Past exam of the laser science and quantum information processing course of the University of Oxford / 2017 / 4 / a / Solution

Use the time-ordered network ${\mathrm{CNOT}}_{1\to 2}$, ${\mathrm{CNOT}}_{2\to 1}$, ${\mathrm{CNOT}}_{1\to 2}$. On bits $(a,b)$ its successive values areThus the network exchanges the two computational-basis states. Since the gates are linear, this basis proof also establishes SWAP gate on arbitrary quantum states.

The Bell states $|{\Phi }^{\pm }⟩=(|00⟩\pm |11⟩)/\sqrt{2}$ and $|{\Psi }^{\pm }⟩=(|01⟩\pm |10⟩)/\sqrt{2}$ obeyThe antisymmetric spin singlet state alone has eigenvalue $-1$. Its standalone sign is a global phase; a Fredkin gate can make that sign observable relative to the control branch, as in part d.