Efficient quantum walk on the grid with multiple marked elements

TL;DR

This paper introduces a quantum algorithm that efficiently finds marked elements on a grid with multiple targets, outperforming classical random walks by reducing the number of steps needed.

Contribution

It presents the first quantum walk algorithm that finds marked elements faster than the square root of the extended hitting time, with a new bound on this quantity.

Findings

Quadratically fewer steps than classical random walk

First quantum walk to beat the square-root of extended hitting time

New upper bound on extended hitting time for marked subsets

Abstract

We give a quantum algorithm for finding a marked element on the grid when there are multiple marked elements. Our algorithm uses quadratically fewer steps than a random walk on the grid, ignoring logarithmic factors. This is the first known quantum walk that finds a marked element in a number of steps less than the square-root of the extended hitting time. We also give a new tighter upper bound on the extended hitting time of a marked subset, expressed in terms of the hitting times of its members.