The quantum Fourier transformation has been demonstrated to be a useful tool in dealing with quantummechanical problems. This paper discusses how quantum Fourier transformation is applied in finding the period of functions within Shor’s algorithm, allowing the algorithm to be much more efficient in factorizing large integers than classic algorithms. This paper also discusses how Grover’s algorithm uses the concept of harmonic process, where amplitude amplification is likened to harmonic oscillation for improving the efficiency of searching unsorted data base. The paper further explores the application of harmonic analysis in quantum error correction, particularly in stabilizer codes. The paper also introduces wavelet analysis as an alternative approach for detecting and correcting localized quantum errors. Finally, the paper extends the discussion into the potential future directions of harmonic analysis in quantum computing, such as extending Fourier transformation to non-abelian groups and using spherical harmonics in quantum algorithms which may optimize current quantum algorithms and solve currently unsolved quantum questions.
The quantum Fourier transformation has been demonstrated to be a useful tool in dealing with quantummechanical problems. This paper discusses how quantum Fourier transformation is applied in finding the period of functions within Shor’s algorithm, allowing the algorithm to be much more efficient in factorizing large integers than classic algorithms. This paper also discusses how Grover’s algorithm uses the concept of harmonic process, where amplitude amplification is likened to harmonic oscillation for improving the efficiency of searching unsorted data base. The paper further explores the application of harmonic analysis in quantum error correction, particularly in stabilizer codes. The paper also introduces wavelet analysis as an alternative approach for detecting and correcting localized quantum errors. Finally, the paper extends the discussion into the potential future directions of harmonic analysis in quantum computing, such as extending Fourier transformation to non-abelian groups and using spherical harmonics in quantum algorithms which may optimize current quantum algorithms and solve currently unsolved quantum questions.
The quantum Fourier transformation has been demonstrated to be a useful tool in dealing with quantum-mechanical problems. This paper discusses how quantum Fourier transformation is applied in finding the period of functions within Shor’s algorithm, allowing the algorithm to be much more efficient in factorizing large integers than classic algorithms. This paper also discusses how Grover’s algorithm uses the concept of harmonic process, where amplitude amplification is likened to harmonic oscillation for improving the efficiency of searching unsorted data base. The paper further explores the application of harmonic analysis in quantum error correction, particularly in stabilizer codes. The paper also introduces wavelet analysis as an alternative approach for detecting and correcting localized quantum errors. Finally, the paper extends the discussion into the potential future directions of harmonic analysis in quantum computing, such as extending Fourier transformation to non-abelian groups and using spherical harmonics in quantum algorithms which may optimize current quantum algorithms and solve currently unsolved quantum questions.
The quantum Fourier transformation has been demonstrated to be a useful tool in dealing with quantum-mechanical problems. This paper discusses how quantum Fourier transformation is applied in finding the period of functions within Shor’s algorithm, allowing the algorithm to be much more efficient in factorizing large integers than classic algorithms. This paper also discusses how Grover’s algorithm uses the concept of harmonic process, where amplitude amplification is likened to harmonic oscillation for improving the efficiency of searching unsorted data base. The paper further explores the application of harmonic analysis in quantum error correction, particularly in stabilizer codes. The paper also introduces wavelet analysis as an alternative approach for detecting and correcting localized quantum errors. Finally, the paper extends the discussion into the potential future directions of harmonic analysis in quantum computing, such as extending Fourier transformation to non-abelian groups and using spherical harmonics in quantum algorithms which may optimize current quantum algorithms and solve currently unsolved quantum questions.
This survey presents a comprehensive analysis of the quantum Fourier transform (QFT), a fundamental tool in quantum computing, in comparison to its classical counterpart, the fast Fourier transform (FFT). The study begins with an introduction to the classical Fourier transform, which is widely used in different disciplines such as signal and image processing. The article introduces the QFT as a quantum version of the Fourier transform, detailing how it leverages quantum parallelism and superposition to reduce the time complexity of the operation, and highlighting its crucial role in quantum algorithms like Shor’s algorithm for integer factorization. The analysis also addresses the mathematical foundations of the QFT, its implementation in quantum circuits, and the key advantages and challenges associated with its use, such as measurement precision and quantum decoherence. Finally, it concludes with an exploration of the current and potential applications of QFT in quantum computing.
