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This recent press release claiming that Improved measurements bring final proof of Majorana particles closer than ever, which summarizes the results of a recent paper in Nature simply entitled "Quantized Majorana conductance" claims that

Thanks to their unique physical characteristics, Majorana particles are much more stable than most other qubits.

Why would this be the case (in theory, at least). Is the approach to qubits with Majorana particles considered to be valid, or are they surrounded by skepticism?

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Majoranas are anyons (a type of quasiparticles wich behave differently from fermions and bosons), and so are related to the idea of topological quantum computation. This means that a good implementation should have properties that help deal with noise built in. Their main problem is that it is difficult to prepare physical systems which behave as Majorana particles.

One way of building Majoranas is with superconducting nanowires. This is the kind that the press release and paper are referring to. Will these actually work well? We shall see. Will they be better than other qubits? We shall see.

Another way of building Majoranas is by performing code deformation in surface codes (a well studied family of quantum error correction codes). Examples can be found in this paper (of which I am an author): Poking holes and cutting corners to achieve Clifford gates with the surface code. These will probably work pretty well. They won't have much in the way of advantages over more mainstream methods though, because using defects in surface codes is the most mainstream method (whether they are Majoranas or not).

There are other ways we could trick Majoranas into existing. But as far as I know, none are being actively pursued.

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I heard an interesting analogy that shed some light on the situation for me, so I'll share it here. Majorana fermions are topologically based; let's look at what topology sort of "means".

Topology looks at the bigger picture. If you have a balloon, no matter how much you blow it up, take air out, or tie it up in knots (if you're a balloon artist), it still doesn't have holes in it. To have holes would make it fundamentally different. You can stretch and shrink and twist a sphere all you want, but it's never going to turn into a donut. If you take a donut, though, you can twist that into all sorts of things with holes - but you can never make something without holes, like a sphere, or with two or more holes.

Another example of topology looking at the bigger picture. Take a balloon (again) and zoom in on its surface. Even though the balloon is curved when you zoom out, when you're zoomed in, it looks like a 2-d Euclidean plane. If you zoom in on a circle, it looks like a 1-d Euclidean plane. The little twists and turns don't matter in topology.

Let's bring this back towards Majorana fermions. Let's picture a system where we're registering if the electron goes all the way around a tree or not. It doesn't matter whether the electron has a really squiggly nutty path or a just a simple circular path - it still goes around.

The noise introduced in these systems might make the electron's path squiggly or it might not, but it doesn't actually matter. It still goes around. That's where the advantage in Majorana fermions lies - the noise doesn't affect it.

Obviously this isn't rigorous; I'll try to add more that sheds light on that as I have time.

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    $\begingroup$ I think a better way of putting it is that in a regular quantum computer information is stored and manipulated in localized degrees of freedom (like an electron's spin or a photon's polarization); but, in a topological quantum computer it is stored and manipulated in topological degrees of freedom, which are more resistant to noise. These "topological degrees of freedom" can be realized via the braiding of Majoranas. If you want to learn more of the math, I highly recommend the recent survey: Mathematics of Topological Quantum Computing. Also, see the above answer. $\endgroup$ – Sanketh Menda Apr 13 '18 at 9:39

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