Quantum Walk Sampling by Growing Seed Sets
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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^1/3δ^-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 provides 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.
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