Circuit of Quantum Fractional Fourier Transform

Abstract

In this paper, we first use the quantum Fourier transform (QFT) and quantum phase estimation (QPE) to realize the quantum fractional Fourier transform (QFrFT). As diverse definitions of the discrete fractional Fourier transform (DFrFT) exist, the relationship between the QFrFT and a classical algorithm is then established; that is, we determine the classical algorithm corresponding to the QFrFT. Second, we observe that many definitions of the multi-fractional Fourier transform (mFrFT) are flawed: when we attempt to propose a design scheme for the quantum mFrFT, we find that there are many invalid weighting terms in the definition of the mFrFT. This flaw may have very significant impacts on relevant algorithms for signal processing and image encryption. Finally, we analyze the circuit of the QFrFT and the reasons for the observed defects.

1. Introduction

In recent years, research into and the application of quantum information technology—represented by quantum computing, quantum communication, and quantum measurement—have accelerated globally, with many countries increasing investments and broadening project layouts. Quantum computing is expected to fundamentally improve the speed of information processing, quantum communication will greatly improve communication security, and quantum precision measurement and sensing technology will have extensive applications in the future digital age and the era of the Internet of Things. Quantum computing is a novel computation mode that follows the laws of quantum mechanics to regulate the calculation of quantum information units, utilizing the characteristics of quantum mechanics [1] (e.g., superposition and entangled states) for the storage, processing, and transmission of information. Since Feynman proposed the concept of quantum computing [2], researchers have shown great interest in the high parallelism and speed provided by such an approach. The discovery of Shor’s efficient decomposition algorithm [3] aroused interest in the quantum implementation of modular arithmetic operations, which are the basis of quantum decomposition circuits. The core of Shor’s algorithm is the quantum Fourier transform (QFT) [4]. In the field of quantum computing, involving topics such as quantum information processing [5], quantum machine learning [6], quantum neural networks [7], and so on, a series of significant theoretical improvements and encouraging experimental results have recently been achieved, in which the QFT plays an important role. Quantum image processing plays an important role in satisfying the demands of machine vision, and quantum image representation is the basis of quantum image processing. The representation of quantum images has been developed from the initial Qubit Lattice [8], Real Ket [9], and Entangled image [10] to the common representation forms FRQI [11] and NEQR [12]. Based on these characterizations, various methods for quantum image processing have also been proposed [13], such as local feature point extraction of quantum images [14], weighted filtering of quantum images in the spatial domain [15], and restoration of quantum image noise removal [16]. Among them, the QFT plays an important role in frequency domain filtering [17,18], and the quantum arithmetic operations associated with the QFT are equally important to consider [19]. It is clear that the QFT plays an important role in quantum algorithm design and information processing.

As an extended definition of the Fourier transform (FT), the fractional Fourier transform (FrFT) has become one of the most actively researched subjects in signal processing and has been widely used in optics, image and signal processing, and communication [20,21,22,23,24]. However, at present, there are few studies on the quantum fractional Fourier transform (QFrFT) [25,26,27,28]. The only existing studies have also been conducted from the perspective of quantum mechanics, in order to study the theory underlying the QFrFT, and have not presented circuits for realization of the QFrFT. In this paper, the QFrFT is designed by means of the QFT and quantum phase estimation (QPE) [1]. The idea is to retain the eigenvector of the QFT, realize the fractional power of the eigenvalue by introducing two phase gates, and finally obtain the QFrFT by combining the eigenvalue of the fractional power with the eigenvector.

The remainder of this paper is organized as follows. The QFrFT is proposed in Section 2. Section 3 explains the correspondence between the QFrFT and a classical algorithm. The flaws of the mFrFT are detailed in Section 4. In Section 5, circuits for the QFrFT and mFrFT are discussed. Finally, the conclusions are presented in Section 6.

2. Quantum Fractional Fourier Transform

At present, some related studies on the QFrFT have been proposed [25,26,27,28], but these studies have not presented specific quantum circuits. Parasa et al. have even stated that the QFrFT is impossible to achieve [25]. In this section, we use the QFT and QPE to design the QFrFT.

For a unitary matrix $U$, assuming that it has an eigenvector $|u\rangle$ and the corresponding eigenvalue $e^{2\pi i\phi }$, then $U|u\rangle =e^{2\pi i\phi }|u\rangle$ is satisfied. Therefore, we can calculate $\phi$ through the QPE. The circuit is shown in Figure 1. It is not difficult to find that the QFT is the key of phase estimation, and phase estimation is the key of many quantum algorithms.

We know the relationship between the eigenvalues and eigenvectors of the discrete Fourier transform (DFT), as shown in Equation (1):

$$F|u\rangle =D|u\rangle ,$$

(1)

where F denotes the Fourier transform, $D$ is the eigenvalue, and $|u\rangle$ is the eigenvector. The discrete fractional Fourier transform (DFrFT) has the same eigenvector $|u\rangle$ as the DFT, and the correspondence between the eigenvalue and eigenvector is shown in Equation (2):

$$F^{\alpha }|u\rangle =D^{\alpha }|u\rangle ,$$

(2)

where $F^{\alpha }$ denotes the FrFT, and $D^{\alpha }$ is the eigenvalue. Thus, we can determine the eigenvalue of the QFrFT by means of the QPE, as depicted in Figure 2.

The process is as follows:

$$\begin{array}{c}|{\psi }_1\rangle =\frac{1}{\sqrt{2}}\left(|0\rangle +|1\rangle \right)\otimes \frac{1}{\sqrt{2}}\left(|0\rangle +|1\rangle \right)\otimes |u\rangle \\ =\frac{1}{2}\left(|00\rangle +|01\rangle +|10\rangle +|11\rangle \right)\otimes |u\rangle \end{array}.$$

(3)

Thus, we can obtain

$$|{\psi }_2\rangle =\frac{1}{2}\left(|00\rangle I|u\rangle +|01\rangle F|u\rangle +|10\rangle F^2|u\rangle +|11\rangle F^3|u\rangle \right).$$

(4)

Here, $|u\rangle$ is the eigenvector of the FT. From Equation (1), we can obtain

$$|{\psi }_2\rangle =\frac{1}{2}\left(|00\rangle D^0|u\rangle +|01\rangle D^1|u\rangle +|10\rangle D^2|u\rangle +|11\rangle D^3|u\rangle \right).$$

(5)

The eigenvalue D can be expressed as:

$$D=\left(\begin{array}{cccc}{\lambda }_0& & & 0\\ & {\lambda }_1& & \\ & & \ddots & \\ 0& & & {\lambda }_{n-1}\end{array}\right),$$

