**2. Preparation**

For a unitary matrix *U*, assuming that it has an eigenvector |*u* and the corresponding eigenvalue *<sup>e</sup>*2*πiϕ*, *<sup>U</sup>*|*u* = *e*2*πiϕ*|*u* is satisfied. Therefore, we can calculate *ϕ* through the phase estimation algorithm. The circuit of phase estimation is shown in Figure 1. It is not difficult to find that the quantum Fourier transform (*QFT*) is the key to phase estimation, and phase estimation is the key of many quantum algorithms.

**Figure 1.** A circuit for phase estimation.

The importance of the *QFT* goes without saying. However, little is known about the report of the *QFRFT*. In 2012, Parasa et al. proposed a *QPFRFT* using multiple-valued logic [32]. The reason why researchers call it "pseudo" is that the FRFT used did not include unitarity. The FRFT was proposed by Bailey et al. [33], and its definition is as follows:

$$F^{a}[k] = \sum\_{j=0}^{N-1} f[j] \cdot \exp\left(2\pi i \cdot \frac{kj}{N} \cdot a\right). \tag{1}$$

Parasa et al. pointed out: "It must be noted that unlike the discrete Fourier transform, the *FRFT* is not a unitary operation. More formally, this means that there exists no unitary operator which can implement the following quantum computational operation".

$$\sum\_{j=0}^{N-1} f(j)|j\rangle \stackrel{NOTPOSSLBLE}{\rightarrow} \sum\_{k=0}^{N-1} F^{\mathfrak{a}}(k)|k\rangle. \tag{2}$$

Therefore, Parasa et al. explicitly state that it is not possible to define the *QFRFT*. However, the definitions of the *FRFT* are diverse, and the definition of one class of *WFRFT* includes unitarity. Hence, Parasa et al.'s statement that there is no *QFRFT* is not rigorous.

In 1995, Shih proposed the definition of a *WFRFT* [30]. The alpha-order *FRFT* of the function *f*(*t*) can be expressed as

$$F^{\mathfrak{a}}[f(t)] = \sum\_{l=0}^{3} A\_l(\mathfrak{a}) f\_l(t). \tag{3}$$

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 Fourier transform). The weighting coefficient *Al*(*α*) is expressed as

$$A\_{I}(\boldsymbol{a}) = \cos\left(\frac{(\boldsymbol{a} - l)\pi}{4}\right) \cos\left(\frac{2(\boldsymbol{a} - l)\pi}{4}\right) \exp\left(\frac{3(\boldsymbol{a} - l)i\pi}{4}\right),\tag{4}$$

where *l* = 0, 1, 2, 3.

#### **3. Unitarity of Weighted Fractional Fourier Transform**

A complex matrix *U* satisfies

$$
\mathcal{U}\mathcal{U}I^H = \mathcal{U}^H \mathcal{U} = I,\tag{5}
$$

where *H* denotes the conjugate transpose, and *I* is the identity matrix. Then, matrix *U* is called a unitary matrix.

The discrete form of the *WFRFT* (Equation (3)) can be expressed as

$$DWFRFT = A\_0(\mathbf{a}) \cdot I + A\_1(\mathbf{a}) \cdot DFT + A\_2(\mathbf{a}) \cdot DFT^2 + A\_3(\mathbf{a}) \cdot DFT^3,\tag{6}$$

where *Al*(*α*) is Equation (4), and *DFT* is the discrete Fourier transform. It is not easy to prove the unitarity of Equation (6). Therefore, we present the reformulation of the *WFRFT* and prove its unitarity. First, Equation (4) can be written as

$$\begin{split} A\_{I}(a) &= \cos\left(\frac{(a-l)\pi}{4}\right)\cos\left(\frac{2(a-l)\pi}{4}\right)\exp\left(\frac{3(a-l)i\pi}{4}\right) \\ &= \frac{1}{2}\times\left[\exp\left(\frac{(a-l)\pi i}{4}\right) + \exp\left(\frac{-(a-l)\pi i}{4}\right)\right] \\ &\times\frac{1}{2}\times\left[\exp\left(\frac{2(a-l)\pi i}{4}\right) + \exp\left(\frac{-2(a-l)\pi i}{4}\right)\right] \times \exp\left(\frac{3(a-l)i\pi}{4}\right) \\ &= \frac{1}{4}\left(1 + \exp\left(\frac{2(a-l)\pi i}{4}\right) + \exp\left(\frac{4(a-l)\pi i}{4}\right) + \exp\left(\frac{6(a-l)\pi i}{4}\right)\right) \\ &= \frac{1}{4}\sum\_{k=0}^{3} \exp\left(\frac{2\pi i}{4}(a-l)k\right) \\ &= \frac{1}{4}\sum\_{k=0}^{3} \exp\left(\frac{2\pi ik}{4}\right)\exp\left(\frac{-2\pi ik}{4}\right). \end{split} \tag{7}$$

