What exactly are anyons and how are they relevant to topological quantum computing?

Question

I have been trying to get a basic idea of what anyons are for the past couple of days. However, the online articles (including Wikipedia) seem unusually vague and impenetrable as far as explaining topological quantum computing and anyons goes.

The Wiki page on Topological quantum computer says:

A topological quantum computer is a theoretical quantum computer that employs two-dimensional quasiparticles called anyons, whose world lines pass around one another to form braids in a three-dimensional spacetime (i.e., one temporal plus two spatial dimensions). These braids form the logic gates that make up the computer. The advantage of a quantum computer based on quantum braids over using trapped quantum particles is that the former is much more stable. Small, cumulative perturbations can cause quantum states to decohere and introduce errors in the computation, but such small perturbations do not change the braids' topological properties.

This sounded interesting. So, on seeing this definition I tried to look up what anyons are:

In physics, an anyon is a type of quasiparticle that occurs only in two-dimensional systems, with properties much less restricted than fermions and bosons. In general, the operation of exchanging two identical particles may cause a global phase shift but cannot affect observables.

Okay, I do have some idea about what quasiparticles are. For example, as an electron travels through a semiconductor, its motion is disturbed in a complex way by its interactions with all of the other electrons and nuclei; however, it approximately behaves like an electron with a different mass (effective mass) travelling unperturbed through free space. This "electron" with a different mass is called an "electron quasiparticle". So I tend to assume that a quasiparticle, in general, is an approximation for the complex particle or wave phenomenon that may occur in matter, which would be difficult to mathematically deal with otherwise.

However, I could not follow what they were saying after that. I do know that bosons are particles which follow the Bose-Einstein statistics and fermions follow the Fermi-Dirac statistics.

Questions:

• However, what do they mean by "much less restricted than fermions and bosons"? Do "anyons" follow a different kind of statistical distribution than what bosons or fermions follow?

• In the next line, they say that exchanging two identical particles may cause a global phase shift but cannot affect the observables. What is meant by global phase shift in this context? Moreover, which observables are they actually talking about here?

• How are these quasiparticles i.e. anyons actually relevant to quantum computing? I keep hearing vague things likes "The world-lines of anyons form braids/knots in 3-dimensions (2 spatial and 1 temporal). These knots help form stable forms of matter, which aren't easily susceptible to decoherence". I think that this Ted-Ed video gives some idea, but it seems to deal with restricting electrons (rather than "anyons") to move on a certain closed path inside a material.

I would be glad if someone could help me to connect the dots and understand the meaning and significance of "anyons" at an intuitive level. I think a layman-level explanation would be more helpful for me, initially, rather than a full-blown mathematical explanation. However, I do know basic undergraduate level quantum mechanics, so you may use that in your explanation.

Answer 1

The first thing to do is to think topologically: make sure you understand why a coffee cup is the same thing topologically as a donut.

Now, imagine we swap two identical particles, and do it again, so that we are back where we started. Apply this topological thinking to the paths taken by the particles: it is the same as doing nothing.

Here I show a picture of this, where one particle is dragged around another particle. Topologically, the path taken can be deformed back to the "do nothing" path.

The square root of this operation is a swap:

Since the square root of 1 is either +1 or -1, a swap affects the state by multiplying by either +1 (for bosons) or -1 (for fermions.)

To understand anyons, we are going to do the same analysis, but with one less dimension. So now a particle winding around another particle is not topologically the same as the "do nothing" operation:

We need the extra third dimension to untangle the path of the anyon, and since we can't do this topologically, the state of the system could be modified by such a process.

Things get more interesting as we add particles. With three anyons, the paths taken can get tangled, or braided in arbitrary ways. To see how this works it helps to use three dimensions: two space dimensions and one time dimension. Here is an example of three anyons wandering around and then returning back where they started:

Long before physicists started to think about anyons, the mathematicians already worked out how these braiding processes combine to form new braids or undo braids. These are known as "braid groups" in work that dates back to Emil Artin in 1947.

Like the distinction between Bosons and Fermions above, different anyon systems will behave differently when you do these braid operations. One example of anyon, known as the Fibonacci anyon, are able to approximate any quantum operation just by doing these kinds of braids. And so theoretically we could use these to build a quantum computer.

I wrote an introductory paper on anyons, which is where I got these pictures from: https://arxiv.org/abs/1610.05384. There's more mathematics there, as well as a description of a close cousin of anyon theory known as a "modular functor".

Here is another good reference, with more Fibonacci anyon goodness: Introduction to topological quantum computation with non-Abelian anyons

EDIT: I see that I didn't say anything about the observables. The observables of the system measure the total anyon content within a region. In terms of anyon paths we can think of this as bringing all the anyons in some region together and "fusing" them into one anyon, which may be the "no anyon" aka vacuum state. For a system supporting Fibonacci anyons there will only ever be two outcomes for such a measurement: fibonacci anyon or vacuum. Another example is the toric code where there are four anyon outcomes.

Answer 2

You are right, it does look like the Wikipedia page needs work, so I will have to update it. But for now I will answer all five questions:

1) What do they mean by "much less restricted than fermions and bosons?

