May 11, 2010

Errors as Anyons

[Picture taken from:]

As promised, I will explain here how anyons are related to the errors in the toric code. I will rely heavily on what is in chapter 9 of Preskill's online lecture notes on quantum computation that can be found here:

[1] Chapter 9 - Topological Quantum Computation, Preskill, J.

Actually, I found that the introduction to anyons in Preskill's lectures is much more pedagogical than in the previous paper from Rao that I linked to. If you do not remember what the heck I am talking about, I suggest you to go back to the posts Kitaev's Toric Code and Anyons to refresh your mind. In the first of these posts, I explained that the toric code was defined by a set of stabilizers defined as plaquette and star operators on a square lattice in the surface of a torus. The spins, each one being just the mental image of a two-level quantum system, live on the edges of the lattice and are responsible for storing the quantum state we want to preserve.

Due to the characteristics of the set of stabilizers, we saw that the whole torus can only store 2 qubits, no matter what size it is. These 2 qubits correspond to the 2 non-trivial topological cycles of the torus, i.e., closed loops that cannot be continuously shrunk to points.

Actually, the way I described it, our torus is what is called a quantum memory. Its aim is to keep the two qubits stored for as long as we want, as if it was a quantum hard disc. It turns out that errors in the encoded message occur when the spins are changed from their original configuration. As the encoded message is defined as the ground state of the Hamiltonian, which itself is defined as minus the sum of all star and plaquette operators, when one qubit suffers an error the energy is increased by two for each operator that changes its measured value. This is because being products of the spins, the operators have only two possible eigenvalues, which are also the possible measured values, +1 and -1.

When an error occur, be it a Z or X error, the measured value of the two operators of the relevant kind (A or B) that contain that edge becomes -1 and the energy increases by 2×{-[(-1)-(+1)]}=4. Well, many times the Hamiltonian is rescaled such that the energy difference has a specific value, but that is not important now.

Before I continue there is one more piece of physics that I need to talk about. It's the Aharonov-Bohm effect. This is a quantum effect where a charged particle's wavefunction gains a phase when the particles goes around an infinite solenoid where an electric current is flowing. This infinite solenoid is just the physical way of constructing what is called a flux tube, which means that inside the solenoid there is a magnetic field that never goes outside. It has an effect on the particle because nowadays we know that the vector potential, which is the spatial part of the gauge field associated with electromagnetism, is more fundamental than the electromagnetic field and the vector potential exists outside the flux tube.

Back to the toric code, the beautiful thing is that whenever a Z-error occurs, which means that the |0> in the quantum state of the qubit becomes |1> and the |1> becomes |0>, this can be seen as a pair of "charged" particles being created in the two vertices connected by the link. Similarly, when an X-error occurs, which changes the sign of the relative phase between |0> and |1>, we can see this as the creation of flux tubes in the plaquettes which have this link as a common boundary. Flux tubes and charges are not anyons if considered isolated, but as in the Aharonov-Bohm effect, when a charge goes around a flux tube in a closed path, a loop, it turns out that its wave function is multiplied by -1. Bosons' and fermions' wavefunctions should not gain a phase if we move the particle in a closed loop taking it back to the initial place. Conclusion: the errors are actually anyons (when one kind is compared to the other).

Of course I am cheating a lot because I am not explaining the details, but for those who want to follow them, Preskill's notes and Kitaev's original paper are the recommended readings.

But the story does not end here. What about error-correction? That is the central aim, right? How does it work?

Error-correction is accomplished in the toric code by bringing two anyons together. What happens is that when they encounter each other, they annihilate and depending how this is done, the error is corrected. The important thing here is that anyons need to be annihilated in such a way that the path traced by them is a trivial cycle on the torus. If they annihilate after going around a non-trivial cycle, then the error remains! You can even annihilate anyons from different initial pairs as long as their total paths do not involve non-trivial loops.

You may quickly perceive that the solution to control errors in our code is: do not let the anyons go to far away from each other once they are formed! This way, they will not have gone around dangerous cycles on the torus and everything will be alright. So, you only need to increase the size of the torus, right? Then the anyons would take too long to travel through a non-trivial loop.

It turns out that it's not so simple (in quantum mechanics, it never is...).  In the following very nice (but articleless) paper

[2] Can one build a quantum hard drive? A no-go theorem for storing quantum information in equilibrium systems, Alicki and Horodecki [arXiv:quant-ph/0603260v1]

the authors argue that at a non-zero temperature it does not matter the size of your system, you will not be able to get rid of the errors. They give many arguments, in particular they say that if this was possible, the system would violate the second law of thermodynamics. In essence, the only states that could store some information would be classical states, which cannot store quantum superpositions.

Their conclusion is that you cannot have a quantum hard drive, store information in it and leave it there. The information will always leak unless you spend energy to keep it that way. This is dismaying as it shatters the dream of quantum information storage. However, we know that being mortals we only need to store information for a reasonable amount of time. For instance, I am pretty sure I don't care about what is gonna happen to my files after the next 300 years. So the idea is to try to increase the storage time reasonably. I've been talking to the quantum information group in Leeds about the subject and we had some nice ideas, but that is something I will write about in another opportunity. 

No comments: