# How to implement quantum circuit for this operation

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.

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, $$U\equiv HSH.$$