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Quantum Windowed Fourier Transform and its Application to Quantum Signal Processing

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Abstract

In classical information processing, 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 information, 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 quantum window state, and further prove that the quantum Fourier transform of a quantum window state is also a quantum window state. Based on the definition of the quantum window state the local information of a quantum signal is extracted and the corresponding quantum circuits are also given. And then, by applying the quantum Fourier transform to the windowed quantum superposition states, we propose a novel concept called quantum windowed Fourier transform (QWFT). Finally, an application of quantum signal processing is given where QWFT is employed.

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Acknowledgements

The authors express their gratitude to the anonymous referees for their kind suggestions and useful comments on the original manuscript, which resulted in this final version.

Funding

This work is supported by the National Natural Science Foundation of China (No. 41771375)

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Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University, 475001, Kaifeng, China

    Haiting Yin, Dayong Lu & Rui Zhang

  2. Henan Engineering Research Center for Artificial Intelligence Theory and Algorithms, Henan University, 475001, Kaifeng, China

    Dayong Lu

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  1. Haiting Yin
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  2. Dayong Lu
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Correspondence to Dayong Lu.

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Yin, H., Lu, D. & Zhang, R. Quantum Windowed Fourier Transform and its Application to Quantum Signal Processing. Int J Theor Phys 60, 3896–3918 (2021). https://doi.org/10.1007/s10773-021-04933-7

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  • DOI: https://doi.org/10.1007/s10773-021-04933-7

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