(6)

where ${\lambda }_j\in \left\{1,i,-1,-i\right\}$; $j=0,1,\cdots ,n-1$. The 2-qubit inverse quantum Fourier transform (IQFT) is used in the control register, and its matrix can be expressed as:

$$IQFT=\frac{1}{\sqrt{4}}\left(\begin{array}{cccc}1& 1& 1& 1\\ 1& i& -1& -i\\ 1& -1& 1& -1\\ 1& -i& -1& i\end{array}\right).$$

(7)

After 2-qubit IQFT, we obtain

$$|{\psi }_3\rangle =\frac{1}{4}\left(Q|00\rangle +W|01\rangle +E|10\rangle +R|11\rangle \right)\otimes |u\rangle$$

(8)

where

$$\{\begin{array}{l}Q=D^0+D^1+D^2+D^3\\ W=D^0+i{D}^1-D^2-i{D}^3\\ E=D^0-D^1+D^2-D^3\\ R=D^0-i{D}^1-D^2+i{D}^3\end{array}.$$

(9)

Moreover, from Equations (6) and (9), we have

$$\begin{array}{ll}Q& =D^0+D^1+D^2+D^3\\ & =\left(\begin{array}{cccc}{\lambda }_0^0+{\lambda }_0^1+{\lambda }_0^2+{\lambda }_0^3& & & \\ & {\lambda }_1^0+{\lambda }_1^1+{\lambda }_1^2+{\lambda }_1^3& & \\ & & \ddots & \\ & & & {\lambda }_{n-1}^0+{\lambda }_{n-1}^1+{\lambda }_{n-1}^2+{\lambda }_{n-1}^3\end{array}\right)\end{array},$$

(10)

$$\begin{array}{ll}W& =D^0+i{D}^1-D^2-i{D}^3\\ & =\left(\begin{array}{cccc}{\lambda }_0^0+i{\lambda }_0^1-{\lambda }_0^2-i{\lambda }_0^3& & & \\ & {\lambda }_1^0+i{\lambda }_1^1-{\lambda }_1^2-i{\lambda }_1^3& & \\ & & \ddots & \\ & & & {\lambda }_{n-1}^0+i{\lambda }_{n-1}^1-{\lambda }_{n-1}^2-i{\lambda }_{n-1}^3\end{array}\right)\end{array},$$

(11)

$$\begin{array}{ll}E& =D^0-D^1+D^2-D^3\\ & =\left(\begin{array}{cccc}{\lambda }_0^0-{\lambda }_0^1+{\lambda }_0^2-{\lambda }_0^3& & & \\ & {\lambda }_1^0-{\lambda }_1^1+{\lambda }_1^2-{\lambda }_1^3& & \\ & & \ddots & \\ & & & {\lambda }_{n-1}^0-{\lambda }_{n-1}^1+{\lambda }_{n-1}^2-{\lambda }_{n-1}^3\end{array}\right)\end{array},$$

(12)

and

$$\begin{array}{ll}R& =D^0-i{D}^1-D^2+i{D}^3\\ & =\left(\begin{array}{cccc}{\lambda }_0^0-i{\lambda }_0^1-{\lambda }_0^2+i{\lambda }_0^3& & & \\ & {\lambda }_1^0-i{\lambda }_1^1-{\lambda }_1^2+i{\lambda }_1^3& & \\ & & \ddots & \\ & & & {\lambda }_{n-1}^0-i{\lambda }_{n-1}^1-{\lambda }_{n-1}^2+i{\lambda }_{n-1}^3\end{array}\right)\end{array}.$$

(13)

Here, the eigenvalues are ${\lambda }_j\in \left\{1,i,-1,-i\right\}$; $j=0,1,\cdots ,n-1$. Thus, Equations (10)–(13) can be written as:

$$Q=\{\begin{array}{l}{\lambda }_j^0+{\lambda }_j^1+{\lambda }_j^2+{\lambda }_j^3=4\hspace{1em}if\hspace{1em}{\lambda }_j=1\\ {\lambda }_j^0+{\lambda }_j^1+{\lambda }_j^2+{\lambda }_j^3=0\hspace{1em}if\hspace{1em}{\lambda }_j=i,-1,-i\end{array},$$

(14)

$$W=\{\begin{array}{l}{\lambda }_j^0+i{\lambda }_j^1-{\lambda }_j^2-i{\lambda }_j^3=4\hspace{1em}if\hspace{1em}{\lambda }_j=-i\\ {\lambda }_j^0+i{\lambda }_j^1-{\lambda }_j^2-i{\lambda }_j^3=0\hspace{1em}if\hspace{1em}{\lambda }_j=1,i,-1\end{array},$$

(15)

$$E=\{\begin{array}{l}{\lambda }_j^0-{\lambda }_j^1+{\lambda }_j^2-{\lambda }_j^3=4\hspace{1em}if\hspace{1em}{\lambda }_j=-1\\ {\lambda }_j^0-{\lambda }_j^1+{\lambda }_j^2-{\lambda }_j^3=0\hspace{1em}if\hspace{1em}{\lambda }_j=1,i,-i\end{array},$$

(16)

and

$$R=\{\begin{array}{l}{\lambda }_j^0-i{\lambda }_j^1-{\lambda }_j^2+i{\lambda }_j^3=4\hspace{1em}if\hspace{1em}{\lambda }_j=i\\ {\lambda }_j^0-i{\lambda }_j^1-{\lambda }_j^2+i{\lambda }_j^3=0\hspace{1em}if\hspace{1em}{\lambda }_j=1,-1,-i\end{array}.$$

(17)

Then, the eigenvalues of the FT are preserved in the quantum state, as shown in Equation (18):

$$\{\begin{array}{l}{\lambda }_j=1\to |00\rangle \\ {\lambda }_j=-i\to |01\rangle \\ {\lambda }_j=-1\to |10\rangle \\ {\lambda }_j=i\to |11\rangle \end{array}.$$

(18)

We know that the eigenvector $|u\rangle$ can be written in standard orthogonal basis form, as shown in Equation (19):

$$|u\rangle ={\displaystyle \sum _j{b}_j|u_j\rangle },$$

(19)

where $b_j$ is the projection length. Thus, Equation (8) can also be written as:

$$|{\psi }_3\rangle ={\displaystyle \sum _j|{\varphi }_j\rangle }{b}_j|u_j\rangle ,$$

