= Quantum circuit gate-error telescoping bound
{title2=$\|C'-C\|\leq\sum_j\|U'_j-U_j\|$}
For two <quantum circuits> with corresponding unitary gates, their difference is a sum of terms containing one gate difference and exact or perturbed products on either side. Unitary invariance and the <triangle inequality> for the <operator norm> therefore bound the circuit difference by the sum of individual gate errors. Tensoring a gate with an identity leaves that norm unchanged. On a normalized input, the same bound controls the output vector distance with gate phases retained. This is a worst-case coherent bound; cancellation can make a particular circuit less sensitive, but cannot be assumed without further information.
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