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