Finding Is as Easy as Detecting for Quantum Walks

We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. In the case when the marked set M consists of a single vertex, the number of steps of the quantum walk is quadratically smaller than the classical hitting time HT(P,M) of any reversible random walk P on the graph. In the case of multiple marked elements, the number of steps is given in terms of a related quantity HT(P ,M ) which we call extended hitting time. Our approach is new, simpler and more general than previous ones. We introduce a notion of interpolation between the random walk P and the absorbing walk P ′, whose marked states are absorbing. Then our quantum walk is simply the quantum analogue of this interpolation. Contrary to previous approaches, our results remain valid when the random walk P is not statetransitive. We also provide algorithms in the cases when only approximations or bounds on parameters pM (the probability of picking a marked vertex from the stationary distribution) and HT(P ,M ) are known.