Trading inverses for an irrep in the Solovay-Kitaev theorem

The Solovay-Kitaev theorem states that universal quantum gate sets can be exchanged with low overhead. More specifically, any gate on a fixed number of qudits can be simulated with error epsilon using merely polylog(1/epsilon) gates from any finite universal quantum gate set G. One drawback to the theorem is that it requires the gate set G to be closed under inversion. Here we show that this restriction can be traded for the assumption that G contains an irreducible representation of any finite group G. This extends recent work of Sardharwalla et al. [Sardharwalla et al., 2016], and applies also to gates from the special linear group. Our work can be seen as partial progress towards the long-standing open problem of proving an inverse-free Solovay-Kitaev theorem [Dawson and Nielsen, 2006; Kuperberg, 2015].