"Topological quantum computation", John Preskill, Walter Ogburn, 1998

Reviewed July 26, 2023

Citation: Walter Ogburn, R., and John Preskill. "Topological quantum computation." NASA International Conference on Quantum Computing and Quantum Communications. Berlin, Heidelberg: Springer Berlin Heidelberg, 1998.

Web: https://link.springer.com/chapter/10.1007/3-540-49208-9_31

Tags: Expository, Pedagogical

This is the first expository paper on topological quantum computation. In fact, this is the paper which introduces the term topological quantum computation. John Preskill sure does like to coin terms.

This paper is very pedagogical in the sense that it does not introduce the quantum double model exactly. All it does is give some general philosophy, and state the minimal amount of facts about the model to make the reader comfortable with the philosophy. This makes it a good paper, but it does come with a problem. Namely, many people reference this paper as being the first paper to demonstrate universality of quantum computation by anyons. Specifically, people say that this paper proves that the alternating group A5 gives universal computation by braiding. This is problematic because

This paper does not rigorously define the quantum double model.
Properties about the quantum double model are asserted with little reasoning.
The key part of the paper says "We have found that a Toffoli gate can be constructed from Eq. (4) if G = A5, the group of even permutations on five objects". There is no proof. It is just asserted, along with some pulled-out-the-hat numerical details about this braiding implementation of the Toffoli gate.

All this is not to say that the contribution is meaningless. The strategy presented for showing that A5 gives universal TQC is sound. Namely, one first constructs universal classical computation, then the single qubit Pauli Z gate, then Pauli measurements. This is universal by the main theorem of

> Gottesman, Daniel. "Theory of fault-tolerant quantum computation." Physical Review A 57.1 (1998): 127.

The large size of A5 leads to the TQC not only being universal, but being "easily" universal. This is one end of a general tradeoff: you can get universal TQC with arbitrarily simple anyons, you just have to do more work by adding more difficult/unprotected steps.

Another fun thing about this paper: Walter Ogburn, John Preskill's co-author for this paper, was an undergraduate. This work was part of a summer undergraduate research fellowship!