Spatial search for a general multi-vertex state on graph by continuous-time quantum walks

TL;DR

This paper investigates the use of continuous-time quantum walks for spatial search of general multi-vertex states on graphs, deriving conditions for optimality based on Laplacian spectra and applying them to specific graph classes.

Contribution

It introduces a spectral condition for optimal quantum spatial search on graphs and applies it to various graph types, extending understanding beyond uniform states.

Findings

Optimal search for two-vertex uniform states on hypercubes.

A Laplacian spectrum condition determines search optimality.

Certain graphs can be optimized for search with parameter adjustments.

Abstract

In this work, we consider the spatial search for a general marked state on graphs by continuous time quantum walks. As a simplest case, we compute the amplitude expression of the search for the multi-vertex uniform superposition state on hypercube, and find that the spatial search algorithm is optimal for the two-vertex uniform state. However, on general graphs, a common formula can't be obtained for searching a general non-uniform superposition state. Fortunately, a Laplacian spectrum condition which determines whether the associated graph could be appropriate for performing the optimal spatial search is presented. The condition implies that if the proportion of the maximum and the non-zero minimum Laplacian eigenvalues is less or equal to 1+sqrt(1/2), then the spatial search is optimal for any general state. At last, we apply this condition to three kind graphs, the induced complete graph, the strongly regular graph and the regular complete multi-partite graphs. By the condition, one can conclude that these graphs will become suitable for optimal search with properly setting their graph parameters.