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Assume that we know a quantum state and the result of applying an unknown unitary $U$ on it. For example, if the quantum states are pure qubits, we know $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$ and $U|\psi\rangle=\gamma|0\rangle+\delta|1\rangle$. Then how can we compute the unknown operator $U$?

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If $U$ acts on a $d$-dimensional Hilbert space, then you need the result $U|\psi\rangle$ for a set of $d$ linearly independent vectors $|\psi\rangle$.

So, if you're talking about a single qubit unitary, you need two different states $|\psi\rangle$. If you have these, then by linearity you can find out $$ U|0\rangle=\alpha|0\rangle+\beta|1\rangle\qquad U|1\rangle=\gamma|0\rangle+\delta|1\rangle. $$ Then, your $U$ is just $$ U=\left(\begin{array}{cc} \alpha & \gamma \\ \beta & \delta \end{array}\right) $$ (the columns are just the individual outcomes for the computational basis states.)

You might wonder if you can get away with fewer. For example, we know that $U|0\rangle$ and $U|1\rangle$ must be orthogonal. However, there is still some freedom of parameters (such as a global phase) that you cannot determine, and hence you need to know both outcomes.

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