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arXiv:quant-ph/9511007 · Physical Review Letters

Semiclassical Fourier Transform for Quantum Computation

Robert B. Griffiths, Chi-Sheng Niu

Published Apr 22, 1996• 430 citations
arXivHTMLPDF

Abstract

It is shown that the Fourier transform preceding the final measurement in Shor's algorithm for factorization on a quantum computer can be carried out in a semiclassical way by using the ``classical'' (macroscopic) signal resulting from measuring one bit to determine the type of measurement carried out on the next bit, and so forth. In this way all the two-bit gates in the Fourier transform can be replaced by a smaller number of one-bit gates controlled by classical signals. This suggests that it may be worthwhile looking for other uses of semiclassical methods in quantum computing.

DOI
10.1103/physrevlett.76.3228

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It is shown that the Fourier transform preceding the final measurement in Shor's algorithm for factorization on a quantum computer can be carried out in a semiclassical way by using the ``classical'' (macroscopic) signal resulting from measuring one bit to determine the type of measurement carried out on the next bit, and so forth. In this way all the two-bit gates in the Fourier transform can be replaced by a smaller number of one-bit gates controlled by classical signals. This suggests that it may be worthwhile looking for other uses of semiclassical methods in quantum computing.
Quantum Fourier transform (QFT) plays an eminent role in quantum computation.It creates a superposition of different quantum states, allowing simultaneous calculation, which would take many steps were the same program implemented on a classical computer.The computational speed of a quantum computer is thus boosted dramatically.In this article, we explain the application of QFT in Shor's algorithm, which was proposed by Peter Shor to factor large integers on quantum computers.Specifically, the principle and design of quantum Fourier transform are explained.We stress the subtle distinction between QFT and its inverse.Since former articles did not emphasize it, we hope it could be a supplement to former articles.Our next work is to verify the existing articles on executing Shor's algorithm by conducting two experiments on IBMQ of factoring N = 15 in two ways (a = 7 and a = 11).We find that the effect of quantum entanglement might be crucial to the speed boost of factoring large integers in Shor's algorithm.
Because of the difficulty of building a high-dimensional quantum register, this paper presents an implementation of the high-dimensional quantum Fourier transform (QFT) based on a low-dimensional quantum register. First, we define the t-bit semiclassical quantum Fourier transform. In terms of probability amplitude, we prove that the transform can realize quantum Fourier transformation, illustrate that the requirement for the two-qubit gate reduces obviously, and further design a quantum circuit of the transform. Combining the classical fixed-window method and the implementation of Shor’s quantum factorization algorithm, we then redesign a circuit for Shor’s algorithm, whose required computation resource is approximately equal to that of Parker’s. The requirement for elementary quantum gates for Parker’s algorithm is O(⌈ log N⌉3, and the quantum register for our circuit requires t−1 more dimensions than Parker’s. However, our circuit is t 2 times as fast as Parker’s, where t is the width of the window.
Abstract Expanding on the factoring and periodicity in Chapter 14, the Shor algorithm is explained in detail, with a warning that this is one of the toughest chapters in the book. While the steps of Shor algorithm may look just like the RSA in reverse, the quantum computer is needed to find the periodicity of the factors and thereby the prime factors themselves. Fourier analysis is the tool for this. The quantum Fourier transform is described, including the quantum circuit, after first showing what the classical Fourier transform accomplishes. Extensive examples of all the steps are given.
For classical simulation, the quantum Fourier transform (QFT) requires very large matrix operations. Previous work has shown that the semi-classical quantum Fourier transform (SCQFT) can use these individual coefficients to perform the QFT using only single-quantum bit (qubit) unitary gates and measurement operators. However, the SCQFT requires these individual decomposed qubits of the quantum system as input to the algorithm. We devise two methods to find separable approximations of quantum systems to serve as inputs to the SCQFT. We introduce an application of the approach on classical radio frequency signals represented through a quantum model. The resulting decomposition and QFT are computed on several simulated results, and an example is given using an experimental signal.
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.