  As experimental platforms for quantum information processing continue to
mature, characterization of the quality of unitary gates that can be applied to
their quantum bits (qubits) becomes essential. Eventually, the quality must be
sufficiently high to support arbitrarily long quantum computations. Randomized
benchmarking already provides a platform-independent method for assessing the
quality of one-qubit rotations. Here we describe an extension of this method to
multi-qubit gates. We provide a platform-independent protocol for evaluating
the performance of experimental Clifford unitaries, which form the basis of
fault-tolerant quantum computing. We implemented the benchmarking protocol with
trapped-ion two-qubit phase gates and one-qubit gates and found an error per
random two-qubit Clifford unitary of $0.162 \pm 0.008$, thus setting the first
benchmark for such unitaries. By implementing a second set of sequences with an
extra two-qubit phase gate at each step, we extracted an error per phase gate
of $0.069 \pm 0.017$. We conducted these experiments with movable,
sympathetically cooled ions in a multi-zone Paul trap - a system that can in
principle be scaled to larger numbers of ions.
