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Let $G \subseteq SU(2)$ be a finite set, and let $E \in SU(2^k)$ be any entangling gate (so $k$ is some number strictly greater than 1). We often say $G$ is universal iff $\langle G \rangle$ is dense in $SU(2)$, because then for any $n$, $U \in SU(2^n)$, and $\epsilon > 0$, there is an $n$-qubit circuit $Q$ over $G \cup \{E\}$ that approximates the output statistics of $U$ to within $\epsilon$ (w.r.t. total variation distance and computational basis inputs and measurements).

I am curious about a slight variation to this, which is more of a group theory question, but it's quantum-inspired nonetheless. Suppose $G' \subseteq SL(2;\mathbb{C})$ is a finite set, and let $E \in SU(2^k)$ be any entangling gate. If $\langle G' \rangle$ is dense in $aSU(2)a^{-1}$ for some $a \in SL(2; \mathbb{C}) \backslash SU(2)$, is it still the case that $G' \cup \{E\}$ is universal? That is, is it still the case that for any $n$, $U \in SU(2^n)$, and $\epsilon > 0$, there is an $n$-qubit circuit $Q'$ over $G' \cup \{E\}$ that approximates the output statistics of $U$ to within $\epsilon$ (again w.r.t. total variation distance and computational basis inputs and measurements)?

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