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How do I use quantum gates offered on IBM quantum to construct either eigenvector of the Hadamard gate?

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Notice that the vectors you're trying to create are real. A great way of creating a real vector is to apply a $Y$ rotation, $$ R_Y(\theta)|0\rangle=\cos\frac{\theta}{2}|0\rangle+\sin\frac{\theta}{2}|1\rangle. $$ So, take the state you want to create and normalize it. For example, $$ |\psi\rangle=\frac{1}{\sqrt{4+2\sqrt{2}}}((1+\sqrt{2})|0\rangle+|1\rangle). $$ Now solve for $\theta$, i.e. $$ \cos\frac{\theta}{2}=\frac{1+\sqrt{2}}{\sqrt{4+2\sqrt{2}}}. $$ Thus, $\theta=\pi/4$. (If that's not so obvious, use the double angle formula, $$ \cos\theta=\cos^2\frac{\theta}{2}-\sin^2\frac{\theta}{2}=\frac{(1+\sqrt{2})^2-1}{4+2\sqrt{2}}=\frac{1}{\sqrt{2}}.) $$

Hence, the circuit you want to run is simply $R_Y(\pi/4)|0\rangle$. I'm assuming you can convert this into qiskit commands.

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