Quantum physics these days is all about weird materials and what we can do with them. In this article we’ll take a look at one particular example.

The Nobel prize this year went to three guys for some topological stuff. The Buckley prize went to two other guys, also for topological stuff. But what is this topological nonsense? Why didn’t these prizes get won for something cool, like lasers!

Firstly, it’s because lasers already got the Nobel prize in 1964. But more importantly, it’s because topological stuff is cool! It let’s us build new universes from weird materials. Universes with exotic laws of physics. Universes where the impossible becomes possible.

But let’s not just talk about it. Let’s go and actually build one!

The best physics begins with magnets

Quantum physicists, like everyone else, prefer things to be simple. So one of their favourite types of particle is the spin-1/2. These are the simplest kind of magnet you can ever get. Simple is good. Magnets are awesome. What’s not to like?

As I’m sure you know, magnets interact with each other. They are famous for it. Put two next to each other, and they won’t be at all happy unless they’re pointing the same way. They’ll probably even jump out your hands to make it happen. These are the kinds of interactions we’ll be use using today.

We’re not going to let the spin-1/2 particles do whatever they like, though. They’ll probably fly around getting in a right old mess. So we are going to get a honeycomb and stick them to the vertices. Like the circles on the picture below.

It’s not your eyes playing tricks: The blue lines are thicker.

They’ll still be able to point in whatever direction they like, but they can’t move.

Let’s think about the two spin-1/2 particles called A and B, for a bit. These are next to each other, and so they’ll interact. They want to point the same way.

To make things interesting, we are going to mess about a bit with their interactions. Specifically, we’ll make them not only want to point the same direction as each other, they’ll want that direction to be either up (i.e. towards the top of the page) or down.

So what does this mean? If they are both pointing up, they’ll be happy. If they are both pointing down, they’ll also be happy. If one is up and the other is down, they won’t be entirely content. Nor will they be satisfied if they are pointing left or right, even if they are both doing the same thing.

For B and C, we are going to engineer a different interaction. These will again want to point in the same direction, but this time they’ll also want to point either left or right. For B and D, it’ll be the same again. But now the preferred directions will be the ones pointing out of the page, with either their north or south pole sticking right in your direction.

It’s not just A, B, C and D that are enjoying this party. All the others will be there too. All pairs separated by the blue lines will interact in the same way as A and B. All the red ones will interact like B and C, and the green ones will all be like B and D.

We have now set up something called the Honeycomb lattice model, the study of which has probably used up far more of your tax money than you’d want. But still much less than an Apache helicopter, so pretty good value.

The spins of the Honeycomb lattice model will never be completely happy. Spin B, for example, wants to point either up or down with its friend A. But it also wants to point left or right with C, and either towards or away from us with D. It can’t satisfy everyone.

You might wonder if this is where quantumness is going to come in. Maybe it can quantumly do all these things at once?

Nope. I’m afraid not. It can quantumly disappoint many different partners at once. But even quantum can’t keep everyone happy.

The magnets will try to jiggle into some sort of compromise position, in which each is as comfortable as possible. But it’ll be a mess. In fact, it’ll be a quantum superposition of many different messes, which is even more of a mess.

How to find a quantum compromise

Understanding how the best compromise works is not straightforward. At first glance, it looks like the kind of problem that is typically too hard for even the best of us to solve exactly. But actually there are some mathematical tricks we can use to make this honeycomb lattice reveal all its secrets. This is one reason why physicists like it so much.

These mathematical tricks are beyond the scope of this article. So let’s try to take a course that doesn’t need them too much.

The best way to start is to turn off everything but the interactions along vertical lines. Which looks like this.

Here each spin interacts with only one other spin: A with B, for example. Now everyone is perfectly clear on what they want: A and B should point in the same direction, and that direction should be up or down. No compromises required.

The same is true for C and E. They are also completely free to choose whether to point the same way as A and B both do, or to point the other way. Since these pairs don’t interact, they don’t care what the others are doing. This means there is a huge number of possible ways for everyone to be happy.

Now let’s turn the other interactions back on, but we’ll still keep them much weaker than the vertical ones. Each vertical pair now interacts a little with its neighbouring pairs. And those interactions demand at least a little compromise to their dream of the spins pointing strange directions like left and right.

To understand how this compromise might be brokered, there’s something you need to know about quantum magnets. Spin-1/2 particles are such simple things, there is no room in their tiny little minds to understand both the concept of up and down, and the concept of left and right. They pick the pair of directions they like best and think of everything in terms of those.

Our spin-1/2s are dominated by the desire to point up or down, so up and down will be the directions that the spins like to think about. They then think of left as just being some weird quantum superposition of being up and down. They think of right as a completely different kind of superposition. And similarly also for other directions

So how to make the weak interactions feel like they are being listened to, at least a bit? We have to bust out some superpositions!

