Quantum circuit gate-error telescoping bound ($\Vert C^{\prime }-C\Vert \le {\sum }_j\Vert U_j^{\prime }-U_j\Vert$)

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.