Citation: Sarma, Sankar Das, Michael Freedman, and Chetan Nayak. "Majorana zero modes and topological quantum computation." npj Quantum Information 1.1 (2015): 1-13.
Web: https://www.nature.com/articles/npjqi20151
Tags: Expository, Majorana-fermions, Universal-scheme
This paper is a review-article on the state of topological quantum computing in late 2014, with a special emphasis on experimentally-verified and near-term prospects. The main message was that there were two leading prospects at the time: synthetic topological superconductors and the v=5/2 filling level of the fractional quantum Hall system.
The fractional quantum Hall effect has been hypothesized to host non-abelian anyons for a long time. The general wisdom is that odd-denominator filling fractions will be abelian, and that even-denominator fillings will be more powerful. Among the even-denominator fractions, one stands alone its its ability to be created in experiment: the v=5/2 filling level. It is much for stable than other even-denominator fillings, and it is the only even-denominator filling which has been observed in a single 2D layer of topological material. The anyons in the v=5/2 filling level are Ising anyons, and are not natively universal for quantum computing. A fantastic article about how to do universal quantum computing at this filling level is
> Bravyi, Sergey. "Universal quantum computation with the v= 5/ 2 fractional quantum Hall state." Physical Review A 73.4 (2006): 042313.
The only hard gate to implement is that T-gate - all Cliffords can be implemented topologically. This means that if you run a surface code on top of it, the surface code will require substantially less corrections because the continual syndrome measurements will be implemented fault-tolerantly. The T-gate can be implemented by bringing two Ising anyons close to one another. Energy splitting will induce a relative phase between the two states. Leaving them close to each other for a prescribed time this phase can be controlled to make a T-gate. Clearly, there is no topological protection afforded to this gate.
Possibly more exciting is the advent of synthetic topological superconductors. Putting a topological insulator at the surface of a conventional superconductor can change the properties of the superconductor, by proximity effects. This makes the superconductor's properties change, so that it behaves like topological superconductor. Topological superconductors, in turn, are well-known to host Majorana zero modes, which can be used to perform universal quantum computation just like Ising anyons. This sort of proximity effect was first noticed in the seminal work of Fu and Kane:
> Fu, Liang, and Charles L. Kane. "Superconducting proximity effect and Majorana fermions at the surface of a topological insulator." Physical review letters 100.9 (2008): 096407.
Controlling both superconductors and topological insulators is very complicated, so managing this interface is quite subtle. The more modern approach is try to make native topological superconductors, which don't rely on the proximity effect. The hope is that finding the perfect material is very hard, but once chosen engineering the computer will be relatively easy.
Seeing as in 2014 topological quantum computing was a long way away, there is a big focus in this paper on near-term milestones. Namely, what the signatures of Majorana modes are so that we can conclusively identify them once they have been created. Clearly not enough precautions were made in this section, however, because in a few short years microsoft would have its big retraction scandal:
> Zhang, Hao, et al. "Retracted article: Quantized majorana conductance." Nature 556.7699 (2018): 74-79.
All in all, this is a very nice albeit somewhat dated review article of topological quantum computing with Majorana zero modes for materials-minded people.