# Quantum Fourier transform in computational basis

Quantum Fourier transform in computational basis 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 $$1 - \delta$$ 1 - δ and digit accuracy $$\epsilon$$ ϵ for each Fourier coefficient. Its time complexity depends polynomially on $$\log (N)$$ log ( N ) , where N is the problem size, and linearly on $$1/\delta$$ 1 / δ and $$1/\epsilon$$ 1 / ϵ . We also discuss an application of potential practical importance, namely the simulation of circulant Hamiltonians. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

# Quantum Fourier transform in computational basis

, Volume 16 (3) – Feb 10, 2017
19 pages

/lp/springer_journal/quantum-fourier-transform-in-computational-basis-kLgewR8ThM
Publisher
Springer US
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-017-1515-0
Publisher site
See Article on Publisher Site

### Abstract

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 $$1 - \delta$$ 1 - δ and digit accuracy $$\epsilon$$ ϵ for each Fourier coefficient. Its time complexity depends polynomially on $$\log (N)$$ log ( N ) , where N is the problem size, and linearly on $$1/\delta$$ 1 / δ and $$1/\epsilon$$ 1 / ϵ . We also discuss an application of potential practical importance, namely the simulation of circulant Hamiltonians.

### Journal

Quantum Information ProcessingSpringer Journals

Published: Feb 10, 2017

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