  Finding problems that allow for superpolynomial quantum speedup is one of the
most important tasks in quantum computation. A key challenge is identifying
problem structures that can only be exploited by quantum mechanics. In this
paper, we find a class of graphs that allows for exponential quantum-classical
separation for the pathfinding problem with the adjacency list oracle, and this
class of graphs is named regular sunflower graphs. We prove that, with high
probability, a regular sunflower graph of degree at least $7$ is a mild
expander graph, that is, the spectral gap of the graph Laplacian is at least
inverse polylogarithmic in the graph size.
  We provide an efficient quantum algorithm to find an $s$-$t$ path in the
regular sunflower graph while any classical algorithm takes exponential time.
This quantum advantage is achieved by efficiently preparing a $0$-eigenstate of
the adjacency matrix of the regular sunflower graph as a quantum superposition
state over the vertices, and this quantum state contains enough information to
help us efficiently find an $s$-$t$ path in the regular sunflower graph.
  Because the security of an isogeny-based cryptosystem depends on the hardness
of finding an $s$-$t$ path in an expander graph \cite{Charles2009}, a quantum
speedup of the pathfinding problem on an expander graph is of significance. Our
result represents a step towards this goal as the first provable exponential
speedup for pathfinding in a mild expander graph.
