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## "A blueprint for a topologically fault-tolerant quantum computer", Parsa Bonderson Sankar Das Sarma, Michael Freedman, Chetan Nayak, 2010

*Reviewed December 29, 2023*

*Citation:* Bonderson, Parsa, et al. "A blueprint for a topologically fault-tolerant quantum computer." arXiv preprint arXiv:1003.2856 (2010).

*Web:* https://arxiv.org/abs/1003.2856

*Tags:* Majorana-fermions, Ising-computer, Hardware, Quantum-hall-effect, Universal-scheme

This is the paper which formally introduced the idea of making a topological quantum computer
based on Ising anyons, where instead of realizing non-Clifford gates using non-topological means
one instead realizes non-Clifford gates using Dehn twists.

This paper is very short, is very accessible, and deals mainly with big ideas. The primary contribution
is a new way of implemented Dehn twists by depleting topological liquid parts of a high-genus material to dynamically change
its genus, resulting in a procedure they describe as "dynamical topological changing (DTC)".
The more technical paper where lots of the details of this approach were established is

> Freedman, Michael, Chetan Nayak, and Kevin Walker. "Towards universal topological quantum computation in the v= 5 2 fractional quantum Hall state." Physical Review B 73.24 (2006): 245307.

This paper describes a potential way to achieve universal Ising computation with Dehn twist,
using interferometric measurements in the fractional quantum Hall regime. This paper also contains
a proof of the fact that braiding + Dehn twist is universal for Ising. This universality result is the non-trivial fact
which underlies this family of approach. However, it is NOT due to Freedman, Nayak, or Walker. It is due, apparently,
to an even older unpublished work of Bravyi and Kitaev:

> Bravyi, Sergey, and Alexei Kitaev. "Quantum invariants of 3-manifolds and quantum computation." preprint (2001).

Another very funny thing about this paper: it contains a diagram depicting what a potentially
topological quantum computer should look like. It is very well done, extremely funny, and could be used for
various presentations or talks as a schematic of what mathematicians can keep in mind which doing
the mathematics behind TQC. Of course, it is completely physically unrealistic and
much follow-up work has been done to make the picture less far-fetched.