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In this work, we describe a method of calculation of the 1-D circular quantum convolution of signals represented by 3-qubit superpositions in the computational basis states. The examples of the ideal low pass and high pass filters are described and quantum schemes for the 3-qubit circular convolution are presented. In the proposed method, the 3-qubit Fourier transformis used and one addition qubit, to prepare the quantum superposition for the inverse quantum Fourier transform. It is considered that the discrete Fourier transform of one of the signals is known and calculated in advance and only the quantum Fourier transform of another signal is calculated. The frequency characteristics of many linear time-invariant systems and filters are well known. Therefore, the described method of convolution can be used for these systems in quantum computation. [ABSTRACT FROM AUTHOR]

Quantum computing algorithms offer the potential to revolutionize computational efficiency across a wide range of applications. However, one significant challenge that has impeded progress in color image processing is the difficulty of representing color relationships in quantum computers. In addition to this challenge, another major hurdle is efficiently representing data in quantum computers and performing quantum operations or arithmetic. To overcome these challenges, researchers have proposed various novel approaches, including using qubit lattices and flexible representations of quantum images. Despite these advancements, the quantum color image processing field is still in its early stages. This paper proposes a new arithmetic for processing quantum color images using novel operations for 2-qubits and the concept of quantum quaternion Fourier transform. We first study arithmetic operations in the 2-qubit setting, such as multiplication, inverse, and division on 2-qubits in quantum computation. The space of 2-qubits with real amplitudes is considered, and prototypes of power and exponent operations are also defined. The presented operations are applied to quantum superpositions of 2-qubits, to enable processing of quaternion images, which are 4-dimensional vectors and can be represented by 2-qubits at each pixel. The proposed methods and tools are designed to be embedded in a quantum device, which allows them to be used in various applications in the 2-qubits domain. To facilitate practical applications, the paper also introduces a tool for processing color images that can be used for color image filtering. Furthermore, this work provides a foundation to explore theoretical and practical aspects of color image processing on quantum computers. [ABSTRACT FROM AUTHOR]

Various methods of representation of signals exist in quantum computation, but it is essential to select the representations which allow to perform the operations, transforms, and methods which are widely used in classical signal processing. In this work, we analyse the quantum representation of one-dimensional signals, which is called the Fourier transform quantum representation (FTQR). This representation describes the signal as the superposition of states with amplitudes lying on the unit circle. The same is for two-dimensional signals or images in grays and colors. The tensor product of FTQRs is described, and examples are given for simple operations on discrete-time signals, such as the sum of signals, parallel addition of a few constants to the signal, and time-scaling. All these operations are performed in the phases of qubits in the Fourier transform quantum representation. [ABSTRACT FROM AUTHOR]

The quantum Fourier transform (QFT) is perhaps the furthermost central building block in creation quantum algorithms. In this work, we present a new approach to compute the standard quantum Fourier transform of the length N = 2 r , r > 1 , which also is called the r-qubit discrete Fourier transform. The presented algorithm is based on the paired transform developed by authors. It is shown that the signal-flow graphs of the paired algorithms could be used for calculating the quantum Fourier and Hadamard transform with the minimum number of stages. The calculation of all components of the transforms is performed by the Hadamard gates and matrices of rotations and all simple NOT gates. The new presentation allows for implementing the QFT (a) by using only the r Hadamard gates and (b) organizing parallel computation in r stages. Also, the circuits for the length-2r fast Hadamard transform are described. Several mathematical illustrative examples of the order the N = 4 , 8 , and 16 cases are illustrated. Finally, the QFT for inputs being two, three and four qubits is described in detail. [ABSTRACT FROM AUTHOR]