Let *Bα k* = exp 2*πikα* 4 ; *k* = 0, 1, 2, 3; then, Equation (7) can be expressed as

$$
\begin{pmatrix} A\_0^a \\ A\_1^a \\ A\_2^a \\ A\_3^a \end{pmatrix} = \frac{1}{4} \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & -i & -1 & i \\ 1 & -1 & 1 & -1 \\ 1 & i & -1 & -i \end{pmatrix} \begin{pmatrix} B\_0^a \\ B\_1^a \\ B\_2^a \\ B\_3^a \end{pmatrix} . \tag{8}
$$

We write Equation (6) as Equation (9).

$$DWFRFT = \left(I\_r DFT, DFT^2, DFT^3\right) \begin{pmatrix} A\_0(a) \\ A\_1(a) \\ A\_2(a) \\ A\_3(a) \end{pmatrix} . \tag{9}$$

Equation (8) is substituted into Equation (9), and we obtain

$$DWFRFT = \frac{1}{4} \begin{pmatrix} I, DFT, DFT^2, DFT^3 \end{pmatrix} \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & -i & -1 & i \\ 1 & -1 & 1 & -1 \\ 1 & i & -1 & -i \end{pmatrix} \begin{pmatrix} B\_0^a \\ B\_1^a \\ B\_2^a \\ B\_3^a \end{pmatrix} . \tag{10}$$

We let

⎧⎪⎪⎨⎪⎪⎩ *Y*0 = *I* + *DFT* + *DFT*<sup>2</sup> + *DFT*<sup>3</sup> *Y*1 = *I* − *i*·*DFT* − *DFT*<sup>2</sup> + *i*·*DFT*<sup>3</sup> *Y*2 = *I* − *DFT* + *DFT*<sup>2</sup> − *DFT*<sup>3</sup> *Y*3 = *I* + *i*·*DFT* − *DFT*<sup>2</sup> − *i*·*DFT*<sup>3</sup> (11)

**Definition 1.** *A reformulation of the DWFRFT.*

$$\begin{array}{rcl} DWFRFT &=& \frac{1}{4} (Y\_0, Y\_1, Y\_2, Y\_3) \begin{pmatrix} B\_0^a \\ B\_1^a \\ B\_2^a \\ B\_3^a \end{pmatrix} \\ &=& \frac{1}{4} \Big( \chi\_0 B\_0^a + Y\_1 B\_1^a + Y\_2 B\_2^a + Y\_3 B\_3^a \Big) \\ &=& \frac{1}{4} \sum\_{k=0}^3 Y\_k B\_k^a. \end{array} \tag{12}$$

*where Bαk* = exp 2*πikα* 4 ; *k* = 0, 1, 2, 3.

**Proposition 1.** *Yk are real symmetric matrices.*

**Proof of Proposition 1.** In Equation (11), *I* is the identity matrix, and *DFT* can be expressed as

$$DFT = \frac{1}{\sqrt{N}} \cdot \begin{pmatrix} u^{0 \times 0} & u^{0 \times 1} & \dots & u^{0 \times (n-1)} \\ u^{1 \times 0} & u^{1 \times 1} & \dots & u^{1 \times (n-1)} \\ \vdots & \vdots & \ddots & \vdots \\ u^{(n-1) \times 0} & u^{(n-1) \times 1} & \dots & u^{(n-1) \times (n-1)} \end{pmatrix} \prime \tag{13}$$

where *u* = exp(−2*πi*/*N*). Here, *DFT* is a symmetric matrix, so that *DFT*2, *DFT*3, and *DFT*<sup>4</sup> are also symmetric matrices. We know that the result of adding symmetric matrices is still a symmetric matrix. Therefore, *Yk* are symmetric matrices (Equation (11)).

