# Quadratic speedup for finding marked vertices by quantum walks

## TL;DR

This paper introduces a quantum algorithm that finds marked vertices in any graph significantly faster than classical methods, extending quadratic speedup beyond special cases.

## Contribution

It presents a general quantum algorithm for locating marked vertices in any graph, achieving near-quadratic speedup over classical random walks.

## Key findings

- Quantum algorithm detects marked vertices faster than classical methods.
- Achieves near-quadratic speedup for general graphs.
- Extends previous results limited to single marked vertices.

## Abstract

A quantum walk algorithm can detect the presence of a marked vertex on a graph quadratically faster than the corresponding random walk algorithm (Szegedy, FOCS 2004). However, quantum algorithms that actually find a marked element quadratically faster than a classical random walk were only known for the special case when the marked set consists of just a single vertex, or in the case of some specific graphs. We present a new quantum algorithm for finding a marked vertex in any graph, with any set of marked vertices, that is (up to a log factor) quadratically faster than the corresponding classical random walk.