 Solution

Under fixed phase representatives, two unitary operators on the same space are $\epsilon$-close when $\Vert U-V{\Vert }_{\mathrm{op}}\le \epsilon$, where the operator norm isThus every input vector is approximated uniformly, not merely one selected input. Under the phase-insensitive convention, replace this bound by ${\min }_{\theta }\Vert U-e^{i\theta }V{\Vert }_{\mathrm{op}}\le \epsilon$. The minimization uses one phase for the whole operator, not a separately chosen phase inside the supremum for each input. This is operator-norm distance between unitary gates with the stated global-phase identification.