(20)

where $|{\varphi }_j\rangle \in \left\{|00\rangle ,|01\rangle ,|10\rangle ,|11\rangle \right\}$; $j=0,1,\cdots ,n-1$. Next, the phase gates $R_Z\left(-\pi \alpha \right)$ and $R_Z\left(-\pi \alpha /2\right)$ are expressed as:

$$R_Z\left(-\pi \alpha \right)=\left(\begin{array}{cc}1& 0\\ 0& e^{-\pi i\alpha }\end{array}\right)$$

(21)

and

$$R_Z\left(-\pi \alpha /2\right)=\left(\begin{array}{cc}1& 0\\ 0& e^{-\pi i\alpha /2}\end{array}\right).$$

(22)

We can obtain the quantum state $|{\psi }_4\rangle$ by

$$|{\psi }_4\rangle ={\displaystyle \sum _j{e}^{-\pi i\alpha {\varphi }_j/2}|{\varphi }_j\rangle \otimes b_j|u_j\rangle }.$$

(23)

The 2-qubit QFT can be expressed as:

$$QFT=\frac{1}{\sqrt{4}}\left(\begin{array}{cccc}1& 1& 1& 1\\ 1& -i& -1& i\\ 1& -1& 1& -1\\ 1& i& -1& -i\end{array}\right).$$

(24)

After the QFT of $|{\varphi }_j\rangle$, the eigenvalues are restored:

$$\begin{array}{ll}|{\psi }_5\rangle & =\frac{1}{2}\left({\displaystyle \sum _j{e}^{-\pi i\alpha {\varphi }_j/2}{\lambda }_j^0{b}_j|u_j\rangle |00\rangle }+{\displaystyle \sum _j{e}^{-\pi i\alpha {\varphi }_j/2}{\lambda }_j^1{b}_j|u_j\rangle }|01\rangle +{\displaystyle \sum _j{e}^{-\pi i\alpha {\varphi }_j/2}{\lambda }_j^2{b}_j|u_j\rangle |10\rangle }+{\displaystyle \sum _j{e}^{-\pi i\alpha {\varphi }_j/2}{\lambda }_j^3{b}_j|u_j\rangle |11\rangle }\right)\\ & =\frac{1}{2}\left({\displaystyle \sum _j{e}^{-\pi i\alpha {\varphi }_j/2}{b}_j|u_j\rangle |00\rangle }+{\displaystyle \sum _j{e}^{-\pi i\alpha {\varphi }_j/2}F{b}_j|u_j\rangle }|01\rangle +{\displaystyle \sum _j{e}^{-\pi i\alpha {\varphi }_j/2}{F}^2{b}_j|u_j\rangle |10\rangle +{\displaystyle \sum _j{e}^{-\pi i\alpha {\varphi }_j/2}{F}^3{b}_j|u_j\rangle |11\rangle }}\right)\end{array},$$

(25)

where ${\varphi }_j\in \left\{0,1,2,3\right\}$. Then,

$$\begin{array}{ll}|{\psi }_6\rangle & =\frac{1}{2}\left({\displaystyle \sum _j{e}^{-\pi i\alpha {\varphi }_j/2}{\lambda }_j^0{b}_j|u_j\rangle |00\rangle }+{\displaystyle \sum _j{e}^{-\pi i\alpha {\varphi }_j/2}IFF{b}_j|u_j\rangle }|01\rangle +{\displaystyle \sum _j{e}^{-\pi i\alpha {\varphi }_j/2}I{F}^2{F}^2{b}_j|u_j\rangle |10\rangle +{\displaystyle \sum _j{e}^{-\pi i\alpha {\varphi }_j/2}I{F}^3{F}^3{b}_j|u_j\rangle |11\rangle }}\right)\\ & =\frac{1}{2}\left({\displaystyle \sum _j{e}^{-\pi i\alpha {\varphi }_j/2}{b}_j|u_j\rangle |00\rangle }+{\displaystyle \sum _j{e}^{-\pi i\alpha {\varphi }_j/2}{b}_j|u_j\rangle }|01\rangle +{\displaystyle \sum _j{e}^{-\pi i\alpha {\varphi }_j/2}{b}_j|u_j\rangle |10\rangle +{\displaystyle \sum _j{e}^{-\pi i\alpha {\varphi }_j/2}{b}_j|u_j\rangle |11\rangle }}\right)\\ & ={\displaystyle \sum _j{e}^{-\pi i\alpha {\varphi }_j/2}{b}_j|u_j\rangle \frac{1}{\sqrt{2}}}\left(|0\rangle +|1\rangle \right)\frac{1}{\sqrt{2}}\left(|0\rangle +|1\rangle \right)\end{array},$$

(26)

where IF denotes the inverse Fourier transform. Finally, through the H gate, we obtain $|{\psi }_7\rangle$:

$$|{\psi }_7\rangle =|0\rangle |0\rangle {\displaystyle \sum _j{e}^{-\pi i\alpha {\varphi }_j/2}{b}_j|u_j\rangle }.$$

(27)

From Equations (1) and (2), we know that

$${\displaystyle \sum _j{e}^{-\pi i{\varphi }_j/2}{b}_j|u_j\rangle }=F|u\rangle ,$$

(28)

$${\displaystyle \sum _j{e}^{-\pi i\alpha {\varphi }_j/2}{b}_j|u_j\rangle }=F^{\alpha }|u\rangle .$$

(29)

Then, Equation (27) can be expressed as:

$$|{\psi }_7\rangle =|0\rangle |0\rangle F^{\alpha }|u\rangle .$$

(30)

Therefore, the QFrFT is obtained.

In Figure 2, the upper half is the control register, and the lower half is the target register. As the eigenvalues of QFT are only 4, the scale of the QFT and IQFT in the control register is 2-qubit, while the scale of the QFT and IQFT in the target register can be arbitrary. The QFrFT uses two additional auxiliary qubits (i.e., control registers) in space. In time, when the target register is large, only the quantum gate complexity is considered. The complexity of the QFT is $O\left(n^2\right)$, where n is the number of qubits of the target register [1].

