Expansion Testing using Quantum Fast-Forwarding and Seed Sets

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.