With these superpositions will mean that the vertical pairs can no longer freely choose what to do. A and B cannot freely either both point up or both down. Instead they are forced into a superposition of the two.

In fact, all these pairs are forced into a big correlated superposition thing. Only by doing this will the weaker interactions get a taste of what they wanted, and stop complaining about being ignored.

Bringing back some freedom

The cost of the compromise is freedom. There is now only one way for all the spins to be happy, rather than the multitude possible when only the vertical interactions were turned on. Let’s now try and get some of that freedom back.

If turning on interactions lost us freedom, let’s turn some off. Spins A and B are forced into a weird superposition by the weak interactions, and forced to point the same way as each other by their strong one. Rather than arguing with the former, let’s deal with the latter. Let’s turn their vertical interaction down so it also is nice and weak.

The interaction between A and B gets turned down, so we don’t draw it with a thick line any more. It doesn’t deserve it.

Once we do this, spins A and B are no longer like the others. They are still interacting with their neighbours, and so they are still part of the quantum party, but they don’t have so many rules to follow. They no longer need to point in the same direction as each other

This means we now have two ways to make the great compromise: One is where A and B choose to point in the same direction, and the other is where they choose to be opposite.

Now it’s time to drop another quantum fact on you: Information cannot be destroyed by quantum processes. It can be moved. It can be smeared out. It can be hidden. But it can never be destroyed. The only way to defuse it is to find it out.

So suppose spins A and B have made their choice. They are either pointing in the same direction, or they aren’t. They have definitely chosen one way or the other, but we don’t know what they chose.

Suppose that we also turn off the vertical interaction between spins C and E. They can also choose whether to align or not. In this story, they choose to align and we know that for some reason.

Now let’s turn up the interaction between B and C. Let’s make this weak interaction strong. With no other strong interactions on B and C to oppose it, it gets its way: spins B and C are forced to align with each other, either both pointing left or both right.

But what happened to the information that we didn’t know? Whatever B was doing with A before, has now been overruled by its new loyalty to C. But that information can’t be lost, so it must be somewhere.

There’s only one option: It has moved to E. The job previously done by B is now done by E. The unknown information now governs the way that A and E are correlated with each other.

There are two ways this could happen: the easy way, or the quantum way. Spoiler: it’s going to be the quantum way!

The easy way would mean that E simply starts doing whatever B did before. If this were true, the unknown information would be easy to find. We’d just need to look at whether A and E are pointing the same way or not, and then we’d know that the same had been true of A and B.

The quantum way would be that the information would get spread out. You’d need to look at more than what A and E are doing, you’d need to also look at how they were correlated with the spins around around and between them.

That is what happens. To find out what was originally going on with A and B, we now need to pick a path that runs between A and E, look at all spins along that path and do some maths. The information has been spread out.

If we use the shortest path, this is perhaps not so surprising. That path includes B and C, the old friends of A and E. So it makes sense that they know part of the secret.

But we can use any path! We can use ones that avoid B and C entirely! We just need some way of hooking A and E together. The way we do it doesn’t matter.

Oh brave new world, that has such particles in it

Everything so far has been set-up. Now the pay off starts. Now I’ll show that the honeycomb lattice is its own little universe. A universe in which strange particles, impossible in our universe, can live, and dance, and die.

Any universe worth the name needs particles to populate it. It should be possible to move those particles around and smash them into each other. Sometimes smashing two particles together will make them annihilate: they’ll disappear. But sometimes they’ll combine to form a new particle.

A universe should also let the particles do the opposite. It should be possible for one to fall apart and become a few. And it should also be possible for nothing to occasionally turn into something. As long as that something is always a particle and its antiparticle, which will return to nothingness when combined, this is not a problem.

A universe should also prevent any particle appearing out of nothingness, or disappearing into nothingness. They have to team up with an antiparticle if they want to enter or leave. Otherwise it’s not a universe, it’s just a room.

That’s what a universe is. So how can we interpret the honeycomb lattice as a universe.

We need particles. The spin-1/2 particles don’t count. For one thing, they don’t get moved. They also don’t go around annihilating with things. They are the pillars on which our universe is built, not the particles that live in it.

So remember what happens when we turn one of the vertical interactions off. The two spin-1/2s nearby become something a bit more free than all the other. They are special. So we can do something and get two special things as a result. That’s important.

Once we have these special spots, we can mess about with the interactions nearby to move them. This is what we did when spin E inherited the job of B. So specialness can be moved. That’s important too.

When we get two special spots and put them next to each other, we know that they won’t interact strongly: That’s basically the definition of what makes them special. So if we turned up the interaction between them, the specialness of both would disappear. They’re freedom would be gone.

