Even though it is hard work, it is quite satisfying to drill holes in concrete. Apparently, my degree covers everything from construction to theoretical quantum physics. Good to know that I’m getting a well-rounded education.
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 
If I interchange the particles twice, I should end up with the same function. That is interchange once: 
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 
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.
On the ArXiv today, there’s a paper that answers my earlier question about the classical limit of the Jaynes–Cummings model. Apparently, if I just tried a little harder, I could have had another paper. Such is life.
Basically, in the Jaynes–Cummings model, the dynamics of a qubit and a single photon is given by
That is, the energy of the qubit, the energy of the photon, and the interaction between the two. This predicts energy exchange between the photon and qubit. The qubit will oscillate between zero and one. But the theory predicts a decay of this oscillation as well as revivals.
A classical field interacting is described by 
It seems that the timescale for the oscillation in both cases is inversely proportional to the electric field strength. The decay time is independent of the field strength. So, in the large photon limit, we expect there to still be a decay of the oscillation, but we should still see many oscillations before.
The paper also has another correspondence, that I haven’t quite finished yet. But it involves the Lindblad equation, which should make it fun.
It seems as if I’m spending far too much time dealing with spam comments and malicious hacks into my site. In fact, I feel as if I spend most of my time maintaining the integrity of this site than actually posting on it. Therefore, I have decided to no longer host and maintain my blog. Instead, I have moved it over to wordpress.com. Granted, it may not be as featureful, but at least I won’t have to be spending a majority of my time dealing with the annoying gnats of the internet.
The new location of my blog is now http://photicus.wordpress.com. Please update your links and RSS feeds.
I’m sitting in a coffee shop in Takoma Park, enjoying some espresso. I have to say that to the north of me, the neighborhood is, well, colorful and diverse. Driving down with my car, I accidentally was in an exit-only lane, and I couldn’t merge, due to the traffic. So I took surface streets to my place. First, went through downtown Bethesda—very expensive. Then, through downtown Silver Spring—a rather nice area. Finally, through the diverse neighborhood near me, which is…not so nice. As I drove east, the area became progressively worse. At least I’m not in College Park. It seems like I get a crime report from the University every day.
I will definitely miss Michigan. Lots of good friends. Lots of good memories. It was a somber drive south. I’m glad to have spent some time with old friends at Michigan. It was nice to see you again. For my friends who I didn’t get to see, I’m sorry we weren’t able to meet up. Hopefully I’ll be around soon. To my new friends in Ann Arbor/Detroit, I’m sorry to have left you so soon. My last Saturday in Michigan was fun—drinks at the Rathskeller with my friend M__ (whose ex is dating my ex), and then to Downtown Detroit with S__ (who cuts my hair) for a rave.