Next, we prove that *Yk* are real matrices. The integer powers of the Fourier transform are shown in Figure 2. Here, *DFT*<sup>2</sup> and *DFT*<sup>4</sup> are real matrices; the matrix of *DFT*<sup>2</sup> is shown in Equation (14), and *DFT*<sup>4</sup> is the identity matrix *DFT*<sup>4</sup> = *DFT*<sup>0</sup> = *I*.

$$DFT^2 = \begin{pmatrix} 1 & 0 & \dots & 0 \\ 0 & 0 & \dots & 1 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 1 & \dots & 0 \end{pmatrix} . \tag{14}$$

**Figure 2.** Time–frequency representation of Fourier transform.

Obviously, *I* and *DFT*<sup>2</sup> are real matrices. In Equation (13), each element of the *DFT* can be expressed as

$$u\_{lk} = \exp(-2\pi ilk/N),\tag{15}$$

where *l* = 0, 1, ... , *n* − 1;*k* = 0, 1, ... , *n* − 1. Therefore, *DFT*<sup>3</sup> is an inverse Fourier transform, and each element of its matrix can be expressed as

$$w\_{lk} = \exp(2\pi ilk/N),\tag{16}$$

(18)

where *l* = 0, 1, ... , *n* − 1; *k* = 0, 1, ... , *n* − 1. Thus, the result of *DFT* + *DFT*<sup>3</sup> is a real number,

$$\begin{array}{lcl}w\_{lk} + u\_{lk} &= \exp(-2\pi ilk/N) + \exp(2\pi ilk/N) \\ &= \cos(2\pi lk/N) - i\sin(2\pi lk/N) + \cos(2\pi lk/N) + i\sin(2\pi lk/N) \\ &= 2\cos(2\pi lk/N). \end{array} \tag{17}$$

The result for −*iDFT* + *iDFT*<sup>3</sup> is

$$\begin{array}{rcl} -i\nu\_{lk} + i\mu\_{lk} & = & -i \exp(-2\pi ilk/N) + i \exp(2\pi ilk/N) \\ & = & -i \cos(2\pi lk/N) - \sin(2\pi lk/N) + i \cos(2\pi lk/N) - \sin(2\pi lk/N) \\ & = & -2\sin(2\pi lk/N). \end{array}$$

The result for −*DFT* − *DFT*<sup>3</sup> is

$$\begin{aligned} -w\_{lk} - u\_{lk} &= -\exp(-2\pi ilk/N) - \exp(2\pi ilk/N) \\ &= -\cos(2\pi lk/N) + i\sin(2\pi lk/N) - \cos(2\pi lk/N) - i\sin(2\pi lk/N) \\ &= -2\cos(2\pi lk/N). \end{aligned} \tag{19}$$

The result for *iDFT* − *iDFT*<sup>3</sup> is

$$\begin{array}{rcl} i\nu\_{lk} - i\mu\_{lk} &= i \exp(-2\pi ilk/N) - i \exp(2\pi ilk/N) \\ &= i \cos(2\pi lk/N) + \sin(2\pi lk/N) - i \cos(2\pi lk/N) + \sin(2\pi lk/N) \\ &= 2\sin(2\pi lk/N). \end{array} \tag{20}$$

Therefore, for Equation (11), *Yk* are real symmetric matrices. 

**Proposition 2.** *The weighted fractional Fourier transform is unitary.*

**Proof of Proposition 2.** By the proof of Proposition 1, we know that *Yk* are real symmetric matrices; that is, (*Yk*)*<sup>H</sup>* = *Yk*. Therefore, the conjugate transpose of the *DWFRFT* is

$$\begin{array}{rcl} \left(DWFRFP\right)^{H} &=& \frac{1}{4} \Big(\Upsilon\_{0}B\_{0}^{a} + \Upsilon\_{1}B\_{1}^{a} + \Upsilon\_{2}B\_{2}^{a} + \Upsilon\_{3}B\_{3}^{a}\Big)^{H} \\ &=& \frac{1}{4} \Big(\Upsilon\_{0}B\_{0}^{-a} + \Upsilon\_{1}B\_{1}^{-a} + \Upsilon\_{2}B\_{2}^{-a} + \Upsilon\_{3}B\_{3}^{-a}\Big). \end{array} \tag{21}$$

