Thursday, 29 April 2010

Statphys 24

For those who are interested in, there is still time to register to Statphys 24. Statphys is considered the most important international conference of Statistical Physics and this year it is going to take place on Cairns, Australia, from 19 to 23 of July. This is the official website: 

Every conference, the Boltzmann medal is awarded to scientists with outstanding contributions to statistical physics. This year's award is going to two persons that do not need many introductions: John Cardy and Bernard Derrida. Although their entries in the Wikipedia are small, their contributions are not and those who don't know them can do a quick search of their names in the internet to see the extension of their work.

Although there is still time to participate in the conference, abstract submission closes tomorrow (30/Apr) and so those who are still thinking about submitting their work should rush. Many outstanding scientist will give talks, for instance, R. Baxter, S. Sachdev and C. N. Yang (!!!). The complete list of plenary talks with brief bios of the speakers is here:

In addition, many other related satellite meetings will be happening in the far east. This page has a list of them:

If you have the time, the interest and the funding to go, that should not be missed.

Wednesday, 21 April 2010

Error-Correcting Bug

When you start to work with something, you soon begin to see that thing everywhere. In 2006 I started to work with statistical mechanics of error-correcting codes in the NCRG at Aston University and suddenly I noticed things I haven't before.

I always liked extremophiles. They are really interesting organisms. I guess that what fascinates me is how many extraordinary solutions to the survival problem they present. The guy in the picture above mix my interest in extremophiles with my present work. It is called Deinococcus radiodurans. It is probably the organism with the highest error-correcting ability in nature, and I believe that the exact mechanism is still unclear. Any biologist around please clarify if it is true.

This bacterium can stand radiation levels 500 times larger than a human being can survive to with no effort and even get to 3000 times sometimes. The way it does that is ingenuous. It blends hardware error-correction with software error-correction.

The first part is a pure hardware correction. Its DNA is arranged in a torus that is very tightly packed. As the main effect of radiation is to cut the DNA in parts, this configuration guarantees that most of the parts do not flow away in the cell, staying almost in the same place, which facilitate the correction by the relevant proteins. 

Now, if you look at the picture above, you will see that our star is formed by 4 compartments. In the Wikipedia article [1] I am linking below it is said that each part is a different bacterium, although in a second article [2] it is said that these compartments are of the same bacterium, and then of the same cell I suppose as, as far as I know, bacteria are unicellular organisms. In any case, I think that [2] is probably more correct as the next part of the correction uses DNA from all compartments.

It works actually as the simplest error-correction code of all, a repetition code. Repetition codes are simply codes where symbols are repeated a number of times so that if the number of errors are not that big, you can correct them by just following what the majority of the symbols in the neighborhood are. In the case of our guy, it has one copy in each compartment and (that's fantastic!) after the first hardware error-correction the DNA in one compartment unfolds, migrates to another and melds with the other one to correct its errors! Well, it is not exactly a conventional repetition code in the sense that there seems to be no global comparison between all four copies to decide on the correction based on the majority rule. However, it is still kind of.

I could not find the details of the process. For example, does it migrate and then come back to the original compartment? Does it do it in some specific order? Which is the first DNA to migrate? Again, if there is a biologist that can clarify those questions, please feel free.

Isn't that cool? I must confess that the first time I heard about it I was expecting some very smart kind of error-correcting code embedded in the DNA construction. I was a little disappointed when I learned that from the software point of view, it is as simple as it can get. However, the whole process is quite interesting and not less ingenious because of that.

Tuesday, 20 April 2010

Solving Rubik's Cube

I am experimenting with a platform to write small one-page websites called Squidoo. It seems very interesting and very elegant as it is somewhere between a blog post and a website. As my first tentative, I wrote a lens (which is how they call it) with a method to solve Rubik's Cube I have learned a long time ago:

The method is explained in details with pictures (made by myself) for every move, although I haven't reached the point of making a video yet. Maybe in the future. 

