Expansion Testing using Quantum Fast-Forwarding and Seed Sets
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Expansion testing aims to decide whether an n-node graph has expansion at least Φ, or is far from any such graph. We propose a quantum expansion tester with complexity O(n^1/3Φ^-1). This accelerates the O(n^1/2Φ^-2) classical tester by Goldreich and Ron [Algorithmica '02], and combines the O(n^1/3Φ^-2) and O(n^1/2Φ^-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 borrow a so-called evolving set process from the graph clustering literature, which allows to grow an appropriately local seed set.
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