Optimized quantum random-walk search algorithms

Shenvi, Kempe and Whaley’s quantum random-walk search (SKW) algorithm [2003 Phys. Rev. A 67 052307] is known to require O( √ N) number of oracle queries to find the marked element, where N is the size of the search space. This scaling is thought to be the best achievable on a quantum computer. We prove that the final state of the quantum walk in the SKW algorithm yields the nearest neighbours of the marked element with a similarly high probability as it yields the marked element itself, i.e. probability close to 1/2. Therefore, the total probability of finding the marked element or any of its nearest neighbours is asymptotically 1. We point out that the concentration of the position probability density around the marked element can be exploited for optimizations. Without changing the fundamental scaling property regarding the number of oracle queries, optimization in a wider perspective is possible. Specifically, we take into account the cost of initializing the system, and the cost of verification of the measurement outcome. We present a number of proposals to modify the original SKW algorithm, each representing minimalization of the total cost for physically different realizations. We discuss which optimizations can be used if there is more than one marked element to find. PACS numbers: 03.67.Ac, 03.67.Lx, 42.50.Ex, 05.40.Fb Optimized quantum random-walk search algorithms 2