## Who’s afraid of Majorana qubits?

by Guanzhong Wang

Don’t hide it; don’t deny it. I know every time you, my dear friend in or related to the quantum computing community, hear about the words “topological qubits”, you raise your eyebrows slightly and say to yourself, “weird”… Pretend no more! We know you are puzzled why anyone would want to embark on the journey of making a topological qubit and how on earth they go about doing it. In this new series on topological qubits, we will try to explain to you why building such a seemingly unconventional qubit is rather fun and is even one of the natural choices when it comes to quantum computing.

I will start in this post with a virtual lab tour, hoping to give you an overview on where and how we look for the basic building blocks of a topological qubit—a Majorana bound state in condensed matter systems. From the particle that Ettore Majorana envisioned on a piece of paper to the nanowire devices and then back to the blueprints for a topological qubit, this will be a journey linking seemingly strange ideas to real, tangible chips in cryostats. If you’ve ever got curious about a Majorana qubit, gone through some reviews and tutorials but still wonder how experimentalists try to build them, this article is totally for you. If you haven’t, I hope it will arouse your interest in doing so!

## Majorana states in nanowires

A topological qubit is, most simply speaking, made of several Majorana bound states in solids. What are these Majorana bound states, you ask? Although they are named after and intimately related to the fundamental particles the Italian physicist Ettore Majorana (whose life story is no less mystic than the particle) famously studied, the topic here in the context of quantum computing is about its cousin in the condensed matter world, the quasiparticle states made of certain excitation modes of electrons in solids. The Majorana bound states are a special type of such excitations in that they have a group of interesting and mutually related properties. For the purpose of this post, it is enough to know that the electron wavefunction of such an electron state has a highly non-local character, meaning that its probability density has two peaks at very distant locations. (The next post will go into the details of some other properties of Majorana states, for example that they have zero excitation energies.)

OK sounds cool. But where do we find such exotic electronic states? Original literatures and reviews on the matter abound, so I will make this more of a gallery of what mathematical concepts come to look like after they are translated into real-world physical systems. For the more daring souls, there’s also the full set of math that takes you step by step, all the way from a toy model that illustrates the concept of a Majorana state to the real system where we might find them.

Here in Delft, the biggest efforts on Majorana qubits are centered around the so-called quantum wire proposals. This is the system in which the first Majorana signatures were observed in Delft in 2012. Building up such a system involves four essential ingredients:

• A one-dimensional quantum nanowire (in plain language: thin and narrow in two directions and long in the third one);
• Strong spin-orbit coupling in the nanowire (in plain language: electrons with spin-direction up mostly travel to the right and those spinning down mostly travel to the left);
• $latex s$-wave superconductivity (in plain language: the simplest type of superconductors imaginable, the everyday ones like aluminum, no exoticness needed);
• Magnetic field (in plain language: well, just a very, very big magnet).

Illustration of a Majorana nanowire model. Image: L. Stek.

Let’s now look at the ingredients one by one and see what they look like. If you happen to live in the Netherlands, there is a documentary that captures daily lives in our lab as I’m about to describe below. Check it out for free!

### Nanowires + spin-orbit interaction

I combine the first two ingredients because they are both satisfied once we select the right material—nanowires made of heavy-element III-V semiconductors such as InAs (indium arsenide) and InSb (indium antimonide). They have the favorable properties of high crystal quality, strong spin-orbit interaction and a large $latex g$-factor, meaning that they require a smaller magnetic field to reach the same level of energetic effects.

These nanowires typically have diameters around 100nm or so and a length of several micrometers. Their cross sections come in various shapes: hexagons, triangles, rectangles, parallelograms, or some so-called rooftop shapes.

Nanowires growing on a flat surface with various cross sectional shapes. Source: https://pubs.acs.org/doi/abs/10.1021/acs.nanolett.8b03733

Example images of InAs nanowires grown by molecular beam epitaxy (left) and metal-organic vapor phase epitaxy (right). Image source: http://www.fz-juelich.de/pgi/pgi-9/EN/Forschung-alt/01-Nanowires/_node.html

A network of nanowires lying on a flat substrate. Image: P. Aseev.

They can also be synthesized in various ways: grown bottom-up vertically so that they become a forest (as in the image above), bend and cross each other, or simply horizontally lying on a flat surface. Those that are grown vertically will need to be transferred and laid down for us to be able to make devices on them, and here you can see such a video showing picking up and laying down nanowires with a thin needle tip in action.

