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"Transversal Clifford gates on folded surface codes", Jonathan Moussa, 2016

Reviewed August 18, 2023

Citation: Moussa, Jonathan E. "Transversal Clifford gates on folded surface codes." Physical Review A 94.4 (2016): 042316.

Web: https://arxiv.org/abs/1603.02286

Tags: Abelian-anyons

This paper is a follow-up to

>Moussa, Jonathan E. "Quantum circuits for qubit fusion." arXiv preprint arXiv:1512.06132 (2015).

The author considers "folded" surface codes, where transversal Clifford gates are available. This folding is based on a construction from

> Kubica, Aleksander, Beni Yoshida, and Fernando Pastawski. "Unfolding the color code." New Journal of Physics 17.8 (2015): 083026.

Which establishes a formal connection between surface codes and color codes. The point is that color codes have transversal Clifford gates but surface codes cannot, so you can transfer the transversal Clifford gates from one model to another using the method presented here.

The author's motivation for this method is as follows. He wants to put a d=2 version of the code together with a d=4 version, which together will give universal quantum computation by results from the first work. While certainly interesting, there is a key part missing: the author doesn't know how to convert between the d=2 and d=4 cases! Here's a quote from the conclusion:

"This framework still lacks efficient operations to switch orders either directly with code conversion or indirectly with resource distillation. We are following a 'keystone' design principle whereby the most essential part is missing, and its future development is motivated by a compelling but incomplete design. To compete with existing quantum computing proposals, we will need basic principles for converting between qubit and qudit error correcting codes"

Perhaps this can be remedied with the, quote, "well established" theory of domain walls given in

> Laubscher, Katharina, Daniel Loss, and James R. Wootton. "Universal quantum computation in the surface code using non-Abelian islands." Physical Review A 100.1 (2019): 012338.

Where they implement universal TQC by mixing Z2 anyons and S3 anyons.