Quantum Weighted Fractional-Order Transform

Abstract

Quantum Fourier transform (QFT) transformation plays a very important role in the design of many quantum algorithms. Fractional Fourier transform (FRFT), as an extension of the Fourier transform, is particularly important due to the design of its quantum algorithm. In this paper, a new reformulation of the weighted fractional Fourier transform (WFRFT) is proposed in order to realize quantum FRFT; however, we found that this reformulation can be applied to other transformations, and therefore, this paper presents the weighted fractional Hartley transform (WFRHT). For the universality of application, we further propose a general weighted fractional-order transform (WFRT). When designing the quantum circuits, we realized the quantum WFRFT via QFT and quantum phase estimation (QPE). Moreover, after extending our design to the WFRHT, we were able to formulate the quantum WFRHT. Finally, in accordance with the research results, we designed the quantum circuit of the general WFRT, and subsequently proposed the quantum WFRT. The research in this paper has great value as a reference for the design and application of quantum algorithms.

1. Introduction

The human desire for computing power is endless, and improvements in computing power are closely related to the progress of civilization. Quantum computing comprises parallel computations; this shows that quantum computing quantum algorithms have great advantages over traditional, classical algorithms. Computers are composed of the following four elements: electron tubes, transistors, integrated circuits, and large-scale integrated circuits. As electronic computers continue to develop, chip integration is becoming more common, and a single transistor will eventually be measured on the nanoscopic scale; however, as transistors continue to shrink in size, the physical and chemical properties of the material will qualitatively change, which will lead to the gradual disappearance of the Molar law [1], and the emergence of the quantum effect. Therefore, quantum computing will be important for the next generation of computer research. In 1980, Benioff proposed the quantum Turing machine model, which opened new avenues for quantum computing research [2]. Furthermore, Feynman pointed out that quantum computing has the ability to complete tasks that classical computers cannot [3]. Moreover, the Deutsch–Jozsa algorithm was one of the first to prove the superiority of quantum computing; this algorithm also conceived the physical implementation of the quantum Turing machine and the general quantum computer [4,5]. In 1994, Shor’s algorithm was proposed. Shor’s algorithm can effectively solve the problem of order finding and factorization using quantum Fourier transform, quantum phase estimation, and other basic operations that involve quantum computing. Compared with classical algorithms, Shor’s algorithm can achieve exponential acceleration; undoubtedly, this is a great challenge for public-key cryptography [6]. In 1996, Grover proposed a square root accelerated quantum search algorithm for N unstructured databases with disordered records [7]. The Grover algorithm aims to use quantum computing technology to solve search problems, and it can achieve square-level acceleration for classical search algorithms. With the introduction of two efficient quantum algorithms, researchers have paid an increasing level of attention to quantum computing; as a result, many improved Grover algorithms appeared [8,9,10,11,12], which has further promoted the development of quantum computing. In addition, applied quantum algorithms have emerged in large numbers; these algorithms are also based on the Grover algorithm [13,14,15,16,17,18]. In 2009, the emergence of the HHL algorithm once again proved the superiority of the quantum algorithm [19], and improvements to the HHL algorithm have also been proposed [20,21,22]. In 2012, Preskill proposed the term “quantum supremacy”; this term highlights that quantum computers may easily answer a class of problems that are complex for classical computers [23]. Moreover, quantum supremacy can be demonstrated on quantum computers using random quantum circuits of more than 50 qubits [24]. A series of quantum computing techniques, such as the quantum Fourier transform (QFT) [25], quantum phase estimation (QPE) [26], and HHL algorithm [19], play a key role in quantum algorithm design; hence, they are called quantum basic linear algebra assemblies [27]. Moreover, these basic operations may be applied to quantum singular value decomposition problems [28] and quantum gradient solving problems [29]. In addition, quantum algorithms play an important role in solving linear equations and differential equations [30,31,32,33,34]. At present, quantum computing has been widely used and it shows great potential.

Out of the abovementioned quantum algorithms, QFT and QPE play an important role in quantum computing; hence, they are called quantum toolbox algorithms. Similar quantum algorithms include the quantum wavelet transform [35], quantum cosine transform [36], quantum Hartley transform [37], quantum black-box [38,39], and quantum random walk [40,41] algorithms, among others. As an extension of the Fourier transform, fractional Fourier transform (FRFT) has a wide range of applications; therefore, the proposal of the quantum fractional Fourier transform (QFRFT) is particularly important. Recently, some studies concerning QFRFT have been conducted. Unfortunately, these studies do not present a complete quantum circuit [42,43,44,45]. Parasa et al. proposed a quantum pseudo-fractional Fourier transform (QPFRFT) using multi-valued logic [42]; however, it is not a proper QFRFT because it is not unitary. Recently, we presented a QFRFT using a quantum artificial neural network, which required using more qubits [46]. There are several diverse definitions of FRFT [47,48]. In this paper, the weighted fractional Fourier transform (WFRFT) was studied because it is unitary. First, we propose a new reformulation of the WFRFT, and we designed its quantum circuit using QFT and QPE; secondly, we propose a definition for the weighted fractional Hartley transform (WFRHT), and the design for the quantum circuit is presented; and finally, the general weighted fractional-order transform (WFRT) is defined and the design for its quantum circuit is proposed.

