Discrete Quantum Walks on Graphs and Digraphs

Discrete quantum walks are quantum analogues of classical random walks, and are an important tool in quantum computing. This book studies the mathematical problems that arise from this connection, and the different classes of walks that arise. It is written at a level suitable for graduate students in mathematics, with no prior knowledge of physics required.

Format: Paperback / softback
Length: 300 pages
Publication date: 12 January 2023
Publisher: Cambridge University Press

Discrete quantum walks, remarkable quantum counterparts of classical random walks, hold immense significance in the realm of quantum computing. These intricate processes, characterized by their discrete steps, have found applications in various algorithms, notably Grover's search algorithm. These walks are built upon an underlying graph, establishing a profound connection between the properties of the walks and the characteristics of the graph. This captivating book delves into the mathematical challenges that arise from this interplay, exploring the diverse classes of walks that emerge. Written with a focus on graduate students in mathematics, the text assumes a foundational understanding of linear algebra and basic graph theory. However, it goes beyond mere prerequisites, offering a comprehensive introduction to this rapidly evolving field for mathematicians and serving as a valuable resource for computer scientists and physicists engaged in quantum information theory.

The study of discrete quantum walks revolves around the exploration of these intricate processes on graphs. Graphs, mathematical structures composed of vertices and edges, provide a framework for representing and analyzing complex systems. In the context of quantum walks, the graph serves as the underlying structure upon which the walks are constructed. The properties of the graph, such as its connectivity, structure, and edge weights, significantly influence the behavior and characteristics of the quantum walks.

One of the key insights of discrete quantum walks is the concept of quantum tunneling. Unlike classical random walks, which move along straight lines or curves, quantum walks can exhibit sudden jumps or "tunneling" through barriers. This phenomenon is a result of the wave-like nature of quantum particles and the interference effects that occur when they interact with the graph. Quantum tunneling allows walks to explore a broader range of states and vertices, leading to efficient search algorithms and optimization problems.

Another important aspect of discrete quantum walks is their relation to quantum entanglement. Quantum entanglement is a fundamental property of quantum systems, where two or more particles become intertwined in such a way that their states cannot be described independently. Discrete quantum walks can exhibit entanglement, which can enhance the performance of certain algorithms and enable the manipulation of complex quantum systems.

Discrete quantum walks have found applications in various fields, including quantum information theory, quantum computing, and statistical physics. In quantum information theory, quantum walks are used to simulate the behavior of quantum systems and to perform quantum communication protocols. They have also been used to study the properties of quantum chaos and to develop algorithms for quantum error correction.

In quantum computing, discrete quantum walks have been used to simulate the behavior of quantum computers and to solve complex problems that are intractable for classical computers. For example, Grover's search algorithm is a discrete quantum walk that is used to search for a specific element in a unsorted list efficiently.

In statistical physics, discrete quantum walks have been used to model the dynamics of complex systems, such as the behavior of particles in a magnetic field or the evolution of a population of organisms. They have also been used to study the properties of random walks and to develop new models for complex systems.

Despite the significant advancements in the study of discrete quantum walks, there are still many open questions and challenges that need to be addressed. One of the key challenges is the development of efficient algorithms for simulating and analyzing large-scale quantum walks on complex graphs. Another challenge is the understanding of the quantum dynamics of discrete quantum walks in the presence of noise and decoherence.

In conclusion, discrete quantum walks are a fascinating and rapidly developing area of research in mathematics and physics. These quantum analogues of classical random walks have the potential to revolutionize our understanding of complex systems and enable the development of new technologies. The study of discrete quantum walks has opened up new avenues for research in quantum information theory, quantum computing, and statistical physics, and has the potential to have a profound impact on our understanding of the universe. As we continue to explore the mysteries of quantum mechanics, discrete quantum walks will undoubtedly play an increasingly important role in our quest for knowledge.

Weight: 232g
Dimension: 152 x 229 x 13 (mm)
ISBN-13: 9781009261685