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 widetildeO(n1/3Phi-1). This accelerates the widetildeO(n1/2Phi-2) classical tester by Goldreich and Ron [Algorithmica '02], and combines the widetildeO(n1/3Phi-2) and widetildeO(n1/2Phi-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.