Past exam of the laser science and quantum information processing course of the University of Oxford / 2018 / 4 / c

For $d$-dimensional unitaries, $\mathrm{tr}(U^{†}U)=d$. The relative propagator $W=U^{†}V$ is unitary, and its eigenvalues lie on the unit circle. The triangle inequality gives $|\mathrm{tr}W|\le d$, with equality precisely when all eigenvalues have the same phase, or $V=e^{i\gamma }U$. Therefore the specified trace-overlap gate fidelity lies between zero and one, is insensitive to global phase, and equals one exactly when the implemented gate has the same physical action as the target. It compares propagators independently of any particular input state; it is not the identically defined quantity as average state or average gate fidelity.

For target $R_x(\pi )$ and actual $R_x[\pi (1+ϵ)]$, the relative propagator is $R_x(\pi ϵ)$. Its trace is $2\cos (\pi ϵ/2)$, soThe absolute value is unnecessary in a sufficiently small neighbourhood of zero error, but belongs in the exact expression.