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I am wondering how to construct a gate for some operation like

$$U = \frac{1}{\sqrt{2}}(I - i \sigma_{x})$$

I don't know how to add two $I$ with $-i\sigma_{x}$ specifically, any help would be appreciated.

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Let's start with the fact that $U$, as you've written it, is not unitary. I'm going to assume that you mean $U=(I-i\sigma_X)/\sqrt{2}$.

It's probably not helpful to think of it as an addition. It is probably more helpful to think of it as $$ U=e^{-i\pi \sigma_X/4}. $$ Now, what you don't say is what you're allowed to construct the gate out of, and this makes a big difference. For example, if you have arbitrary single-qubit gates, you can just implement it directly.

If not, you might find it helpful to realise that $\sigma_X=H\sigma_ZH$ where $H$ is the Hadamard gate. So, $$ U=e^{-i H\sigma_ZH\pi/4}=He^{-i\sigma_Z\pi/4}H. $$ How does this help? $$ e^{-i\sigma_Z\pi/4}=\left(\begin{array}{cc} e^{-i\pi/4} & 0 \\ 0 & e^{i\pi/4} \end{array}\right)=e^{-i\pi/4}\left(\begin{array}{cc} 1 & 0 \\ 0 & i \end{array}\right). $$ Up to an irrelevant global phase, this is just the gate $S$. Hence, $$ U\equiv HSH. $$

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