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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 so-called Hamming distance measures the difference between two binary strings A and B. In simplified form, it measures the number of changes in A to get B. This type of distance is very useful in classical computing in applications such as error correction. It is also advantageous in quantum computing, being for example widely used in quantum machine learning. Since current quantum computers have limited resources, this type of distance is particularly attractive because it can be computed using fewer qubits and operations than other distances such as Euclidean or Manhattan distances. In this paper, two circuits for calculating Hamming distances using exclusively Clifford+T gates are presented. The aim of both circuits is to reduce the quantum cost and number of T gates needed to compute the Hamming distance. The T gate is more expensive than the other gates, so this reduction will have a significant impact on the total cost of the circuits. Furthermore, the proposed circuits are implemented using only Clifford+T gates. The circuits implemented exclusively with this group of gates are compatible with proven error detection and correction codes. [ABSTRACT FROM AUTHOR]

Current quantum computers have a limited number of resources and are heavily affected by internal and external noise. Therefore, small, noise-tolerant circuits are of great interest. With regard to circuit size, it is especially important to reduce the number of required qubits. Concerning to fault-tolerance, circuits entirely built with Clifford+T gates allow the use of error correction codes. However, the T-gate has an excessive cost, so circuits with a high number of T-gates should be avoided. This work focuses on optimising in such terms an operation that is widely used in larger circuits and algorithms: the calculation of the absolute-value of two's complement encoded integers. The proposed circuit halves the number of required T gates with respect to the best circuit currently available in the literature. Moreover, our circuit requires at least 2 qubits less than the other circuits for such an operation. [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]

Random numbers are of vital importance in fields such as cyptography and scientific simulations. However, it is well known how difficult it is for classical computers to generate random numbers. This is not the case for quantum computers, which are able to genuinely generate random numbers thanks to the property of superposition and their counter-intuitive concept of measurement. However, despite the simplicity of designing a circuit that generates a random number between 0 and 2 N − 1 (being N the number of available qubits), designing a quantum circuit to generate a number within a specific interval is far from trivial. This paper proposes a customizable circuit design to generate random numbers. The circuit is non- hardware dependent, it allows fault-tolerance, and it can be used by current quantum devices. Therefore, it is a valuable tool for all those quantum applications and algorithms that need to work with random numbers. Moreover, a comparator circuit has also been designed as part of this work. This comparator is the best currently available in the literature in terms of qubits, T-count, and T-depth. It is therefore a useful tool for any other circuit or algorithm where this operation is needed. [ABSTRACT FROM AUTHOR]

The algorithms that best demonstrate the potential of quantum computing are Shor's algorithm and Grover's algorithm. To this day, new evidence continues to emerge in the form of algorithms or ingenious applications that increase the field of application of this type of computing. However, given the limited number of qubits in current quantum computers, and also the noise problems they currently suffer from, implementing optimized circuits that allow us to take full advantage of the available resources, as well as detecting and correcting the errors caused by this noise, is a priority. In this work we present several leading zero detector circuits for quantum computers and simulators, optimized in terms of noise tolerance and number of qubits. These circuits are a fundamental part in major circuits that perform operations as important and basic in computation as addition and division. [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]

A severe drought has affected central Chile since 2009. Various adaptation responses have been developed, and a participatory process is required to learn from them. To enable this, a transdisciplinary approach was adopted to achieve two objectives: first, to test an approach for assessing the effectiveness of existing measures to respond to drought, specifically to distil strengths and weaknesses of implementation, and developing recommendations; second, to reflect on results from a pilot project conducted to ascertain its potential for scalability in terms of processes employed. The research was organized per the three types of knowledge needed to address complex problems through transdisciplinarity: systems, target and transformation knowledge. Using the recent drought as a boundary object, we conducted the pilot in two locations in Chile where we carried out literature reviews, interviews and focus group discussions were carried out. We identified adaptation measures at national and local scale, a set of which were evaluated applying the Index for the Usefulness of Adaptation Practices (IUPA). Results indicate that through IUPA, we could systematically account for the perceived effectiveness of applied measures. Strengths such as autonomy in the decision-making process emerged as key factors that could also be applied in other contexts, whereas weaknesses such as lack of integration with other policy domains, programs or projects were identified. To address weaknesses, key recommendations were proposed, which are congruent with context-specific expectations, capacities, experiences and knowledge, given that they were articulated by local actors. Results present empirical evidence on the important utility of transdisciplinary approaches in the evaluation of adaptation measures and can support the development of metrics related to adaptation process at the local scale. [ABSTRACT FROM AUTHOR]