  Spatial search by discrete-time quantum walk can find a marked node on any
ergodic, reversible Markov chain $P$ quadratically faster than its classical
counterpart, i.e.\ in a time that is in the square root of the hitting time of
$P$. However, in the framework of continuous-time quantum walks, it was
previously unknown whether such general speed-up is possible. In fact, in this
framework, the widely used quantum algorithm by Childs and Goldstone fails to
achieve such a speedup. Furthermore, it is not clear how to apply this
algorithm for searching any Markov chain $P$. In this article, we aim to
reconcile the apparent differences between the running times of spatial search
algorithms in these two frameworks. We first present a modified version of the
Childs and Goldstone algorithm which can search for a marked element for any
ergodic, reversible $P$ by performing a quantum walk on its edges. Although
this approach improves the algorithmic running time for several instances, it
cannot provide a generic quadratic speedup for any $P$. Secondly, using the
framework of interpolated Markov chains, we provide a new spatial search
algorithm by continuous-time quantum walk which can find a marked node on any
$P$ in the square root of the classical hitting time. In the scenario where
multiple nodes are marked, the algorithmic running time scales as the square
root of a quantity known as the extended hitting time. Our results establish a
novel connection between discrete-time and continuous-time quantum walks and
can be used to develop a number of Markov chain-based quantum algorithms.
