7014
1. circuit = H1, C1X0
2. \( |\Psi_{CAB} \rangle = \frac{\alpha}{\sqrt{2}}(|000\rangle+|011\rangle) + \frac{\beta}{\sqrt{2}}(|100\rangle+|111\rangle) \)
3. \( |\Psi_{CAB} \rangle = \frac{\alpha}{\sqrt{2}}(|000\rangle+|011\rangle) + \frac{\beta}{\sqrt{2}}(|110\rangle+|101\rangle) \)
4. \( |\Psi_{CAB} \rangle = \frac{\alpha}{2}(|000\rangle+|100\rangle+|011\rangle+|111\rangle) + \frac{\beta}{2}(|010\rangle-|110\rangle+|001\rangle-|101\rangle) \)
5. \( |\psi_a\rangle = \alpha|0\rangle + \beta|1\rangle \\ |\psi_b\rangle = \beta|0\rangle + \alpha|1\rangle \\ |\psi_c\rangle = \alpha|0\rangle - \beta|1\rangle \\ |\psi_d\rangle = - \beta|0\rangle + \alpha|1\rangle \)
6. a. nothing, b. NOT gate c. Z gate d. NOT gate and then Z gate