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"Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes", Hector Bombin, 2015

Reviewed August 18, 2023

Citation: Bombín, Héctor. "Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes." New Journal of Physics 17.8 (2015): 083002.

Web: https://arxiv.org/pdf/1311.0879.pdf

Tags: Abelian-anyons, Universal-scheme, Foundational

This paper introduces the technique of "gauge fixing" in color codes. In this method, the codespace is decomposed into a space of logical qubits and a space of gauge qubits. The idea is that, depending on the states of the gauge qubits, the set of states which can be implemented transversely is different. By "gauge fixing" (initializing the gauge qubits to the correct state) the possible gate set implementable transversely on the logical qubits can be made universal.

Generally, the smaller code will be simpler and the larger code will be more complicated. A canonical example is a lower dimension (generally 2 dimensional, non-universal) living inside a larger dimensional (generally 3 dimensional, universal) system. The dimension jump lets you spend most of your time in the smaller model, and only go to your bigger model when you need to implement costly gates. This idea was introduced in the following paper:

> Bombín, Héctor. "Dimensional jump in quantum error correction." New Journal of Physics 18.4 (2016): 043038.

This sort of scheme in which you introduce errors on a non-computational space on purpose to allow yourself a wider space of computations was already known, introduced in the following paper:

> Paetznick, Adam, and Ben W. Reichardt. "Universal fault-tolerant quantum computation with only transversal gates and error correction." Physical review letters 111.9 (2013): 090505.

This technique has since seen a lot of success, and lots of methods with this sort of principle are now known as gauge-fixing techniques.