How can I implement the following unitary operation using CNOT and single qubit gates?
$$U =\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & e^{i \theta} \end{bmatrix}$$
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2$\begingroup$ This looks like a CU1 gate -> qiskit.org/documentation/stubs/… . Its closely related to the controlled-Rz or CRz gate $\endgroup$– CallumCommented Sep 8, 2023 at 22:33
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1 Answer
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As I mentioned in the comments the gate is a CU1 gate. For ease of visualisation I've used $\theta=2a$ as the rotation angle.
This $\mathrm{CU1}$ gate is equivalent to a controlled-Rz up to a global phase. An Rz gate is defined to be equal to $\exp\big(-\frac{i a}{2} Z\big)$ where $Z$ is the Pauli $Z$ operator.
A $\mathrm{CU1}(2a)$ can be decomposed into the following CX and Rz operations.
If we remove the Rz from the q[0] control qubit we have an implementation of Controlled-Rz instead of CU1.
To see that this is correct consider what happens when the control qubit is in the $|0\rangle$ or $|1\rangle$.
If the control qubit is $|0\rangle$ the first Rz will leave the state unchanged. The CX gates will also not flip the target qubit and the opposing Rz rotations on the q[1] target qubit will cancel out.
Can you work out what happens when the control is in the $|1\rangle$ state?
