12 - Swapping Toric Code Anyons

This is part of a project to get people involved with quantum error correction. See here for more info.

Last time we showed that broken rules in the toric code behave like particles. The white squares have one type of particle, that are created, moved around and annihilated using bit flips. The blue squares have a different type of particle, which are manipulated by phase flips instead.

Since then, we also published an article about anyons. These are strange particles that do strange things when they dance around each. Anyons are impossible in our 3D universe. But the surface code is a 2D universe of particles. Could they behave as anyons? It turns out that they do!

In the following picture, some of the qubits are given different colours. One is half blue. Doing a bit flip here creates a couple of particles on the white squares either side. These are shown in orange, so I'll call them orange particles.

Another qubit  is light green. Doing a phase flip there creates the other kind of particle on the blue squares either side. These are shown in red, so I'll call them red particles.

What about the dark green qubits? Doing phase flips on these, one after the other, would move the top red particle in a loop around the rightmost orange particle. 

For example, if we first do a couple of them, we move the red particle two steps to the left.

The next one, which is on the same qubit as we did a bit flip earlier, moves it up past the orange particle.

The rest then take it back to where it started.

In 3D universes, as we learned here, having a particle loop around another does absolutely nothing. But with particles living in 2D, interesting things can happen. Let's see what happens here.

Firstly, let's see what we mean by a loop that does nothing. Say you never did the bit flip, and never had any orange particles. Then all we'd have is a loop around nothing.

This is what boring looks like.

So let's compare the following two situations.

1. First we do the blue bit flip, then the light green phase flip, then the dark green phase flips.
2. First we do the light green phase flip, then the dark green phase flips, then the blue bit flip.

Both start with nothing and end with a pair of orange particles and a pair of red ones. Both had one of the red particles do a little loop. The difference is that the first case has a red particle doing the loop around an orange one The second has a boring loop that won't do anything, because the orange particles haven't been created yet.

If these two orders of doing things end up the same, we know that moving one of these particles around the other does nothing. If they are different, it does something. But which is it?

Most qubits have just one thing happen to them, and don't talk to any other qubits. So they don't really even know if their phase flip is happening before or after other stuff. They won't give us much insight.

The important qubit is the one where both a bit and phase flip happen. In the first case, the bit flip happens first. In the second, it happens after the phase flip.

To see if this makes a difference, we have to remember what bit and phase flips do.

• Bit flips flip a 0 to a 1, and an 1 to a 0.
• Phase flips flip a + to a -, and a - to a +

But what if you have a qubit that is 0 or 1, and do a phase flip to that? Understanding what happens here would take a wee bit of maths. The maths from this post in fact. But let's keep it simple. It turns out that it does nothing to 0, but turns 1 to -1.

What the flipping heck is -1? It's basically 1, but with a -. If you asked the qubit whether it was 0 or 1, it would still tell you that it's 1. But if you had superpositions, the - might do some stuff with the maths. Like turning a + to a -, for example.

Now, let's suppose that this qubit was initially 0 and do the flips according to order number 1. When we first do the bit flip, we get a 1. When we then do the phase flip, we get -1. But if did order number 2, doing the phase flip first, it would do nothing. Then when we did the bit flip we'd get a 1.

So when we do bit and phase flips to the same qubit, it does matter which order we do them in! The results differ by some weird mathsy -1! The same would happen whether the qubit started as a 1, or a +, or a -, or anything.

So there is a difference between an empty loop of a red particle, and one that goes round an orange one. Their dancing does things that are impossible for particles in our universe. They are anyons!