I am trying to understand a part from the article "Restrictions on Transversal Encoded Quantum Gate Sets" (arxiv link) by Eastin and Knill.
In the article they talk about the importance of transversal encoded gates for fault tolerant quantum computing. They prove that there can't be a gate set that works on a non trivial quantum code space $\mathcal C$ that is also universal and the gates are transversal.
I understand the claim, but I have tried to understand the proof for a couple of weeks and just cant wrap my mind around it.
Can someone help me understand the following paragraph from the article?
An outline of the argument is as follows: The set of logical unitary product operators, $\mathcal G$, is a Lie subgroup of the Lie group of unitary product operators, $\mathcal T$. As a Lie group, $\mathcal G$ can be partitioned into cosets of the connected component of the identity, $\mathcal C$; these cosets form a discrete set, $\mathcal Q$. Using the fact that the Lie algebra of $\mathcal C$ is a subalgebra of $\mathcal T$, it can be shown that the connected component of the identity acts trivially for any local-error-detecting code. This implies that the number of logically distinct operators implemented by elements of $\mathcal G$ is limited to the cardinality of $\mathcal Q$. Due to to the compactness of $\mathcal T$, this number must be finite. A finite number of operators can approximate infinitely many only up to some fixed accuracy; thus, $\mathcal G$, the set of logical unitary product operators, cannot be universal. Transversal operators may be viewed as product operators with respect to a transversal partitioning of the code, so the ability to detect an arbitrary error on a transversal part implies the nonexistence of a universal, transversal encoded gate set.
Thank you for any help :)