The remainder of this paper is organized as follows. The theories behind WFRFT, WFRHT, and WFRT are analyzed in Section 2. The quantum WFRFT is proposed in Section 3. The quantum WFRHT is proposed in Section 4. Section 5 discusses the quantum circuit of the WFRT. Finally, the conclusions are presented in Section 6.

2. Weighted Fractional-Order Transform

In 1995, the WFRFT was proposed [49]. We have studied this algorithm in great detail, and in this paper, we propose a new reformulation of this algorithm; we found that this new reformulation can be used with many new algorithms, such as the WFRHT, weighted fractional sine transform (WFRST), weighted fractional cosine transform (WFRCT), and weighted fractional Hadamard transform (WFRHaT) algorithms, among others. Furthermore, we found that when a periodic matrix is given, we can define a new algorithm; thus, we present the definition of the general WFRT algorithm in Section 2.3.

2.1. Weighted Fractional Fourier Transform

In 1995, Shih proposed the WFRFT using the integer power operation of the Fourier transform [49]. As a generalized version of the Fourier transform, the algorithm can be widely used in signal processing and image encryption, among other fields [50,51]; it is defined as follows:

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

(1)

Here, $f_0(t)=f(t)$, $f_1(t)=(F{f}_0)(t)$, $f_2(t)=(F{f}_1)(t)$ and $f_3(t)=(F{f}_2)(t)$ (F denotes the Fourier transform). The weighting coefficient $A_l^{\alpha }$ can be expressed as:

$$A_l^{\alpha }=\cos \left(\frac{(\alpha -l)\pi }{4}\right)\cos \left(\frac{2(\alpha -l)\pi }{4}\right)\exp \left(-\frac{3(\alpha -l)i\pi }{4}\right).$$

(2)

For Equation (1), we can also write

$$\begin{array}{ll}\hfill F^{\alpha }\left[f\left(t\right)\right]& ={\displaystyle \sum _{l=0}^3{A}_l^{\alpha }{f}_l\left(t\right)}\hfill \\ & =A_0^{\alpha }{f}_0\left(t\right)+A_1^{\alpha }{f}_1\left(t\right)+A_2^{\alpha }{f}_2\left(t\right)+A_3^{\alpha }{f}_3\left(t\right)\\ & =A_0^{\alpha }If\left(t\right)+A_1^{\alpha }Ff\left(t\right)+A_2^{\alpha }{F}^{2}f\left(t\right)+A_3^{\alpha }{F}^{3}f\left(t\right)\\ & =\left(A_0^{\alpha }I+A_1^{\alpha }F+A_2^{\alpha }{F}^2+A_3^{\alpha }{F}^3\right)f\left(t\right)\\ & =\left(I,F,F^2,F^3\right)\left(\begin{array}{c}{A}_0^{\alpha }\\ A_1^{\alpha }\\ A_2^{\alpha }\\ A_3^{\alpha }\end{array}\right)f\left(t\right).\end{array}$$

(3)

From [49], we can obtain Equation (4),

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

(4)

where $B_k^{\alpha }=\exp \left(\frac{2\pi ik\alpha }{4}\right)$; $k=0,1,2,3$. Via inverse transformation, we can obtain Equation (5),

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

(5)

Next, Equation (5) is substituted into Equation (3), so that Equation (6) is obtained.

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

(6)

Let

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

(7)

Thus, a new reformulation of WFRFT is obtained, as shown in Equation (8).

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

(8)

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

Unitarity is a prerequisite for quantum algorithm design. We can easily prove that the WFRFT is unitary using Equation (8) [52]. Moreover, in Section 3, Equation (8) was used to design quantum circuits. The new reformulation that we propose can easily be used with other new algorithms.

2.2. Weighted Fractional Hartley Transform

The abovementioned WFRFT uses the Fourier transform as the base function. We know that the discrete Fourier transform (DFT) is a periodic matrix with eigenvalues (1, i, −1, −i). Similar to the WFRFT, we can also try to define new fractional-order transforms for other periodic matrices; therefore, we used the Hartley transform as an example. The discrete Hartley transform (DHT) has been proposed [53], and its definition is as follows:

$$H=\frac{1}{\sqrt{N}}\left[\cos \left(\frac{2\pi mn}{N}\right)+\sin \left(\frac{2\pi mn}{N}\right)\right].$$

(9)

The DHT comprises period 2; hence, its eigenvalues are 1 and −1. Therefore, we present the definition of WFRHT as:

$$(H^{\alpha }f)(t)={\displaystyle \sum _{l=0}^1{A}_l^{\alpha }{f}_l(t)}.$$

(10)

Here, $f_0(t)=f(t)$ and $f_1(t)=(H{f}_0)(t)$ (H denotes the Hartley transform). The weighting coefficient $A_l^{\alpha }$ can be obtained using Equation (11),

$$\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\end{array}\right)=\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)\left(\begin{array}{c}{A}_0^{\alpha }\\ A_1^{\alpha }\end{array}\right),$$

