When does tail-risk optimization change a quantum gate? An equivalence boundary for mean and CVaR control

Control pulses for quantum gates are normally optimized for the error averaged over noise. Fault tolerance, however, is sensitive to the worst gates, which suggests optimizing the tail of the error distribution instead. Doing so is costly, since only a small fraction of the samples lies in the tail, and it is not known in advance whether the resulting pulse will differ from the one that minimizes the mean. We show that for Gaussian noise entering the gate linearly it does not: the error distributions of all pulses differ only by a scale factor, so the pulse with the smallest mean error is also optimal for every monotone measure of tail risk. In simulations of a transmon qubit, hundreds of candidate pulses per setting contain essentially no clearly separated pair that the mean and the tail rank differently below an accumulated phase variance of about 0.05. The two objectives separate once the optimized pulse retains error of second order in the noise. The tail-optimal gate then carries a coherent offset that lowers its nominal fidelity but cancels the quadratic error of the worst noise realizations. Its advantage scales as the square of the ratio of quadratic to linear error, and we give the coefficient in closed form. Noise memory and rare relaxation events change the shape of the tail, in quantitative agreement with our predictions, but alter the optimal pulse only when the pulse can act on the channel involved. Because the size of the advantage can be estimated from the mean-optimal gate, one can decide whether tail optimization is worth its cost before running it.