Of shadows and gaps in spatial search

Spatial search occurs in a connected graph if continuous-time quantum walk on the adjacency matrix of graph, suitably scaled, plus rank-one perturbation induced by any vertex will unitarily map principal eigenvector to characteristic vector vertex. This phenomenon is natural analogue Grover search. The spatial said be optimal it with constant fidelity and time inversely proportional shadow target eigenvector. Extending result Chakraborty \etal ({\em Physical Review A}, {\bf 102}:032214, 2020), we prove simpler characterization Based this characterization, observe that some families distance-regular graphs, such as Hamming Grassmann have We also show matching lower bound for fidelity, which extends due Farhi Gutmann perfect fidelity. Our elementary proofs employ standard tools, Weyl inequalities Cauchy determinant formula.