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Suppose we have a one-qubit unitary U that maps |0⟩ to 13|0⟩+2−2i3|1⟩ and |1⟩ to 2+2i3|0⟩−13|1⟩. What is U?

Explicitly indicate multiplication with a * symbol. For instance, you should write 3*i+2 instead of 3i+2.

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