= Solution
Two <unitary transformations> $U,V$ on the same <Hilbert space> are $\varepsilon$-close in <operator-norm distance between unitary gates> when
$$
\boxed{\|U-V\|_{\mathrm{op}}
=\sup_{\|\psi\|_2=1}\bigl\|U|\psi\rangle-V|\psi\rangle\bigr\|_2\leq\varepsilon.}
$$
Thus their outputs are $\varepsilon$-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 $\min_\gamma\|U-e^{i\gamma}V\|_{\mathrm{op}}$ instead, with one phase chosen for the operator rather than separately for each input.
Back to article page
