Is SO(2) with any entangling gate universal?

Question

Let $E \in SU(2^k)$ be any entangling gate (for some $k \geq 2)$. Then my question is simply whether or not it is known that $SO(2) \cup \{ E \}$ is universal for $\mathsf{BQP}$?

Clearly it seems that this ought to be true, in light of the standard facts that any entangling gate with arbitrary single-qubit gates is universal, and that "real quantum computation" (computation with gates having only real entries) can also realize universality for $\mathsf{BQP}$.

I am just curious if anyone has seen the result made explicit that $SO(2) \cup \{ E \}$ is universal for any entangling gate $E$, or if this is something I'd have to prove myself?

Answer 1

No. With the conventional notation of the Pauli basis, $SO(2) \cong U(1)$ rotations just constitute rotations around $Y$, i.e., these gates are parameterized by the set $\{e^{i \theta Y} : \theta \in [0, 2\pi)\}$. For ease, let us redefine our basis so that we instead have rotations around the $Z$ axis. One can choose $E$ to be $CZ$, which is entangling. However, it commutes trivially with the above representation of $SO(2)$ rotations. Given that $\langle \{e^{i \theta Z} : \theta \in [0, 2\pi)\}, E \rangle$ is Abelian, it cannot be universal.