@kylegulshen.bsky.social
PhD student at Caltech studying quantum info / error correction
I am looking forward to continue tackling the remaining challenges surrounding this work—there are many interesting problems!
10.10.2025 02:47 — 👍 0 🔁 0 💬 0 📌 0
A visual representation of a chain map that maps a logical X operator on a small triangulation of the 2D torus into a 'shrunk lattice' and then into an X logical in the color code on the same lattice.
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.
10.10.2025 02:47 — 👍 5 🔁 0 💬 1 📌 0On 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.
10.10.2025 02:47 — 👍 2 🔁 0 💬 1 📌 0There 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.
10.10.2025 02:47 — 👍 2 🔁 0 💬 1 📌 0The self-duality and strictly-transversal single-qubit phase gates also offer advantages over the more complicated fault tolerant gates of related sheaf codes.
10.10.2025 02:47 — 👍 1 🔁 0 💬 1 📌 0We 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.
10.10.2025 02:47 — 👍 1 🔁 0 💬 1 📌 0First 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.
10.10.2025 02:47 — 👍 1 🔁 0 💬 1 📌 0Our 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.
10.10.2025 02:47 — 👍 2 🔁 0 💬 1 📌 0Second, 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.
10.10.2025 02:47 — 👍 1 🔁 0 💬 1 📌 0First, 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.
10.10.2025 02:47 — 👍 1 🔁 0 💬 1 📌 0The 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.
10.10.2025 02:47 — 👍 1 🔁 0 💬 1 📌 0
I’m very excited to advertise my first paper! scirate.com/arxiv/2510.0...
Joint work with Tali Kaufman. This paper highlights progress in an effort to build the best qLDPC codes by using highly symmetric simplicial high dimensional expanders (HDX). There are also broader results about sheaf codes.
