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 (\widetilde{O}(m^{1/3} \delta^{-1/3})), with (m) the number of edges and (\delta) 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 (\widetilde{O}(2^{n/3})), surpassing the (Ω(2^{n/2})) barrier set by index erasure.