# Expansion Testing using Quantum Fast-Forwarding and Seed Sets

## TL;DR

This paper introduces a quantum algorithm for expansion testing in graphs that significantly outperforms classical methods by utilizing quantum fast-forwarding and seed set growth techniques from graph clustering.

## Contribution

It presents a novel quantum expansion tester with improved complexity, combining quantum fast-forwarding with seed set growth via evolving set processes.

## Key findings

- Quantum expansion tester complexity: O(n^{1/3}\u03a6^{-1})
- Accelerates classical and previous quantum algorithms for expansion testing
- Uses evolving set process to grow seed sets for local graph exploration

## Abstract

Expansion testing aims to decide whether an $n$-node graph has expansion at least $\Phi$, or is far from any such graph. We propose a quantum expansion tester with complexity $\widetilde{O}(n^{1/3}\Phi^{-1})$. This accelerates the $\widetilde{O}(n^{1/2}\Phi^{-2})$ classical tester by Goldreich and Ron [Algorithmica '02], and combines the $\widetilde{O}(n^{1/3}\Phi^{-2})$ and $\widetilde{O}(n^{1/2}\Phi^{-1})$ quantum speedups by Ambainis, Childs and Liu [RANDOM '11] and Apers and Sarlette [QIC '19], respectively. The latter approach builds on a quantum fast-forwarding scheme, which we improve upon by initially growing a seed set in the graph. To grow this seed set we use a so-called evolving set process from the graph clustering literature, which allows to grow an appropriately local seed set.