3. Classical Algorithm Corresponding to the QFrFT

There exist diverse definitions for the DFrFT [29]. This section analyses the correspondence between the proposed QFrFT and classical algorithms. The design idea underlying the proposed QFrFT is the fractional power of the eigenvalues. We know that the DFT can be expressed as eigenvalues and eigenvectors, as shown in Equation (31):

$$F=VD{V}^T,$$

(31)

where $D$ is the eigenvalue matrix and $V=\left[v_0|v_1|\cdots v_{N-2}|v_{N-1}\right]$ is the eigenvector. Thus, the DFrFT can be expressed as:

$$F^{\alpha }=V{D}^{\alpha }{V}^T.$$

(32)

The eigenvalues of the FT are 1, i, −1, and −i. Therefore, Equations (31) and (32) can be further expressed as:

$$F={\displaystyle \sum _{k\in E_1}\left(1\right)v_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}\left(-i\right)v_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}\left(-1\right)v_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}\left(i\right)v_k{v}_k^T}$$

(33)

and

$$F^{\alpha }={\displaystyle \sum _{k\in E_1}{\left(1\right)}^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(-i\right)}^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{\left(-1\right)}^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{\left(i\right)}^{\alpha }{v}_k{v}_k^T},$$

(34)

where $E_1$, $E_2$, $E_3$, and $E_4$ represent the set of subscripts corresponding to the respective eigenvalues and eigenvectors. For the QFrFT in Figure 2, the important calculation steps are

$$\begin{array}{ll}IQFT\left(I|00\rangle +F|01\rangle +F^2|10\rangle +F^3|11\rangle \right)& =IQFT\left(\begin{array}{l}I\\ F\\ F^2\\ F^3\end{array}\right)\\ & =\frac{1}{\sqrt{4}}\left(\begin{array}{cccc}1& 1& 1& 1\\ 1& i& -1& -i\\ 1& -1& 1& -1\\ 1& -i& -1& i\end{array}\right)\left(\begin{array}{l}I\\ F\\ F^2\\ F^3\end{array}\right)\end{array}.$$

(35)

Let

$$\{\begin{array}{l}{Y}_0=I+F+F^2+F^3\\ Y_1=I+iF-F^2-i{F}^3\\ Y_2=I-F+F^2-F^3\\ Y_3=I-iF-F^2+i{F}^3\end{array}.$$

(36)

At this point, we can obtain

$$\{\begin{array}{l}{Y}_0|u\rangle =\left(D^0+D^1+D^2+D^3\right)|u\rangle \\ Y_1|u\rangle =\left(D^0+i{D}^1-D^2-i{D}^3\right)|u\rangle \\ Y_2|u\rangle =\left(D^0-D^1+D^2-D^3\right)|u\rangle \\ Y_3|u\rangle =\left(D^0-i{D}^1-D^2+i{D}^3\right)|u\rangle \end{array}.$$

(37)

Then,

$$\{\begin{array}{l}{Y}_0|u\rangle =Q|u\rangle \\ Y_1|u\rangle =W|u\rangle \\ Y_2|u\rangle =E|u\rangle \\ Y_3|u\rangle =R|u\rangle \end{array}.$$

(38)

From the above results in Equations (14)–(17) for Q, W, E, and R, Equation (36) can be expressed as:

$$\{\begin{array}{l}{Y}_0={\displaystyle \sum _{k\in E_1}{v}_k{v}_k^T}\\ Y_1={\displaystyle \sum _{k\in E_2}{v}_k{v}_k^T}\\ Y_2={\displaystyle \sum _{k\in E_3}{v}_k{v}_k^T}\\ Y_3={\displaystyle \sum _{k\in E_4}{v}_k{v}_k^T}\end{array}.$$

(39)

From the phase gates shown in Figure 2, we can obtain

$$\left(\begin{array}{cc}1& 0\\ 0& e^{-\pi i\alpha }\end{array}\right)\otimes \left(\begin{array}{cc}1& 0\\ 0& e^{-\pi i\alpha /2}\end{array}\right)=\left(\begin{array}{cccc}1& & & 0\\ & e^{-\pi i\alpha /2}& & \\ & & e^{-2\pi i\alpha /2}& \\ 0& & & e^{-3\pi i\alpha /2}\end{array}\right)=\left(\begin{array}{cccc}{\left(1\right)}^{\alpha }& & & 0\\ & {\left(-i\right)}^{\alpha }& & \\ & & {\left(-1\right)}^{\alpha }& \\ 0& & & {\left(i\right)}^{\alpha }\end{array}\right).$$

(40)

Thus, Equation (34) can be obtained by combining Equations (39) and (40). Thus, the mathematical principle of the QFrFT is explained.

Considering the many definitions of the DFrFT [29], we found that Shih’s FrFT is consistent with the proposed QFrFT. In 1995, Shih proposed the FrFT using the weighted combination of four functions [30]. Its definition can be expressed as:

$$(F^{\alpha }f)(t)={\displaystyle \sum _{l=0}^3{A}_l^{\alpha }{f}_l(t)},$$

(41)

where $f_l\left(t\right)=F^l\left[f\left(t\right)\right];l=0,1,2,3$. The weighting coefficient $A_l^{\alpha }$ can be further written as:

$$\begin{array}{ll}{A}_l^{\alpha }& =\cos \left(\frac{(\alpha -l)\pi }{4}\right)\cos \left(\frac{2(\alpha -l)\pi }{4}\right)\exp \left(-\frac{3(\alpha -l)i\pi }{4}\right)\\ & =\frac{1}{2}\times \left[\exp \left(\frac{(\alpha -l)\pi i}{4}\right)+\exp \left(\frac{-(\alpha -l)\pi i}{4}\right)\right]\\ & \hspace{1em}\times \frac{1}{2}\times \left[\exp \left(\frac{2(\alpha -l)\pi i}{4}\right)+\exp \left(\frac{-2(\alpha -l)\pi i}{4}\right)\right]\times \exp \left(-\frac{3(\alpha -l)i\pi }{4}\right)\\ & =\frac{1}{4}\left(1+\exp \left(-\frac{2(\alpha -l)\pi i}{4}\right)+\exp \left(-\frac{4(\alpha -l)\pi i}{4}\right)+\exp \left(-\frac{6(\alpha -l)\pi i}{4}\right)\right)\\ & =\frac{1}{4}{\displaystyle \sum _{k=0}^3\exp \left(-\frac{2\pi i}{4}(\alpha -l)k\right)}\\ & =\frac{1}{4}{\displaystyle \sum _{k=0}^3\exp \left(-\frac{2\pi i\alpha k}{4}\right)}\exp \left(\frac{2\pi ikl}{4}\right)\end{array}.$$

(42)