(11)

where $B_k^{\alpha }=\exp \left(\pi ik\alpha \right)$; $k=0,1$. Thus, Equation (12) is obtained,

$$\left(\begin{array}{c}{A}_0^{\alpha }\\ A_1^{\alpha }\end{array}\right)=\frac{1}{2}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\end{array}\right).$$

(12)

Then, Equation (10) can be re-expressed as:

$$\begin{array}{ll}\hfill F^{\alpha }\left[f\left(t\right)\right]& =\left(I,H\right)\left(\begin{array}{c}{A}_0^{\alpha }\\ A_1^{\alpha }\end{array}\right)f\left(t\right)\hfill \\ & =\frac{1}{2}\left(I,H\right)\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\end{array}\right)f\left(t\right).\end{array}$$

(13)

Let

$$\left\{\begin{array}{l}{Y}_0=I+H\\ Y_1=I-H\end{array}\right.$$

(14)

Therefore, the WFRHT can also be expressed as:

$$H^{\alpha }\left[f\left(t\right)\right]=\frac{1}{2}\left(Y_0,Y_1\right)\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\end{array}\right)f\left(t\right).$$

(15)

Here $B_k^{\alpha }=\exp \left(\pi ik\alpha \right);k=0,1$.

The discrete sine transform, discrete cosine transform, and discrete Hadamard transform, among other discrete matrices, have period 2, and their eigenvalues are 1 and −1; therefore, a new algorithm can be defined by replacing the Hartley transform in Equation (14).

2.3. Weighted Fractional-Order Transform

In the analysis of Section 2.1 and Section 2.2, it is evident that for periodic transforms (that is, when the matrix of a transform is periodic), we can present the definition of its weighted fractional-order. Next, we present the definition of a general WFRT.

Let T denote a periodic matrix (strictly speaking, the matrix should be symmetrical) and $T^M=I$ (M is the period of the matrix T, and I denotes the identity matrix); therefore, the WFRT can be defined as:

$$(T^{\alpha }f)(t)={\displaystyle \sum _{l=0}^{M-1}{A}_l^{\alpha }{f}_l(t)}.$$

(16)

Here, $f_l\left(t\right)=T^l\left(f\left(t\right)\right)$; $l=0,1,\cdots ,M-1$. The weighting coefficient $A_l^{\alpha }$ can be obtained by Equation (17),

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

(17)

where $w=e^{2\pi i/M}$ and $B_k^{\alpha }=e^{2\pi i\alpha k/M}$; $k=0,1,\cdots ,M-1$. Furthermore, we can obtain Equation (18) as follows:

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

(18)

where $u=e^{-2\pi i/M}$. Next, Equation (16) can be re-expressed as:

$$\begin{array}{c}(T^{\alpha }f)(t)={\displaystyle \sum _{l=0}^{M-1}{A}_l^{\alpha }{f}_l(t)}\\ =\left(I,T,\cdots ,T^{M-1}\right)\left(\begin{array}{c}{A}_0^{\alpha }\\ A_1^{\alpha }\\ ⋮\\ A_{M-1}^{\alpha }\end{array}\right)f\left(t\right).\end{array}$$

(19)

Substituting Equation (18) into Equation (19), we can obtain

$$\begin{array}{ll}(T^{\alpha }f)(t)& ={\displaystyle \sum _{l=0}^{M-1}{A}_l^{\alpha }{f}_l(t)}\hfill \\ & =\frac{1}{M}\left(I,T,\cdots ,T^{M-1}\right)\left(\begin{array}{cccc}{u}^{0\times 0}& u^{0\times 1}& \cdots & u^{0\times (M-1)}\\ u^{1\times 0}& u^{1\times 1}& \cdots & u^{1\times (M-1)}\\ ⋮& ⋮& \ddots & ⋮\\ u^{(M-1)\times 0}& u^{(M-1)\times 1}& \cdots & u^{(M-1)\times (M-1)}\end{array}\right)\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\\ ⋮\\ B_{M-1}^{\alpha }\end{array}\right)f\left(t\right).\end{array}$$

(20)

Let

$$\left\{\begin{array}{l}{Y}_0=u^{0\times 0}I+u^{1\times 0}T+\cdots +u^{\left(M-1\right)\times 0}{T}^{M-1}\\ Y_1=u^{0\times 1}I+u^{1\times 1}T+\cdots +u^{\left(M-1\right)\times 1}{T}^{M-1}\\ Y_2=u^{0\times 2}I+u^{1\times 2}T+\cdots +u^{\left(M-1\right)\times 2}{T}^{M-1}\\ ⋮\\ Y_{M-1}=u^{0\times \left(M-1\right)}I+u^{1\times \left(M-1\right)}T+\cdots +u^{\left(M-1\right)\times \left(M-1\right)}{T}^{M-1}\end{array}\right.$$

(21)

Therefore, the WFRT can also be expressed as:

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

(22)

Here $B_k^{\alpha }=e^{2\pi i\alpha k/M}$ and the WFRT is proposed; thus, the WFRFT and the WFRHT presented above are its special cases. For example, M = 4, and T denotes the Fourier transform in Equation (21), which is defined as the WFRFT. If M = 2, T denotes the Hartley transform in Equation (21), which is the WFRHT. This new reformulation can be acceptably applied to the fractional-order definition of periodic functions. Our findings have important value as they can be used as a reference for future information processing and quantum algorithm designs. With increasing demand for greater computing power, we will eventually enter the quantum age. Next, we designed quantum circuits for these algorithms.

