*Article* **Quantum Weighted Fractional Fourier Transform**

**Tieyu Zhao 1,\*, Tianyu Yang 2 and Yingying Chi 1**


**Abstract:** Quantum Fourier transform (QFT) is an important part of many quantum algorithms. However, there are few reports on quantum fractional Fourier transform (QFRFT). The main reason is that the definitions of fractional Fourier transform (FRFT) are diverse, while some definitions do not include unitarity, which leads to some studies pointing out that there is no QFRFT. In this paper, we first present a reformulation of the weighted fractional Fourier transform (WFRFT) and prove its unitarity, thereby proposing a quantum weighted fractional Fourier transform (QWFRFT). The proposal of QWFRFT provides the possibility for many quantum implementations of signal processing.

**Keywords:** quantum weighted fractional Fourier transform; quantum Fourier transform; quantum algorithm; quantum computing

**MSC:** 81-08
