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Suppose I want to put a qubit whose initial state is $|0\rangle$ to the final state $\frac{1}{\sqrt{3}}|0\rangle + \sqrt{\frac{2}{3}}|1\rangle$.

Well, in that case, the unitary matrix that performs such operation is given by: $$U = \frac{1}{\sqrt{3}}\begin{pmatrix}1&-\sqrt{2}\\ \sqrt{2} & 1 \end{pmatrix}$$ So the question is, how can I build a quantum circuit with the usual quantum gates (X, Y, Z, etc) which reproduces this behavior?

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You can implement this using a

– met927 – 2020-01-20T14:27:42.730`U3`

gate, see how to do this in this question