Search via quantum walk
We propose a new method for designing quantum search algorithms for finding a âmarked â element in the state space of a classical Markov chain. The algorithm is based on a quantum walk Ã la Szegedy  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  and Szegedy . 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.