Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 61 / 3 / a / ii / Solution

For target unitary operator $U_T$ and actual propagator $U_f(T)$ on an $n$-dimensional space, minimize the phase-insensitive unitary gate errorThe Cauchy-Schwarz inequality for the Hilbert-Schmidt inner product gives a value in $[0,1]$, because both unitary matrices have squared norm $n$. The error is zero exactly when the matrices are proportional, namely $U_f(T)=e^{i\varphi }{U}_T$. It therefore tests the full process, not just its action on one state, and ignores a physically irrelevant common phase. If the gate phase itself must be fixed, instead minimize $\Vert U_f(T)-U_T{\Vert }_{\mathrm{HS}}^2/(2n)=1-\mathrm{Re}\mathrm{Tr}(U_T^{†}{U}_f(T))/n$, which vanishes only at exact equality.