Two-qubit parity phase gate ($G_{\theta }=\mathrm{diag}(1,e^{i\theta },e^{i\theta },1)$)

A two-qubit parity phase gate multiplies odd-parity computational vectors by $e^{i\theta }$ and fixes even-parity ones. With $R_{\theta }=\mathrm{diag}(1,e^{i\theta })$,The first controlled-NOT gate computes $x\oplus y$ in the second wire, the phase gate multiplies by $e^{i\theta (x\oplus y)}$, and the last controlled-NOT gate restores the original bits. This compute-phase-uncompute construction leaves the data unchanged except for the desired phases. The more general Pauli-string phase by parity computation uses the same mechanism. In terms of the Pauli Z gate, $G_{\theta }=e^{i\theta /2}\exp [-i\theta (Z\otimes Z)/2]$; the scalar factor ensures equality as a matrix, rather than only up to global phase.