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Quantum comparators hold substantial significance in the scientific community as fundamental components in a wide array of algorithms. In this research, we present an innovative approach where we explore the realm of comparator circuits, specifically focussing on three distinct circuit designs present in the literature. These circuits are notable for their use of T-gates, which have gained significant attention in circuit design due to their ability to enable the utilisation of error-correcting codes. However, it is important to note that T-gates come at a considerable computational cost. One of the key contributions of our work is the optimisation of the quantum gates used within these circuits. We articulate the proposed circuits employing Clifford+T gates, facilitating error correction code implementation. Additionally, we minimise T-gate usage, thereby reducing computational costs and fortifying circuit robustness against errors and environmental disturbances-essential for mitigating the effects of internal and external noise. Our methodology employs a bottom-up examination of comparator circuits, initiating with a detailed study of their gates. Subsequently, we systematically dissect the functions of these gates, thereby advancing towards a comprehensive understanding of the circuit's overall functionality. This meticulous examination forms the foundation of our research, enabling us to identify areas where optimisations can be made to improve their performance. [ABSTRACT FROM AUTHOR]

The intrinsic high parallelism and entanglement characteristics of quantum computing have made quantum image processing techniques a focus of great interest. One of the most widely used techniques in image processing is segmentation, which in one of their most basic forms can be carried out using thresholding algorithms. In this paper, a fault-tolerant quantum dual-threshold algorithm has been proposed. This algorithm has been built using only Clifford+T gates for compatibility with error detection and correction codes. Because fault-tolerant implementation of T gates has a much higher cost than other quantum gates, our focus has been on reducing the number of these gates. This has allowed adding noise tolerance, computational cost reduction, and fault tolerance to the state-of-the-art dual-threshold segmentation circuits. Since the dual-threshold image segmentation involves the comparison operation, as part of this work we have implemented two full comparator circuits. These circuits optimize the metrics T-count and T-depth with respect to the best circuit comparators currently available in the literature. [ABSTRACT FROM AUTHOR]