I also write en passant about the relationship between the cube and group theory, as I learned the solution when I was actually preparing a coursework for a Group Theory class. Check it out and leave a comment.

[The photo above is from Erno Rubik, the creator of the cube. I don't have the credits for it, but I will be happy to put it here if someone can provide it.]

Thursday, 15 April 2010

Kitaev's Toric Code

After 3 years, here I am again. Hard times, but I will neither complain nor explain. Let me go straight to what matters, for time is still short.

I have just come back from the 10th Topological Quantum Computation Symposium in Leeds. I was invited to talk about my present work on statistical physics of classical error-correcting codes and our tentatives to extend it to quantum codes. It was a bit off topic I thought after watching all those talks about anyons, but they insisted to me it was useful anyway. I am not so sure, but I would like to believe so.

I am becoming very interested in TQC, probably because I miss quantum mechanics. Let me explain more or less how it works, or how I understood it. Following John Baez's philosophy, I may be able to clarify my thoughts by doing that.

Topological Quantum Computation is a spinoff of Topological Quantum Codes, which seem to have been first proposed by A. Kitaev in the by now classic paper:

Kitaev's model became known as the Toric Code, because it is designed to correct errors in a quantum system which is constructed in the surface of a torus. The Toric Code belongs to a class of quantum error-correcting codes called Stabilizer Codes. In this kind of code the codewords t correspond to eigenstates |t> with eigenvalue 1 of an Abelian group of operators S called the Stabilizer Group for that particular code such that

S|t> = |t>.

For obvious reasons, |t> is said to be stabilized by the code. A more detailed explanation can be found on Nielsen and Chuang's book:

[2] Quantum Computation and Quantum Information, Nielsen & Chuang

In the toric code, the codewords are states of a system composed by a square N x N lattice with qubits (two-level quantum systems) placed on the edges, not on the vertices. Opposite boudaries of the square lattice are identified, i.e., periodic boundary conditions are assumed resulting in the surface of a torus, which gives the model its name. The total amount of qubits is 2k^2.

The toric code is then defined by a set of star operators A and plaquette operators B. Each star operator A_s is associated with the s-th vertex of the lattice and is composed by the tensor product of the four X operators (2 x 2 Pauli matrix corresponding to the spin in the x direction) acting on the four edges meeting at that vertex. One plaquette operator B_p is associated to each square on the lattice and composed by the tensor product of the four Z operators (spin in the z direction) acting on the edges of the corresponding square. As X and Z commute for different vertices and anticommute for the same vertex and as each star operator and each plaquette operator has only zero or two edges in common, all A_s's commute with all B_p's. Then we call the toric code the set of states stabilized by all A_s and B_p simultaneously.

However, the A's and B's are not completely independent and multiplying all operators of one kind together must give the identity. Therefore, the stabilizer group contains 2k^2-2 independent generators. Now, the handwaving argument is that each stabilizer condition halves the stabilized vector space for each stabilizer has half of the eigenvalues -1 and the other half +1. The rigorous proof can be found in Nielsen & Chuang's book above. This implies that the stabilized vector space has dimension 2^2, which means that it encodes 2 qubits.

Any Pauli operator that commutes with all the stabilizers sends a codeword to another codeword. These operators can be represented by the product of two other Pauli operators containing respectively only I's and X's and only I's and Z's. These can be represented graphically by chains, which are structures composed by the edges in which the Z's act and the edges of the dual lattice that cut the edges of the direct lattice where the X's act. The picture at the side give examples of these chains. It was taken from Kitaev's paper [1]. 

It turns out that topologically trivial loops, those that can be contracted to a point, correspond to stabilizers, while non-trivial ones do not and therefore change one codeword into another.

This basically defines the toric code. In the next post I will explain how errors can be graphically represented and how they correspond to quasiparticles of a specific Hamiltonian that posses fractional spin, which means, they are anyons