# Quantum walks: a comprehensive review

**Authors:** Salvador E. Venegas-Andraca  
**arXiv:** [1201.4780](https://arxiv.org/abs/1201.4780)  
**Date:** 2012-01-23  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/1201.4780v1/

## Key points

- Quantum walks are a powerful tool for developing advanced quantum algorithms
- Quantum walks serve as a universal model of quantum computation, opening up new avenues for research in physics, computer science, mathematics, and engineering
- The review delves into the theoretical foundations of both discrete- and continuous-time quantum walks
- It explores the interplay between randomness and quantum walks, shaping their behavior and outcomes
- The review examines connections between mathematical models of coined discrete quantum walks and continuous quantum walks
- "Quantumness" in quantum walks is discussed to highlight unique features distinguishing them from classical random walks
- Recent advancements in discrete quantum walks and entanglement are showcased through published papers
- Experimental proposals and realizations of discrete-time quantum walks are summarized to show practical applications
- Algorithms based on both discrete- and continuous-time quantum walks are examined for solving complex computational problems
- Both continuous- and discrete-time quantum walks exhibit computational universality, highlighting their potential for designing efficient quantum algorithms

## Summary

In the field of quantum computation, quantum walks have emerged as a powerful tool for developing advanced quantum algorithms. Recently, it has been demonstrated that quantum walks serve as a universal model of quantum computation, opening up new avenues for research and exploration in the realm of physics, computer science, mathematics, and engineering. This comprehensive review delves into the theoretical foundations of both discrete- and continuous-time quantum walks. It explores the intricate interplay between randomness and quantum walks, shedding light on how these elements shape the behavior and outcomes of quantum walk processes. The review also examines the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, providing insights into their similarities and differences. One key aspect discussed in this review is the concept of "quantumness" in quantum walks – highlighting the unique features that distinguish them from classical random walks. The review also offers a summary of papers published on discrete quantum walks and entanglement, showcasing recent advancements in these areas. Furthermore, the review provides a succinct overview of experimental proposals and realizations of discrete-time quantum walks, offering a glimpse into how these theoretical concepts are being translated into practical applications. Several algorithms based on both discrete- and continuous-time quantum walks are also examined in detail, demonstrating their potential for solving complex computational problems. A significant highlight of this review is the revelation of the computational universality of both continuous- and discrete-time quantum walks. This groundbreaking result underscores the immense potential of quantum walks as a fundamental building block for designing efficient and powerful quantum algorithms. Overall, this comprehensive review serves as a valuable resource for researchers and practitioners interested in exploring the diverse facets of quantum walks and their implications for advancing the field of quantum computation. By synthesizing theoretical advances, experimental findings, and algorithmic developments, this review offers a holistic perspective on the current state-of-the-art in this exciting area of research.