  There is currently a significant need for robust and efficient methods for
characterizing quantum devices. While there has been significant progress in
this direction, there remains a crucial need to precisely determine the
strength and type of errors on individual gate operations, in order to assess
and improve control as well as reliably bound the total error in a quantum
circuit given some partial information about the errors on the components. In
this work, we first provide an optimal bound on the total fidelity of a circuit
in terms of component fidelities, which can be efficiently experimentally
estimated via randomized benchmarking. We then derive a tighter bound that
applies under additional information about the coherence of the error, namely,
the unitarity, which can also be efficiently estimated via a related
experimental protocol. This improved bound smoothly interpolates between the
worst-case quadratic and best-case linear scalings for composite error
channels. As an application we show how our analysis substantially improves the
achievable precision on estimates of the error rates of individual gates under
interleaved randomized benchmarking, enabling greater precision for current
experimental methods to assess and tune-up control over quantum gate
operations.
