Universal Quantum Computation and Total Quantum Dimension

Question

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?

Comments

Judging by the very last lines on the very last slide of http://www.uoguelph.ca/quigs/cssqi14/Slides/Guelph1.pdf (see also arXiv:1405.7778 and 1401.7096), this seems to be an open question (with the conjecture being that one anyon with dimension $d^2\notin \mathbb Z$ is required).

Answer 1

These are open questions in the field of topological quantum computations. A conjecture by Zhenghan Wang says that the braid group representation of a modular tensor category (i.e. anyon systems) with total quantum dimensions $D^2$ being an integer (which means that quantum dimension of each anyon also squares to an integer) has a finite image, thus braiding alone is not universal. It is widely believed to be true in the community and no counterexamples are known.

One can try to get universal gate sets by supplying some other topologically-protected operations, such as measurement of topological charges. For Ising anyons, even with measurement one can not get universal gates. But for SU(2)$_4$ anyons (dimensions $1, \sqrt{3}, 2, \sqrt{3}, 1$), by adding measurement universal computation can be achieved, as shown in arXiv:1405.7778.

Comments

Hi Meng Cheng,

thanks for this interesting answer. I hope you dont mind that I edited in the link to the ArXiv paper.