Galois rings are regarded as "building blocks" of a finite commutative ring with identity. There have been many papers on classical error correction codes over Galois rings published. As an important warm-up before exploring quantum algorithms and quantum error correction codes over Galois rings, we study the quantum Fourier transform (QFT) over Galois rings and prove it can be efficiently preformed on a quantum computer. The properties of the QFT over Galois rings lead to the quantum algorithm for hidden linear structures over Galois rings.
The application of the quantum Fourier transform (QFT) within the field of quantum computation has been manifold. Shor’s algorithm, phase estimation and computing discrete logarithms are but a few classic examples of its use. These initial blueprints for quantum algorithms have sparked a cascade of tantalizing solutions to problems considered to be intractable on a classical computer. Therefore, two main threads of research have unfolded. First, novel applications and algorithms involving the QFT are continually being developed. Second, improvements in the algorithmic complexity of the QFT are also a sought after commodity. In this work, we review the structure of the QFT and its implementation. In order to put these concepts in their proper perspective, we provide a brief overview of quantum computation. Finally, we provide a permutation structure for putting the QFT within the context of universal computation.
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.
The quantum Fourier transform (QFT) is a fundamental primitive in quantum computation and quantum information. In this work, we extend the framework for the QFT from finite groups to finite-dimensional semisimple algebras, and give efficient QFTs for the partition algebra
Pn(d), Brauer algebra
Bn(d), and walled Brauer algebra
Br,s(d). These algebras play important roles in generalized Schur-Weyl duality, statistical physics and many-body systems, and have recently found several applications in quantum algorithms.
Unlike the group case, the Fourier transform over a semisimple algebra can be non-unitary. Nevertheless, we show that when the parameter
d is sufficiently large, the Fourier transform is well approximated by a unitary operator. Furthermore, we show that for each of the algebras
A from above, such an approximate Fourier transform can be implemented efficiently: we give a quantum algorithm with gate complexity
poly(n,logd,log(1/ε)) for approximating the Fourier transform to error
(d−1/2+ε)⋅poly(∣A∣). Along the way, we establish several properties of the Fourier basis of semisimple algebras that may be of independent interest.
Their explicit forms are In this article we present a concrete form of the spectral decomposition of the quantum Fourier transform (QFT). QFT is one of the most useful mathematical tools in applied mathematics and mathematical sciences such as signal processing and quantum computing') . For example, in the famous quantum algorithm of factorization of a large integer due to Shor2}, QFT plays an essential role. Thus studying QFT is an important subject in various fields of applied mathematics. Spectral decomposition of QFT provides us with a very convenient way of theoretical treatment of QFT. As an application, we compute the logarithm of QFT.