papers about searches on star graphs several patterns have been made apparent; the speed up only occurs when graphs are "tuned" so that their time step operators have degenerate eigenvalues, and only certain initial states are effective. More than that, the searches are never faster than order root N time. In this paper the problem is defined rigorously, the causes for all of these patterns are identified, sufficient and necessary conditions for quadratic-speed searches for any connected subgraph are demonstrated, the tolerance of these conditions is investigated, and it is shown that (unfortunately) we can do no better than order root N time. Along the way, a useful formalism is established that may be useful in future work involving highly symmetric graphs.