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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]

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]