Wait, didn’t I say we wanted a 1D wire? And yet if you click the links you will see all kinds of funny crosses and hashtags like these below:

Nanowire crosses and “hashtags”. Source: https://www.nature.com/articles/nature23468

That’s because although Majorana bound states are readily found in 1D systems, if we want to perform gate operations on a Majorana qubit, we would have to bring them around each other and exchange them (i.e. the action called braiding) and this can only happen on a 2D surface. Think about two cars meeting in a narrow alley – there is no way for them to go past each other and swap locations unless one of them makes use of a second dimension, e.g. an intersecting alley, to wait for the other to pass first.

A simple illustrations of how to exchange two Majorana states in a nanowire by making use of a second branch.

### Superconductors

The second part of the material platform is a thin layer (sometimes only several nanometers) of superconducting film in contact with the semiconductor nanowire. You can see the cross-section of such a hybrid nanowire made of a thin Al film on top of an InSb nanowire here, although in theory the superconductor can also wrap around the entire wire:

InSb nanowires with a thin Al shell on top. Source: https://www.nature.com/articles/nature23468

Such hybrid nanowire devices are usually fabricated in a cleanroom. Because of their small size, nanowires can only sustain very low amounts of electrical current running through them. Cleanrooms, as the name suggests, provide an environment free of charged dust particles that could potentially fly around in the air and land on our precious nanowires to burn them off. (Teaser: the sights are not pretty when such events happen—)

The aftermath of nanowires and metal leads burnt by large currents running through them.

If you want to know more about cleanroom nanofabrication, here’s a great post on our blog to help you understand it better.

### Magnetic field in a dilution fridge

After cleanroom fabrication, we have our Majorana devices with the necessary electrical connections (ohmic contacts), tuning knobs (gates) and maybe more elaborate sensors (resonators etc). They look something like this after we attach them to printed circuit boards (PCBs) and make electrical bonds between the chip and the PCB:

A chip with nanowire devices, together with the connected circuit board that can be attached to the dilution fridge.

Next, it’s time to cool them down to low temperature and add the last ingredient: the magnetic field. This usually happens in a dilution refrigerator. If you have followed news on quantum computing, mostly likely you have seen images of dilution refrigerators, either in a (sometimes very fancy) enclosure or opened up with its internals exposed for all to see.

The inside of a BlueFors dilution fridge.

You can also see some of them in the new Microsoft Quantum lab in Delft that we work in, recently opened by the Dutch King. Such dilution fridges are not the quantum devices themselves, but the tool we use to cool down the actual devices to very low temperatures. At such low temperatures, we can run massive amounts of currents through a superconducting coil surrounding the device without generating any heat and therefore produce a very strong magnetic field around the quantum devices. The order of magnitude of the magnetic field needed to observe Majorana signatures in nanowires is somewhere around or less than 1 Tesla, some 20000 times stronger than earth’s own magnetic field.

## Looking for Majoranas in nanowires

Suppose now you have a proper hybrid device made of nanowires with thin superconducting films inside a dilution fridge with a strong magnet, how do you go about searching for Majorana bound states in such a system? And how will you know when you have such an exotic state? There are hundreds, if not more, of papers trying to answer this question. Here I will only mention a few.

The most famous signature of Majorana states is probably the quantized zero-bias conductance peak. To observe such a peak, we apply a small voltage between the two ends of the nanowire and measure the current running through it, not very dissimilar from the experiments we conducted to verify Ohm’s law in our early teens. Distinctly different from the classical Ohm’s law, though, is that the quantum transport of electrons involve tunneling processes. When there are Majorana states, tunneling through it becomes much easier and a conductance peak shows up in the signal. The fact that Majorana states reside at zero energy (that’s why they’re called Majorana zero modes) means we have a conductance peak at zero bias voltage, hence the name zero-bias peaks. Furthermore, Majorana states are different from just any zero-energy electron state because at each end of the nanowire there is, in fact, “half an electron”. This practically means that, at zero temperature, the height of the Majorana zero-bias peak is restricted to take on only one possible value—one conductance quantum. However, such a simple prediction was not convincingly observed until very recently here in Delft with state-of-the-art materials and fabrication.

