  Randomized benchmarking (RB) is widely used to measure an error rate of a set
of quantum gates, by performing random circuits that would do nothing if the
gates were perfect. In the limit of no finite-sampling error, the exponential
decay rate of the observable survival probabilities, versus circuit length,
yields a single error metric $r$. For Clifford gates with arbitrary small
errors described by process matrices, $r$ was believed to reliably correspond
to the mean, over all Cliffords, of the average gate infidelity (AGI) between
the imperfect gates and their ideal counterparts. We show that this quantity is
not a well-defined property of a physical gateset. It depends on the
representations used for the imperfect and ideal gates, and the variant
typically computed in the literature can differ from $r$ by orders of
magnitude. We present new theories of the RB decay that are accurate for all
small errors describable by process matrices, and show that the RB decay curve
is a simple exponential for all such errors. These theories allow explicit
computation of the error rate that RB measures ($r$), but as far as we can tell
it does not correspond to the infidelity of a physically allowed (completely
positive) representation of the imperfect gates.