Therefore, the weighting coefficient $A_l^{\alpha }$ can be written in matrix form

$$\left(\begin{array}{l}{A}_0^{\alpha }\\ A_1^{\alpha }\\ A_2^{\alpha }\\ A_3^{\alpha }\end{array}\right)=\frac{1}{4}\left(\begin{array}{cccc}1& 1& 1& 1\\ 1& i& -1& -i\\ 1& -1& 1& -1\\ 1& -i& -1& i\end{array}\right)\left(\begin{array}{l}{B}_0^{\alpha }\\ B_1^{\alpha }\\ B_2^{\alpha }\\ B_3^{\alpha }\end{array}\right),$$

(43)

where $B_k^{\alpha }=\exp \left(-\frac{2\pi ik\alpha }{4}\right)$; $k=0,1,2,3$. Thus, Shih’s FrFT can be expressed as:

$$\begin{array}{ll}{F}^{\alpha }\left[f\left(t\right)\right]& =\left(I,F,F^2,F^3\right)\left(\begin{array}{c}{A}_0^{\alpha }\\ A_1^{\alpha }\\ A_2^{\alpha }\\ A_3^{\alpha }\end{array}\right)f\left(t\right)\\ & =\frac{1}{4}\left(I,F,F^2,F^3\right)\left(\begin{array}{cccc}1& 1& 1& 1\\ 1& i& -1& -i\\ 1& -1& 1& -1\\ 1& -i& -1& i\end{array}\right)\left(\begin{array}{l}{B}_0^{\alpha }\\ B_1^{\alpha }\\ B_2^{\alpha }\\ B_3^{\alpha }\end{array}\right)f\left(t\right)\end{array}.$$

(44)

From Equation (36), Shih’s FrFT can be further expressed as:

$$F^{\alpha }\left[f\left(t\right)\right]=\frac{1}{4}\left(Y_0,Y_1,Y_2,Y_3\right)\left(\begin{array}{l}{B}_0^{\alpha }\\ B_1^{\alpha }\\ B_2^{\alpha }\\ B_3^{\alpha }\end{array}\right)f\left(t\right),$$

(45)

where $B_k^{\alpha }=\exp \left(-\frac{2\pi ik\alpha }{4}\right)$; $k=0,1,2,3$ and

$$\left(\begin{array}{l}{B}_0^{\alpha }\\ B_1^{\alpha }\\ B_2^{\alpha }\\ B_3^{\alpha }\end{array}\right)=\left(\begin{array}{c}{\left(1\right)}^{\alpha }\\ {\left(-i\right)}^{\alpha }\\ {\left(-1\right)}^{\alpha }\\ {\left(i\right)}^{\alpha }\end{array}\right).$$

(46)

Thus, we obtain Equation (34). Therefore, Shih’s FrFT coincides with the proposed QFrFT.

4. Implementation of Quantum Multi-Fractional Fourier Transform

Shih’s FrFT has many extended definitions [31,32,33,34], which are collectively referred to as the multi-fractional Fourier transform (mFrFT). Next, we focus on the design of quantum algorithms for the mFrFT. The implementation of its quantum algorithm will be of great help in the fields of quantum computing and quantum information security. In 2000, Zhu et al. proposed the definition of the mFrFT and applied it to image encryption [31]. This definition can be expressed as:

$$F_M^{\alpha }\left[f\left(t\right)\right]={\displaystyle \sum _{l=0}^{M-1}{A}_l^{\alpha }{f}_l\left(t\right)},$$

(47)

where the basic functions are $f_l\left(t\right)=F^{4l/M}\left[f(t)\right];l=0,1,\cdots ,M-1$. The weighting coefficient $A_l^{\alpha }$ can be expressed as:

$$\begin{array}{ll}{A}_l^{\alpha }& =\frac{1}{M}{\displaystyle \sum _{k=0}^{M-1}\exp \left[-\frac{2\pi ik\left(\alpha -l\right)}{M}\right]}\\ & =\frac{1}{M}{\displaystyle \sum _{k=0}^{M-1}\exp \left(\frac{2\pi ikl}{M}\right)\exp \left(-\frac{2\pi i\alpha k}{M}\right)}\\ & =IDFT{\left[e^{\left(-2\pi i\alpha k/M\right)}\right]}_{k=0,1,2,\cdots \cdots ,M-1}\end{array}.$$

(48)

where IDFT is the inverse discrete Fourier transform. Furthermore, the weighting coefficients $A_l^{\alpha }$ can be expressed in matrix form as:

$$\left(\begin{array}{c}{A}_0^{\alpha }\\ A_1^{\alpha }\\ ⋮\\ A_{M-1}^{\alpha }\end{array}\right)=\frac{1}{M}\left(\begin{array}{cccc}{w}^{0\times 0}& w^{0\times 1}& \cdots & w^{0\times (M-1)}\\ w^{1\times 0}& w^{1\times 1}& \cdots & w^{1\times (M-1)}\\ ⋮& ⋮& \ddots & ⋮\\ w^{(M-1)\times 0}& w^{(M-1)\times 1}& \cdots & w^{(M-1)\times (M-1)}\end{array}\right)\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\\ ⋮\\ B_{M-1}^{\alpha }\end{array}\right),$$

(49)

where $w=\exp \left(2\pi i/M\right)$ and $B_k^{\alpha }=\exp \left(-\frac{2\pi ik\alpha }{M}\right),k=0,1,\cdots ,M-1$. Therefore, Equation (47) can be expressed as:

$$\begin{array}{ll}{F}_M^{\alpha }\left[f\left(t\right)\right]& =A_0^{\alpha }{f}_0\left(t\right)+A_1^{\alpha }{f}_1\left(t\right)+\cdots +A_{M-1}^{\alpha }{f}_{M-1}\left(t\right)\\ & =A_0^{\alpha }{F}^{\frac{0}{M}}\left[f\left(t\right)\right]+A_1^{\alpha }{F}^{\frac{4}{M}}\left[f\left(t\right)\right]+\cdots +A_{M-1}^{\alpha }{F}^{\frac{4\left(M-1\right)}{M}}\left[f\left(t\right)\right]\\ & =\left(I,F^{\frac{4}{M}},\cdots ,F^{\frac{4\left(M-1\right)}{M}}\right)\left(\begin{array}{c}{A}_0^{\alpha }\\ A_1^{\alpha }\\ ⋮\\ A_{M-1}^{\alpha }\end{array}\right)f\left(t\right)\end{array},$$

(50)