3. Quantum Weighted Fractional Fourier Transform

Recently, studies concerning QFRFT have been conducted [42,43,44,45]; however, these research methods describe the QFRFT from the perspective of quantum mechanics, and they do not mention quantum circuits. FRFT definitions are diverse, and some definitions are not unitary [47,48]; this causes issues when designing the QFRFT. As a result of such difficulties, Parasa et al. showed that QFRFT cannot be realized [42]. Recently, we proposed a method to help realize the QFRFT using a quantum artificial neural network; however, this method uses more qubits, and thus, it requires more resources [46]. In this section, we will use QFT and QPE to redesign the QFRFT quantum circuit. Studies by Gidney et al. assisted with our research, as follows [54,55].

First, we used QPE to obtain the eigenvalues of Equation (7), and the quantum circuit of Equation (7) is shown in Figure 1. The upper part represents the control register, the lower part represents the target register, and $\left|u\right.〉$ represents the feature vector that corresponds with the operator in the target register. Here, we used QFT instead of the inverse quantum Fourier transform (IQFT) in the control register, as the purpose of this design is to obtain the same results as Equation (7).

From Figure 1, we know

$$\left|{\psi }_0\right.〉=\left|0\right.〉\otimes \left|0\right.〉\otimes \left|u\right.〉.$$

(23)

Via the H gate, we obtain

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

(24)

Then, the QFT operation of the target register is,

$$\left|{\psi }_2\right.〉=\frac{1}{2}\left(\left|00\right.〉\left|u\right.〉+\left|01\right.〉F\left|u\right.〉+\left|10\right.〉\left|u\right.〉+\left|11\right.〉F\left|u\right.〉\right),$$

(25)

and

$$\left|{\psi }_3\right.〉=\frac{1}{2}\left(\left|00\right.〉I\left|u\right.〉+\left|01\right.〉F\left|u\right.〉+\left|10\right.〉F^2\left|u\right.〉+\left|11\right.〉F^3\left|u\right.〉\right).$$

(26)

Here, $\left|u\right.〉$ is the eigenvector of Fourier transform; therefore, $F\left|u\right.〉=D\left|u\right.〉$ (D is the eigenvalue of Fourier transform). The eigenvalue D can be expressed as:

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

(27)

where ${\lambda }_j\in \left\{1,i,-1,-i\right\}$; $j=0,1,\cdots ,n-1$. Therefore, Equation (26) can be expressed as:

$$\left|{\psi }_3\right.〉=\frac{1}{2}\left(D^0\left|00\right.〉+D^1\left|01\right.〉+D^2\left|10\right.〉+D^3\left|11\right.〉\right)\otimes \left|u\right.〉.$$

(28)

The eigenvector $\left|u\right.〉$ can be written in accordance with standard orthogonal basis conditions, as shown in Equation (29),

$$\left|u\right.〉={\displaystyle \sum _j{b}_j\left|u_j\right.〉},$$

(29)

where $b_j$ is the projection length. Thus, we can obtain

$$D\left|u\right.〉={\displaystyle \sum _j{\lambda }_j{b}_j\left|u_j\right.〉}.$$

(30)

Equation (28) can be expressed as:

$$\left|{\psi }_3\right.〉=\frac{1}{2}\left({\displaystyle \sum _j{\lambda }_j^0{b}_j\left|u_j\right.〉\left|00\right.〉}+{\displaystyle \sum _j{\lambda }_j^1{b}_j\left|u_j\right.〉}\left|01\right.〉+{\displaystyle \sum _j{\lambda }_j^2{b}_j\left|u_j\right.〉\left|10\right.〉}+{\displaystyle \sum _j{\lambda }_j^3{b}_j\left|u_j\right.〉\left|11\right.〉}\right).$$

(31)

The two-qubit QFT is used in the control register, and its matrix can be expressed as:

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

(32)

Therefore, we can obtain the following quantum state $\left|{\psi }_4\right.〉$,

$$\left|{\psi }_4\right.〉={\displaystyle \sum _j\left|{\varphi }_j\right.〉\otimes b_j\left|u_j\right.〉},$$

(33)

where $\left|{\varphi }_j\right.〉\in \left\{\left|00\right.〉,\left|01\right.〉,\left|10\right.〉,\left|11\right.〉\right\}$; the results are shown in Figure 2.

