Question: 4. Prove that for any two unitary matrices U, and U2 representing single qubit gates, (a) (U_U_)* =UJU (b) (kU, +U2)* = k*U +U (c)

4. Prove that for any two unitary matrices U, and U2

4. Prove that for any two unitary matrices U, and U2 representing single qubit gates, (a) (U_U_)* =UJU (b) (kU, +U2)* = k*U +U (c) The product UUz is also a unitary matrix 5. Find the unitary matrix which performs a 30. rotation around the axis 2 1 2 (3'3'3 Verify that the matrix is unitary. Verify what it does to a qubit pointing in the y-direction on the Bloch sphere. 6. Calculate the rotation on the Bloch sphere (angle of rotation and axis of rotation) corresponding to: (a) The Pauli Y matrix (b) The Pauli Z matrix (c) The matrix el cos -e i sin -sin - cos la

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