Now we have everything we need. In the honeycomb lattice we can get little spots of specialness. They can only be created in pairs. They can only be annihilated in pairs. They can be moved around. They behave just like particles, with each acting as the antiparticle of the other.

But their story is not yet at an end. There are two things that can happen when a pair of special spots are combined, depending on whether the particles were aligned or not. If they were, the spins are just be integrated back into the same normality as all the others. Their specialness would be gone without a trace. This is annihilation.

If they weren’t aligned, the new strong interaction between them will be pretty unhappy. So though the special spots would be gone, they’d leave something in their place. They would have combined to form a new particle: a bundle of frustrated energy. We’ll call these fermions, because that’s what they are.

There’s one last basic property of these particles to tell you about. I’m afraid that it will just seem to emerge from mathemagics. So you’ll have to trust me on this one

Say you have two pairs of special spots that you created from nothing, and so know will return to nothing again. Now take one spot from one pair, and move it in a big circle around one spot from the other pair until it’s back where it started.

This Majorana is drunk. But it doesn’t matter. So long as it goes around the one it wanted to, the effects are the same as for a perfect circle.

At no point did any of these particles touch. And no point did they meet up and throw fermions at each other. So you’d be forgiven for thinking that the end result of this process is nothing.

But it’s not. If you combine the pairs, you’ll find that they don’t annihilate any more. They will both combine to form a fermion. Just dancing the particles around each other changed what happens when they combine. Which is a bit odd, if you ask me.

We’ve now seen all the basic properties of these new particles that live in our honeycomb universe. They aren’t usually called ‘special spots’, though. They are called ‘Majorana modes’ and they are one of the big fashions in physics at the moment.

Building quantum computers from impossible particles

No particles that behave like Majorana modes exist in our universe. Not as proper particles like electrons, anyway. The particles of our universe always have a definite fate when combined. If you put a particle with its antiparticle, they’ll annihilate. Always. For certain.

But the Majoranas in the honeycomb universe are different. Combining a particle and antiparticle might cause them to annihilate. Or they might combine to form another particle. There are no obvious clues as to which will happen if the Majoranas are well separated. Just looking at them alone is not enough. You only find out by actually bringing them together¹.

Particles like this only exist in 2D universes, like our honeycomb lattice. There are many other ways to get weird 2D universes, with their own types of impossible particle. In general, these particles are called non-Abelian anyons. And they are awesome.

One reason for their awesomeness is that they could be used to build a completely new kind of computer.

How? Well you need to put an input in a computer for it to do anything. The input is usually in binary: a bunch of 0s and 1s.

To represent this input in the honeycomb universe, we could make as many Majorana pairs as we need bits. For each 0, we make a pair that would annihilate when combined. For each 1 we make one that would form a fermion. The Majoranas can then all be split up. That will make it hard for us to see what the input is. But it also makes it hard for noise to mess about with it². Which is good.

Our computer then needs to process the information. The input bits have to be messed about with to make the ouput bits. We’ve already seen that dancing Majoranas make things happen. So let’s use that. Our computer program turns into choreography.

It’s important to note that the effects of the dance don’t depend on the exact steps used. The only thing that matters is who passes around whom. So even if noise gives our Majoranas two left feet, everything will still do what we want it to.

Finally, we need to get the output. This is done by combining pairs of Majoranas. Annihilation means 0. Turning into a fermion means 1.

So there we go. Everything we need for a computer. Made out of impossible particles in an artificial universe. Does this imbue this new computational device with magical powers, that would outperform the world’s greatest supercomputer?

Yes! We get a quantum computer³. And quantum computers do have magical powers.

Now let’s build one

Earlier I said that we should stop talking about these new universes and start building one. And then I just talked more. A lot more.

But you can build one! I did, and I’m a theorist. Theorists aren’t usually allowed to make anything more complex than a mess on their desks. But I built a universe just like the one I’ve been talking about here. It had Majoranas in. I moved them around. Things happened. It was awesome! And If I can do it, so can you. Check out my video on how I did it here.

This article was based on my paper “A family of stabilizer codes for D(Z_2) anyons and Majorana modes”, but it was told more in the style of the great “Anyons in an exactly solved model and beyond” by Alexei Kitaev. It is also the basis for my recent work “Braiding Majoranas in a five qubit experiment”.

¹ Or, as we discussed earlier, looking all the way along a path that stretches between them, as we said earlier. But that turns out to be equivalent to bringing them together, seeing what happens when you combine them, turning the result back into two Majoranas and them putting them back where they were?

² If you want to know more about error correction for non-Abelian anyons (and who wouldn’t?) we have an app for that.

³ Or, more accurately, no. Not without coming up with a few more tricks, anyway. Like bringing Majoranas close together, letting them interact a little and then separating them again. But details like that are what footnotes were made for.