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"Qudit color codes and gauge color codes in all spatial dimensions", Fern Watson, Earl Campbell, Hussain Anwar, Dan Browne, 2015

Reviewed August 18, 2023

Citation: Watson, Fern HE, et al. "Qudit color codes and gauge color codes in all spatial dimensions." Physical Review A 92.2 (2015): 022312.

Web: https://arxiv.org/abs/1503.08800

Tags: Abelian-anyons, Error-correcting-codes, Higher-dimensional

This paper introduces the qudit variant of color codes. It is shown that they can transversally implement the qudit Hadamard, S, and CX gates. Note that their construction of logical Hadamard is a bit sketchy, since it requires the "gauge fixing" method introduced in

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

They repeatedly talk about how this saturates the Bravyi-Konig bound, in reference to the result in

>Bravyi, Sergey, and Robert Konig. "Classification of topologically protected gates for local stabilizer codes." Physical review letters 110.17 (2013): 170503.

Which says that two dimensional topological codes can only transversally implement the Clifford group. It seems like the authors are implicitly assuming a qudit variant of the Bravyi-Konig theorem.

The authors conclude by saying "We desire a better understanding of universality and the Clifford hierarchy when using systems of non-prime dimension". They say this in reference to the fact that they are unsure whether Hadamard, S, and CX generate the qudit Clifford group. Clearly, they are not well-versed in the relevant literature, where papers such as

>De Beaudrap, Niel. "A linearized stabilizer formalism for systems of finite dimension." arXiv preprint arXiv:1102.3354 (2011).

deal with generators for the qudit Clifford group explicitly for all d. This paper generalizes its results to arbitrary dimensions.