# Quantum 3D FFT in Tomography
## 1 Introduction  
The Fast Fourier Transform (FFT) gives the means for frequency analysis and filtering avoiding the operation of convolution. The Quantum version of the Fourier Transform (QFT) provides the opportunity to obtain very fast the results derived by the FFT. The use of the QFT has been investigated for various applications. Accordingly, in [1], it is proved that a quantum version for the filtering operation can be achieved, even though the quantum convolution of two sequences is physically impossible [2]. There are important differences between the classical and the quantum implementations for image filtering. These differences are analyzed in [1], and it is shown that the major advantage of the quantum approach lies in the exploitation of the efficient implementation of the QFT. In [3], a survey is provided with some topics on the properties of quantum gates and their assembly into interesting quantum circuits. The application of the QFT within the field of quantum computation has been extensively presented in [4]. [PERSON]'s algorithm, phase estimation and computing discrete logarithms are some classic examples of its use. These special properties of quantum algorithms have resulted in novel solutions to problems difficult to be solved on a classical computer. The QFT has been used in several applications [5; 6; 7]. Since QFT is the core to a lot of quantum algorithms, current research is mainly focused to its effective realization [7; 8; 9; 10; 11; 12; 13]. A review on quantum image processing is presented in [14], revealing the possibilities for intensive image processing procedures due to the powerful parallel computing capabilities of quantum computers. In [15], a quantum implementation of the FFT algorithm composed of a combination of quantum gates is proposed. A QFT has been implementedin [16] on a three-qubit nuclear magnetic resonance quantum computer to extract the periodicity of an input state. A fast quantum image component labeling algorithm was proposed in [17], which is the quantum counterpart of classical local operator technique. A quantum color image encryption algorithm was designed in [18] based on geometric transformation and intensity channel diffusion. A framework of quantum image filtering in the spatial domain was proposed in [19]. A quantum image median filtering approach is proposed, and its corresponding quantum circuit is designed in [20]. The main idea of the approach is that the classical image is converted into a quantum version based on the novel enhanced quantum representation of digital images, and then a unique quantum module is designed to realize the median calculation of neighborhood pixels for each pixel point in the image. In [21], the authors considered QFT-based color image filtering operations and their applications in image smoothing, sharpening, and selective filtering using quantum frequency domain filters. The proposed quantum filters use the principle of the quantum Oracle to implement the filter function. The stroboscopic approach to quantum tomography originated in 1983 when [PERSON] published a theorem on the minimal number of distinct observables required for tomography of systems evolving according to the von Neumann equation [22]. The approach was developed in subsequent articles and applied to open quantum systems with evolution given by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generators [23; 24; 25]. Among others, a general formula was proved for the minimal number of distinct observables for quantum tomography of systems with evolution given by the GKSL generator [26]. Quantum state tomography plays an important role in the field of quantum information. It can determine the mathematical representation of an unknown quantum system by measuring a large number of replicated quantum states to estimate them in real time [27].