In this chapter, we will study the quantum Fourier transform and its application in different quantum algorithms. Problems such as factoring an integer into prime numbers or period finding are computationally intractable problems for a classical computer because of the exponentially large number of operations involved. Integer factoring and period finding can be efficiently solved using the quantum phase estimation algorithm that is heavily based on the quantum Fourier transform. Alternately, since quantum phase estimation aims to find the eigenvalue corresponding to an eigenvector of a unitary operator, it is backbone of important algorithms in optimization such as the HHL algorithm (named for Hassim, Harrow, and Lloyd), which serves as the matrix inversion routine in quantum computing. We start this chapter by revising our concepts of the Fourier transform and its discrete counterpart, the discrete Fourier transform , and then move on to the exciting domain of the quantum Fourier transform and the quantum phase estimation algorithm. We follow this up with a discussion and implementation of the few quantum Fourier transform–related algorithms such as factoring a number and period finding. At the end of the chapter, we briefly introduce the basics of group theory with an attempt to explain the hidden subgroup problem and how it relates to several of the Fourier transform–based algorithms.
Quantum computing, based on the fundamental principles of quantum mechanics, has become a breakthrough that has the potential to revolutionize computing. At the forefront of this quantum revolution are quantum algorithms that use the origin of qubits to solve computational problems much faster than classical algorithms. This article provides a review of quantum algorithms to introduce their principles, examine famous examples such as the Grover and Shor algorithms, and delve into their applications, including quantum machine learning. Exploring the fundamentals of quantum computing begins with an overview that highlights the unique features of quantum computing compared to classical computing. We then examine several different algorithmic paradigms to highlight the unique advantages that quantum parallelism and interference bring to computing. The best knowledge of quantum algorithms focuses on Grover's blind search, Shor's efficient factorization algorithm, and the quantum phase approach for solving quantum chemistry and eigenvalue problems. As well as celebrating the achievements of quantum algorithms, this research also looks at the challenges and limitations that hinder their widespread use. Topics such as decoherence, error correction, and finding errors in quantum computing are discussed in the context of overcoming obstacles to the use of practical quantum algorithms. In addition, the latest developments are clarified by showing the successes of the experiments and their implications for the future. As we stand at the forefront of a new era in computing, this article not only provides a snapshot of the current state of quantum algorithms but also speculates on future directions. The potential applications of quantum algorithms in many fields and the ongoing quest to overcome current limitations offer exciting opportunities for further research. This research aims to contribute to the ongoing debate about quantum algorithms, gaining a deeper understanding of their impact on the future of computing.
Quantum computing represents a transformative shift in computational science by exploiting the principles of quantum mechanics to solve problems that are computationally infeasible for classical systems. At the core of this transformation lie mathematical algorithms that govern how quantum information is represented, manipulated, and measured. This paper examines the role of mathematical algorithms in quantum computing by analysing their theoretical foundations, structural properties, and algorithmic behaviour. The study explores major quantum algorithms, including Shor's algorithm, Grover's algorithm, the Quantum Fourier Transform, Quantum Phase Estimation, Hamiltonian simulation techniques, and NISQ-era hybrid algorithms. Through qualitative theoretical analysis and literature-based synthesis, the research highlights how linear algebra, number theory, Fourier analysis, and operator theory enable quantum algorithms to achieve computational advantages. The paper also discusses the limitations imposed by noise, decoherence, and scalability constraints, as well as the role of error correction and hybrid approaches in addressing these challenges. The findings underscore that mathematical algorithms are central to realizing quantum advantage and will continue to guide the evolution of quantum computing from theoretical promise to practical application.