Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,873 answers , 20,701 comments
1,470 users with positive rep
502 active unimported users
More ...

Universal Quantum Computation and Total Quantum Dimension

+ 5 like - 0 dislike
222 views

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?

asked Dec 26, 2014 in Theoretical Physics by RKKY (320 points) [ revision history ]

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).

1 Answer

+ 4 like - 0 dislike

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.

answered Jan 10, 2015 by Meng (550 points) [ revision history ]
edited Jan 10, 2015 by Meng
Hi Meng Cheng,

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

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysic$\varnothing$Overflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...