par Krovi, Hari;Magniez, Frédéric;Ozols, Maris;Roland, Jérémie 
Référence Algorithmica, 74, 2, page (851-907)
Publication Publié, 2015
Référence Algorithmica, 74, 2, page (851-907)
Publication Publié, 2015
Article révisé par les pairs
| Résumé : | 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 $HT^+(backslash mathitP,M)$ 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 state-transitive. We also provide algorithms in the cases when only approximations or bounds on parameters $p_M$ (the probability of picking a marked vertex from the stationary distribution) and $HT^+(backslash mathitP,M)$ are known. |
