Abstract
We investigate a periodicity of Grover walk on complete graphs with a self-loop at each vertex. We study an evolution matrix which steps forward a state of probability amplitude vector by using algebraic method. Then we find a periodic behaviour that the probability amplitude vector at each vertex gets back to initial state after some steps. It is shown that Grover walk on complete graphs on n vertices with a self-loop at each vertex is periodic with period 2n.
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