  Standard randomized benchmarking protocols entail sampling from a unitary 2
design, which is not always practical. In this article we examine randomized
benchmarking protocols based on subgroups of the Clifford group that are not
unitary 2 designs. We introduce a general method for analyzing such protocols
and subsequently apply it to two subgroups, the group generated by
controlled-NOT, Hadamard, and Pauli gates and that generated by only
controlled-NOT and Pauli gates. In both cases the error probability can be
estimated to within a factor of two or less where the factor can be arranged to
be conservative and to decay exponentially in the number of qubits. For
randomized benchmarking of logical qubits even better accuracy will typically
be obtained. Thus, we show that sampling a distribution which is close to a
unitary 2 design, although sufficient, is not necessary for randomized
benchmarking to high accuracy.
