Operator-norm distance between unitary gates ($d_{\mathrm{op}}(U,V)=\Vert U-V{\Vert }_{\mathrm{op}}$)

The largest pure-state norm distance between outputs of two unitary operators on the same normalized input. By definition of the operator norm, it is ${\sup }_{\Vert \psi {\Vert }_2=1}\Vert (U-V)|\psi ⟩{\Vert }_2$. If it is at most $\epsilon$, every input is approximated to that accuracy. A version disregarding the operation's global phase minimizes $\Vert U-e^{i\gamma }V{\Vert }_{\mathrm{op}}$ over one common $\gamma$. Error in a finite quantum circuit accumulates at most additively by the quantum circuit gate-error telescoping bound.