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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.

Two unitary transformations $U,V$ on the same Hilbert space are $\epsilon$-close in operator-norm distance between unitary gates whenThus their outputs are $\epsilon$-close in pure-state norm distance for every normalized input. This is an operator norm condition, stronger than checking only one chosen input. If global phase is disregarded for the whole operation, use ${\min }_{\gamma }\Vert U-e^{i\gamma }V{\Vert }_{\mathrm{op}}$ instead, with one phase chosen for the operator rather than separately for each input.