Past exam of the laser science and quantum information processing course of the University of Oxford / 2021 / 4 / c

Track the $-i$ factors only when the corresponding transition is addressed. They are common within the addressed two-state subspace, but become physically relevant relative phases between computational basis states. Using a semicolon to separate the shared motional number state, the complete paths areThe middle arrow comprises both Y pulses. In the last row those individual steps are $-i|ge;1⟩\to -|gf;0⟩\to +i|ge;1⟩$. The two X pulses contribute $(-i{)}^2=-1$ for either initially excited control; the Y pair supplies a second minus sign only when the second ion is also excited.

The shared motion returns to its ground state for all four inputs, and the auxiliary internal state is empty. Quantum superposition therefore gives the two-qubit gateDiscarding the X-pulse factors would incorrectly identify $G$ with a Controlled-Z gate itself.