A continuous-time quantum walk on a graph G is given by the unitary matrix U (t) = exp(−itA), where A is the Hermitian adjacency matrix of G. We say G has pretty good state transfer between vertices a and b if for any ǫ > 0, there is a time t, where the (a, b)-entry of U (t) satisfies |U (t) a,b | ≥ 1 − ǫ. This notion was introduced by . The state transfer is perfect if the above holds for ǫ = 0. In this work, we study a natural extension of this notion called universal state transfer. Here, state transfer exists between every pair of vertices of the graph. We prove the following results about graphs with this stronger property:• Graphs with universal state transfer have distinct eigenvalues and flat eigenbasis (where each eigenvector has entries which are equal in magnitude).• The switching automorphism group of a graph with universal state transfer is abelian and its order divides the size of the graph. Moreover, if the state transfer is perfect, then the switching automorphism group is cyclic.• There is a family of prime-length cycles with complex weights which has universal pretty good state transfer. This provides a concrete example of an infinite family of graphs with the universal property.• There exists a class of graphs with real symmetric adjacency matrices which has universal pretty good state transfer. In contrast, Kay (2011) proved that no graph with real-valued adjacency matrix can have universal perfect state transfer.We also provide a spectral characterization of universal perfect state transfer graphs that are switching equivalent to circulants.
Atomistic molecular dynamics (MD) and a microstructural dislocation density-based crystalline plasticity (DCP) framework were used together across time scales varying from picoseconds to nanoseconds and length scales spanning from angstroms to micrometers to model a buried copper–nickel interface subjected to high strain rates. The nucleation and evolution of defects, such as dislocations and stacking faults, as well as large inelastic strain accumulations and wave-induced stress reflections were physically represented in both approaches. Both methods showed similar qualitative behavior, such as defects originating along the impactor edges, a dominance of Shockley partial dislocations, and non-continuous dislocation distributions across the buried interface. The favorable comparison between methods justifies assumptions used in both, to model phenomena, such as the nucleation and interactions of single defects and partials with reflected tensile waves, based on MD predictions, which are consistent with the evolution of perfect and partial dislocation densities as predicted by DCP. This substantiates how the nanoscale as modeled by MD is representative of microstructural behavior as modeled by DCP.
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