Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 324 / 4 / a / ii / Solution

For the controlled-NOT gate $U={\mathrm{CNOT}}_{12}$, propagation of the Pauli generators givesThus an $X$ on the control propagates forward to the target, while a $Z$ on the target propagates backward to the control.

Suppose $\tilde{V}$ has the same four conjugation rules and put $W=U^{†}\tilde{V}$. Then $W$ commutes with $X_1,X_2,Z_1,Z_2$. These generators span the full two-qubit operator algebra, so its commutant consists only of scalar multiples of the identity. Hence $W=e^{i\varphi }I$ and