Quantum Fourier Transform in Computational Basis

Abstract

The conventional Quantum Fourier Transform, with exponential speedup 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, the 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 algorithm to encode the Fourier coefficients in the computational basis, with success probability $1-\delta$ and desired precision $\epsilon$. Its time complexity %$\mathcal{O}\big((\log N)^2\log(N/\delta)/\epsilon)\big)$ depends polynomially on $\log(N)$, where $N$ is the problem size, and linearly on $\log(1/\delta)$ and $1/\epsilon$. We also discuss an application of potential practical importance, namely the simulation of circulant Hamiltonians.