= Solution
Under fixed phase representatives, two <unitary operators> on the same space are \b[$\varepsilon$-close when $\|U-V\|_{\rm op}\le\varepsilon$], where the <operator norm> is
$$
\|U-V\|_{\rm op}=\sup_{\|\psi\|_2=1}\|(U-V)|\psi\rangle\|_2.
$$
Thus every input <vector> is approximated uniformly, not merely one selected input. Under the phase-insensitive convention, replace this bound by $\min_\theta\|U-e^{i\theta}V\|_{\rm op}\le\varepsilon$. 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.
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