I usually find Wikipedia very useful, so I was interested to read the entry on topological quantum computers. Wikipedia defines a topological quantum computer as “a theoretical quantum computer that employs two-dimensional quasiparticles called anyons, whose world lines cross over one another to form braids in a three-dimensional spacetime.”
Wow, for the non-theoretical physicists out there, that is pretty useless. (Admittedly, Wikipedia does go on to have a much longer article, but it's still aimed at a very knowledgeable audience.)
I'd like to give a basic explanation of topological quantum computing, but first let me address a different question. Why would you care about topological quantum computing?
Let's ignore the topological part for a moment, I'll get to that later. Quantum computation is the focus of a great deal of current research because it explores very basic quantum phenomena which fascinate many physicists. Additionally (and quite fortunately for people writing grants) a quantum computer could perform algorithms that are not possible on standard classical computers. The most exciting of these algorithms is Shor's factoring algorithm that would allow for the factoring in polynomial time [1]. The best classical algorithms factor in exponential time, and this is relevant because all current encryption schemes rely on the near impossibility of factoring large numbers. In other words, if a large quantum computer existed, it could figure out your credit card number next time you made an online purchase.
To understand what a topological quantum computer is, and why they are so fascinating, let's go word by word, starting at the end. Since you presumably reading this article online, you are familiar with computers. For our purposes, the relevant feature of a computer is that all the information is stored in bits which either have the value 0 or 1 (picture the green 0's and 1's scrolling past in The Matrix).
Okay, that wasn't too bad. What about quantum?
When I hear the word quantum, I think small, so it would be natural to think that quantum computers are just incredibly small computers that can hold even more pictures and movies than the computers we have now. Fortunately, quantum computers are much more exciting than that!
To understand what is so different about a quantum computer, we have to understand what makes a quantum bit, or qubit, different from a classical bit. A qubit is a quantum two state system. Exactly what the two states are is not particularly important, but they can be thought of as 0 and 1, like the classical bit. However, unlike a classical bit that is either 0 or 1, a qubit can exist in a combination of 0 and 1 such that when you measure it, you don't know a priori whether the answer will be 0 or 1.
This is weird! It's not that you can figure out whether the answer will be 0 or 1. Let's say you do a calculation and find the answer to be 0. If you repeat the exact same calculation, the answer could turn out to be 1, and no, you did not make a mistake.
How can this be possible? Think for a minute about an electron. The uncertainty principle states that you cannot simultaneously know the exact position and momentum of an electron. Now, let's put that electron in an atom and focus on its position. The exact location of the electron is still unknown, but its position takes on a range of possible values (somewhere in the atom), each with an associated probability. For example, there may be a 10% probability that the electron is located between positions 1 and 2. This probability distribution is determined, even though the exact position of the electron at an given time is unknown.
A qubit is similar, but even simpler because there are only two possible “locations”, or states, for the qubit, 0 and 1, and each state will occur with some probability. In the language of quantum mechanics, the qubit is in a superposition of the states 0 and 1. Performing a computation will change the relative weights between 0 and 1, and measuring the state of a qubit collapses the superposition such that the qubit only exists in the measured value. This additional capability of a qubit over a classical bit allows a quantum computer to perform algorithms that are not possible on a classical computer.
The most important difference between a classical bit and a qubit is the ability of a qubit to exist in a superposition between 0 and 1. I also said above that measurement collapses the superposition. We have now stumbled upon one of the many difficulties with quantum computation. In classical computers, all of the information is stored redundantly. Mistakes happen all the time, but classical computers can just check with the other copies and correct error. A quantum computers would not be able to do this because measuring the state of a qubit destroys the information present in the superposition. Additionally, by the no-cloning theorem you cannot make copies of quantum states, so there is no way to store information redundantly [2].
People have though a great deal about how to do quantum error correction, but it would also be convenient to use a system that naturally does not make that many mistakes. This is where topological quantum computing comes in.
First, what is topology? Topology is a field of mathematics where a donut and a coffee are considered to be identical. This is because a donut can be transformed into a coffee cup without making any cuts in the donut. In other words, each one has a single hole. For another example, consider a tangled necklace. So long as you can return to an untangled circle without unfastening the necklace, it's topology is identical to a circle. All those twits and turns are unimportant to the topologist. However, if when you untangle the necklace there is still a twist that cannot be undone without unfastening the chain, this represents a new topology from the simple circle.
A topological quantum computer uses a similar idea [3]. Imagine particles which live on a two dimensional plane. What if the two states of a qubit could be accesses by winding one particle around another. It does not matter whether the path the particle takes is smooth or some crazy path. So long as it only goes around its partner, the topology is the same and the state of the qubit changes from 0 to 1. In this system, it would be easier to avoid mistakes because just moving a particle a little has no effect. The topology of the system must change in order to change the state of a qubit.
This all sounds great - a quantum computer where errors would hopefully be relatively rare. The only *minor* detail is that creating (not to mention manipulating) these topological particles is still a work in progress. Experimentalists are working on it, and for now, we will have to be satisfied thinking about how a computer like this would work.
[1] P. Shor, Proceedings of the 35th Annual Symposium on Foundations of Computer Science, edited by S. Goldwasser (IEEE Computer Society, Los Alamitos, CA, USA, 1994), pp. 124134.
[2] W.K. Wootters and W.H. Zurek, A Single Quantum Cannot be Cloned, Nature 299 (1982), pp. 802–803
[3] C. Nayak, S. H. Simon, A. Stern. M. Freedman, S. Das Sarma, Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys. 80, 1083 (2008).
