It seems to me that the following must be well-known. Anybody know a reference for it?
Let $U$ be a $d \times d$ unitary matrix. For any $\epsilon > 0$ there exists some positive integer $m$ such that $\|U^m-I\| \leq \epsilon$. Here, $I$ indicates the identity matrix and $\| \cdot \|$ just indicates the standard operator norm induced by the Euclidean norm on vectors.
It would be extra nice to have an upper bound on how big $m$ needs to be as a function of $d$ and $\epsilon$ but I don't actually need this.
The group of unitary matrices is compact, and for any $\epsilon>0$, there are distinct integers $m$ and $n$ such that $\| U^m- U^n\| < \epsilon$. So $\|U^{m-n}-I\| <\epsilon$. This seems so simple that a reference would not be required.