Thus, we obtain

$$\begin{split} \text{DWF} \boldsymbol{F} \boldsymbol{F} \cdot \left( \text{DWF} \boldsymbol{F} \boldsymbol{F} \boldsymbol{I} \right)^{\bar{H}} &= \quad \frac{1}{16} \left( Y\_0 \boldsymbol{B}\_0^a + Y\_1 \boldsymbol{B}\_1^a + Y\_2 \boldsymbol{B}\_2^a + Y\_3 \boldsymbol{B}\_3^a \right) \left( Y\_0 \boldsymbol{B}\_0^{-a} + Y\_1 \boldsymbol{B}\_1^{-a} + Y\_2 \boldsymbol{B}\_2^{-a} + Y\_3 \boldsymbol{B}\_3^{-a} \right) \\ &= \quad \frac{1}{16} \sum\_{k=0}^3 \sum\_{l=0}^3 Y\_k \boldsymbol{Y}\_l \boldsymbol{B}\_k^a \boldsymbol{B}\_l^{-a} . \end{split} \tag{22}$$

Here,

$$\mathcal{Y}\_k \mathcal{Y}\_l = \begin{cases} \ 0, k \neq l \\ \mathcal{Y}\_{k'}^2 k = l \end{cases} \tag{23}$$

Then, Equation (22) is written as

$$\left(DWFRFT \cdot \left(DWFRFT\right)^{H}\right)^{H} = \frac{1}{16} \sum\_{k=0}^{3} \mathcal{Y}\_{k}^{2}.\tag{24}$$

After calculation, we know that *Y*2*k* = 4*Yk*. Equation (25) is obtained.

$$DWFRFT \cdot \left(DWFRFT\right)^{H} = \frac{1}{4} \sum\_{k=0}^{3} \Upsilon\_{k} = \frac{1}{4} (\Upsilon\_{0} + \Upsilon\_{1} + \Upsilon\_{2} + \Upsilon\_{3}) = I. \tag{25}$$

Thus, the unitarity of the *WFRFT* is proved. 

We can also implement the new reformulation with the help of fast Fourier transform (*FFT*), and its implementation module is shown in Figure 3. The weighting coefficients are readjusted *<sup>A</sup><sup>α</sup>l* in Figure 3; so, the computational complexity is *O*(*N* log *<sup>N</sup>*).

**Figure 3.** The reformulation of the *WFRFT* module.

#### **4. Quantum Weighted Fractional Fourier Transform**

In this section, we will present the *QWFRFT* with the help of the *QFT*. The *QFT* is an application of the classical Fourier transform to the amplitude of a quantum state. the vector *x* is transformed into the vector *y* by the classical Fourier transform,

$$y\_k = \frac{1}{\sqrt{N}} \sum\_{j=0}^{N-1} x\_j u^{j\mathbf{k}}; k = 0, 1, 2, \dots, N-1 \tag{26}$$

where *u* = *e*<sup>−</sup>2*πi*/*<sup>N</sup>* and *N* is the signal length.

Similarly, *QFT* is applied to quantum state |*x* = *N*−1 ∑ *j*=0 *xj*|*j* to obtain quantum state |*y* = *N*−1 ∑ *k*=0*yk*|*k* , and we have

$$y\_k = \frac{1}{\sqrt{N}} \sum\_{j=0}^{N-1} x\_j w\_{n\_j}^{jk} \,\,\,\,\tag{27}$$

where *k* = 0, 1, 2, ... , *N* − 1 and *w* = *<sup>e</sup>*2*πi*/*N*. We note that Equation (27) is the inverse of the classical discrete Fourier transform; by convention, the *QFT* has the same effect as the inverse discrete Fourier transform.

In case that |*j* is a basis state, the *QFT* can also be expressed as the map

$$QFT: |j\rangle \mapsto \frac{1}{\sqrt{N}} \sum\_{k=0}^{N-1} w^{jk} |k\rangle. \tag{28}$$

Equivalently, the *QFT* can be viewed as a unitary matrix acting on quantum state vectors, where the unitary matrix *FN* is given by

$$F\_N = \frac{1}{\sqrt{N}} \begin{pmatrix} w^{0 \times 0} & w^{0 \times 1} & \dots & w^{0 \times (n-1)} \\ w^{1 \times 0} & w^{1 \times 1} & \dots & w^{1 \times (n-1)} \\ \vdots & \vdots & \ddots & \vdots \\ w^{(n-1) \times 0} & w^{(n-1) \times 1} & \dots & w^{(n-1) \times (n-1)} \end{pmatrix} . \tag{29}$$

Sine *N* = 2*n* and *w* = *e*2*πi*/2*<sup>n</sup>* . The electronic circuit of the *QFT* is shown in Figure 4.

