I am looking forward to continue tackling the remaining challenges surrounding this work—there are many interesting problems!

Our unfolding proof provides a thorough description of the connection between certain sheaf codes, which generalize the toric code, and quantum Tanner codes, which generalize the color code.

On a technical level, we achieve the results on transversal gates by providing a partial understanding of the logical basis of these generalized color codes through a generalization of the ‘unfolding’ of the color code. The proof of this generalization constitutes the large appendix.

There are many interesting aspects of our construction; for example, we describe a Floquet implementation where the 4-qubit check measurements each round are fixed and geometrically local, but where the data qubits are shuffled in parallel along fixed nonlocal 3-site orbits between each round.

The self-duality and strictly-transversal single-qubit phase gates also offer advantages over the more complicated fault tolerant gates of related sheaf codes.

We provide the first qubit code family on a (2D) simplicial HDX. It has good rate, several transversal gates (S, H, CZ, fold-), and is self-dual. The self-duality and level of symmetry seem hard to achieve with product complexes. The symmetry yields many (generalized) fold-transversal gates.

First we describe a broader framework that generalizes the color code, and we identify a mechanism to achieve certain transversal gates within this framework. We show how to build such codes on HDX and conjecture some can be good; we are still working through some aspects like distance lower bound.

Our proposal is to move away from product complexes and embrace the irreducible fundamental objects of the HDX world: D>=2 simplicial HDX. Specifically, we use highly symmetric coset complexes. This work explores constructions of quantum codes on these complexes.

Second, the current constructions require searching over a random local code, which leaves the quantum code unwieldy and lacking useful structure that could help with things like fault tolerant logic.

First, the 2D HDX for these codes is constructed as a (quotient of a) product of graphs, but it is unclear how to extend this operation optimally to larger dimensions D>2, which seems to be necessary to unlock some applications like transversal non-Clifford gates.

The first asymptotically good qLDPC codes were recently constructed using 2D HDX. But these constructions leave much to be desired; for example, we need to be able to do fault tolerant logic on codes! There are a couple of bottlenecks for existing codes.