Quantum Walks

TL;DR

This tutorial explores various types of quantum walks, highlighting their differences from classical walks, discussing algorithmic applications, and demonstrating potential exponential speedups in quantum computing tasks.

Contribution

It provides a comprehensive overview of quantum walks, their connections to classical processes, and their applications in algorithms and quantum speedups, which is a valuable resource for understanding quantum algorithms.

Findings

Quantum walks differ fundamentally from classical random walks.

Quantum walks can lead to exponential speedups in certain algorithms.

Quantization of Markov chains enhances sampling efficiency.

Abstract

This tutorial article showcases the many varieties and uses of quantum walks. Discrete time quantum walks are introduced as counterparts of classical random walks. The emphasis is on the connections and differences between the two types of processes (with rather different underlying dynamics) for producing random distributions. We discuss algorithmic applications for graph-searching and compare the two approaches. Next, we look at quantization of Markov chains and show how it can lead to speedups for sampling schemes. Finally, we turn to continuous time quantum walks and their applications, which provide interesting (even exponential) speedups over classical approaches.