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:

[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 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.