Randomized Benchmarking with Restricted Gate Sets

Standard randomized benchmarking protocols require sampling from a unitary 2-design, which is not always practical. In this talk I examine randomized benchmarking protocols based on subgroups of the Clifford group that are not unitary 2-designs. I show in a variety of cases that one can benchmark the error probability to within a small factor that rapidly approaches unity as the number of qubits in the benchmarking experiment grows.