Complete randomized benchmarking protocol accounting for leakage errors

Randomized Benchmarking allows to efficiently and scalably characterize the average error of an unitary 2-design such as the Clifford group $\mathcal{C}$ on a physical candidate for quantum computation, as long as there are no non-computational leakage levels in the system. We investigate the effect of leakage errors on Randomized Benchmarking induced from an additional level per physical qubit and provide a modified protocol that allows to derive reliable estimates for the error per gate in their presence. We assess the variance of the sequence fidelity corresponding to the number of random sequences needed for valid fidelity estimation. Our protocol allows for gate dependent error channels without being restricted to perturbations. We show that our protocol is compatible with Interleaved Randomized Benchmarking and expand to benchmarking of arbitrary gates. This setting is relevant for superconducting transmon qubits, among other systems.