2006/08/31 by Frédéric Magniez, Ashwin Nayak, Jérémie Roland +1 · 3 citations
Physics and Astronomy · Computer Science · #quant-ph #cs.CC #cs.DS
paper · pdf · doi:10.1137/090745854
published as SIAM Journal on Computing, 40(1):142-164, 2011 · 21 pages. Various modifications and improvements, especially in Section 4
arxiv created 2011/02/14 · arxiv updated 2018/03/22
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 á 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.