Spatial Search by Continuous-Time Quantum Walk with Multiple Marked Vertices

TL;DR

This paper explores how the position of marked vertices affects the efficiency of continuous-time quantum walk-based spatial search, revealing a sensitivity to parameters that can cause search failure, unlike discrete-time methods.

Contribution

It analytically demonstrates the impact of marked vertex configurations on search success and highlights a unique sensitivity in continuous-time quantum walks.

Findings

Marked vertex location influences the required jumping rate.

Incorrect rate choice can cause search failure.

Sensitivity is unique to continuous-time quantum walks.

Abstract

In the typical spatial search problems solved by continuous-time quantum walk, changing the location of the marked vertices does not alter the search problem. In this paper, we consider search when this is no longer true. In particular, we analytically solve search on the "simplex of $K_M$ complete graphs" with all configurations of two marked vertices, two configurations of $M+1$ marked vertices, and two configurations of $2(M+1)$ marked vertices, showing that the location of the marked vertices can dramatically influence the required jumping rate of the quantum walk, such that using the wrong configuration's value can cause the search to fail. This sensitivity to the jumping rate is an issue unique to continuous-time quantum walks that does not affect discrete-time ones.