Past exam of the computer science course of the University of Oxford / 2020 / Quantum information / 1 / e / ii / Solution

Compute the parity into the second wire, phase that wire with the T gate, then uncompute with the same controlled-NOT gate. The answer isFor computational bits $x,y$, the successive states are $|x,y⟩\mapsto |x,x\oplus y⟩\mapsto e^{i\pi (x\oplus y)/4}|x,x\oplus y⟩\mapsto e^{i\pi (x\oplus y)/4}|x,y⟩$. The parity $x\oplus y$ is one precisely on the two odd-parity inputs. Thus, in the ordered computational basis $00,01,10,11$, the matrix is exactly $\mathrm{diag}(1,e^{i\pi /4},e^{i\pi /4},1)$, with no omitted overall phase. This is a two-qubit parity phase gate.