  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.