**Figure 4.** A circuit for the *QFT*.

Therefore, the *QFT* of the quantum state |*j* = |*j*1 *j*2 ... *jn* can be expressed as

$$QFT(|j\_1j\_2\dots j\_n\rangle) = \frac{1}{2^{n/2}} \left( |0\rangle + \epsilon^{2\pi i[0,j\_n]} |1\rangle \right) \otimes \left( |0\rangle + \epsilon^{2\pi [0,j\_{n-1}j\_n]} |1\rangle \right) \otimes \dots \otimes \left( |0\rangle + \epsilon^{2\pi i[0,j\_1j\_2\dots j\_n]} |1\rangle \right),\tag{30}$$

where the binary of decimals can be expressed as

$$[0.j\_1j\_2\dots j\_m] = \sum\_{k=1}^m j\_k 2^{-k}.\tag{31}$$

.

For instance, [0.*j*1] = *j*1/2 and [0.*j*1 *j*2] = *j*1/2 + *j*2/22. Then, the *QFT* can be further expressed as

$$QFT(\left|\mathbf{j}\,\mathbf{j}\,\mathbf{2}\ldots\mathbf{j}\_{\rm l}\right\rangle) = \frac{1}{2^{n/2}}\left(\left|0\right\rangle + w\_1^{\left[\boldsymbol{j}\_{\rm l}\right]}\left|1\right\rangle\right) \otimes \left(\left|0\right\rangle + w\_2^{\left[\boldsymbol{j}\_{\rm l}\dots\boldsymbol{j}\_{\rm l}\right]}\left|1\right\rangle\right) \otimes \ldots \otimes \left(\left|0\right\rangle + w\_n^{\left[\boldsymbol{j}\_{\rm l}\right]}\left|1\right\rangle\right) . \tag{32}$$

Here, we use [0.*j*1 *j*2 ... *jm*] = [*j*1 *j*2 ... *jn*]/2*<sup>m</sup>*, and *wm* = *w*<sup>−</sup>2*<sup>m</sup>* = *e*2*πi*/2*<sup>m</sup>*

To implement the *QWFRFT*, we first present the integer powers (*QFT*0, *QFT*1, *QFT*2, *QFT*3) of the *QFT*.

1. We know that *QFT*<sup>0</sup> = *I*, and *I* is the identity matrix; obviously, this is a unitary operator. Then, its operation can be expressed as

$$|a\rangle^{-}I^{-}|\beta\_{0}\rangle$$

2. The *QFT* is a unitary operator. The Fourier transform of a quantum state |*α* can be expressed as

$$|a\rangle \text{–} QFT^{-} |\beta\_1\rangle$$

3. The quadratic power of the *QFT* can be expressed as

$$QFT^2 = \begin{pmatrix} 1 & 0 & \dots & 0 \\ 0 & 0 & \dots & 1 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 1 & \dots & 0 \end{pmatrix}$$

For the vector (*<sup>α</sup>*0, *α*1,..., *<sup>α</sup>n*−<sup>1</sup>), the transformation can be expressed as

$$(\mathfrak{a}\_{0\prime}\mathfrak{a}\_{1\prime}, \dots, \mathfrak{a}\_{n-1}) \begin{pmatrix} 1 & 0 & \dots & 0 \\ 0 & 0 & \dots & 1 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 1 & \dots & 0 \end{pmatrix} = (\mathfrak{a}\_{0\prime}\mathfrak{a}\_{n-1\prime}, \dots, \mathfrak{a}\_1)$$

In order to realize the quantum circuit of the above matrix, multiple swap gates are required. The swap gate of two quanta is shown in Figure 5.

**Figure 5.** Swap gate.

Thus, for *QFT*2, we provide quantum circuits of eight quantum states, as shown in Figure 6.

**Figure 6.** A circuit forthe *QFT*2.

For a 2*n* × 2*n* dimensional identity matrix, we can obtain the *QFT*<sup>2</sup> by row transformation, as shown in Figure 7.


