# Quantum 3D FFT in Tomography
## 4 The Radon Transform and the Back Projection by Means of the Inverse 3D QFT
### Quantum 3D Fourier Transform  
A quantum cube can be represented by a quantum register Q constructed so that it encodes all required information, i.e., the position of a pixel in the data cube (x, y, z), as well as the intensity or color (c) of the pixel. We assume that we have \(2^{n}\times 2^{n}\times 2^{n}\) pixels in the data cube, and the color of each pixel requires m bits for its representation. In this case, a register \(\mid\)P\(>\) having 3n qubits is adequate to hold all position information, while a register \(\mid\)C\(>\) with m qubits will represent \(2^{m}\) different colors or grayscale levels. The register \(\mid\)P\(>\) is separated into three sub-registers of \(n\) qubits each containing the row, column, and the level information in the form \(\mid\)y\(>\)\(\mid\)\(\times\)\(\mid\)z\(>\). The quantum register Q, containing all the information of the quantum frame cube, can be expressed as:  
\[Q=\mid\)C\(>_{m}\otimes\)P\(>_{3n}=\sum_{i=0}^{2^{3n}-1}\sum_{j=0}^{2^{3n}-1}a_{ij}\)\(\mid\)j\(>\)\(\mid\)i\(>\) (22)  
In Equation (22) the coefficients \(a_{ij}\) sum up to one:  
\[\sum_{j=0}^{2^{m}-1}\left|a_{ij}\right|^{2}=1\ \forall i\ \ with\ 0\leq i<2^{3n} \tag{23}\]  
and are used to express the color of a pixel with position \(i\) by means of a superposition of all possible colors. For a given pixel \(i\), the coefficients \(a_{ij}\) take the value 1 if the color of the pixel is j and 0 otherwise. This is illustrated in Figure 10 with a simple example of a 2 \(\times\) 2 \(\times\) 2 frame cube with eight colors (0-7), for which the values of the coefficients \(a_{ij}\) are given in the following in a form of a matrix:  
\[ \begin{array}{cccccccc}1&0&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ a_{ij}&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&1&0&0\\ \end{array}  \tag{24}\]  
The quantum register Q, expressed by Equation (22), is experimentally implemented by the circuit of Figure 11.  
Figure 10: This is an example of 2 \(\times\) 2 \(\times\) 2 quantum cube with eight different colors. Three qubits are used to represent color information and three-pixel position information. As shown in the block diagram of Figure 11, at the output of the quantum filtering circuit we actually find the input image cube. Additionally, by exploiting the quantum interference phenomenon, we can use an additional qubit initially in state \(|0\rangle\) to reinterpret the quantum image cube as a superposition of two image cubes. For example, for a high- or low-pass 3D filter, the output image is the sum of the image cube containing the high frequencies and the image cube containing the corresponding low frequencies. The additional qubit can be used to make the distinction between the two image cubes. For implementing the Radon and the Inverse Radon transform as it was exposed in Section 3, the block diagram in Figure 11 is modified so that  
1. A 2D QFT is used in the input stage to provide the spectra of the 256 2D projections of the random taken as input. 2. The projections spectra are radially placed to create the 3D spectrum of the initial data (random). 3. This 3D spectrum is properly interpolated and normalized by multiplying with the function \(\sqrt{1+\omega_{x}^{2}+\omega_{y}^{2}+\omega_{z}^{2}}\). 4. The 3D inverse QFT is applied to provide the required random cube. The whole procedure is depicted in Figure 12.  
In the following, we analyze this process and describe the state of the quantum reconstruction tomosynthesis filtering circuit at each step of the computation, as marked in Figure 12. The input state \(|I_{0}\rangle\) is represented by the input image cube and an additional qubit in state \(|0\rangle\):  
\[|I_{0}>\ =|Q>\ \otimes\ |0>\ =\frac{1}{2^{n}}\ \sum_{d=0}^{2^{n}-1}\sum_{y=0}^{2 ^{n}-1}\sum_{x=0}^{2^{n}-1}\sum_{j=0}^{2^{n}-1}a_{ijk}|j>|t>|y>|x>|0> \tag{25}\]  
Figure 11: Implementation of the quantum circuit for 3D filtering.