In this paper, we also for the first time properly understand tile codes from a mathematical perspective: While the first paper introducing tile codes gave an ad hoc construction, we rigorously explain how tile codes can be understood through the lens of homological algebra and algebraic geometry.

Derived automorphisms are a more general concepts that is useful for CSS codes that have certain symmetries but don't allow for automorphisms (stabilizer preserving permutations of qubits and checks). For example, tile codes don't have automorphisms, but always allow for derived automorphisms.

We call these logical gates "derived automorphisms". We understand the action of these derived automorphisms by constructing a canonical symplectic basis of logical operators supported along lattice boundaries for tile codes, which can be constructed by hand by a simple cellular automaton.

In our new paper, we propose a fault-tolerant protocol applying parallel logical CNOT gates to a tile code by sliding it. Our new protocol is advantageous for implementation, as it does not require additional connectivity needed for syndrome measurement. It also has very low physical qubit overhead.

In the mean time, Yingli Yang et al. have figured out how one can use lattice surgery-style methods to implement logical gates on tile codes: arxiv.org/abs/2506.18061 The cool thing here is that their protocol does not require any additional connectivity compared to using tile codes as a memory!

Tile codes (journals.aps.org/prl/abstract...) generalize surface codes by allowing for slightly higher non-locality and connectivity. In return, these codes give more than an order of magnitude savings in terms of physical qubit overhead per logical qubit at some fixed distance.