The exchange of two fermions or bosons is restricted by: $|\psi_1\psi_2\rangle = \pm|\psi_2\psi_1\rangle$.
The "$+$" corresponds to bosons and the "$-$" corresponds to fermions.

For anyons we have the much less restricted: $|\psi_1\psi_2\rangle = e^{i\theta}|\psi_2\psi_1\rangle$.
Notice that when $\theta=0$ we have bosons, and when $\theta=\pi$ we have fermions (by Euler's formula).

2) Do "anyons" follow a different kind of statistical distribution than what bosons or fermions follow?

Anyons can obey statistics ranging continuously between Fermi-Dirac statistics and Bose-Einstein statistics, because $\theta$ can be $0$ (bosons), $\pi$ (fermions), or anything in between.

3) Exchanging two identical particles may cause a global phase shift but cannot affect the observables. What is meant by global phase shift in this context?

That line from Wikipedia needs to be improved. The "global phase shift" is the $e^{i\theta}$ in the above formula. So it is not specific to anyons, since there is a global phase change of $-1$ when we exchange fermions as well.

What the Wikipedia article should have said was that when you exchange two identical particles twice you still get a global phase shift, which is not true for bosons and fermions. Here the first and second arrows indicate the first and second times we exchange particles 1 and 2:

Bosons: $|\psi_1\psi_2\rangle \rightarrow |\psi_2\psi_1\rangle \rightarrow |\psi_1\psi_2\rangle$ (no global phase)
Fermions: $|\psi_1\psi_2\rangle \rightarrow -|\psi_2\psi_1\rangle \rightarrow -(-|\psi_1\psi_2\rangle)=|\psi_1\psi_2\rangle$ (no global phase)
Anyons: $|\psi_1\psi_2\rangle \rightarrow e^{i\theta}|\psi_2\psi_1\rangle \rightarrow e^{i\theta}(e^{i\theta})=e^{i2\theta}|\psi_1\psi_2\rangle$ (global phase of $e^{i2\theta}$)

4) Moreover, which observables are they actually talking about here?

An observable is anything that can be observed in an experiment. For example, the position of the particle, $x$. When measuring the position of the particle, the probability of finding the particle at position $x$ is given by $\langle \psi|\hat{x}|\psi\rangle $.

Notice that this is unaffected by a global phase, because we have:
$| \psi\rangle=e^{i\theta}|\phi\rangle$
$\langle \psi|=e^{-i\theta}\langle\phi|$
$\langle\psi|\hat{x}|\psi\rangle = \langle\phi|\hat{x}|\phi\rangle $.

So two states $|\phi\rangle$ and $|\psi\rangle$, which differ by a phase of $e^{i\theta}$, as in this case, have the same observations in experiment.

5) How are these quasiparticles i.e. anyons actually relevant to quantum computing?

There are many proposals for building a quantum computer, for example:

• (i) NMR quantum computers make use of fermions (such as the spin of a proton).
  
• (ii) Photonic quantum computers make use of bosons (photons are bosons)
  
• (iii) Topological quantum computers are a proposed type of quantum computer which would make use of anyons.

An advantage of (iii) over (i) is that fidelities should be much greater. The advantage over (ii) is that it should be easier to get the qubits to interact. The disadvantage over both (i) and (ii) is that experiments involving anyons are comparatively rather new. NMR has been around since 1938 and lasers (photonics) have been around since 1960, but experiments with anyons began in the 1980s and are still far away from reaching the maturity of spin science or laser science, not to say that it will never happen in the future.

"I think a layman-level explanation would be more helpful for me, initially, rather than a full-blown mathematical explanation."

A layman definition without mathematics is going to be very difficult because what distinguishes anyons from bosons and anyons is that the exchange of anyons introduces a factor of $e^{i\theta}$ to the wavefunction, which is a mathematical explanation. If I had to explain anyons to someone who knows what a wavefunction is but nothing else, I would say:

"When two particles are switched, the wavefunction of the overall system stays the same for bosons, picks up a negative sign for fermions, and can pick up any factor of the form $e^{i\theta}$ for anyons."

Answer 3

Here would be my laymen explanation.

The big idea of quantum computation is that there are quantum systems which are hard to classically simulate. This means that quantum processes can be viewed as computation devices, which can solve the classically intractable problem "what would happen if this quantum process occurred"?

To build a quantum computer then, all you need to do is find some system which is hard to simulate. The observation of Kitaev is that there are (quasi)particles called anyons which do this. When (quasi)particles are moved around each other without touching, quantum effects (spooky action at a distance) can cause the physical system to change. Even though before and after moving around each other the (quasi)particles are literally in the same places, the overall system is different. The "internal states" of the particles have changed. Bosons are particles whose internal states don't change upon being moved around each other, and Fermions are particles which pick up some predictable -1 signs. Anyons can have other strange beavers when moved around (braided) with each other.

The punchline is this: not only do anyons have unusual behaviors when braided, but their behavior is so unusual that it is unfeasible to simulate classically. That is, if you get the right anyons you automatically have a quantum computer. The only task it can solve a-priori though is the "simulate anyons" problem, but it turns out that's a pretty good first step. If you get the right anyons then every other quantum computation problem can be reduced to asking "if I move anyons around this easy-to-make braid, what result will it have"? To perform the computation, you just move around your anyons in this easy-to-make braid and record to result.