Quantum searches on highly symmetric graphs

Reitzner, Daniel and Hillery, Mark and Feldman, Edgar and Buzek, Vladimir (2009) Quantum searches on highly symmetric graphs. Physical Review A, 79 (1). 012323.1-012323.10. ISSN 1050-2947

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We study scattering quantum walks on highly symmetric graphs and use the walks to solve search problems on these graphs. The particle making the walk resides on the edges of the graph, and at each time step scatters at the vertices. All of the vertices have the same scattering properties except for a subset of special vertices. The object of the search is to find a special vertex. A quantum circuit implementation of these walks is presented in which the set of special vertices is specified by a quantum oracle. We consider the complete graph, a complete bipartite graph, and an M-partite graph. In all cases, the dimension of the Hilbert space in which the time evolution of the walk takes place is small (between three and six), so the walks can be completely analyzed analytically. Such dimensional reduction is due to the fact that these graphs have large automorphism groups. We find the usual quadratic quantum speedups in all cases considered.

Item Type: Article
Keywords: Quantum searches; highly symmetric graphs;
Academic Unit: Faculty of Science and Engineering > Experimental Physics
Faculty of Science and Engineering > Mathematics and Statistics
Item ID: 2178
Depositing User: Prof. Vladimir Buzek
Date Deposited: 12 Oct 2010 13:47
Journal or Publication Title: Physical Review A
Publisher: American Physical Society
Refereed: Yes

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