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If A is the adjacency matrix of a graph X, then the matrix
is the transition matrix of a so-called continuous quantum walk on X.
In this context we say that we have perfect state transfer from a vertex u
to a vertex v at time τ if the uv-entry of H(τ) has
absolute value 1. One fundamental problem is to characterize the pairs of vertices
where perfect state transfer occurs. In all cases where it does, there is an
automorphism of order two of X that swaps u and v, but it is
not clear whether this condition is necessary.
My talk will provide an introduction to this problem, and to some recent work in
the area.
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