Here, we analyzed the evolutionary process of $\left|{\varphi }_3\right.〉$ to $\left|{\varphi }_4\right.〉$ in detail. Equations (28) and (31) are equivalent; therefore, we can continue to use Equation (28). Then, after the application of the two-qubit QFT, we can obtain

$$\left|{\psi }_4\right.〉=\frac{1}{4}\left(W_0\left|00\right.〉+W_1\left|01\right.〉+W_2\left|10\right.〉+W_3\left|11\right.〉\right)\otimes \left|u\right.〉,$$

(34)

where

$$\left\{\begin{array}{l}{W}_0=D^0+D^1+D^2+D^3\\ W_1=D^0-i{D}^1-D^2+i{D}^3\\ W_2=D^0-D^1+D^2-D^3\\ W_3=D^0+i{D}^1-D^2-i{D}^3\end{array}\right.$$

(35)

Equation (35) is consistent with Equation (7); however, we used the description of the eigenvalues in Equation (7). Furthermore, we know that

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

(36)

where ${\lambda }_j\in \left\{1,i,-1,-i\right\}$; $j=0,1,\cdots ,n-1$. When ${\lambda }_j=1$,

$${\lambda }_j^0+{\lambda }_j^1+{\lambda }_j^2+{\lambda }_j^3=4.$$

(37)

When ${\lambda }_j=i$,

$${\lambda }_j^0+{\lambda }_j^1+{\lambda }_j^2+{\lambda }_j^3=0.$$

(38)

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

$${\lambda }_j^0+{\lambda }_j^1+{\lambda }_j^2+{\lambda }_j^3=0.$$

(39)

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

$${\lambda }_j^0+{\lambda }_j^1+{\lambda }_j^2+{\lambda }_j^3=0.$$

(40)

Regarding $W_0$, only in the following circumstance, when ${\lambda }_j=1$, is the corresponding element not zero; notably, the elements at the other positions are zero. For $W_1$,

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

(41)

When ${\lambda }_j=i$,

$${\lambda }_j^0-i{\lambda }_j^1-{\lambda }_j^2+i{\lambda }_j^3=4.$$

(42)

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

$${\lambda }_j^0-i{\lambda }_j^1-{\lambda }_j^2+i{\lambda }_j^3=0.$$

(43)

For $W_2$,

$$\begin{array}{cc}{W}_2& =D^0-D^1+D^2-D^3\hfill \\ & =\left(\begin{array}{cccc}{\lambda }_0^0-{\lambda }_0^1+{\lambda }_0^2-{\lambda }_0^3& & & \\ & {\lambda }_1^0-{\lambda }_1^1+{\lambda }_1^2-{\lambda }_1^3& & \\ & & \ddots & \\ & & & {\lambda }_{n-1}^0-{\lambda }_{n-1}^1+{\lambda }_{n-1}^2-{\lambda }_{n-1}^3\end{array}\right).\end{array}$$

(44)

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

$${\lambda }_j^0-{\lambda }_j^1+{\lambda }_j^2-{\lambda }_j^3=4.$$

(45)

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

$${\lambda }_j^0-{\lambda }_j^1+{\lambda }_j^2-{\lambda }_j^3=0.$$

(46)

For $W_3$,

$$\begin{array}{ll}\hfill W_3& =D^0+i{D}^1-D^2-i{D}^3\hfill \\ & =\left(\begin{array}{cccc}{\lambda }_0^0+i{\lambda }_0^1-{\lambda }_0^2-i{\lambda }_0^3& & & \\ & {\lambda }_1^0+i{\lambda }_1^1-{\lambda }_1^2-i{\lambda }_1^3& & \\ & & \ddots & \\ & & & {\lambda }_{n-1}^0+i{\lambda }_{n-1}^1-{\lambda }_{n-1}^2-i{\lambda }_{n-1}^3\end{array}\right).\end{array}$$

(47)

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

$${\lambda }_j^0+i{\lambda }_j^1-{\lambda }_j^2-i{\lambda }_j^3=4.$$

(48)

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

$${\lambda }_j^0+i{\lambda }_j^1-{\lambda }_j^2-i{\lambda }_j^3=0.$$

(49)

Therefore, in Equation (34), we know that for $W_0$, only in the following circumstance, when the eigenvalue is 1, is the corresponding element not zero. For $W_1$, $W_2$, and $W_3$, the corresponding elements are not zero when the eigenvalues are i, −1, and −i, respectively. The number of eigenvalues is shown in

Table 1. Multiplicities of the DFT eigenvalues.

N	1	−1	−i	i
4n	n + 1	n	n	n − 1
4n + 1	n + 1	n	n	n
4n + 2	n + 1	n + 1	n	n
4n + 3	n + 1	n + 1	n + 1	n

[56,57]. Then, the combined eigenvectors of $W_0$, $W_1$, $W_2$, and $W_3$ comprise the exact eigenvectors of the DFT. It is evident that Equations (33) and (34) are equivalent.

Next, we introduced two phase gates (Equations (50) and (51)) and added them to the control register, as shown in Figure 3.

The phase gates $R_Z\left(\pi \alpha \right)$ and $R_Z\left(\pi \alpha /2\right)$ are expressed as:

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

(50)

and

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

(51)

Thus, we can obtain

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

(52)

The diagonal element of the matrix is exactly $B_k^{\alpha }$ of Equation (8). Then, $\left|{\psi }_5\right.〉$ is obtained

$$\left|{\psi }_5\right.〉={\displaystyle \sum _j{e}^{\pi i\alpha {\varphi }_j/2}\left|{\varphi }_j\right.〉\otimes b_j\left|u_j\right.〉},$$

(53)

where ${\varphi }_j\in \left\{0,1,2,3\right\}$ and $\left|{\varphi }_j\right.〉\in \left\{\left|00\right.〉,\left|01\right.〉,\left|10\right.〉,\left|11\right.〉\right\}$. Next, we used inverse quantum phase estimation (IQPE), as shown in Figure 4.

