**Question 1**: IF the anyonic system can perform the Universal Quantum Computation, THEN the Total Quantum Dimension $D$ of the system must be $D \not\in \mathbb{Z}$. **True or False**?

Here

$$D=\sqrt{\sum_i d_i^2},$$

with $d_i$ as the quantum dimension of individual anyons.

For example, the Ising anyon can-NOT implement the Universal Quantum Computation (unless adding extra phase gate with extra dynamical operations), and $D=\sqrt{1+1+2^2}=2 \in \mathbb{Z}$.

For example, the Fibonacci anyon can implement the Universal Quantum Computation, and $D=\sqrt{1+(\frac{1+\sqrt{5}}{2})^2} \not\in \mathbb{Z}$.

Reverse the statement:

**Question 2**: IF the Total Quantum Dimension $D$ of the anyonic system has $D \not\in \mathbb{Z}$, THEN the anyonic system can perform the Universal Quantum Computation. **True or False**?

**Question 3**: How to show/prove the above two statements? Or what are the counter examples?