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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]

Adders are one of the most interesting circuits in quantum computing due to their use in major algorithms that benefit from the special characteristics of this type of computation. Among these algorithms, Shor's algorithm stands out, which allows decomposing numbers in a time exponentially lower than the time needed to do it with classical computation. In this work, we propose three fault-tolerant carry lookahead adders that improve the cost in terms of quantum gates and qubits with respect to the rest of quantum circuits available in the literature. Their optimal implementation in a real quantum computer is also presented. Finally, the work ends with a rigorous comparison where the advantages and disadvantages of the proposed circuits against the rest of the circuits of the state of the art are exposed. Moreover, the information obtained from such a comparison is summarized in tables that allow a quick consultation to interested researchers. [ABSTRACT FROM AUTHOR]