Further, we can obtain

$$F_M^{\alpha }\left[f\left(t\right)\right]=\frac{1}{M}\left(I,F^{\frac{4}{M}},\cdots ,F^{\frac{4\left(M-1\right)}{M}}\right)\left(\begin{array}{cccc}{w}^{0\times 0}& w^{0\times 1}& \cdots & w^{0\times (M-1)}\\ w^{1\times 0}& w^{1\times 1}& \cdots & w^{1\times (M-1)}\\ ⋮& ⋮& \ddots & ⋮\\ w^{(M-1)\times 0}& w^{(M-1)\times 1}& \cdots & w^{(M-1)\times (M-1)}\end{array}\right)\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\\ ⋮\\ B_{M-1}^{\alpha }\end{array}\right)f\left(t\right).$$

(51)

Here, let

$$\{\begin{array}{l}{Y}_0=w^{0\times 0}I+w^{1\times 0}{F}^{\frac{4}{M}}+\cdots +w^{\left(M-1\right)\times 0}{F}^{\frac{4\left(M-1\right)}{M}}\\ Y_1=w^{0\times 1}I+w^{1\times 1}{F}^{\frac{4}{M}}+\cdots +w^{\left(M-1\right)\times 1}{F}^{\frac{4\left(M-1\right)}{M}}\\ Y_2=w^{0\times 2}I+w^{1\times 2}{F}^{\frac{4}{M}}+\cdots +w^{\left(M-1\right)\times 2}{F}^{\frac{4\left(M-1\right)}{M}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em} ⋮\\ Y_{M-1}=w^{0\times \left(M-1\right)}I+w^{1\times \left(M-1\right)}{F}^{\frac{4}{M}}+\cdots +w^{\left(M-1\right)\times \left(M-1\right)}{F}^{\frac{4\left(M-1\right)}{M}}\end{array}.$$

(52)

In this way, the mFrFT can also be expressed as Equation (53):

$$\begin{array}{ll}{F}_M^{\alpha }\left[f\left(t\right)\right]& =\frac{1}{M}\left(Y_0,Y_1,\cdots ,Y_{M-1}\right)\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\\ ⋮\\ B_{M-1}^{\alpha }\end{array}\right)f\left(t\right)\\ & =\frac{1}{M}{\displaystyle \sum _{k=0}^{M-1}{Y}_k}{B}_k^{\alpha }f\left(t\right)\end{array},$$

(53)

where $B_k^{\alpha }=\exp \left(-\frac{2\pi ik\alpha }{M}\right);k=0,1,\cdots ,M-1$. Next, analyzing the results of Equation (52), from Equation (37), we obtain

$$\{\begin{array}{l}{Y}_0|u\rangle =\left(w^{0\times 0}{D}^0+w^{1\times 0}{D}^{\frac{4}{M}}+\cdots +w^{\left(M-1\right)\times 0}{D}^{\frac{4\left(M-1\right)}{M}}\right)|u\rangle \\ Y_1|u\rangle =\left(w^{0\times 1}{D}^0+w^{1\times 1}{D}^{\frac{4}{M}}+\cdots +w^{\left(M-1\right)\times 1}{D}^{\frac{4\left(M-1\right)}{M}}\right)|u\rangle \\ Y_2|u\rangle =\left(w^{0\times 2}{D}^0+w^{1\times 2}{D}^{\frac{4}{M}}+\cdots +w^{\left(M-1\right)\times 2}{D}^{\frac{4\left(M-1\right)}{M}}\right)|u\rangle \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}⋮\\ Y_{M-1}|u\rangle =\left(w^{0\times \left(M-1\right)}{D}^0+w^{1\times \left(M-1\right)}{D}^{\frac{4}{M}}+\cdots +w^{\left(M-1\right)\times \left(M-1\right)}{D}^{\frac{4\left(M-1\right)}{M}}\right)|u\rangle \end{array}.$$

(54)

Let

$$\{\begin{array}{l}{Q}_0=w^{0\times 0}{D}^0+w^{1\times 0}{D}^{\frac{4}{M}}+\cdots +w^{\left(M-1\right)\times 0}{D}^{\frac{4\left(M-1\right)}{M}}\\ Q_1=w^{0\times 1}{D}^0+w^{1\times 1}{D}^{\frac{4}{M}}+\cdots +w^{\left(M-1\right)\times 1}{D}^{\frac{4\left(M-1\right)}{M}}\\ Q_2=w^{0\times 2}{D}^0+w^{1\times 2}{D}^{\frac{4}{M}}+\cdots +w^{\left(M-1\right)\times 2}{D}^{\frac{4\left(M-1\right)}{M}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}⋮\\ Q_{M-1}=w^{0\times \left(M-1\right)}{D}^0+w^{1\times \left(M-1\right)}{D}^{\frac{4}{M}}+\cdots +w^{\left(M-1\right)\times \left(M-1\right)}{D}^{\frac{4\left(M-1\right)}{M}}\end{array}.$$

(55)

Then,

$$Q_0=w^{0\times 0}{\lambda }_j^0+w^{1\times 0}{\lambda }_j^{{}^{\frac{4}{M}}}+\cdots +w^{\left(M-1\right)\times 0}{\lambda }_j^{{}^{\frac{4\left(M-1\right)}{M}}}={\lambda }_j^0+{\lambda }_j^{{}^{\frac{4}{M}}}+\cdots +{\lambda }_j^{{}^{\frac{4\left(M-1\right)}{M}}},$$

(56)

where ${\lambda }_j\in \left(1,i,-1,-i\right)$. For proof, we set ${\lambda }_j\in \left(e^{\pi i0/2},e^{\pi i1/2},e^{\pi i2/2},e^{\pi i3/2}\right)$.

When ${\lambda }_j=1$,

$${\lambda }_j^0+{\lambda }_j^{{}^{\frac{4}{M}}}+\cdots +{\lambda }_j^{{}^{\frac{4\left(M-1\right)}{M}}}=M.$$

(57)

When ${\lambda }_j=i$,

$$\begin{array}{ll}{\lambda }_j^0+{\lambda }_j^{{}^{\frac{4}{M}}}+\cdots +{\lambda }_j^{{}^{\frac{4\left(M-1\right)}{M}}}& =1+e^{2\pi i1/M}+e^{2\pi i2/M}+\cdots +e^{2\pi i\left(M-1\right)/M}\\ & =\frac{1-{\left(e^{2\pi i/M}\right)}^M}{1-e^{2\pi i/M}}\\ & =\frac{1-1}{1-e^{2\pi i/M}}\\ & =0\end{array}.$$

(58)

