**1. Introduction**

Feynman was the first to present the idea of quantum computing, that is, to directly use the state of microscopic particles to represent quantum information, which is considered to be the early prototype of the concept of quantum computing [1]. Subsequently, Deutsch formalized the concept of quantum computing, proposed the idea of a quantum Turing machine, and designed the first quantum parallel algorithm, which exhibited excellent performance beyond classical computing [2]. The proposal of Shor's algorithm caused researchers to realize that quantum computing had a natural parallel processing capability, which could introduce many disruptive technological innovations. Shor's algorithm states that a large number can be decomposed into the product of two prime factors in polynomial time. This greatly challenged the RSA (Rivest–Shamir–Adleman) encryption system, thus indicating that the RSA encryption system had been cracked in theory [3,4]. Grover's search algorithm convinced researchers of the power of quantum computing. Compared with the traditional search method, this algorithm can achieve the acceleration effect of square level [5]. Therefore, many improved Grover search algorithms have been proposed [6–10]. Meanwhile, quantum-inspired algorithms have also been proposed that can be simulated by classical computing [11–16]. Moreover, the quantum algorithm has been applied to solve linear systems of equations, which introduced new ideas for solving linear equations. This algorithm is also called the *HHL* algorithm [17]. The *HHL* algorithm has been widely used, and its improved algorithms have been continuously proposed [18–20]. Recently, quantum algorithms have been applied to solve differential equations [21–24]. A series of quantum computing technologies, such as quantum Fourier transform [25], quantum phase estimation [26], and the *HHL* algorithm, are called quantum basic linear algebra assembly [27]. At present, quantum computing has been widely used in cryptography, quantum simulation, machine learning, and other fields and shows a strong ability and grea<sup>t</sup> potential.

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

Academic Editors: Fernando L. Pelayo and Mauro Mezzini

Received: 3 May 2022 Accepted: 31 May 2022 Published: 1 June 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

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The Fourier transform plays an important role in the design of quantum algorithms, but little is known about the quantum algorithms of the fractional Fourier transform (*FRFT*). The initial definition of the *FRFT* was proposed in [28]. Its application provides a convenient technique for solving certain classes of ordinary and partial differential equations, which arise in quantum mechanics from classical quadratic Hamiltonians. The theoretical research of the *FRFT* has developed rapidly, and various definitions have been proposed, such as eigenvalue *FRFT* [29], weighted *FRFT* [30], and sampling *FRFT* [31]. These definitions are widely used in various fields of signal processing. So far, little is known about the reports and studies on the quantum fractional Fourier transform (*QFRFT*). The main reason is that the design of quantum algorithms should satisfy unitarity, and some FRFTs do not include unitarity. Thus, a quantum pseudo-fractional Fourier transform (*QPFRFT*) was proposed [32], and the authors showed that there was no *QFRFT*. However, we present a reformulation of the weighted fractional Fourier transform (*WFRFT*) and prove its unitarity, whereupon a quantum weighted fractional Fourier transform (*QWFRFT*) is proposed.

The remainder of this paper is organized as follows. The preliminary knowledge is described in Section 2. The unitarity of the WFRFT is proved in Section 3. Section 4 presents the *QWFRFT*. Finally, the conclusions are presented in Section 5.
