  We present an adiabatic quantum algorithm for the abstract problem of
searching marked vertices in a graph, or spatial search. Given a random walk
(or Markov chain) $P$ on a graph with a set of unknown marked vertices, one can
define a related absorbing walk $P'$ where outgoing transitions from marked
vertices are replaced by self-loops. We build a Hamiltonian $H(s)$ from the
interpolated Markov chain $P(s)=(1-s)P+sP'$ and use it in an adiabatic quantum
algorithm to drive an initial superposition over all vertices to a
superposition over marked vertices. The adiabatic condition implies that for
any reversible Markov chain and any set of marked vertices, the running time of
the adiabatic algorithm is given by the square root of the classical hitting
time. This algorithm therefore demonstrates a novel connection between the
adiabatic condition and the classical notion of hitting time of a random walk.
It also significantly extends the scope of previous quantum algorithms for this
problem, which could only obtain a full quadratic speed-up for state-transitive
reversible Markov chains with a unique marked vertex.
