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

Abstract

We will present a few new generalizations of the multi-controlled X (MCX) gate that uses the quantum Fourier transform (QFT). Firstly, we will optimize QFT-MCX and prove that it is equivalent to a stair MCX gates array. This stair-wise structure will allow us to devise a method for adding an arbitrary phase factor to each qubit. The first MCX generalization into multi-controlled unitary gates (MCU) relies on replacing phase gates acting on the target qubit with controlled unitary gates. We will employ alternative single-qubit gate notation to minimize the complexities of these gates and show how to expand the circuit straightforwardly to the multi-controlled multi-target (MCMT) gate. The second generalization relies on the ZYZ-like decomposition. We will show that by extending one QFT-MCX circuit we implement the two multi-controlled X gates needed for the decomposition. Finally, we will split control wirelines into groups and use iterative ZYZ-like decomposition on QFT-MCU to obtain "deep decomposed" (DD) MCU which employs a lower number of C-NOTs than the previous two, thus making DD-MCU less prone to decoherence and noise. The supremacy of our implementations over the best-known optimized algorithm will be demonstrated by emulating noisy quantum calculations.