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The matrices I've seen are the square-root-of-not-with-extra-phase-factor matrix ##A = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & i \\ i & 1 \end{bmatrix}## here and here, and the hadamard matrix ##H = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}## here.

A major difference between the square root of NOT gate and the Hadamard gate is that ##H^2 = I## whereas ##A^2 \propto X##. Also, if you have a 180-degree phase shift gate ##Z## then ##HZH = X## whereas ##AZA = A^2 e^{i \pi Z/2}##.

Which operation should I be thinking of, when an article says "beam splitter"?