Quantum Walk Sampling by Growing Seed Sets

Simon Apers, Michael Wagner
2019 European Symposium on Algorithms  
This work describes a new algorithm for creating a superposition over the edge set of a graph, encoding a quantum sample of the random walk stationary distribution. The algorithm requires a number of quantum walk steps scaling as O(m 1/3 δ −1/3 ), with m the number of edges and δ the random walk spectral gap. This improves on existing strategies by initially growing a classical seed set in the graph, from which a quantum walk is then run. The algorithm leads to a number of improvements: (i) it
more » ... rovides a new bound on the setup cost of quantum walk search algorithms, (ii) it yields a new algorithm for st-connectivity, and (iii) it allows to create a superposition over the isomorphisms of an n-node graph in time O(2 n/3 ), surpassing the Ω(2 n/2 ) barrier set by index erasure.
doi:10.4230/lipics.esa.2019.9 dblp:conf/esa/Apers19 fatcat:ov7wu3byozbffmg5gfnbsgqh7m