Introduction to multiplicative group on 2-qubits in quantum color image processing

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.