Approximate randomized benchmarking for finite groups

We investigate randomized benchmarking in a general setting with quantum gates that form a representation, not necessarily an irreducible one, of a finite group. We derive an estimate for the average fidelity, to which experimental data may then be calibrated. Furthermore, we establish that randomized benchmarking can be achieved by the sole implementation of quantum gates that generate the group as well as one additional arbitrary group element. In this case, we need to assume that the noise is close to being covariant. This yields a more practical approach to randomized benchmarking. Moreover, we show that randomized benchmarking is stable with respect to approximate Haar sampling for the sequences of gates. This opens up the possibility of using Markov chain Monte Carlo methods to obtain the random sequences of gates more efficiently. We demonstrate these results numerically using the well-studied example of the Clifford group as well as the group of monomial unitary matrices. For the latter, we focus on the subgroup with nonzero entries consisting of n-th roots of unity, which contains T gates.