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.
Highlights
Introduction and SummaryThe expansion of a graph is a measure for how well connected the graph is
In [AS19] we introduced a new quantum walk technique called “quantum fast-forwarding” (QFF) that allows to approximately prepare these quantum samples in the square root of the random walk runtime
In [AS19] we introduced a more involved quantum walks (QWs) technique called quantum fastforwarding (QFF)
Summary
The (vertex) expansion of a graph is a measure for how well connected the graph is. For an undirected graph G = (V, E), with |V| = n and |E| = m, it is defined as. Given graphs G and G with degree bound d, G is -far from G if at least nd edges have to be added or removed from G to obtain G They proved an Ω(n1/2) lower bound on the query complexity of this problem, and proposed an elegant tester based on random walk collision counting with complexity. In later works by Czumaj and Sohler [CS10], Kale and Seshadhri [KS11] and Nachmias and Shapira [NS10] the correctness was unconditionally established The ideas underlying this tester and its analysis were more recently extended towards testing the k-clusterability of a graph [CPS15, CKK+18], which is a multipartite generalization of the expansion testing problem. The gist of their algorithm is to combine an appropriate derandomization of the GR tester with Ambainis’ quantum algorithm for element distinctness [Amb07] The latter allows to count collisions among the set of O(n1/2) random walk endpoints using only O(n1/3) quantum queries.
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