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"Normalizer Circuits and Quantum Computation", Juani Bermejo-Vega, 2016

Reviewed August 17, 2023

Citation: Bermejo-Vega, Juan. "Normalizer circuits and quantum computation." arXiv preprint arXiv:1611.09274 (2016).

Web: https://arxiv.org/pdf/1611.09274.pdf

Tags: Abelian-anyons, Computer-scientific

This thesis gives a generalization of Gottesman's stabilizer formalism to models based on finite abelian groups. People had already generalized from qubits to qudits, but this goes even further because it allows products of cyclic groups. Additionally, since it considers small qubit numbers on large abelian groups, this paper deals with the very interesting subtleties around group isomorphism problems in representing data.

This seems to be the be-all-and-end-all reference for this sort of formalism. This seems to be, however, exactly the formalism which one would naturally arrive on when studying abelian anyons. Hence, it seems like this paper is (unwittingly?) the best reference for computational complexity results about a larger class of anyonic systems.

The main result is that, if you fix a finite group and look at more-and-more qudits, a Gottesman-Knill theorem holds. Namely, the resulting TQC can be efficiently simulated with a classical computer.

There are lots of other results, which are very interesting in their own right. There is also a lot of material on "hypergroups" presented. I had never heard the term hypergroup, but it seems that Juani just means "fusion ring"!

An aside: the opening quote to this thesis is fantastic:

"'This-'' He indicated his sword again, seeing Bellis begin to understand. '-is a sword of possible strikes. A Possible Sword. It's a conductor for a very rare kind of energy. It's a node in a circuit, a possibility machine. This-' He patted the little pack strapped to his waist. '-is the power: a clockwork engine. These,' the wires stitched into his armor, 'draw the power up. And the sword completes the circuit. When I grip it, the engine's whole.

If the clockwork is running, my arm and the sword mine possibilities. For every factual attack there are a thousand possibilities, nigh-sword ghosts, and all of them strike down together.'

Doul sheathed the blade and stared up into the trees' pitch-black canopy. 'Some of the most likely are very nearly real. Some are fainter than mirages, and their power to cut...is faint. There are countless nigh-blades, of all probabilities, all striking together.'

China MiƩville, The Scar."