Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Chapman & Hall/CRC eBooks

Quantum algorithms 1

Helmut Bez, Tony Croft

Published Jan 6, 2023• 0 citations
Publisher

Abstract

The quantum algorithms covered in this chapter, i.e., the Deutsch, Deutsch-Jozsa and Bernstein-Vazirani algorithms, have been chosen for their relative simplicity and theoretical interest. They do not address problems of real practical value, their purpose being to demonstrate, using simple examples, the great potential power of quantum computation relative to the digital case. The most significant breakthrough in quantum computing is Shor's integer factorisation algorithm. Shor's algorithm is beyond the scope of this introductory text, but its significance lies in the fact that it demonstrates the potential to factor large integers efficiently. No efficient digital algorithms for factoring large integers exist, and many current encryption algorithms rely on this being the case. Shor's algorithm therefore threatens current encryption methods, and is responsible for much of the recent accelerated interest in quantum computation.

DOI
10.1201/9781003264569-22

Related papers

Quantum Computing Algorithms for Integer Factorization: A Comparative Analysis

Nadia Ahmed

The field of quantum computing has witnessed remarkable advancements in recent years, particularly in its potential applications to solve computationally hard problems such as integer factorization. a comparative analysis of quantum computing algorithms for integer factorization, focusing on their theoretical foundations, computational complexity, and practical implications. We review prominent algorithms such as Shor's algorithm, which leverages quantum parallelism and period finding to efficiently factor large composite integers, and compare them with classical factorization algorithms like the General Number Field Sieve (GNFS). Through a comprehensive examination of algorithmic strategies, runtime complexities, and quantum circuit implementations, we assess the relative strengths and limitations of quantum computing approaches for integer factorization. Furthermore, we discuss potential challenges and future research directions in harnessing the power of quantum computing to address cryptographic security and algorithmic complexity in the era of post-quantum cryptography.

Modern Dynamics: Mathematical Progressions.•May 25, 2024•8 citations

Quantum Advantage with Deutsch-Jozsa, Bernstein-Vazirani, and Simon’s Algorithms

Vladimir Silva

In this chapter, we study three algorithms of little practical use but important because they were the first to show that quantum computers can solve problems significantly faster than classical ones. Consider the time complexities O(n) for the algorithms: classical vs. quantum in Table 9-1 (where n is the size of the input).

Apress eBooks•Dec 18, 2023•0 citations

Fundamental Structure of Shor’s Quantum Algorithm for Factoring Integers

Turgut Hanoymak, Akram Chehrazi

One of the most well known mathematically hard problems in number theory is the integer factorization problem, roughly stated that decomposition of a composite number into its prime factors. In modern cryptography, RSA encryption algorithm whose security is based on integer factorization problem is highly practical, widespread and up to date no classical algorithm having polynomial running time for the factorization of large numbers is known. In 1994, Peter Shor proposed an efficient algorithm on quantum computer. In this paper, we  mention about the fundamentals of  Shor's quantum algorithm illustrating a concrete example.

Turkish Journal of Mathematics and Computer Science•Dec 31, 2019•0 citations

Quantum algorithms for query problems

Steven Herbert

Abstract In 1985, David Deutsch proposed the original quantum algorithm, a simple yet remarkable demonstration of the enhanced computational power offered by quantum mechanics. Deutsch’s algorithm provides a reduction in the number of queries to an unknown function required to discern some property of that function, relative to the best possible classical counterpart. This is known as an advantage in query complexity, and whilst the attention of the quantum computing community has, by and large, moved on to quantum algorithms that provide an advantage in computational rather than query complexity, it is still instructive to study some of the most important quantum algorithms for query problems. Such algorithms are historically important and, moreover, reveal the source of the computational power of quantum mechanics in a clear and easy to understand manner. This chapter studies four algorithms for query problems: Deutsch’s algorithm, the Deutsch–Jozsa algorithm, the Bernstein–Vazirani algorithm, and Simon’s algorithm.

•Dec 8, 2025•0 citations

An Overview of Regev’s Quantum Factoring Algorithm and Its Recent Developments

Rick Lan Chen

Factoring large integers has long been a computationally difficult problem in classical computing, forming the foundation of widely used encryption methods like RSA. In 1994, Shor’s quantum algorithm introduced a revolutionary method for factoring integers exponentially faster than classical algorithms, laying the groundwork for quantum cryptography. Despite numerous attempts to enhance Shor’s algorithm over the last three decades, significant breakthroughs remained elusive until Regev’s discovery in 2023. Regev introduced a novel, multi-dimensional version of the quantum factoring algorithm, achieving a substantial improvement by reducing the required number of quantum gates to O(n^2 lo g⁡n )(n^(3/2) log⁡n ), compared to Shor’s originalO(n^2 lo g⁡n )original O(n^2 log⁡n ) gate complexity. Although Regev’s approach offers a significant speedup, it comes with increased qubit requirements and relies on an unproven number theory assumption. This paper presents an overview of Regev’s factoring algorithm, including a review of Shor’s work for context, followed by an examination of key recent developments and follow-up research. These include efforts to reduce the qubit count, improve error resilience, generalizegeneralise Regev’s algorithm to related problems, and validate the number theory assumption. This review serves as an accessible entry point for researchers interested in the rapidly evolving field of quantum computing and factoring algorithms.

