Quantum Fourier transform

노트

In this article, we will take a look at QFT.^[1]
To prove that QFT is implementable, we need to prove the transformation is unitary.^[1]
In Fourier Transform, we develop a faster version called Fast Fourier Transform to compute the transformation iteratively.^[1]
An implementation of the Fourier transform as a quantum circuit sometimes plays a crucial role on quantum computing.^[2]
The Fourier transform that we consider in this paper is somewhat different from the QFT: We propose a quantum implementation of the algorithm of the FFT rather than the QFT.^[2]
where the data sequence \(\{X_k\}\) is the Fourier transform of \(\{x_j\}\) as expressed in (1.2).^[2]
Nevertheless, there are following advantages compared to the classical FFT, and even compared to the QFT.^[2]
The quantum Fourier transform can be performed efficiently on a quantum computer, with a particular decomposition into a product of simpler unitary matrices.^[3]
Since there is an efficient quantum circuit implementing the quantum Fourier transform, the circuit can be run in reverse to perform the inverse quantum Fourier transform.^[3]
The quantum Fourier transform can be approximately implemented for any N; however, the implementation for the case where N is a power of 2 is much simpler.^[3]
"An improved quantum Fourier transform algorithm and applications".^[3]