Expansion Testing using Quantum Fast-Forwarding and Seed Sets

Abstract

Expansion testing aims to decide whether ann-node graph has expansion at leastΦ, or is far from any such graph. We propose a quantum expansion tester with complexityO~(n1/3Φ−1). This accelerates theO~(n1/2Φ−2)classical tester by Goldreich and Ron [Algorithmica '02] \cite{goldreich2002property}, and combines theO~(n1/3Φ−2)andO~(n1/2Φ−1)quantum speedups by Ambainis, Childs and Liu [RANDOM '11] and Apers and Sarlette [QIC '19] \cite{apers2019quantum}, 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.