# Optimal Quantum Walk Search on Kronecker Graphs with Dominant or Fixed Regular Initiators. (arXiv:1809.01249v2 [quant-ph] UPDATED)

In network science, graphs obtained by taking the Kronecker or tensor power

of the adjacency matrix of an initiator graph are used to construct complex

networks. In this paper, we analytically prove sufficient conditions under

which such Kronecker graphs can be searched by a continuous-time quantum walk

in optimal $\Theta(\sqrt{N})$ time. First, if the initiator is regular and its

adjacency matrix has a dominant principal eigenvalue, meaning its unique

largest eigenvalue asymptotically dominates the other eigenvalues in magnitude,

then the Kronecker graphs generated by this initiator can be quantum searched

with probability 1 in $\pi\sqrt{N}/2$ time, asymptotically, and we give the

critical jumping rate of the walk that enables this. Second, for any fixed

initiator that is regular, non-bipartite, and connected, the Kronecker graphs

generated by it are quantum searched in $\Theta(\sqrt{N})$ time. This greatly

extends the number of Kronecker graphs on which quantum walks are known to

optimally search. If the fixed, regular, connected initiator is bipartite,

however, then search on its Kronecker powers is not optimal, but is still

better than classical computer's $O(N)$ runtime if the initiator has more than

two vertices.