CRC Press eBooks

Quantum Computing and Quantum Walks

Fei Yan, Liang Wen, Fangyan Dong

Published Nov 24, 2025• 0 citations

Abstract

This chapter primarily introduces the theoretical foundations of quantum computing, provides an overview of notable quantum algorithms, and discusses their connections with classical (non-quantum) algorithms. Additionally, it explores the applications of low-dimensional quantum walks in fields such as information security and spatial search.

DOI: 10.1201/9781003683902-2

Related papers

INTRODUCTION TO QUANTUM COMPUTING

Faisal Khan, Atharva Gambhir, Om Jadhav +2 more

Quantum computing represents a fundamentally new paradigm in information processing, in which information is encoded and manipulated using two-level quantum systems known as qubits, whose evolution is governed by the principles of quantum mechanics. This chapter presents a structured introduction to the theoretical foundations of quantum computation, beginning with the limitations of classical computation and the physical constraints that motivate the need for a quantum model of computation. We develop the mathematical framework required to describe quantum information, introducing qubits as state vectors in Hilbert space and examining the roles of superposition, measurement, and unitary evolution. Single-qubit operations are presented through the Pauli and Hadamard gates, along with their geometric interpretation on the Bloch sphere. Building on this formalism, the discussion is extended to multi-qubit systems using tensor products, and quantum entanglement is introduced as a non-classical correlation fundamental to quantum information processing. The chapter then explores quantum algorithms, focusing on Grover’s unstructured search algorithm, to demonstrate how superposition, interference, and entanglement can be harnessed to achieve computational advantages over classical approaches. Finally, the chapter discusses applications of quantum computation, including quantum simulation, cryptography, search and optimization problems, and near-term hybrid quantum algorithms such as the Variational Quantum Eigensolver (VQE)and the Quantum Approximate Optimization (QAO) Algorithm, highlighting their relevance in the noisy intermediate-scale quantum (NISQ) era.

Iterative International Publishers eBooksJun 1, 20260 citations

Foundational Theory of Quantum Walks

Fei Yan, Liang Wen, Fangyan Dong

This chapter presents the quantum walks on regular graphs (including low-dimensional discrete-time quantum walks, one-dimensional continuous-time quantum walks, and related variants) as well as quantum walks on complex networks. It provides a general framework for describing and defining quantum walk algorithms on complex networks.

CRC Press eBooksNov 24, 20250 citations

An Overview of Quantum Computing for Technology Managers

Eleanor Gilbert Rieffel

Faster algorithms, novel cryptographic mechanisms, and alternative methods of communication become possible when the model underlying information and computation changes from a classical mechanical model to a quantum mechanical one. Quantum algorithms perform a select set of tasks vastly more efficiently than any classical algorithm, but for many tasks it has been proved that quantum algorithms provide no advantage. The breadth of quantum computing applications is still being explored. Major application areas include security and the many fields that would benefit from efficient quantum simulation. The quantum information processing viewpoint provides insight into classical algorithmic issues as well as a deeper understanding of entanglement and other non-classical aspects of quantum physics. This overview is aimed at technology managers who wish to gain a high level understanding of quantum information processing, particularly quantum computing.

arXiv:0804.2264Apr 14, 20083 citations

Quantum walks: a comprehensive review

Salvador Elías Venegas-Andraca

Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entangle-ment as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete-time quantum walks.

Jan 1, 2012436 citations

A Review on Models and Applications of Quantum Computing

Eduard Grigoryan, Sachin Kumar, Plácido Rogério Pinheiro

This manuscript is intended for readers who have a general interest in the subject of quantum computation and provides an overview of the most significant developments in the field. It begins by introducing foundational concepts from quantum mechanics—such as superposition, entanglement, and the no-cloning theorem—that underpin quantum computation. The primary computational models are discussed, including gate-based (circuit) quantum computing, adiabatic quantum computing, measurement-based quantum computing and the quantum Turing machine. A selection of significant quantum algorithms are reviewed, notably Grover’s search algorithm, Shor’s factoring algorithm, and Quantum Singular Value Transformation (QSVT), which enables efficient solutions to linear algebra problems on quantum devices. To assess practical performance, we compare quantum and classical implementations of support vector machines (SVMs) using several synthetic datasets. These experiments offer insight into the capabilities and limitations of near-term quantum classifiers relative to classical counterparts. Finally, we review leading quantum programming platforms—including Qiskit, PennyLane, and Cirq—and discuss their roles in bridging theoretical models with real-world quantum hardware. The paper aims to provide a concise yet comprehensive guide for those looking to understand both the theoretical foundations and applied aspects of quantum computing.

Quantum ReportsSep 4, 20255 citations

Quantum Walk-Based Algorithms for Large-Scale Graph Search

Jincheng Zhang

This paper investigates the application of quantum walk algorithms to large-scale graph search problems. Classical algorithms often struggle with the computational complexity of exploring vast graphs, particularly when searching for specific patterns or nodes. Quantum walks, inspired by the probabilistic nature of random walks, provide a promising alternative by utilizing quantum superposition and interference to accelerate the search process. We explore the theoretical framework and potential advantages of employing quantum walks for graph traversal, focusing on scenarios where exponential speedups are theoretically possible. This work aims to demonstrate the potential of quantum computing to tackle computationally demanding graph search tasks that are currently intractable for classical methods. The core concept leverages the ability to simultaneously explore multiple paths within a graph, drastically reducing the search time compared to sequential exploration. We present a foundational understanding of quantum walk implementation and analyze its suitability for various graph search algorithms.

Aug 28, 20260 citations