Quantum Windowed Fourier Transform And Its Application To Quantum Signal Processing

In classical information processing,Fourier transform has a very wide range of appli-cations.Through Fourier transform,the signal can be Changing to the frequency domain allows the composition and characteristics of the signal to be studied more deeply and quantitatively.The windowed Fourier transform(WFT),or short-time Fourier transform,which is a variant of the Fourier transform by dividing a longer time signal into shorter segments of equal length and then computing the Fourier transform separately on each shorter segment,is proposed to provide a method of signal processing.Up to now the discrete Fourier transform has been successfully applied to the field of quantum infor-mation,but the related short-time discrete Fourier transform of this field has not been developed accordingly.To address this problem,we first introduce the concept of quan-tum window states,and further prove that the quantum Fourier transform of a quantum window state is also a quantum window state.Next,the quantum window states is applied to the quantum signals,and the long quantum signals are divided into shorter segments to extract the local information of the quantum signals.Then,each shorter segment is studied separately and the corresponding quantum circuit is given.Then by applying the quantum Fourier transform to the windowed quantum superposition state,we propose a new concept called the quantum windowed Fourier transform(QWFT).Finally,a specific application of quantum window Fourier transform(QWFT)for quantum signal processing is given where QWFT is employed.