Citation: Anderson, Jonas T., Guillaume Duclos-Cianci, and David Poulin. "Fault-tolerant conversion between the steane and reed-muller quantum codes." Physical review letters 113.8 (2014): 080501.
Web: https://arxiv.org/abs/1403.2734
Tags: Abelian-anyons, Universal-scheme
This paper gives the clearest example of the general idea "switch between codes where different things are easy/hard". Namely, in the 7-qubit Steane codes one can fault-tolerantly implement the Clifford group, and in the 15-qubit Reed-Muller code one can fault-tolerantly implement the T-gate. This paper shows that there is a fault-tolerant way of switching between these two codes. Hence, there is a scheme for universal quantum computation where you switch between these two codes to apply whatever gate you want.
To quote the authors: "our scheme can be seen as a subsystem encoding with... gauge fixing". Here, gauge fixing refers to the technique presented in
>Bombín, Héctor. "Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes." New Journal of Physics 17.8 (2015): 083002.
While certainly interesting, the non-topological nature of these codes make them unfavorable. As you go to higher-and-higher qubit numbers the connections get too large, and this proposal is un-tenable for the same reason all Reed-Muller solutions are un-tenable.