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Randomized benchmarking (RB) is a powerful method for determining the errorrate of experimental quantum gates. Traditional RB, however, is restricted togatesets, such as the Clifford group, that form a unitary 2-design. Therecently introduced character RB can benchmark more general gates usingtechniques from representation theory; up to now, however, this method has onlybeen applied to "multiplicity-free" groups, a mathematical restriction on thesegroups. In this paper, we extend the original character RB derivation toexplicitly treat non-multiplicity-free groups, and derive several applications.First, we derive a rigorous version of the recently introduced subspace RB,which seeks to characterize a set of one- and two-qubit gates that aresymmetric under SWAP. Second, we develop a new leakage RB protocol that appliesto more general groups of gates. Finally, we derive a scalable RB protocol forthe matchgate group, a group that like the Clifford group is non-universal butbecomes universal with the addition of one additional gate. This exampleprovides one of the few examples of a scalable non-Clifford RB protocol. In allthree cases, compared to existing theories, our method requires similarresources, but either provides a more accurate estimate of gate fidelity, orapplies to a more general group of gates. In conclusion, we discuss thepotential, and challenges, of using non-multiplicity-free character RB todevelop new classes of scalable RB protocols and methods of characterizingspecific gates.

Character randomized benchmarking for non-multiplicity-free groups with applications to subspace, leakage, and matchgate randomized benchmarking