Applied and Computational Engineering•Nov 28, 2024•1 citation

Quantum Algorithms

A.Jeyamurugan, M. Anita, A. Revathi

Quantum algorithms have emerged as powerful computational tools that leverage the principles of quantum mechanics to address complex problems beyond the scope of classical computing. This chapter explores the foundational concepts, classifications, implementation challenges, and future prospects of quantum algorithms. Key breakthroughs such as Grover's and Shor's algorithms are explored alongside recent developments in quantum machine learning and optimization. The chapter also compares quantum and classical algorithmic complexity, highlighting areas of potential speedup. Practical limitations including hardware constraints, error correction, and resource estimation are critically analyzed. Finally, the chapter outlines emerging research directions and ethical considerations essential for responsible quantum computing development.

IGI Global eBooks•Oct 9, 2025•1 citation
The quantum algorithms covered in this chapter, i.e., the Deutsch, Deutsch-Jozsa and Bernstein-Vazirani algorithms, have been chosen for their relative simplicity and theoretical interest. They do not address problems of real practical value, their purpose being to demonstrate, using simple examples, the great potential power of quantum computation relative to the digital case. The most significant breakthrough in quantum computing is Shor's integer factorisation algorithm. Shor's algorithm is beyond the scope of this introductory text, but its significance lies in the fact that it demonstrates the potential to factor large integers efficiently. No efficient digital algorithms for factoring large integers exist, and many current encryption algorithms rely on this being the case. Shor's algorithm therefore threatens current encryption methods, and is responsible for much of the recent accelerated interest in quantum computation.
The field of quantum computing has witnessed remarkable advancements in recent years, particularly in its potential applications to solve computationally hard problems such as integer factorization. a comparative analysis of quantum computing algorithms for integer factorization, focusing on their theoretical foundations, computational complexity, and practical implications. We review prominent algorithms such as Shor's algorithm, which leverages quantum parallelism and period finding to efficiently factor large composite integers, and compare them with classical factorization algorithms like the General Number Field Sieve (GNFS). Through a comprehensive examination of algorithmic strategies, runtime complexities, and quantum circuit implementations, we assess the relative strengths and limitations of quantum computing approaches for integer factorization. Furthermore, we discuss potential challenges and future research directions in harnessing the power of quantum computing to address cryptographic security and algorithmic complexity in the era of post-quantum cryptography.
One of the most well known mathematically hard problems in number theory is the integer factorization problem, roughly stated that decomposition of a composite number into its prime factors. In modern cryptography, RSA encryption algorithm whose security is based on integer factorization problem is highly practical, widespread and up to date no classical algorithm having polynomial running time for the factorization of large numbers is known. In 1994, Peter Shor proposed an efficient algorithm on quantum computer. In this paper, we  mention about the fundamentals of  Shor's quantum algorithm illustrating a concrete example.
Abstract In 1985, David Deutsch proposed the original quantum algorithm, a simple yet remarkable demonstration of the enhanced computational power offered by quantum mechanics. Deutsch’s algorithm provides a reduction in the number of queries to an unknown function required to discern some property of that function, relative to the best possible classical counterpart. This is known as an advantage in query complexity, and whilst the attention of the quantum computing community has, by and large, moved on to quantum algorithms that provide an advantage in computational rather than query complexity, it is still instructive to study some of the most important quantum algorithms for query problems. Such algorithms are historically important and, moreover, reveal the source of the computational power of quantum mechanics in a clear and easy to understand manner. This chapter studies four algorithms for query problems: Deutsch’s algorithm, the Deutsch–Jozsa algorithm, the Bernstein–Vazirani algorithm, and Simon’s algorithm.
Factoring large integers has long been a computationally difficult problem in classical computing, forming the foundation of widely used encryption methods like RSA. In 1994, Shor’s quantum algorithm introduced a revolutionary method for factoring integers exponentially faster than classical algorithms, laying the groundwork for quantum cryptography. Despite numerous attempts to enhance Shor’s algorithm over the last three decades, significant breakthroughs remained elusive until Regev’s discovery in 2023. Regev introduced a novel, multi-dimensional version of the quantum factoring algorithm, achieving a substantial improvement by reducing the required number of quantum gates to O(n^2 lo g⁡n )(n^(3/2) log⁡n ), compared to Shor’s originalO(n^2 lo g⁡n )original O(n^2 log⁡n ) gate complexity. Although Regev’s approach offers a significant speedup, it comes with increased qubit requirements and relies on an unproven number theory assumption. This paper presents an overview of Regev’s factoring algorithm, including a review of Shor’s work for context, followed by an examination of key recent developments and follow-up research. These include efforts to reduce the qubit count, improve error resilience, generalizegeneralise Regev’s algorithm to related problems, and validate the number theory assumption. This review serves as an accessible entry point for researchers interested in the rapidly evolving field of quantum computing and factoring algorithms.
Quantum algorithms have emerged as powerful computational tools that leverage the principles of quantum mechanics to address complex problems beyond the scope of classical computing. This chapter explores the foundational concepts, classifications, implementation challenges, and future prospects of quantum algorithms. Key breakthroughs such as Grover's and Shor's algorithms are explored alongside recent developments in quantum machine learning and optimization. The chapter also compares quantum and classical algorithmic complexity, highlighting areas of potential speedup. Practical limitations including hardware constraints, error correction, and resource estimation are critically analyzed. Finally, the chapter outlines emerging research directions and ethical considerations essential for responsible quantum computing development.