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"Dissecting Topological Quantum Computation", James Wootton, 2010

Reviewed August 17, 2023

Citation: Wootton, James Robin. Dissecting topological quantum computation. University of Leeds, 2010.

Web: https://ncatlab.org/nlab/files/WoottonDissectingTQC.pdf

Tags: Abelian-anyons, Expository, Universal-scheme

This PhD thesis is best summarized in the author's words: "This thesis is about anyons, Abelian and non-Abelian, and their use in quantum computation. I aim to demonstrate the power of Abelian anyons, showing that these humble quasiparticles have the same potential for fault-tolerant universal quantum computation as their non-Abelian counterparts. Not only this, they do so without demanding as much from the experimentalist.

In order to achieve this aim I take a spin lattice model used to realize non-Abelian anyons, pull it apart to see how it works, and then put it back together again. In doing so I find that the non-Abelian group structure underlying these models is an unnecessary complication. It can be removed without compromising fault-tolerance or computational power, while making the models more tractable theoretically and experimentally."

Of course, this claim is a bit over-hyped. Wootton requires arbitrary single-spin measurements, which in general is a lot to ask out of an abelian model. I would be hard pressed in saying that this thesis shows that abelian anyons have the same power as non-abelian anyons. I see it as part of the general observation that every model can give you universal quantum computation if you throw in enough cheating.

This gives a state-of-the-art look at what is known about computing with abelian anyons, circa 2010. In particular, which they show that for D(Z2) allowing Pauli measurements gives the Clifford group, they are unsure about what happens with you allow Pauli measurements for larger abelian groups: "For all other D(Zd) models the measurements of [Pauli] operations... do not lead to sqrt(Z) or sqrt(X). The gate set realized does not seem to correspond to one that is well studied, and so the computational power remains an open question. It is not known whether they implement the Clifford group or not."

Of course, asking whether they implement the qubit Clifford group is the wrong question (they don't). The correct question is whether they implement the qudit Clifford group (they do) and then what the power of that qudit Clifford group is (classical).

This paper is also a joy to read because of how many funny quotes there are, especially in the acknowledgements at the start of the paper. Here is one of the more informative of this quotes: "But what topology giveth, it also taketh away. Though it supplies us with intrinsically resilient quantum computation, it does not allow anyons to exist in our three-dimensional universe. Hence, rather than just catching a few wild anyons and harnessing them for the next generation of computers, we must endeavour to create them as quasiparticles in two-dimensional systems. The fractional quantum Hall effect was the first means found to do this, with spin lattice models proposed some years later."