# Quantum Error Correction

1. Apr 13, 2005

### phmzk

Hello people,

Not sure if I'm posting in the right place, but I have read quite a few posts since joining and have to say this forum is of a very high standard, well done.

I'm a final year student working on my dissertation on Quantum Error Correction and was just wondering if anyone had any good articles or views on the subject. Also if anyone has any ideas about how to plan out a piece on this subject would be greatly appreciated. Currently my plan is to speak about how computers have developed to a stage where we've had to look at computing on a quantum level. Then a bit about Quantum world of physics to try and link the topics.

I've looked at classical error correction, and replicated a repetition code in excel. Any ideas about other codes I could try and produce and how to deal with them? Once classical error correction has been delt with how can I deal with quantum error correction? Any feedback/suggestions would be greatly appreciated :)

2. Apr 16, 2005

### slyboy

I have also written a dissertation on quantum error correction, although that was in graduate school, so maybe mine was more technical than yours has to be. As I see it, the main topics you could choose to cover are:

1. The standard model'' of quantum error correction and how it leads to the requirements and definitions of quantum error correction codes.

2. Simple examples of error correction codes - quantum repetition code, Shor's nine-qubit code, CSS codes and the 5 qubit code.

3. Bounds for quantum codes.

4. Gottesman's stabilizer formalism for quantum error correcting codes and its connection to classical error correction codes.

5. Fault tolerance - how error correction can be used to prove the threshold theorem for quantum computation.

6. Experimental implementations of error correction codes (these have mainly been done in NMR).

7. Nonstandard methods of error correction, e.g. collective decoherence and encoding in decoherence free subspaces.

Of course, you probably only want to cover two or three of these topics.

A good starting point is Dan Gottesman's Ph.D. thesis http://www.arxiv.org/abs/quant-ph/9705052 and the references therein. This will give you a good picture of the field up to about 1997.

Also recommended is his introductory article http://www.arxiv.org/abs/quant-ph/0004072.

That should cover you for 1-5. I am not an expert on 6 or 7, but doing a search on arXiv.org for those topics should throw up some interesting papers.

Let me know if you need any more help.

3. Apr 20, 2005

### phmzk

Wow! Thank you so much for all that! Thats so much more help than our project tutor was willing to dish out. Don't understand whats wrong with him...must be a game to him to see us struggle.

We've replicated some code for classical error correction using excel...is there any way we can produce some kind of simulation for quantum error correction on a classical computer without it becoming an extremely long process?

Many thanks again

4. Apr 21, 2005

### H-bar None

Last edited: Apr 21, 2005
5. Apr 23, 2005

### slyboy

Generally, trying to simulate any nontrivial quantum computation in a straightforward way will require exponential overhead and quickly become impractical. That's why quantum computing is supposed to be more powerful than classical computing in the first place.

There are a few things you can do though. It may be possible to simulate small quantum codes, such as the 5-qubit code, without too much effort. Another trick is to restrict attention to stabilizer states and quantum gates in the Clifford group. These can be simulated efficiently on a classical compute. See "The Heisenberg Representation of Quantum Computing" by Gottesman and a recent paper by Gottesman and Aaronson, which are both on arXiv. I believe the latter paper may even direct you to a website with some code for doing this. Of course, whilst this is mathematically nice, it defeats the main point of quantum error correction, which is that it works for all states and all independednt error models.

6. Jul 4, 2011

### explorer_qbit

Dear experts,
May I ask how do you calculate the QER (Qbit Error Rate) for a given probability of error Pe? I mean for a normal computer you have BER (Bit Error Rate) versus SNR curve. I have seen some QER versus Pe curves in some research papers, but I am not sure how to calculate the QER. For example, a qbit is given by phi = a|0> + b|1> , where the probability of it being |0> is given by a^2 and the prob of it being |1> is b^2. The qbit itself is not hard-decision based, then, how do you decide whether it is errorneous or not?

Thank you.