**Figure 7.** Matrix of the *QFT*2.

> Therefore, the quantum circuit of Figure 6 can be simplified as Figure 8.

*n*

**Figure 8.** A circuit for the *QFT*2.

Thus, the *QFT*<sup>2</sup> for quantum state |*α* can be expressed as

$$|\alpha\rangle^- QFT^{2-} |\beta\_2\rangle$$

1. The third power of the *QFT*, which is equivalent to the inverse operation of the *QFT*, is also a unitary operator.

$$|\alpha\rangle^{-}QFT^{3-}|\beta\_{3}\rangle$$

Therefore, the *QWFRFT* of the quantum state by Equation (10) can be expressed as

$$\begin{array}{rcl} QWFFT(|\boldsymbol{\kappa}\rangle) &=& \frac{1}{4} \Big( I(|\boldsymbol{\kappa}\rangle), QFT(|\boldsymbol{\kappa}\rangle), QFT^{2}(|\boldsymbol{\kappa}\rangle), QFT^{3}(|\boldsymbol{\kappa}\rangle) \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & -i & -1 & i \\ 1 & -1 & 1 & -1 \\ 1 & i & -1 & -i \end{pmatrix} \begin{pmatrix} B\_{0}^{x} \\ B\_{1}^{x} \\ B\_{2}^{x} \\ B\_{3}^{x} \end{pmatrix} \\ &=& \frac{1}{4} (|\beta\_{0}\rangle, |\beta\_{1}\rangle, |\beta\_{2}\rangle, |\beta\_{3}\rangle) \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & -i & -1 & i \\ 1 & -i & -1 & i \\ 1 & -1 & 1 & -1 \\ 1 & i & -1 & -i \end{pmatrix} \begin{pmatrix} B\_{0}^{x} \\ B\_{1}^{x} \\ B\_{2}^{x} \\ B\_{3}^{x} \\ B\_{3}^{x} \end{pmatrix}. \end{array} \tag{33}$$

Equation (33) can be further written as

$$\begin{array}{rcl} QWPERT(|\mu\rangle) &=& \frac{1}{4}(|\beta\_{0}\rangle, |\beta\_{1}\rangle, |\beta\_{2}\rangle, |\beta\_{3}\rangle) \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & -i & 1 & i \\ 1 & -1 & 1 & -1 \\ 1 & i & -1 & -i \end{pmatrix} \begin{pmatrix} B\_{0}^{k} \\ B\_{1}^{k} \\ B\_{2}^{k} \\ B\_{3}^{k} \end{pmatrix} \\ &=& \frac{1}{4}(|\beta\_{0}\rangle, |\beta\_{1}\rangle, |\beta\_{2}\rangle, |\beta\_{3}\rangle) \begin{pmatrix} \exp\left(-\frac{2\pi i \beta \otimes 1}{4}\right) & \exp\left(-\frac{2\pi i \otimes 1}{4}\right) & \exp\left(-\frac{2\pi i \otimes 2}{4}\right) \\\\ \exp\left(\frac{-2\pi i \otimes 1}{4}\right) & \exp\left(-\frac{2\pi i \otimes 1}{4}\right) & \exp\left(\frac{-2\pi i \otimes 2}{4}\right) & \exp\left(\frac{-2\pi i \otimes 2}{4}\right) \\\\ \exp\left(\frac{-2\pi i \otimes 0}{4}\right) & \exp\left(\frac{-2\pi i \otimes 1}{4}\right) & \exp\left(\frac{-2\pi i \otimes 2}{4}\right) & \exp\left(\frac{-2\pi i \otimes 2}{4}\right) \\\\ \exp\left(\frac{-2\pi i \otimes 0}{4}\right) & \exp\left(\frac{-2\pi i \otimes 1}{4}\right) & \exp\left(\frac{-2\pi i \otimes 2}{4}\right) & \exp\left(\frac{-2\pi i \otimes 3}{4}\right) \\\\ \end{array} \tag{34}$$

where *Bαk* = exp 2*πikα* 4 ; *k* = 0, 1, 2, 3. Then, Equation (34) can be written again as

$$\begin{split} QWFRFT(|\alpha\rangle) &= \ & \frac{1}{4} \sum\_{l=0}^{3} \sum\_{k=0}^{3} |\beta\_{l}\rangle \exp\left(\frac{-2\pi ik}{4}\right) B\_{k}^{\alpha} \\ &= \ & \frac{1}{4} \sum\_{l=0}^{3} \sum\_{k=0}^{3} |\beta\_{l}\rangle \exp\left(\frac{-2\pi ik}{4}\right) \exp\left(\frac{2\pi ika}{4}\right) \\ &= & \frac{1}{4} \sum\_{l=0}^{3} \sum\_{k=0}^{3} |\beta\_{l}\rangle \exp\left(\frac{2\pi ik(a-l)}{4}\right). \end{split} \tag{35}$$

