Multi-controlled single-qubit unitary gates based on the quantum Fourier transform

Multi-controlled (MC) unitary (U) gates are widely employed in quantum algorithms and circuits. Few state-of-the-art decompositions of MCU gates use non-elementary $C-R_x$ and $C-U^{1/2^{m-1}}$ gates resulting in a linear function for the depths of an implemented circuit on the number of these gates. Our approach is based on two generalizations of the multi-controlled X (MCX) gate that uses the quantum Fourier transform (QFT) comprised of Hadamard and controlled-phase gates. For the native gate set used in a genuine quantum computer, the decomposition of the controlled-phase gate is twice as less complex as $C-R_x$, which can result in an approximately double advantage of circuits derived from the QFT. The first generalization of QFT-MCX is based on altering the controlled gates acting on the target qubit. These gates are the most complex and are also used in the state-of-the-art circuits. The second generalization relies on the ZYZ decomposition and uses only one extended QFT-based circuit to implement the two multi-controlled X gates needed for the decomposition. Since the complexities of this circuit are approximately equal to the QFT-based MCX, our MCU implementation is more advanced than any known existing. The supremacy over the best-known optimized algorithm will be demonstrated by comparing transpiled circuits assembled for execution in a genuine quantum device. One may note that our implementations use approximately half the number of elementary gates compared to the most efficient one, potentially resulting in a smaller error. Additionally, we elaborated optimization steps to simplify the state-of-the-art linear-depth decomposition (LDD) MCU circuit to one of our implementations.