# Quantum Weighted Fractional Fourier Transform

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preparation

## 3. Unitarity of Weighted Fractional Fourier Transform

**Definition**

**1.**

**Proposition**

**1.**

**Proof**

**of**

**Proposition**

**1.**

**Proposition**

**2.**

**Proof**

**of**

**Proposition**

**2.**

## 4. Quantum Weighted Fractional Fourier Transform

**x**is transformed into the vector

**y**by the classical Fourier transform,

- We know that ${QFT}^{0}=I$, and $I$ is the identity matrix; obviously, this is a unitary operator. Then, its operation can be expressed as$$|\alpha \rangle \u2015I\u2015|{\beta}_{0}\rangle $$
- The QFT is a unitary operator. The Fourier transform of a quantum state $|\alpha \rangle $ can be expressed as$$|\alpha \rangle \u2015QFT\u2015|{\beta}_{1}\rangle $$
- The quadratic power of the QFT can be expressed as$${QFT}^{2}=\left(\begin{array}{cccc}1& 0& \dots & 0\\ 0& 0& \dots & 1\\ \vdots & \vdots & \u22f0& \vdots \\ 0& 1& \dots & 0\end{array}\right)$$

- The third power of the QFT, which is equivalent to the inverse operation of the QFT, is also a unitary operator.$$|\alpha \rangle \u2015{QFT}^{3}\u2015|{\beta}_{3}\rangle $$

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 9.**The output $|{\alpha}_{k}^{\left(i\right)}\rangle $ of a QANN, where $i=1,2$; $k=1,2,\dots ,M$.

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Zhao, T.; Yang, T.; Chi, Y.
Quantum Weighted Fractional Fourier Transform. *Mathematics* **2022**, *10*, 1896.
https://doi.org/10.3390/math10111896

**AMA Style**

Zhao T, Yang T, Chi Y.
Quantum Weighted Fractional Fourier Transform. *Mathematics*. 2022; 10(11):1896.
https://doi.org/10.3390/math10111896

**Chicago/Turabian Style**

Zhao, Tieyu, Tianyu Yang, and Yingying Chi.
2022. "Quantum Weighted Fractional Fourier Transform" *Mathematics* 10, no. 11: 1896.
https://doi.org/10.3390/math10111896