A quantum walk algorithm can detect the presence of a marked vertex on a graph quadratically faster than the corresponding random walk algorithm (Szegedy, FOCS 2004). However, quantum algorithms that actually find a marked element quadratically faster than a classical random walk were only known for the special case when the marked set consists of just a single vertex, or in the case of some specific graphs. We present a new quantum algorithm for finding a marked vertex in any graph, with any set of marked vertices, that is (up to a log factor) quadratically faster than the corresponding classical random walk, resolving a question that had been open for 15 years.

Markov chains, Quantum algorithms, Quantum search, Quantum walks, Search by random walk
dx.doi.org/10.1145/3357713.3384252
Annual ACM SIGACT Symposium on Theory of Computing
Centrum Wiskunde & Informatica, Amsterdam, The Netherlands

Ambainis, A, Gilyén, A.P, Jeffery, S, & Kokainis, M. (2020). Quadratic speedup for finding marked vertices by quantum walks. In Proceedings of the Annual ACM SIGACT Symposium on Theory of Computing (pp. 412–424). doi:10.1145/3357713.3384252