The two-qubit IQFT of the control register can be expressed as:

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

(54)

After the IQFT of $\left|{\varphi }_j\right.〉$ is calculated, the eigenvalues are restored,

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

(55)

Next,

$$\left|{\psi }_7\right.〉=\frac{1}{2}\left({\displaystyle \sum _j{e}^{\pi i\alpha {\varphi }_j/2}{\lambda }_j^0{a}_j\left|u_j\right.〉\left|00\right.〉}+{\displaystyle \sum _j{e}^{\pi i\alpha {\varphi }_j/2}F{b}_j\left|u_j\right.〉}\left|01\right.〉+{\displaystyle \sum _j{e}^{\pi i\alpha {\varphi }_j/2}I{F}^2{F}^2{b}_j\left|u_j\right.〉\left|10\right.〉+{\displaystyle \sum _j{e}^{\pi i\alpha {\varphi }_j/2}I{F}^2{F}^3{b}_j\left|u_j\right.〉\left|11\right.〉}}\right),$$

(56)

where IF denotes the inverse Fourier transform.

$$\begin{array}{ll}\hfill \left|{\psi }_8\right.〉& =\frac{1}{2}\left({\displaystyle \sum _j{e}^{\pi i\alpha {\varphi }_j/2}{\lambda }_j^0{b}_j\left|u_j\right.〉\left|00\right.〉}+{\displaystyle \sum _j{e}^{\pi i\alpha {\varphi }_j/2}IFF{b}_j\left|u_j\right.〉}\left|01\right.〉+{\displaystyle \sum _j{e}^{\pi i\alpha {\varphi }_j/2}I{F}^2{F}^2{b}_j\left|u_j\right.〉\left|10\right.〉+{\displaystyle \sum _j{e}^{\pi i\alpha {\varphi }_j/2}I{F}^3{F}^3{b}_j\left|u_j\right.〉\left|11\right.〉}}\right)\hfill \\ & =\frac{1}{2}\left({\displaystyle \sum _j{e}^{\pi i\alpha {\varphi }_j/2}{b}_j\left|u_j\right.〉\left|00\right.〉}+{\displaystyle \sum _j{e}^{\pi i\alpha {\varphi }_j/2}{b}_j\left|u_j\right.〉}\left|01\right.〉+{\displaystyle \sum _j{e}^{\pi i\alpha {\varphi }_j/2}{b}_j\left|u_j\right.〉\left|10\right.〉+{\displaystyle \sum _j{e}^{\pi i\alpha {\varphi }_j/2}{b}_j\left|u_j\right.〉\left|11\right.〉}}\right)\\ & ={\displaystyle \sum _j{e}^{\pi i\alpha {\varphi }_j/2}{b}_j\left|u_j\right.〉\frac{1}{\sqrt{2}}}\left(\left|0\right.〉+\left|1\right.〉\right)\frac{1}{\sqrt{2}}\left(\left|0\right.〉+\left|1\right.〉\right).\end{array}$$

(57)

Finally, through the H gate, we obtain $\left|{\psi }_9\right.〉$

$$\left|{\psi }_9\right.〉=\left|0\right.〉\left|0\right.〉{\displaystyle \sum _j{e}^{\pi i\alpha {\varphi }_j/2}{b}_j\left|u_j\right.〉},$$

(58)

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

$${\displaystyle \sum _j{e}^{\pi i{\varphi }_j/2}{b}_j\left|u_j\right.〉}=F\left|u\right.〉,$$

(59)

so

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

(60)

Then, Equation (58) is expressed as:

$$\left|{\psi }_9\right.〉=\left|0\right.〉\left|0\right.〉F^{\alpha }\left|u\right.〉.$$

(61)

Therefore, the quantum WFRFT (QWFRFT) is obtained.

The QWFRFT was designed as shown in Figure 4. This design provides the foundation for subsequent quantum WFRHT designs (QWFRHT).

4. Quantum Weighted Fractional Hartley Transform

The quantum Hartley transform (QHT) has been proposed [37]. Next, we use QPE and QHT to design the circuit of the QWFRHT, as shown in Figure 5. The eigenvalues of the Hartley transform are 1 and −1; therefore, the control register only needs one qubit to complete the circuit.

