Quantum Fourier transform in computational basis

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• 作者：S. S. Zhou ; T. Loke ; J. A. Izaac ; J. B. Wang
• 关键词：Quantum algorithm ; Quantum Fourier transform ; Computational basis state ; Controlled quantum gates
• 刊名：Quantum Information Processing
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：16
• 期：3
• 全文大小：
• 刊物类别：Physics and Astronomy
• 刊物主题：Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics;
• 出版者：Springer US
• ISSN：1573-1332
• 卷排序：16

文摘

The quantum Fourier transform, with exponential speed-up compared to the classical fast Fourier transform, has played an important role in quantum computation as a vital part of many quantum algorithms (most prominently, Shor’s factoring algorithm). However, situations arise where it is not sufficient to encode the Fourier coefficients within the quantum amplitudes, for example in the implementation of control operations that depend on Fourier coefficients. In this paper, we detail a new quantum scheme to encode Fourier coefficients in the computational basis, with fidelity \(1 - \delta \) and digit accuracy \(\epsilon \) for each Fourier coefficient. Its time complexity depends polynomially on \(\log (N)\), where N is the problem size, and linearly on \(1/\delta \) and \(1/\epsilon \). We also discuss an application of potential practical importance, namely the simulation of circulant Hamiltonians.