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Microsoft has its own agenda regarding quantum computer - it is topological quantum computer being invented by the team lead by Michael Freedman https://www.microsoft.com/en-us/research/project/topological-quantum-computing/

While this idea is very efficient implementation, it still required experimental proof of anyons. At last such proof has arrived, see New evidence that the quantum world is even stranger than we thought (original publication in Nature Physics, article Direct observation of anyonic braiding statistics).

My question is this: Does this experimental observation and manipulation of anyons solve the main obstacles towards topological quantum computer (e.g. as being implemented by Microsoft)? Or is any fundamental (not technical) challenge remain towards topological quantum computer?

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This is not the first time that there is found 'experimental proof' of (non-Abelian) anyons. Note also that the article does not use the word proof but rather evidence - that's why I also used the quotation marks above.

Back in 2012 there supposedly was found the first experimental evidence of Majorana bound states by an observation of the zero bias peak. I say supposedly, because there were people who were actually explaining the results as something else, something less 'quantum' - I won't go into the details, partly because it is out of scope and partly because there are other people on this website with much more expertise regarding the subject.

Anyway, there have been new experiments by various groups, and many of these have always had some 'controversy' - other people in the field didn't always share the optimism and sometimes explained the results differently. Also (for instance) QuTech voiced concerns about their own results.

The result you have linked definitely looks like a very nice result on the road to topologically protected quantum computers, but I don't expect it to be the 'main' result or in the end even the only result that determines if we can make a topological quantum computer or not.

I haven't read the paper, but it looks like they show evidence of the thing what makes a single anyon an 'anyon' - the non-trivial phase it picks up under conjugation. This is a very good result, and definitely a necessary property of a topological qubit. However, there are other fundamental things that this system must do before it makes a quantum computer. Computations are performed using braiding of multiple anyons 'through' each other - that's at least one fundamental step above making just a single qubit (as a rule of thumb, two anyons make one qubit, but I believe that's an oversimplification). That means that to perform computations you need to braid more than two anyons together - I would say that that is equally a fundamental step as is braiding of two anyons.

Of course, something being shown in the lab does not mean it is at all feasible - don't gross over these technical challenges, they will prove at least as though I presume.

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