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Cambridge University Press eBooks

Quantum Walk Algorithms

Robert Hundt

Published Nov 20, 2025• 0 citations
Publisher

Abstract

A quantum walk algorithm is the quantum analog to a classical random walk with potential applications in search problems, graph problems, quantum simulation, and even machine learning. In this section, we describe the basic principles of this class of algorithms on a simple one-dimensional topology.
DOI
10.1017/9781009548519.013

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Using numerical simulation, we measured the performance of several potential quantum algorithms, based on quantum random walks, to solve Boolean satisfiability (SAT) problems. We develop the fundamentals of quantum computing and the theory of classical algorithms to indicate how these algorithms could be implemented. We also discuss the development of quantum random walks and the principles that go into creating them, presenting existing work on search algorithms using coined discrete-time quantum random walks. Although our quantum algorithms do not solve SAT problems faster than may be done classically, as a step toward that objective, we present the first example of a quantum walk on a directed graph that has a significant increase in speed over the analogous classical walk.
Abstract In this section, we introduce the basics of quantum walk algorithm and its applications. Quantum walk is a natural extension of the concept of random walk in quantum way; therefore, the results obtained from the discussion are considered as the results by Quantum Computation. In principle, we can expect a certain type of the computation would be boosted. There are a variety of phenomena in Quantum walks, and much broader outcomes are often obtained than those from classical random walk. Such famous examples include quantum search algorithms and quantum simulations. In this article, we introduce a quantum simulation of QCD parton shower algorithm appearing in particle physics.
Discrete quantum walks are quantum analogues of classical random walks. They are an important tool in quantum computing and a number of algorithms can be viewed as discrete quantum walks, in particular Grover's search algorithm. These walks are constructed on an underlying graph, and so there is a relation between properties of walks and properties of the graph. This book studies the mathematical problems that arise from this connection, and the different classes of walks that arise. Written at a level suitable for graduate students in mathematics, the only prerequisites are linear algebra and basic graph theory; no prior knowledge of physics is required. The text serves as an introduction to this important and rapidly developing area for mathematicians and as a detailed reference for computer scientists and physicists working on quantum information theory.
This paper explores the potential of quantum-inspired algorithms for graph search problems. Traditional random walk approaches often suffer from inefficiencies, particularly in complex graphs. We propose a novel framework that leverages concepts from quantum walks – specifically interference and tunneling – within a classical computational environment. Our algorithms dynamically adjust the probability distribution of node visits based on graph connectivity and the history of previous steps, mimicking the probabilistic nature of quantum mechanics. We demonstrate that this approach can, in certain scenarios, outperform standard random walks, offering a more adaptable and potentially faster search strategy. The core of the method lies in approximating the quantum effects through carefully designed classical simulations, without requiring actual quantum hardware. We present a mathematical formulation of the core algorithm and discuss its key parameters, focusing on the balance between classical simulation fidelity and computational cost. The results suggest a promising direction for graph search optimization, drawing inspiration from the principles of quantum mechanics.
This paper investigates the application of quantum walk algorithms to graph traversal problems. Traditional graph traversal algorithms often suffer from performance limitations when dealing with large and complex graphs due to their inherent sequential nature. We propose utilizing quantum walks, quantum analogs of random walks, as a potential solution. Quantum walks leverage quantum superposition and interference to enable more efficient exploration of the graph. We present a theoretical framework for designing quantum walk-based algorithms for graph traversal, focusing on key aspects such as Hamiltonian construction, walker evolution, and measurement strategies. Our work demonstrates the potential for exponential speedups for specific graph algorithms compared to their classical counterparts. We analyze the core principles underpinning this approach and highlight the advantages of quantum walk based algorithms for graph traversal.
This paper explores the application of quantum walk algorithms to graph traversal and search problems. Classical algorithms often struggle with the computational complexity of traversing and searching large graphs, leading to significant time and resource limitations. Quantum walks, utilizing superposition and interference, offer a potential pathway to achieve exponential speedups. This work details the theoretical framework for applying quantum walks to graph problems, specifically focusing on traversal and search tasks. We present a generalized approach for mapping graph structures onto quantum walk systems and discuss the key mathematical formulations involved. The core of our investigation revolves around the construction of quantum walk operators that mimic the movement and interaction rules inherent in graph traversal. We analyze the evolution of the quantum walk state and derive expressions for the probability of reaching a specific node in the graph. Furthermore, we examine the impact of graph topology on the performance of the quantum walk, highlighting the potential for tailored algorithms to exploit specific graph structures. The results demonstrate the theoretical feasibility of quantum walk algorithms for significantly accelerating graph traversal and search operations, contributing to advancements in areas such as network analysis, database management, and artificial intelligence.