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arXiv:quant-ph/9601018 · Physical Review A

Approximate quantum Fourier transform and decoherence

Adriano Barenco, ARTUR K. EKERT, Kalle‐Antti Suominen, Päivi Törmä

Published Jul 1, 1996• 180 citations
arXivHTMLPDF

Abstract

We discuss the advantages of using the approximate quantum Fourier transform (AQFT) in algorithms which involve periodicity estimations. We analyze quantum networks performing AQFT in the presence of decoherence and show that extensive approximations can be made before the accuracy of AQFT (as compared with regular quantum Fourier transform) is compromised. We show that for some computations an approximation may imply a better performance. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.139

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We discuss the advantages of using the approximate quantum Fourier transform (AQFT) in algorithms which involve periodicity estimations. We analyze quantum networks performing AQFT in the presence of decoherence and show that extensive approximations can be made before the accuracy of AQFT (as compared with regular quantum Fourier transform) is compromised. We show that for some computations an approximation may imply a better performance. \textcopyright{} 1996 The American Physical Society.
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Quantum Fourier transform is of primary importance in many quantum algorithms. In order to eliminate the destructive effects of decoherence induced by couplings between the quantum system and its environment, we propose a robust scheme for quantum Fourier transform over the intrinsic decoherence-free subspaces. The scheme is then applied to the circuit design of quantum Fourier transform over quantum networks under collective decoherence. The encoding efficiency and possible improvements are also discussed.
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) can calculate the Fourier transform of a vector of size NNNN with time complexity O(log⁡2 N)\mathcal {O}(\log ^2~N)O(log2 N)O(log2N) as compared to the classical complexity of O(N  log⁡N)\mathcal {O}(N \;\log N)O(NlogN)O(NlogN). However, if one wanted to measure the full output state, then the QFT complexity becomes O(N  log⁡2 N)\mathcal {O}(N \;\log ^2~N)O(Nlog2 N)O(Nlog2N), thus losing its apparent advantage, indicating that the advantage is fully exploited for algorithms when only a limited number of samples is required from the output vector, as is the case in many quantum algorithms. Moreover, the computational complexity worsens if one considers the complexity of constructing the initial state. In this article, this issue is better illustrated by providing a concrete implementation of these algorithms and discussing their complexities as well as the complexity of the simulation of the QFT in matlab.