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System Analysis in Science and Education

Инструментарий проектирования квантовых алгоритмов. Ч. 3.

А. Решетников, Olga Tyatyushkina, С. В. Ульянов, Д.А. Джиованни

Published Sep 16, 2021• 0 citations
OpenAlex

Abstract

The universality of the quantum Fourier transform in forming the basis of quantum computing algorithms is considered. The unique universal fundamental properties of quantum computing concerning quantum superposition, entanglement and interference are all explicitly represented in terms of quantum multiparticle interferometry.

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В статье рассматриваются принципы квантовых вычислений на основе спиновых состояний. Анализируются фундаментальные принципы квантовой физики, такие как суперпозиция, запутанность и корпускулярно-волновой дуализм. Описаны математические модели кубитов в сфере Блоха, уравнение Шрёдингера и процесс коллапса волновой функции при измерении. Показана роль спиновой запутанности в реализации квантовых алгоритмов. Проведен анализ возможностей применения квантовых алгоритмов в области криптографии, моделирования сложных систем и др. The article examines the principles of quantum computing based on spin states. It analyzes the fundamental principles of quantum physics, such as superposition, entanglement, and particle-wave duality. Mathematical models of qubits on the Bloch sphere, the Schrödinger equation, and the wave function collapse during measurement are described. The role of spin entanglement in implementing quantum algorithms is demonstrated. The possibilities of using quantum algorithms in cryptography, complex systems modeling, and optimization.
Quantum Computing represents a transformative shift in the way we process information. Rooted in the principles of quantum mechanics—superposition, entanglement, and interference—it promises computational capabilities far beyond the reach of classical machines. Problems once considered intractable, such as large-scale optimization, cryptographic analysis, and complex system simulations, now appear within conceivable reach. This book is designed to provide a clear and structured introduction to the foundations and frontiers of quantum computing. It bridges the gap between theory and practice, guiding readers from the fundamental concepts of qubits and quantum gates to advanced topics such as quantum algorithms, error correction, and emerging quantum architectures. Mathematical rigor is balanced with intuitive explanations, ensuring accessibility for students, researchers, and professionals from diverse backgrounds in science, engineering, and computing.
A methodology is presented to obtain the basis of qudits which are admissible to quantum Fourier transform (QFT) in the sense that the set of such kets are related by the QFT in the same way as the kets of the computational basis. We first study this method for qubits to characterize the ensemble that works for the Hadamard transformation (QFT for two dimension). In this regard we identify certain incompleteness in the result of Maitra and Parashar (Int. J. Quantum Inform. 4 (2006) 653). Next we characterize the ensemble of qutrits for which QFT is possible. Further, some theoretical results related to higher dimensions are also discussed. Considering the unitary matrix U n related to QFT, the issue boils down to the problem of characterizing matrices that commute with U n .
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.