With the help of the quantum artificial neural network (*QANN*), we are inspired to design a *QWFRFT*. Here, we first introduce the *QANN* [34,35]. If we use {|*<sup>e</sup>*1 , |*<sup>e</sup>*2 ,..., |*eM* } to denote the canonical basis for C*M*, then the quantum artificial neural network above can be rewritten as

$$Q(|\mathbf{x}\rangle) = \sum\_{k=1}^{M} \sum\_{j=1}^{N} \left( a\_{j,k}^{(1)} \sigma\_k \left( \left\langle w\_{j,k}^{(1)} \right| T | \mathbf{x}\rangle + \theta\_{j,k}^{(1)} \right) + i a\_{j,k}^{(2)} \sigma\_k \left( \left\langle w\_{j,k}^{(2)} \right| T | \mathbf{x}\rangle + \theta\_{j,k}^{(2)} \right) \right) |\mathbf{e}\_k\rangle. \tag{36}$$

$$\begin{aligned} \text{Put } y\_{j,k}^{(i)} &= \sigma\_k \left( \sum\_{t=1}^n \left< w\_{j,k}^{(i)}(t) \right| T | \mathbf{x}\_t \rangle + \theta\_{j,k}^{(i)} \right) \text{ and } \left| \mathbf{a}\_k^{(i)} \right> = \sum\_{j=1}^N a\_{j,k}^{(i)} y\_{j,k}^{(i)} | \mathbf{c}\_k \rangle. \text{ Then, a QANN problem is to find } \mathbf{x}\_t &= \sigma\_k \mathbf{x}\_t + \sum\_{j=1}^n a\_{j,k}^{(i)} y\_{j,k}^{(i)} | \mathbf{c}\_k \rangle. \end{aligned}$$

can be illustrated by Figures 9 and 10 below.

**Figure 9.** The output 333*α*(*i*) *k*4of a *QANN*, where *i* = 1, 2; *k* = 1, 2, . . . , *M*.

**Figure 10.** The output *Q*(|*x* ) of a *QANN*.

> Thus, we can present the circuit of the *QWFRFT*, as shown in Figure 11.

**Figure 11.** A circuit for the *QWFRFT*.

So far, we have completed the *QWFRFT* and circuit implementation. The work of this paper is a supplement to the work of Parasa et al. At one point, researchers pointed out that there is no quantum-weighted fractional Fourier transform [32]. However, our study illustrates the diversity of *FRFT* and proposes *QWFRFT*. Due to the characteristics of quantum parallelism, we believe that the *QWFRFT* has a wider application space.

At present, our method is only applicable to closed systems. The standard quantum theory has shown its limit to describe successfully experimental results. Counterintuitive results are obtained in different experiments [36,37]. The open system effects need to be further analyzed.

## **5. Conclusions**

Unitarity is a prerequisite for the realization of quantum algorithms. In this paper, we proposed the reformulation of the *WFRFT*. The unitarity of the *WFRFT* was proved by means of the proposed reformulation. The *QFT* is an important part of the *QWFRFT*. Furthermore, we presented the integer power operation and quantum circuit of the *QFT*, which lays the foundation for the *QWFRFT*. Finally, we designed the circuit of the *QWFRFT* with the help of a quantum artificial neural network and proposed the electronic circuit of the *QWFRFT*. The results of this paper show that there is a *QFRFT* algorithm, which lays the foundation for further research.

**Author Contributions:** Conceptualization, T.Z.; methodology, T.Z.; validation, T.Z., T.Y. and Y.C.; formal analysis, T.Z.; investigation, T.Z.; resources, T.Y.; writing—original draft preparation, T.Z.; writing—review and editing, T.Z.; project administration, T.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This study was supported by the Fundamental Research Funds for the Central Universities (N2123016); and the Scientific Research Projects of Hebei colleges and universities (QN2020511).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
