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:

[1]

*Fault tolerant quantum computation by Anyons*, A. Y. Kitaev
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**correspond to eigenstates |***t**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 2*k*^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 2*k*^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*.
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