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On Quantum Walks on Finite Graphs(2) : Some Examples and its Properties

章 根岸, Akira Negishi

Published Dec 17, 2013• 0 citations
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Graphs that exhibit perfect state transfer (PST), by which a quantum walk beginning at one vertex is guaranteed to end at another after a nonzero displacement in time, are of great interest to researchers in the field of quantum computing. The conditions imposed by PST are strong, and many of the graphs that are known to exhibit PST are highly symmetric, which provides motivation to relax the restrictions imposed by perfect state transfer. This project introduces the property of group state transfer (GST), by which quantum walks are guaranteed to transfer states between nontrivial vertex sets. We define GST, examine its implications for finite simple graphs, and provide examples of graphs that exhibit GST.
Quantum walks determined by the coin operator on graphs have been intensively studied. The typical examples of coin operator are the Grover and Fourier matrices. The periodicity of the Grover walk is well investigated. However, the corresponding result on the Fourier walk is not known. In this paper, we present a necessary condition for the Fourier walk on regular graphs to have the finite period. As an application of our result, we show that the Fourier walks do not have any finite period for some classes of regular graphs such as complete graphs, cycle graphs with selfloops, and hypercubes.
Several inequalities are proved for the mixing time of discrete-time quantum walks on finite graphs. The mixing time is defined differently than in Aharonov et al (2002 Proc. 33rd STOC ( 2001 ) (New York: ACM) pp 50–9 arXiv:quant-ph/0012090v2) and it is found that for particular examples of walks on a cycle, a hypercube and a complete graph, quantum walks provide no speedup in mixing over the classical counterparts. In addition, non-unitary quantum walks (i.e. walks with decoherence) are considered and a criterion for their convergence to the unique stationary distribution is derived.