13 - The Planar Code

So far we've been looking at the toric code which, as the name suggests, needs to be wrapped around in a doughnut shape. This isn't really the most practical thing in the world. So how about something that can be done on an ordinary flat surface. Something like the following, perhaps?

As before, the little white squares are qubits. For each big white square there is a rule that all the qubits must obey. Usually they have four qubits around them, but the ones on the edges only have three and the ones at the corners have just two. The blue squares also have a rule that their qubits must follow. All of those have four qubits around them. When the rules are broken on any square, we can think of it as a particle: an anyon.

The whole point of a quantum code is to store a qubit, and to protect it from bit flips and unwanted measurements. For the toric code, this was done with big loops around the doughnut. But there are no loops like that any more. We'll have to work out a different way.

One thing we can do is to take the big white square at the top left, and stop enforcing its rule. We'll also do the same for the big white square at the bottom right. Now these squares are allowed to both have an anyon in, or both not (because these anyons always come in pairs). They could also have some superposition of the two.

These squares can now be used to store our qubit. We can have them both empty for a qubit that is 0, and both with anyons for a qubit 1. This is then nicely protected against logical bit flips, as a line of bit flips would have to stretch from top left to bottom right to do any harm.

Unfortunately, the qubit won't be protected from unwanted measurements. To see if there is an anyone on the top left square, you only need to look at two qubits. The same is true for the bottom right. So it's quite easy to measure our logical qubit.

To make it harder, let's knock down the wall between the top left square and its neighbour. We'll also do the same at the borrom left. Specifically, let's look at the code below.

This has a 'double square' at the top left, which is just two of the old squares stuck together. Otherwise it is exactly like a normal square. A rule can be set for the three qubits around it, and an anyon can live there. The same is true for the bottom left. Now the Gremlins need to measure three qubits to destroy our superpositions. So it's a bit harder than before.

To protect the qubit from unwanted measurements even more, we can just carry on knocking walls down. Now we have a big multisquare at the top and bottom. Both can hold an anyon, but they will be smeared across the whole edge, and so need lots of measurements to work out which you have. Now we have a good way of storing a logical qubit, this finally deserves the 'code' part of the name 'planar code'. 

Now that's basically it. You could check out what is happening from the perspective of the other kind of anyons, the ones living on the blue squares, but I'll leave that as an exercise to the reader.