Of Shadows and Gaps in Spatial Search

TL;DR

This paper characterizes optimal spatial search in quantum walks on graphs, extending previous results, and identifies specific graph families where optimal search is achievable, providing bounds on search time.

Contribution

It offers a simpler characterization of optimal spatial search and demonstrates its applicability to certain distance-regular graphs, with new lower bounds on search time.

Findings

Hamming and Grassmann graphs have optimal spatial search.

A matching lower bound on search time for constant fidelity.

Elementary proofs using Weyl inequalities and Cauchy determinants.

Abstract

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