Quantum Fourier transform in computational basis

  • S. S. Zhou
  • T. Loke
  • J. A. Izaac
  • J. B. WangEmail author


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.


Quantum algorithm Quantum Fourier transform Computational basis state Controlled quantum gates 



The authors would like to thank Ashley Montanaro for constructive comments and Jeremy O’Brien, Jonathan Matthews, Xiaogang Qiang, Lyle Noakes, Chuheng Zhang and Hanwen Zha for helpful discussions.


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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of PhysicsThe University of Western AustraliaCrawleyAustralia
  2. 2.Department of PhysicsYale UniversityNew HavenUSA

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