This page will contain a list of an exhaustive list of unitary matrices decomposition.

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List of known unitary matrices decompositions. Simple factorization of unitary transformations by Hubert de Guise, Olivia Di Matteo, and Luis L. Sánchez-Soto. https://journals.aps.org/pra/abstract/10.1103/PhysRevA.97.022328.

The canonical coset decomposition of unitary matrices through Householder transformations by Renan Cabreraa), Traci Strohecker, and Herschel Rabitz. https://aip.scitation.org/doi/10.1063/1.3466798 Note: this decomposition only requires $N$ terms. Note: There is an apparent error in matrix $\vert R\rangle_{\vert U_2\rangle}$: entry $3,1$ must be $$ \frac{2+\sqrt{2}}{4}. $$

Factorization of unitary matrices by P Dita. https://iopscience.iop.org/article/10.1088/0305-4470/36/11/309 Note: I used this one when searching unistochastic channels. My numerical method worked better with this one.

Random Unitary matrices by K Zyczkowski and M Kus https://iopscience.iop.org/article/10.1088/0305-4470/27/12/028/meta Note: In this paper there is a reference to another one by Hurwitz. However, I was not able to find it, so I quote it. Also, the explanation is good.

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