Toric codes all the way down: decoupling translation-invariant topological CSS codes in higher dimensions

Abstract

Topological Calderbank-Shor-Steane (CSS) codes in more than two spatial dimensions offer features useful for fault-tolerant quantum computation, such as transversal non-Clifford gates and single-shot decoding, but a systematic understanding of them is still lacking. One approach is to decouple a code: map it to a stack of higher-form toric codes by a finite-depth local CSS-preserving Clifford circuit, possibly after adding ancillas. For translation-invariant $p$-ary CSS codes, where $p$ is prime ($p=2$ for qubits), we require the circuit to commute with the translations of a superlattice. In two dimensions, such a decoupling exists once these translations act trivially on every anyon type. In higher dimensions, excitations are sorted by their dimension into superselection layers, which can remain glued together even when translations act trivially on each layer. We show that this gluing is the only remaining obstruction: a code can be decoupled on a superlattice exactly when, after coarse-graining to it, translations act trivially on every layer and the gluing vanishes. Moreover, for a code with finitely many excitation types in each layer and $m$ nonzero layers, once translations act trivially on every layer, further coarse-graining by a factor of $p^{m-1}$ in every direction removes the aforementioned gluing. Physically, the same criterion determines when the code encodes the maximal number of logical qudits on a finite torus. Finally, we give an explicit algorithm that constructs the decoupling as a circuit of SUM (qudit CNOT) and SWAP gates.