  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 $\Omega(2^{n/2})$ barrier set by index
erasure.
