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"Fault tolerant quantum computation by anyons", Alexei Kitaev, 1997

Reviewed July 21, 2023

Citation: Kitaev, A. Yu. "Fault-tolerant quantum computation by anyons." Annals of physics 303.1 (2003): 2-30.

Web: https://arxiv.org/abs/quant-ph/9707021

Tags: Foundational, Kitaev-quantum-double

This paper is the birth of topological quantum computation. The key observation of the paper is that not only are the braiding patterns of anyons neither Bosonic nor Fermionic, but they can be very complicated. So complicated, in fact, that their braiding is hard to simulate classically. Braiding anyons will induce representations of the braid group, which by their topological nature are fault-tolerant. If the anyonic model is complicated enough, this gives universal quantum computation.

To demonstrate this idea, Kitaev constructs his "quantum double" model based on a finite group. He discusses how to think about stabilizer codes in terms of ground spaces of gapped Hamiltonians. He describes the anyons of quantum double models, and the "ribbon operators" one uses to move them. It is this sort of formal algebra which takes up the majority of the paper.

An important point to note is that this is not the paper which introduces the toric code. This code was developed by Kitaev in an earlier paper:

> Kitaev, A. Yu. "Quantum error correction with imperfect gates." Quantum communication, computing, and measurement. Boston, MA: Springer US, 1997. 181-188.

The difference is that in this earlier paper Kitaev does not connect his toric code to a physical model. To quote the paper: "A new suggestion is to perform error correction at the physical level". It is the understanding that the codespace of the toric code is the ground state of a Hamiltonian which is new.

After defining the toric code Hamiltonian, Kitaev says "this Hamiltonian is more or less realistic because it involves only local interactions". To me, that's a pretty weak argument - the Hamiltonian feels like it's pulled out of the air. The Hamiltonian appears naturally if you think about Z2 gauge theories, but that pushes the question back to what sort of physical system would have a Z2 gauge group. Of course, Kitaev ended up being right. This Hamiltonian is physically realistic, and can appear in naturally accruing Z2 gauge theoretic contexts. For instance, the toric code naturally appears in quantum dimer models where the Z2 symmetry is the fact that a given site will either be filled or it won't. A good paper to get started is here:

> Misguich, G., D. Serban, and V. Pasquier. "Quantum dimer model on the kagome lattice: Solvable dimer-liquid and ising gauge theory." Physical review letters 89.13 (2002): 137202.

The sorts of quantum dimers can be created, for instance, using Rydberg atoms:

> Verresen, Ruben, Mikhail D. Lukin, and Ashvin Vishwanath. "Prediction of toric code topological order from Rydberg blockade." Physical Review X 11.3 (2021): 031005.

From a modern perspective, this paper is a bit hard to read. It's greatest utility is in giving a sense of where Kitaev's head was at during the birth of this theory. His treatment of the toric code is fine, but I find his discussion of the general theory very opaque. He is very heuristic, and the gaps he left in this paper have been filled in over the years. I'd highly recommend using more mordern references like the ones below:

> Cowtan, Alexander, and Shahn Majid. "Quantum double aspects of surface code models." Journal of Mathematical Physics 63.4 (2022).
> Yan, Bowen, Penghua Chen, and Shawn X. Cui. "Ribbon operators in the generalized Kitaev quantum double model based on Hopf algebras." Journal of Physics A: Mathematical and Theoretical 55.18 (2022): 185201.

As for Kitaev's motivation when writing this paper, I find the story to be quite funny. Kitaev was on the fence about quantum computing, but was convinced by it's potential when he learned about Shor's factoring paper:

> Shor, Peter W. "Algorithms for quantum computation: discrete logarithms and factoring." Proceedings 35th annual symposium on foundations of computer science. Ieee, 1994.

However, the continental barrier was somehow strong enough that Kitaev was not able to access a copy of Shor's work. So, he tried to come up with his own algorithm, and succeeded:

> Kitaev, A. Yu. "Quantum measurements and the Abelian stabilizer problem." arXiv preprint quant-ph/9511026 (1995).

The natural next step was to try to make fault-tolerant quantum computation work - this was what led him to the discovery of the toric code. However, he was once again beaten to the punch by Shor:

> Shor, Peter W. "Fault-tolerant quantum computation." Proceedings of 37th conference on foundations of computer science. IEEE, 1996

Happily for Kitaev, Shor did not know much about anyons so Kitaev beat him to the punch for this discovery.