Here, $\left|u\right.〉$ is the eigenvector of the Hartley transform; thus, we can obtain

$$\left|{\psi }_0\right.〉=\left|0\right.〉\otimes \left|u\right.〉,$$

(62)

$$\left|{\psi }_1\right.〉=\frac{1}{\sqrt{2}}\left(\left|0\right.〉+\left|1\right.〉\right)\otimes \left|u\right.〉,$$

(63)

and

$$\begin{array}{c}\left|{\psi }_2\right.〉=\frac{1}{\sqrt{2}}\left(I\left|0\right.〉\left|u\right.〉+QHT\left|1\right.〉\left|u\right.〉\right)\\ =\frac{1}{\sqrt{2}}\left(D^0\left|0\right.〉+D\left|1\right.〉\right)\otimes \left|u\right.〉.\end{array}$$

(64)

We know $QHT\left|u\right.〉=D\left|u\right.〉$. Here, D is the eigenvalue of the Hartley transform and it can be expressed as:

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

(65)

where ${\lambda }_j\in \left\{1,-1\right\}$; $j=0,1,\cdots ,n-1$. This is because

$$\left|u\right.〉={\displaystyle \sum _j{r}_j\left|u_j\right.〉},$$

(66)

therefore,

$$D\left|u\right.〉={\displaystyle \sum _j{\lambda }_j{r}_j\left|u_j\right.〉}.$$

(67)

Equation (64) can be expressed as:

$$\left|{\psi }_2\right.〉=\frac{1}{\sqrt{2}}\left({\displaystyle \sum _j{\lambda }_j^0{r}_j\left|u_j\right.〉\left|0\right.〉}+{\displaystyle \sum _j{\lambda }_j^1{r}_j\left|u_j\right.〉}\left|1\right.〉\right).$$

(68)

Similarly to the abovementioned QWFRFT, after passing through the H gate, we obtain

$$\left|{\psi }_3\right.〉={\displaystyle \sum _j\left|{\varphi }_j\right.〉}\otimes r_j\left|u_j\right.〉,$$

(69)

where $\left|{\varphi }_j\right.〉\in \left\{\left|0\right.〉,\left|1\right.〉\right\}$; thus, the phase gate of Equation (50) is used,

$$\left|{\psi }_4\right.〉={\displaystyle \sum _j{e}^{\pi i\alpha {\varphi }_j}\left|{\varphi }_j\right.〉\otimes r_j\left|u_j\right.〉}.$$

(70)

Next, the IQPE is used, so that the eigenvalues can be recovered:

$$\left|{\psi }_5\right.〉=\frac{1}{\sqrt{2}}\left({\displaystyle \sum _j{e}^{\pi i\alpha {\varphi }_j}{\lambda }_j^0{r}_j\left|u_j\right.〉\left|0\right.〉}+{\displaystyle \sum _j{e}^{\pi i\alpha {\varphi }_j}{\lambda }_j^1{r}_j\left|u_j\right.〉}\left|1\right.〉\right),$$

(71)

and

$$\begin{array}{ll}\hfill \left|{\psi }_6\right.〉& =\frac{1}{\sqrt{2}}\left({\displaystyle \sum _j{e}^{\pi i\alpha {\varphi }_j}{r}_j\left|u_j\right.〉\left|0\right.〉}+{\displaystyle \sum _j{e}^{\pi i\alpha {\varphi }_j}{r}_j\left|u_j\right.〉}\left|1\right.〉\right)\hfill \\ & ={\displaystyle \sum _j{e}^{\pi i\alpha {\varphi }_j}{r}_j\left|u_j\right.〉}\frac{1}{\sqrt{2}}\left(\left|0\right.〉+\left|1\right.〉\right).\end{array}$$

(72)

Finally, we obtain

$$\left|{\psi }_7\right.〉=\left|0\right.〉{\displaystyle \sum _j{e}^{\pi i\alpha {\varphi }_j}{r}_j\left|u_j\right.〉}.$$

(73)

This is because

$${\displaystyle \sum _j{e}^{\pi i{\varphi }_j}{r}_j\left|u_j\right.〉}=H\left|u\right.〉,$$

(74)

where H denotes the Hartley transform; therefore,

$${\displaystyle \sum _j{e}^{\pi i\alpha {\varphi }_j}{r}_j\left|u_j\right.〉}=H^{\alpha }\left|u\right.〉.$$

(75)

Then, Equation (73) is expressed as

$$\left|{\psi }_7\right.〉=\left|0\right.〉H^{\alpha }\left|u\right.〉.$$

(76)

Thus, the QWFRHT is obtained.

Similarly to the QWFRHT, the quantum sine transform, quantum Hadamard transform, quantum cosine transform, and so on, are periodic functions, and their eigenvalues are 1 and −1; therefore, the designs of these functions are consistent with the findings in this section.

5. Discussion

With the help of the designs in Section 3 and Section 4, in this section, we present the quantum circuit of the WFRT (QWFRT); that is, the quantum circuit design of the WFRT in Section 2.3. Moreover, in this section, the unitary matrix $T^M=I$ (M is the period and I is the identity matrix) is determined. In order to fulfil the design requirements, the period should satisfy $M=2^q$. We also used QPE to design quantum circuits, as shown in Figure 6.

