  Rigorously establishing that the error in an experimental quantum operation
is beneath the threshold for fault-tolerant quantum computation currently
requires considering the worst-case error, which can be orders of magnitude
smaller than the average gate infidelities routinely reported in experiments.
We show that an improved bound on the worst-case error can be obtained by also
considering the recently-introduced unitarity of the noise where the upper and
lower bounds differ by a factor of $\approx 2.45$ for unital qubit channels. We
prove that the contribution from the nonunital part of any noise map is at most
on the order of the average gate infidelity and so is negligible relative to
any coherent contribution. We also show that the "average" error rate when
measurements are not restricted to an eigenbasis containing the state of the
system exhibits the same scaling as the worst-case error, which, for coherent
noise, is the square-root of the infidelity. We also obtain improved bounds for
the diamond distance when the noise map is known (or approximately known).
