Topological Quantum Computing

For some time, I’ve heard theorists give talks about topological quantum computing. Every one of them jumps right in and never discusses what the hell this is. And it has never made sense. They talk about braiding of particles and somehow that’s equivalent to quantum gates. What the hell is braiding? Nothing ever made sense.

But as of yesterday, it finally does. Or at least some small part of it does. I am not a solid-state/condensed-matter physicist. So I understand absolutely nothing about proposed solid-state quantum computing, and the proposed architecture for topological quantum computing either. But at least, I now have some vague idea of this. Sankar Das Sarma gave a talk yesterday, and, well I think things are starting to make sense.

Either that, or I’ve heard these terms far too many times.

Here’s the main idea:

Local geometry is a redundant way of encoding topological information. Slight denting or stretching of a torus does not change its genus.

What does that mean? Provided I understood this correctly, if we can find a physical quantum system that is insensitive to slight perturbations in its topology, then the quantum states are, in a sense, immune to decoherence. This insensitivity cannot be due to symmetry, as we can break the symmetry. Somehow, this magically leads to “non-Abeliean quasiparticle braiding statistics.”

Briefly on quantum statistics and identical particles: (some lecture notes here) If we have two identical particles (for example, electrons), we can describe the state of both of them by the wavefunction , where are the positions of the particles. Since these particles are identical, then the dynamics of the system is the same if I swap the positions of the two particles. So, the dynamics of particle A at and particle B at is exactly the same as the dynamics of particle A at and particle B at .

If I interchange the particles twice, I should end up with the same function. That is interchange once: . Interchange a second time: . Because the dynamics are the same, then in three spatial dimensions, we have two types of wavefunctions:

This leads to bosons and fermions, which obey different statistics, and all the wonderful things that we know (BEC, fermi-degenerate gasses, etc.). Fermions obey Pauli’s Exclusion Principle, while bosons do not.

However, strange things happen in two dimensions. It turns out that you can have another type of particle, an anyon. Here, an interchange (?) picks up an additional phase. You can view the two particles as points on a sheet, and the interchange as winding one around the other.

Here’s where the topology comes in. I believe that because the fundamental group of is trivial, and thus you get these bosons and fermions. However, the fundamental group of is non-trivial.

What this means is that any loop in three dimensions (with some points removed) can be pulled together into a point. But a loop restricted to a plane (minus some points) cannot. If I had a loop around one of those points on the plane, I cannot pull it past the hole.

This is what’s meant by braiding. By moving particles around each other, these lines get braided together and knotted up. Somehow (magically) this is equivalent to gates and such, although, it makes no sense as to how that is done.

So, it seems like I’m starting to get the idea, but I don’t have a clue as to how the actual ‘computing’ comes into play here.