When ${\lambda }_j=-1$,

$$\begin{array}{ll}{\lambda }_j^0+{\lambda }_j^{{}^{\frac{4}{M}}}+\cdots +{\lambda }_j^{{}^{\frac{4\left(M-1\right)}{M}}}& =1+e^{4\pi i1/M}+e^{4\pi i2/M}+\cdots +e^{4\pi i\left(M-1\right)/M}\\ & =\frac{1-{\left(e^{4\pi i/M}\right)}^M}{1-e^{2\pi i/M}}\\ & =\frac{1-1}{1-e^{2\pi i/M}}\\ & =0\end{array}.$$

(59)

When ${\lambda }_j=-1$,

$$\begin{array}{ll}{\lambda }_j^0+{\lambda }_j^{{}^{\frac{4}{M}}}+\cdots +{\lambda }_j^{{}^{\frac{4\left(M-1\right)}{M}}}& =1+e^{6\pi i1/M}+e^{6\pi i2/M}+\cdots +e^{6\pi i\left(M-1\right)/M}\\ & =\frac{1-{\left(e^{6\pi i/M}\right)}^M}{1-e^{2\pi i/M}}\\ & =\frac{1-1}{1-e^{2\pi i/M}}\\ & =0\end{array}.$$

(60)

So, we can obtain

$$Q_0=\{\begin{array}{l}M,\hspace{1em}{\lambda }_j=1\\ 0,\hspace{1em}\hspace{0.17em}{\lambda }_j=i,-1,-i\end{array}.$$

(61)

Next, we calculate $Q_1$ as

$$Q_1=w^{0\times 1}{D}^0+w^{1\times 1}{D}^{\frac{4}{M}}+\cdots +w^{\left(M-1\right)\times 1}{D}^{\frac{4\left(M-1\right)}{M}}={\displaystyle \sum _{k=0}^{M-1}{e}^{2\pi ik/M}{\lambda }_j^{4k/M}},$$

(62)

when ${\lambda }_j\in \left(e^{\pi i0/2},e^{\pi i1/2},e^{\pi i2/2},e^{\pi i3/2}\right)$, we have $Q_1=0$. Moreover, our calculation shows that, for Equation (55), $Q_2=Q_3=\cdots =Q_{M-4}=0$.

So, we calculate $Q_{M-3}$ as

$$Q_{M-3}=w^{0\times \left(M-3\right)}{D}^0+w^{1\times \left(M-3\right)}{D}^{\frac{4}{M}}+\cdots +w^{\left(M-1\right)\times \left(M-3\right)}{D}^{\frac{4\left(M-1\right)}{M}}={\displaystyle \sum _{k=0}^{M-1}{e}^{2\pi ik\left(M-3\right)/M}{\lambda }_j^{4k/M}},$$

(63)

where ${\lambda }_j\in \left(e^{\pi i0/2},e^{\pi i1/2},e^{\pi i2/2},e^{\pi i3/2}\right)$. Then, we obtain

$$Q_3={\displaystyle \sum _{k=0}^{M-1}{e}^{2\pi ik\left(M-3\right)/M}{\lambda }_j^{4k/M}}={\displaystyle \sum _{k=0}^{M-1}{e}^{\left(2\pi ikM-6\pi ik+2\pi ikh\right)/M}},$$

(64)

where $h=0,1,2,3$. For Equation (64), when $h=3$, the result is non-zero and the eigenvalue is $e^{\pi i3/2}=-i$,

$$Q_{M-3}=\{\begin{array}{l}M\hspace{1em}\lambda =-i\\ 0\hspace{1em}\hspace{0.17em}\lambda =1,i,-1\end{array}.$$

(65)

For $Q_{M-2}$,

$$\begin{array}{ll}{Q}_{M-2}& =w^{0\times \left(M-2\right)}{D}^0+w^{1\times \left(M-2\right)}{D}^{\frac{4}{M}}+\cdots +w^{\left(M-1\right)\times \left(M-2\right)}{D}^{\frac{4\left(M-1\right)}{M}}\\ & ={\displaystyle \sum _{k=0}^{M-1}{e}^{2\pi ik\left(M-2\right)/M}{\lambda }_j^{4k/M}}\\ & ={\displaystyle \sum _{k=0}^{M-1}{e}^{\left(2\pi ikM-4\pi ik+2\pi ikh\right)/M}}\end{array}.$$

(66)

For Equation (66), when $h=2$, the result is non-zero and the eigenvalue is $e^{\pi i2/2}=-1$,

$$Q_{M-2}=\{\begin{array}{l}M\hspace{1em}\lambda =-1\\ 0\hspace{1em}\hspace{0.17em}\lambda =1,i,-i\end{array}.$$

(67)

For $Q_{M-1}$,

$$\begin{array}{ll}{Q}_{M-1}& =w^{0\times \left(M-1\right)}{D}^0+w^{1\times \left(M-1\right)}{D}^{\frac{4}{M}}+\cdots +w^{\left(M-1\right)\times \left(M-1\right)}{D}^{\frac{4\left(M-1\right)}{M}}\\ & ={\displaystyle \sum _{k=0}^{M-1}{e}^{2\pi ik\left(M-1\right)/M}{\lambda }_j^{4k/M}}\\ & ={\displaystyle \sum _{k=0}^{M-1}{e}^{\left(2\pi ikM-2\pi ik+2\pi ikh\right)/M}}\end{array}.$$

(68)

For Equation (68), when $h=1$, the result is non-zero and the eigenvalue is $e^{\pi i/2}=i$,

$$Q_{M-1}=\{\begin{array}{l}M\hspace{1em}\lambda =i\\ 0\hspace{1em}\hspace{0.17em}\lambda =1,-1,-i\end{array}.$$

(69)

Therefore, for Equation (52), only $Y_0$, $Y_{M-3}$, $Y_{M-2}$, and $Y_{M-1}$ are non-zero, while the other terms are zero. If the eigenvalue ${\lambda }_j\in \left(e^{-\pi i0/2},e^{-\pi i1/2},e^{-\pi i2/2},e^{-\pi i3/2}\right)$, in Equation (55), only $Q_0$, $Q_1$, $Q_2$, and $Q_3$ are non-zero, while the other results are zero. For Equation (53), there are only four effective weighting terms for the mFrFT.

