What is topological quantum computing?

Quantum, quantum, quantum

There is one word which seemingly appears everywhere nowadays – quantum – in contexts where you would never expect it. From quantum deodorants to quantum therapy, it has turned into a buzzword. There arehowever, some instances where something is truly quantum, such as quantum bits, or qubits. As it is widely known, the building blocks of all electronic devices we use everyday, like smartphones or laptops, are bits – the unit of information, which takes on two values: 0 or 1. In the quantum world we are not limited to purely these values, as a qubit has the chance, or probability, of being in any one of the two states. We refer to this property as superposition. This means that, when I check my qubit there will be a certain chance that I will see a 0 or 1. It is not a matter of not knowing which of the two values the qubit has, because it can have both by its nature. 

One might naturally ask: What makes qubits useful at all? Why bother making qubits in the lab? The answer lies in exploiting their quantum nature. Let’s say that you want to open the lock on your house door, but you have 100 keys in your pocket. The natural solution is to try each key one-by-one, which on average will take you 50 tries. If you have 1,000,000 keys, it will take on average 500,000 tries and so on. The point is that the required number of tries will increase at the same pace as the number of keys, N. Now imagine you have a special quantum key, which can be simultaneously all possible N keys, one of which opens the lock. Instead of trying to open the lock with each key individually, we apply the quantum key in a special way, which will require square root of N number of “tries” to open the lock. What this means, is that the quantum computer will roughly need only 1,000 tries to find the right key out of 1,000,000. It’s like testing all keys at once and homing in on the right one almost instantly! 

As suggested above, realising quantum computers with many qubits would revolutionise digital security, or cybersecurity. It would also speed up solving other types of problems, such as simulating complicatedmolecules for drug discovery, or complex systems, such as climate models, global supply chains, air traffic control, etc. However, the underlying problem in constructing such machines is that qubits are inherently fragile and can easily lose their superposition. This effect is called decoherence. Companies which are actively building real-world quantum computers try to handle this problem by improving their qubits or correcting the errorsthat appear. A promising solution to decoherence is offered by topological qubits. 

Pouring electrons into wine glasses

Before trying to understand what a topological qubit is, let’s consider the popular experiment demonstrating resonance: when singing the right note, or frequency, a wine glass can be shattered. There is a term called an eigenstate, which physicists use to describe the way the glass deforms and bends at the right pitch. The concept of an eigenstate appears everywhere in physics, describing an inherent behaviour of a system, and is also at the basis of topological qubits. While in the case of the wine glass the eigenstate describes how the wine glass vibrates, in the case of a system of electrons it describes the way electrons “slosh” around. In essence, thebasis of a topological qubit is the eigenstate of a special electron system. What makes it special, is the fact that it is topological.  

The term topology is used in mathematics to describe certain properties of an object – such as number of holes – that cannot be changed by smoothly deforming it. Imagine a very stretchy doughnut – we do not care about the material it’s made from, only about its surface – we can stretch it into a pipe without closing the hole in the middle. However, if we want to work it into a sphere, we must close the hole in the middle. Meanwhile if we consider a stretchy wine glass, its surface can be deformed into a sphere as it has no holes. On the other hand, the wine glass cannot be stretched into a doughnut as its surface has no holes. A doughnut and a wineglass are in this sense topologically distinct.  

Now if we want to create our very own topological qubit, we must talk about electrons. Let’s create a special electron system by “filling” our wine glass with electrons. The important detail is that our electron wine glass is topologically distinct from other systems. There will be a special way the electrons can “slosh” around in the wine glass, which is possible as long as topological properties of the system do not change. It is this special way the electrons behave which is the basis of the topological qubit. This is called a Majorana state. 

A visual representation of an eigenstate.

The promise of Majoranas

So, we have our topological qubit, but what makes it special compared to other qubits? A good analogy is to imagine that if the wine glass is manufactured carefully, with a perfectly symmetric circular lip and foot, it is guaranteed that its surface will be very tough. This is a very special property, called topological protection, which other qubit systems do not have. In reality, our “perfectly symmetric electron wine glass” is a wire 500 times thinner than a human hair. The electrons in the nano-scale wire, or nanowire have very specific symmetries, which will ensure the “tough surface”, or topological protection. The special symmetries are achieved bycombining different materials such as semiconductors, superconductors and strong magnetic fields to create a single franken-device. The obtained nanowire is able host Majoranas, in theory. 

To understand the benefits of using Majoranas for quantum computing, it is useful to think of the initial example of the wine glass vibrating at the right frequency. Topological protection ensures that there are no other pitches close to the desired one which might cause the electrons to “slosh” around in an undesired way. Thus, the Majorana is protected from other states which helps against decoherence. Another important property is that Majoranas always come in pairs, with one at each opposing end of the nanowire. The information is split between the two Majoranas, and if one end of the nanowire is disturbed, the Majorana at the other endunaffected. Therefore, the Majoranas benefit from additional protection.

Right now, scientists have yet to confirm the true topological nature of Majoranas found in the devices we’ve built. Thus, the million-dollar problem is to create devices with actual Majoranas, as the quantum computers that could be constructed with them promise to leave existing ones in the dust, surpassing them in scale and performance. Countless researchers are working day and night to solve this very problem, and it is also one of the goals of the Theory of Mesoscopic Quantum Systems at the University of Luxembourg. If you are interested in learning more about topological quantum computing, feel free to contact Prof. Thomas SCHMIDT.