Sheaf Codes: An Expository Introduction via the Toric Code
An expository introduction to sheaf-theoretic coding theory, with the toric code as a running example. After fixing the basic vocabulary of linear codes — rate, distance, locality — we describe the toric code in pictorial language and reinterpret every piece of it as local linear data on the cellular chain complex of the torus. This rereading motivates the definitions of a cellular sheaf and a sheaf code, of which the toric code is the simplest example. We then explain why the formalism is forced on us as soon as one seeks quantum LDPC codes with constant rate and linear distance: the Bravyi–Poulin–Terhal bound rules out fixed-dimensional Euclidean constructions, and the modern qLDPC breakthroughs combine an expanding cell complex with nontrivial local-algebra stalks linked by restriction maps — precisely the structure of a cellular sheaf. A short tour reads each major construction (classical Tanner, DELLM, Leverrier–Zémor quantum Tanner, Dinur–Lin–Vidick higher cubical) as a cellular sheaf code on this common template.
Topics
- Linear codes, parity checks, distance, and locality
- The toric code: pictorial setup and $X$/$Z$ stabilizers
- Cellular chain complex of the torus and stabilizers as boundary maps
- Logical operators as (co)homology classes
- Cellular sheaves: stalks, restriction maps, and the constant sheaf
- Classical and quantum (CSS) sheaf codes
- The Bravyi–Poulin–Terhal Euclidean distance ceiling
- Modern qLDPC constructions (Tanner, DELLM, Leverrier–Zémor, Dinur–Lin–Vidick) as sheaf codes