Majorana signatures seen in conductance measurements in nanowires. The red line in the bottom-left panel demonstrates the quantized conductance peak. Source: https://www.nature.com/articles/nature26142

Following the reasoning that Majorana states reside at zero energy, more variants of such signatures can be found in electron transport experiments where people apply a voltage and measure the current or vice versa. For example, when a small floating island on a hybrid nanowire hosts Majorana states, the conductance oscillation period is halved compared to the case where no Majorana is present.

If we take the idea of measuring the presence of electronic states and their energy to extremes, performing scanning tunneling spectroscopy (STS) experiments is arguably the most direct demonstration of the Majorana wavefunction. Such experiments have been performed, although not in the quantum wire system I have been talking about, due to practical hardware constraints.

Another famous theoretical prediction is that Josephson junctions hosting Majorana states exhibit halved current-oscillation frequencies in its dependence on magnetic fields, an effect that can be hand-wavingly understood as a signature that a Majorana is “half an electron”.

To sum it up, since the day of the first report on Majorana signatures, more and more experiments are conducted, evidence accumulated, but controversies still exist. Biggest of them all, perhaps, concerns the so-called quasi-Majoranas that are almost indistinguishable from “real” ones in many of the aforementioned experiments. Will they be a problem going forward? Some theorists say maybe not. As an experimentalist myself, my answer is “we don’t know so let’s go do it and find out”. That uncertainty, in my humble opinion, is precisely the charm of this field.

## From observing to manipulating Majoranas

As will be explained in the second post in this series by our fellow student-author Jaap, the way of computing with Majorana states is surprisingly simple: you bring two of them around each other and voila, you have performed a 90-degree rotation on the Bloch sphere. However, such a task is easily described but can prove very hard to execute. A reasonable plan is to use a series of gates like a piano keyboard and gradually shift the voltages applied to them one by one to move the Majorana. Doing this both smoothly and quickly is very difficult, not to mention other more detailed technical problems.

Later, theorists came up with another type of braiding proposals without relying on physically moving the Majoranas around, but rather by devising a Hamiltonian whose time evolution produces the same effect as doing so. Now gradually changing one voltage or magnetic field surely sounds simpler than having to do so on a dozen of piano gates, but it still doesn’t solve one issue: practically we can only produce a uniform, strong magnetic field along one single direction near the nanodevices. This is in conflict with the requirement of having several nanowires pointing along different directions all having Majoranas.

This does not stop the theorists from trying again. Eventually they came up yet another type of brilliant proposals to braid Majoranas involving only nanowires aligned all in one direction—the “measurement-based” braiding experiments. The operation of bringing Majoranas around each other is now achieved by simply performing quantum measurements on certain combinations of them following a (rather complicated) protocol. In the following images you can see some illustrations of the nanowire devices we can use to verify the concept of and even perform such prototypical qubit operations.

Illustrations of the nanowire device geometries for verifying and performing measurement-based Majorana braiding operations. Images: S. Heedt and F. Borsoi.

It’s worth noting that braiding operations only constitute a subset of rotations on the single-qubit Bloch sphere. Such operations are topologically protected and thus are expected to have much higher fidelity than the quantum gates realized in non-Majorana qubits, but they cannot cover the whole Bloch sphere. To perform arbitrary qubit rotations, we will need to perform the rest of the gates without topological protection just as with any other qubit type. This is, when viewed at a higher level, the same approach to fault-tolerant quantum computation using non-topological qubits: there, by using a large number of physical qubits as a logical qubit, a subset of quantum gates can be effectively error-corrected but not all of them. Both methods aim at building a large-scale quantum computer with this subset of very good gates and imperfect realizations of other gates. In this sense, our often-received question of “why do you guys work on such a strange implementation of qubits” has a simple answer—the topological qubit is not that different from other fault-tolerant logical qubit structures, at least mathematically. We try to build an error-protected, albeit harder-to-realize, qubit now so that we can afford the same fidelity with way fewer physical qubits later. As for the people working in the field, practically this means research on this front has a larger component of exploring unknown scientific questions as opposed to meticulous engineering than other qubit types.

In the next post you will hear more about why Majorana qubits are expected to have so much better performance than a non-topological qubit, so stay tuned and find out soon!

Guan is a PhD student in the Kouwenhoven lab. He came to QuTech for a Master’s project with the ambition of exploring all the various types of qubits being studied here, but chose to stay for a PhD in the lab with the least number of qubits. The exact reason for that decision is about as mysterious as a Majorana particle.

### Happy Easter: looking for Quantum Eggs

It’s that time of the year again: the Easter Bunny comes by and hides his eggs. Everywhere you look ...