Trading inverses for an irrep in the Solovay-Kitaev theorem

Understand

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 $\mathrm{polylog}(1/\epsilon)$ gates from any finite universal quantum gate set $\mathcal{G}$.
• One drawback to the theorem is that it requires the gate set $\mathcal{G}$ to be closed under inversion.
• Here we show that this restriction can be traded for the assumption that $\mathcal{G}$ contains an irreducible representation of any finite group $G$.