However, for Equation (53), when $B_k^{\alpha }$ takes different values, we obtain different definitions [35]. For example, $B_k^{\alpha }=\exp \left(-\frac{2\pi ik\alpha }{4}\right)$; $k=0,1,2,3$ gives Shih’s FrFT [30]; $B_k^{\alpha }=\exp \left(-\frac{2\pi ik\alpha }{M}\right)$; $k=0,1,\cdots ,M-1$ gives the mFrFT of Zhu et al. [31]; $B_k^{\alpha }=\exp \left[-\frac{2\pi i\left(m_{k}M+1\right)\alpha \left(n_{k}M+k\right)}{M}\right]$; $k=0,1,\cdots ,M-1$ gives the multi-parameter fractional Fourier transform of Tao et al. [32]; $B_k^{\alpha }=\exp \left[-\frac{2\pi i\alpha \left(r_{k}M+l\right)}{M}\right]$; $k=0,1,\cdots ,M-1$ gives the modified multi-parameter fractional Fourier transform of Ran et al. [33]; and $B_k^{\overline{\alpha }}=\exp \left[-\frac{2\pi i{\alpha }_l\left(r_{l}M+l\right)}{M}\right]$; $k=0,1,\cdots ,M-1$ gives the vector power multi-parameter fractional Fourier transform of Ran et al. [34].

Obviously, all of these extended definitions also have only four effective weighting terms, which has a great impact on the application of these algorithms in signal processing and image encryption.

5. Discussion

5.1. Phase Gates

For the QFrFT shown in Figure 2, we select the phase gates $R_Z\left(-\pi \alpha \right)$ and $R_Z\left(-\pi \alpha /2\right)$. If the phase gates $R_Z\left(\pi \alpha \right)$ and $R_Z\left(\pi \alpha /2\right)$ are selected, the QFrFT circuit is as shown in Figure 3.

From Figure 3, we can see that the phase gates have changed, and the order of the QFT and the IQFT regarding the 2-qubits in the control register has also changed. This is to ensure that the eigenvalues correspond to the eigenvectors. The correspondence between the eigenvalues of the control register is:

$$\{\begin{array}{l}{\lambda }_j=1\to |00\rangle \\ {\lambda }_j=i\to |01\rangle \\ {\lambda }_j=-1\to |10\rangle \\ {\lambda }_j=-i\to |11\rangle \end{array}.$$

(70)

From the phase gates $R_Z\left(\pi \alpha \right)$ and $R_Z\left(\pi \alpha /2\right)$, we obtain

$$\left(\begin{array}{cc}1& 0\\ 0& e^{\pi i\alpha }\end{array}\right)\otimes \left(\begin{array}{cc}1& 0\\ 0& e^{\pi i\alpha /2}\end{array}\right)=\left(\begin{array}{cccc}1& & & 0\\ & e^{\pi i\alpha /2}& & \\ & & e^{2\pi i\alpha /2}& \\ 0& & & e^{3\pi i\alpha /2}\end{array}\right)=\left(\begin{array}{cccc}{\left(1\right)}^{\alpha }& & & 0\\ & {\left(i\right)}^{\alpha }& & \\ & & {\left(-1\right)}^{\alpha }& \\ 0& & & {\left(-i\right)}^{\alpha }\end{array}\right).$$

(71)

This ensures the correspondence between eigenvalues and eigenvectors. Thus, we have

$$F={\displaystyle \sum _{k\in E_1}\left(1\right)v_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}\left(i\right)v_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}\left(-1\right)v_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}\left(-i\right)v_k{v}_k^T}$$

(72)

and

$$F^{\alpha }={\displaystyle \sum _{k\in E_1}{\left(1\right)}^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(i\right)}^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{\left(-1\right)}^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{\left(-i\right)}^{\alpha }{v}_k{v}_k^T}.$$

(73)

The QFrFT and Shih’s FrFT are the same. To ensure that the eigenvalues correspond to the eigenvectors, when the eigenvalues become fractional powers, they also correspond to the eigenvectors. Therefore, the key idea of the algorithm’s design is to retain the eigenvector of the FT, such that the eigenvalue takes a fractional power.

Considering Figure 2, if we only change the phase gates, the result is shown in Figure 4. The relevant correspondence between the eigenvalues and eigenvectors is shown in Equations (33) and (73). In this way, the obtained results do not correspond to the feature vector of the FT and, so, the design scheme shown in Figure 4 is not feasible.

5.2. Flaws of the mFrFT

From the design of the quantum mFrFT, we find that there are many invalid weighting terms in the definition of classical mFrFT. The reason for this problem is due to the eigenvalues of the algorithm. We know that the eigenvalues of the FT are 1, i, −1 and −i, and the time–frequency spatial distribution is shown in Figure 5. For Equation (54), the sum of eigenvalues can be expressed as a change in angle in Figure 5.

For angle $\theta$, we set

$$w=e^{i\theta },$$

(74)

where $\theta =2\pi /M$. We can obtain

$$1+w+w^2+\cdots +w^{M-1}=0.$$

(75)

Furthermore, we have

$$1+w^h+w^{2h}+\cdots +w^{\left(M-1\right)h}=\{\begin{array}{l}M\hspace{1em}h=0\\ 0\hspace{1em} h\ne 0\end{array}.$$

(76)

Thus, Equation (64) can be expressed as

$${\displaystyle \sum _{k=0}^{M-1}{e}^{\left(2\pi ikM-6\pi ik+2\pi ikh\right)/M}}={\displaystyle \sum _{k=0}^{M-1}{e}^{2\pi ik}{e}^{\frac{2\pi ik}{M}\left(h-3\right)}}.$$

(77)

Here, $e^{2\pi ik}=1$. Therefore, the result of Equation (77) is non-zero only when $h=3$. The other results are also confirmed.

Since 2000, Zhu’s mFrFT [31] and previous results [36], as well as some extended definitions, have been shown to have flaws [32,33,34], which we discussed in terms of the eigenvalues of the QFT. In the extended definition, the number of effective weighting terms depends on the period of the matrix; for example, the period of the FT matrix is 4, so its effective weighting is 4, while the period of the Hadamard transform is 2, so the effective weighting term is defined as 2 accordingly.

6. Conclusions

In this paper, we used the QFT and QPE to design the QFrFT. We retained the eigenvectors of the QFT and forced its eigenvalues to take fractional powers. As diverse definitions for the DFrFT exist, we determined the correspondence between the proposed QFrFT and classical algorithms, which lays the foundation for future application of the QFrFT to solve practical problems. Regarding the design of the quantum mFrFT, we observed a flaw in the definition of classical algorithms—namely, there are only four effective weighting terms in a series of mFrFT definitions—which has a significant impact on the applicability of the algorithms. Finally, we discussed the correspondence between eigenvalues and eigenvectors in the QFrFT circuit and analyzed the definition of the algorithm in depth. Furthermore, the defects of the mFrFT were mathematically explained.