We propose a new method for designing quantum search algorithms for finding a "marked" element in the state space of a classical Markov chain. The algorithm is based on a quantum walkà la Szegedy [25] that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis [6] and Szegedy [25]. Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chain. In addition, it is conceptually simple, avoids several technical difficulties in the previous analyses, and leads to improvements in various aspects of several algorithms based on quantum walk.

We propose a new method for designing quantum search algorithms for finding a "marked" element in the state space of a classical Markov chain. The algorithm is based on a quantum walk \'a la Szegedy (2004) that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis (2004) and Szegedy (2004). Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chains. In addition, it is conceptually simple and avoids some technical difficulties in the previous analyses of several algorithms based on quantum walk.Comment: 21 pages. Various modifications and improvements, especially in Section

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 p M (the probability of picking a marked vertex from the stationary distribution) and HT + (P , M ) are known.