Continuous time quantum walks on graphs: Group state transfer

We introduce the concept of group state transfer on graphs, summarize its relationship to other concepts in the theory of quantum walks, set up a basic theory, and discuss examples. Let X be a graph with adjacency matrix A and consider quantum walks on the vertex set V(X) governed by the continuous time-dependent unitary transition operator U(t) = exp(itA). For S, T subset of V(X), we say X admits "group state transfer"from S to T at time tau if the submatrix of U(tau) obtained by restricting to columns in S and rows not in T is the all-zero matrix. As a generalization of perfect state transfer, fractional revival and periodicity, group state transfer satisfies natural monotonicity and transitivity properties. Yet non-trivial group state transfer is still rare; using a compactness argument, we prove that bijective group state transfer (the optimal case where |S| = |T|) is absent for almost all tau. Focusing on this bijective case, we obtain a structure theorem, prove that bijective group state transfer is "monogamous", and study the relationship between the projections of S and T into each eigenspace of the graph. Group state transfer is obviously preserved by graph automorphisms and this gives us information about the relationship between the setwise stabilizer of S subset of V(X) and the stabilizers of certain vertex subsets F(S, t) and I(S, t). The operation S -> F(S, t) is sufficiently well-behaved to give us a topology on V(X); this is simply the topology of subsets for which bijective group state transfer occurs at time t. We illustrate non-trivial group state transfer in bipartite graphs with integer eigenvalues, in joins of graphs, and in symmetric double stars. The Cartesian product allows us to build new examples from old ones. (c) 2023 Elsevier B.V. All rights reserved.