From Expander Codes to Higher Cubical Complexes in Quantum Coding Theory
A self-contained expository note tracing the geometric progression underlying modern quantum error-correcting codes. The central question is: what combinatorial geometry is needed so that many small local checks force strong global coding properties? The answer evolves through three stages — expanding graphs, left-right Cayley square complexes, and high-dimensional cubical complexes with sheaf coefficients — each adding a new controlled direction of overlap among local constraints. The note ends at the Dinur–Lin–Vidick construction of almost-good quantum locally testable codes at dimension $t=4$.
Topics
- Tanner codes and Sipser–Spielman expander codes
- Left-right Cayley square complexes (Dinur–Evra–Livne–Lubotzky–Mozes)
- Tensor product codes and quantum Tanner codes (Leverrier–Zémor)
- CSS codes, stabilizer formalism, and the toric code
- Graded incidence posets and sheaf coefficients on cubical complexes
- Cycle and cocycle expansion; product expansion (Kalachev–Panteleev)
- Almost-good qLTCs via high-dimensional cubical complexes (Dinur–Lin–Vidick)