Here, $\left|u\right.〉$ is the eigenvector of matrix T; thus, we can obtain

$$\left|{\psi }_0\right.〉={\left|0\right.〉}^{\otimes q}\left|u\right.〉,$$

(77)

where $q={\log }_2^M$. Thus

$$\left|{\psi }_1\right.〉=\frac{1}{\sqrt{2^q}}{\left(\left|0\right.〉+\left|1\right.〉\right)}^{\otimes q}\left|u\right.〉,$$

(78)

and

$$\left|{\psi }_2\right.〉=\frac{1}{\sqrt{2^q}}\left(\left|0\right.〉+T^{2^{q-1}}\left|1\right.〉\right)\left(\left|0\right.〉+T^{2^{q-2}}\left|1\right.〉\right)\otimes \cdots \otimes \left(\left|0\right.〉+T\left|1\right.〉\right)\otimes \left|u\right.〉.$$

(79)

This is because $M=2^q$ and $\left|u\right.〉={\displaystyle \sum _j{c}_j\left|u_j\right.〉}$; therefore, Equation (79) can be expressed as:

$$\begin{array}{ll}\hfill \left|{\psi }_2\right.〉& =\frac{1}{\sqrt{2^q}}{\displaystyle \sum _{k=0}^{M-1}\left|k\right.〉T^k}\left|u\right.〉\hfill \\ & =\frac{1}{\sqrt{2^q}}{\displaystyle \sum _{k=0}^{M-1}\left|k\right.〉{\displaystyle \sum _j{\lambda }_j^k{c}_j\left|u_j\right.〉}}.\end{array}$$

(80)

Then, after the q-qubit IQFT of ${\displaystyle \sum _{k=0}^{M-1}{\lambda }_j^k\left|k\right.〉}$, we obtain

$$\left|{\psi }_3\right.〉={\displaystyle \sum _j\left|{\varphi }_j\right.〉c_j\left|u_j\right.〉}.$$

(81)

Via the phase gates, we can obtain

$$\left|{\psi }_4\right.〉={\displaystyle \sum _j{e}^{2\pi i\alpha {\varphi }_j/M}\left|{\varphi }_j\right.〉c_j\left|u_j\right.〉}.$$

(82)

Next, the IQPE is used. First, for the q-qubit QFT of $\left|{\varphi }_j\right.〉$, the eigenvalues are recovered,

$$\left|{\psi }_5\right.〉=\frac{1}{\sqrt{2^q}}{\displaystyle \sum _j{e}^{2\pi i\alpha {\varphi }_j/M}}{\displaystyle \sum _{k=0}^{M-1}{\lambda }_j^k\left|k\right.〉}{c}_j\left|u_j\right.〉.$$

(83)

Hence,

$$\left|{\psi }_6\right.〉=\frac{1}{\sqrt{2^q}}{\displaystyle \sum _j{e}^{2\pi i\alpha {\varphi }_j/M}}{\displaystyle \sum _{k=0}^{M-1}\left|k\right.〉}{c}_j\left|u_j\right.〉.$$

(84)

where ${\varphi }_j\in \left(0,1,\cdots ,M-1\right)$. Finally, we obtain

$$\left|{\psi }_7\right.〉={\left|0\right.〉}^{\otimes q}{\displaystyle \sum _j{e}^{2\pi i\alpha {\varphi }_j/M}}{c}_j\left|u_j\right.〉.$$

(85)

Thus

$$\left|{\psi }_7\right.〉={\left|0\right.〉}^{\otimes q}{T}^{\alpha }\left|u\right.〉.$$

(86)

Ergo, the QWFRT is obtained.

In the quantum circuit of Figure 6, the upper part represents the control register, and the lower part represents the target register. The quantum bits required in the control register are closely related to the period of the unitary matrix T, which is $q={\log }_2^M$. In the quantum circuit of Figure 4, the period of the Fourier transform is 4; that is, M = 4; therefore, in the circuit design, the control register needs two qubits. Similarly, in the quantum circuit of Figure 5, the period of the Hartley transform is 2; that is, M = 2; therefore, we used a qubit in the control register. Therefore, it is evident that the schemes proposed in Section 3 and Section 4 are special cases of the scheme proposed in Section 5.

Due to the complexity of the designed quantum algorithm, in space, in order to store eigenvalues, we use an additional q qubits (control registers). The qubits required for the control registers of different algorithms may also be different. For example, for the QWFRFT in Figure 4, we use two auxiliary qubits to store eigenvalues; for the QWFRHT in Figure 5, we use only one auxiliary qubit to store the eigenvalues. That is to say, the qubit of the control register depends on the number of eigenvalues of the algorithm. In time, when the target register is large, we only consider the complexity of the quantum gate in the circuit. Taking the QWFRFT in Figure 4 as an example, we use three QFTs and three IQFTs, and the complexity of each QFT and IQFT is $O\left(n^2\right)$. Therefore, the time complexity of the QWFRT depends on the complexity of quantum algorithm T.

6. Conclusions

As a generalized Fourier transform algorithm, the quantum circuit design of FRFT is particularly important. In order to design a QFRFT, we first analyzed the definitions for WFRFT; then, we proposed a new reformulation. Based on the new reformulation, we presented the WFRHT, and this fractional-order definition may also be applicable to the sine transform, cosine transform, and Hadamard transform algorithms; therefore, we proposed an additional weighted fractional-order definition based on the periodic matrix, which has wide applicability. Moreover, the proposed new reformulation contributes to the design of quantum algorithms. We first used QFT and QPE to design the quantum circuit of WFRFT; then, we designed the QWFRHT, a design which has wide applicability. Moreover, by designing the quantum circuit of the WFRT, we were able to realize the QWFRT. The quantum algorithms proposed in this paper will have great value in terms of their ability to be used as a reference for future quantum information processing.